Text

Forwards Summary of 821117 Working Meeting Re Seismic/ Structure/Cell Liner
	ML20077K543
Person / Time
Site:	Clinch River
Issue date:	01/12/1983
From:	Longenecker J; ENERGY, DEPT. OF, CLINCH RIVER BREEDER REACTOR PLANT
To:	Check P; Office of Nuclear Reactor Regulation
References
	HQ:S:83:172, NUDOCS 8301130102
	Download: ML20077K543 (184)
	v • d • e

{{#Wiki_filter:@ Department of Energy Washington, D.C. 20545 Docket No. 50-537 HQ:S:83:172 JAN ? ? 1983 Mr. Paul S. Check, Director CRBR Program Office Office of Nuclear Reactor Regulation U.S. Nuclear Regulatory Commission Washington, D.C. 20555

Dear Mr. Check:

NOVEMBER 17, 1982, SEISMIC / STRUCTURE / CELL LINER ANALYSIS WORKING MEETING REGARDING THE CLINCH RIVER BREEDER REACTOR PLANT A working meeting was held on November 17, 1982, to discuss the Nuclear Regulatory Commission (NRC) concerns related to the seismic / structure / cell liner analysis effort. Attachment 1 contains project responses to the NRC's (draft) Structural Audit Report dated July 20, 1982. The other project responses were provided in the December 29, 1982, submittal (HQ:S:82:169). Attachment 2 contains additional information supporting project responses that were handed out at the November 17, 1982, meeting and a summary of the meeting presentations made by the project. is the meeting agenda. Attachment 4 is a list of meeting attendees. Questions regarding this subject may be referred to Mr. V. Fayne (FTS 626-6394) of the Oak Ridge Project Office staff. l Sincerely, y JoRn R. Longene@ Office of i er Acting Director, Breeder Demonstration Projects Office of Nuclear Energy Attachments l cc: Service List l Standard Distribution S 1 Licensing Distribution f I(40 C301130102 030112 DR ADOCK 05000337 PDR ,~, , the project response to NRC's (draft) Structural Audit Report, dated July 20, 1982, includes the following: Audit Finding I.A.1 Cell Liner Design Reanalysis I.A.l.a Wall Liner at a Circular Penetration I.A.l.b Wall Liner at a Rectangular Penetration I.A.l.c Wall Liner Near Embedded Plates I. A.l.d Wall Liner at a Stud Connection I.A.2 Cell Liner Design Buckling I.A.3 Cell Liner Design Corner Analysis I.B.2 Containment Design I.C.3 Seismic Analysis, Torsional Moments I.C.4 Seismic Analysis, Interface with Structural Design I.C.5 Seismic Analysis, SRSS Rule I.C.6 Seismic Analysis, Combined Stresses General Finding I.D.1 Formula for Shear Strength I.D.2 Locations of Compressive Material I.D.3 Description of Computer Codes Concern II.1 Failure Criterion for Cell Liners II.3.a Basemat Model II.3.b Rock Layer Angle and Reference 1: " Surface Motion of a Layered Medium Having an Irregular Interface due to Incident Plane SH Waves," by R. Aki and K. L. Larner of M.I.T., Journal of Geophysical Research, Feb.1970. The December 29, 1982, project submittal (HQ:S:82:169) provided responses to the remaining Audit Findings.

I f t i ATTACHMENT I 1 l 1 l 1

I. FINDINGS A. CELL LINER DESIGN (3.8.3) la. Wall Liner at a Circular Penetration

RESPONSE

The mathemagical model discussed in the finding had a plane of symmetry at an angle of 45 (1/8 segment). It was considered by the audit team that this could prevent the non-symmetrical buckling modes of the plate. The revised analysis has considered a 1/4 segment of the penetration. The purpose of this analysis is to study the response of the wall liner and a penetration to the D.B. A. sodium spill accident. An analysis of a 24" diameter penetration composed of a Schedule 80 slegve reinforced by a 6" x ik" collar, with ik" thick gusset plates spaced at 45 around the penetration (Figures 1.A-1 through 1.A-3) will be carried out by using a finite element analysis. The wall liner consists of 3/8" thick plate supported by-1/2" diameter studs i~ at 15.0" on center. The elasto-plastic analysis will be performed using the computer program ANSYS. The liner, collar, sleeve and the gusset plates are representsd by plastic triangular plate elements (STIF. 48) whereas the liner anchors are represented by plastic pipe elements (STIF. 20) and spar elements (STIF. 8). The concrete interface is modeled with 3D-interface elements (STIF. 52). ~ The model of the penetration sleeve, collar and gusset plates, together with i the cell liner, is shown in Figures 1.A-1 through 1.A-3. The strains calculated by this analysis will be compared with the strain limits specified in PSAR. e I I i l

,J.y sX , N/N/N/N/\\/N/N/' / N/VN / VN / N/ N X/\\/\\/\\ /N/ N/N / /N/N/ N/N/ N/N/N N/\\/N/N/\\/\\/ N/ /N/VN/VN / V N VVNNX/N/N/N /N/ N/ N/N/ N/ N/N =5-N/ N/'N/\\/\\/\\/ AN/ \\ \\ N/N/N/N/N s . \\/ \\ 'N g NN/N/N / /N /N/'N < ~ X/x ado"'^" ?p /vW/ /AN /VN ,m N/N/N/ j 7 y, L j,4,= q ~ 1 "# si.ps' 1 @ 15 " C, C, EA, WAY 3.6I _ 3 "_ 3 @. $ " _$" G Q 3.75" Q = ll 19 i 1 WALL LINER AT P.M!r.TRATIC"S

+-sYkf M, I 403 4fi A SCHER,60 ~3 SLEEVE 1 G s . ;" TWICK ' A L U S S E T fE.- "o 2,.43 #7 r- ,r i r A c /N/'N/\\/\\/\\/NY a .O u d l =__ i 0 4 ,b LD c. 3LT 55\\ f l 5 " _ j S.G l "_ _ 8 c0 ll,13 ' = 30 ' S.G l" _ 5" _ 'I l' I ~ ~' (RADIUS = 11.59) l 1 1 DEVI'CPED ELEVATION OF S*.EEVE AND GUSSET FIGU?2 1.?.-2

f GUSSET f?_ c3 6 = 45' - $l# 1 1 a, .u, .o a, r C D 4 a e ll 2.J /3 1v K, l l. q 3 ll e 11 3 = l s g,&l 11 y GUSSET PLATE AT PE::ETFAT Cli SI.EEVE G..-. ,.n,-

A u n: 1 r

I. FINDINGS A. CELL LINER DESIGN (3.8.3) lb. Wall Liner at a Rectangular Penetration

RESPONSE

0 The analysis of the finding consisted of a mathematical model with a 45 symmetry plane (similar to Item la). This analysis will be perfonmed in a format similar to the format adopted in the circular penetration analysis (ItemI.A.la). e i e O g

~ I. FINDINGS A. CELL LINER DESIGN (3.8.3) Ic. Wall Liner Near Embedded Plates

RESPONSE

The analysis of the finding had considered a strip which included the embedded plate and liner plate with studs. The auditors considered that the boundary conditions utilized could prevent non-symmetrical buckling. The revised analysis considers a model with a 1/4 segment of embedment, liner plate and stud. A 1/4 model was developed to analyze the liner in the vicinity of 30. inch x 30 inch embedded plates with 45 inches of ' liner between adjacent embedded plates. Figure 1.C-1 shows the details of the model. The liner elements in the model described above were represented by plastic triancular elements (STIF. 48). The liner stud anchor elements were represented by plastic pipe elements (STIF. 20) and spar elements (STIF. 8). The concrete interface is modeled with 3-D interface elements (STIF. 52). The calculated strains and displacements for the cell liner, studs and embedded plate are presented in Figures 1.C-1 through 1.C-3. The maximum ~ strain of 2.4% (M + B) for the liner is below the strain allowable of 13.4% (M + B) at 700 F based upon stress / strain allowables of PSAR Section 3.8-B. l l 1 =

FIGURE l'.C-1 LINER AT K EDDED M TE LINER' STRAINS (IN/IN) Allowable Strain 0.1000 M = Z. 0.1340 1 M+B = STRAIN .0237 (M+B) .0239 (M+3) .0235 (M+B) .0205 (M+B) ' 0060 (M) f.0069(M) [.0072(M) [.0071(M) N/N N/ / .0205 (M+B) / m =g .0071 (M) "x. b l / \\ I i VN 'K/N/N/N S N/N/N/X /\\/ .0235 (M+3) s i Y1qgnwx my .0239 (M+3) ~ s E;3, ^T \\/'N/\\/

',l:,)

e i NAN /'x4 W Y J"> X 5dD4"=IT J _5"] e O cD 5.7 5'" = 27 5" 'l

YMMETRIC

= 4 _ 5 e" LINER. 3 " EMS PL ATE l = n - EMSEOMENT PLA i5 0 12" C' STUD , _ 2 C.: '2" O STU D . ANCHOR A -R GiD L!Ni(( 3CC3NTR! CITY CF m

~ FIGURE 1.% 2- - LINER AT EMBEDDED PLATE _. STUD STRAINS (IN/IN) ' Z. MAXIMUM STRAIN Allowable Strain 0.1000 M = 0.1340 M+B = .082 (M+B) .099 (M+3) .100 (M+B) .041 (M) .049 (M) .051 (M) N/\\/N/\\/Y/ /N /VN/ VN s. s e N/\\/N/\\/N/ ! i /N/N/N/N/N N/N/N/N/N/ = /N/N/N/ VN ~ N NZEMX/\\/'N/ i J XN4M/N/N/N ~ / o 1 (M 3, 3,

1:;;c '/ N/ V N A

d lSO dI L 5'r MMETRIC I ,,L'" 3 gMS ptA7g ~ l 4 , 3g tin g g, a I O12" @ STUD n - EM5EDM5KT PLATE . ANCHOR CF 'G' O S UD A -R:C-lO LIN X. ( 5CCiNTR! CITY OF ? ; LINER

Za NOTE: Positive Displacements Represent Displacement / Deflection Towards the Concrete. \\. \\ N SYletETH IC \\ (Al.I, ED(;ES)\\p J v CONTOUR VAI,UES ~ ~ 'N ~ ~ '~f (TNCllES) ~ -1.51" es[ D--I yg I '50 ~ !!AXI!!Uti E-.e ok" F-.e DISPI.ACEffENT ~ C*1 I[ its.2 qm B C pE F CH I*O g j M

D

/ E ~ 0.26" ig I 3/8" I, h ' kf # 'i f PIAT k e.U' t. ,?.: q *. 1 i J l O - 1/2" 4 STUD EMITEDI)EI) i ) ?'f..h.h,;c.': 1; $ (SilADEI)) l ' fll I A'- 1/2 of 1/2" d STUD !$I k

e

',,88h th f l{,, cf. "g,y';;.gd.p .ti.

nlW;at${ bkl l77..lfl,,?,','.,

15" 15" 7-1/ 2 g ~ g

I. FINDINGS A. . CELL LINER DESIGN (3.8.3) Id. Typical Wall Liner at a Stud Connection

RESPONSE

0 The analysis of the finding 'had considered a plane of symmetry at 45 angle (1/8 segment). The auditors considered that this boundary condition could prevent non-symmetrical buckling. The revised analysis considers a 1/2 segment model. The analysis of the typical wall / ceiling cell liner panel was performed with the mathematical model arranged to maximize the asymmetrical bucklin,g of the liner plate in order to artificially induce the worst case buckling of the stud. The mathematical model is shown in Figure 1.D-1. A plane of symmetry was assumed through the stud. To induce buckling in the desired pattern an initial force of 1 lb. perpendicular to the plane of the liner and directed away from the wall was applied on Node 77. The resultant deformations and strains are shown in Figures 1.0-1 and 1.D-2 respectively. The maximum displacements are shown in Figure 1.0-1 and the maximum strain values are shown in Figure 1.0-2.- The cell liner and stud strains are within the strain allowables per PSAR Section 3.8-B. The liner elements in the model described above were represented by plastic triangular elements (STIF. 48). This liner stud anchor elements were represented by plastic pipe elements (STIF. 20) and spar elements (STIF. 8). The concrete interface was modeled with 3-D interface elements (STIF. 52). l l 1 l I

7 j pp m e s.47 SYM if .207 .221 .242 .250 .250 s S h 1 5 .138 157 4207 .243 .251 j gRl ii 3/8 Liner (Typ.) .056 '.015 .089 .195 .237 ll N q .323 .258 .104 0.67 .152 11 Nq ii smo .61i .536 .352 .144- .044 N 37 41 D 1/2" dia. ~O Stud f .901 .822 .642 .448 .355 l.l.' N g N -1.163.-l.102 .951 .803 .741 g k -1.366 -1.321 -1.225 -1.132 -1.095 ii, N y q y -1.444 -1.410 -r.340 -1.280 g N ~ 73

7. c tl 77 i

SYM NOTE: Positive displace-cents represent displace:ents/. deflection towards ~ the cencrete. FICUPI 1.D-1 DISP 1.ACEMENTS i

8 (MEMBRANE, MIM3RANE + BENDING) T ] '-SYF (0.0094, 0.012) \\ \\5 (M) (M+3) 'l 1i A ii q 1 (- f f (0.0056, 0.0113)> (M) (M+3) N

i. l q

i Il h -(0.006h,0.0195) (3 (M) (M+B) b ) tud 41 , 0.0 ) 37 Q 3)= N -(0.0054, 0.0207) p (M) (M+B) (0.0050, 0.0146) ~ (M) (M43), I N S II N Y q, -(0.0100, 0.0243) (M) (M3) I 7 Jk N 'X R /

7. y j

77 73 i ( 'i ~ SYM l - - (0.0062, 0.0162) (M) (M+3) Allevable Strain FIGURE 1.D,-2 0.0703 M = NON-SMTRIC SUCKLING M+3 = 0.0942 STRAINS

NRC AUDIT FINDINGS I. FINDINGS A. CELL LINER DESIGN (3.8.3) 2. The' finite element me'sh near discontinuities, such as anchor studs, is much too coarse to properly predict high stress gradients in these areas. Areas of discontinuity need to be reanalyzed with refined meshes or another accept-able method needs to be used for accurately predicting peak stresses and strains at discontinuities.

RESPONSE

Two main patterns of liner plate buckling should be considered in evaluating ~ local stresses at the stud / liner plate interface, symmetrical and non-symmetrical. Detailed models near the stud /liaer plate interface will be developed to evaluate stresses in these areas for both buckling conditions. The boundary conditions of the detailed models will be obtained~from previous analyses of the plate-liner system with larger and coarser models. In the case of symmetrical bucklinp, the liner buckled against the concrete and improved high tensile 'orces on the stud. The larger model from which the detailed model was developed is shown in Figure 2-1. The shaded area represents the region considered in the detailed model. Figure.2-2 shows the mathematical model developed to evaluate stresses.in the case of symmetrical buckling. An axisymmetric option of isoparametric elements ( ANSYS - STIFF. 42) was used. Boundary conditions obtained from the larger model were imposed on Nodes 1 through 6,153,133 and 113. The cell liner and stud strains for the symmetric buckling case will be provided upon completion. Another analysis for the case of non-symmetrical buckling will also be per-l formed using a three dimensional mathematical model which will be similar to the attached model. The case to be considered will represent the worst case l condition from the non-symmetrical buckling cases. i i f l l [ t l

MAXIMUM GI'iI?JLIZID STRAr1S M= P__M3?J_NI ST?AIN M + 3 =.9_3?ANI PLUS BINDING STPAIN NOTE: Positive displacements represent displacements away from the Concrete (Reference Q220.40(a) ) s MAITMUM I.n x ST?>.IN /ia/i-1 h_k 7*)

272 OF S'fMMITRY sp 1

0 .0 . 3.0 " .15" . 0 5 "\\. 0 " ._ __ l,, 5"".7D STPJIN (i 7 _, 0,, NI \\ .-(% (' .021 (M + \\ l %'J. 4l ~ .021 (M) i l I \\ / 1N s,., ~ - <j \\g \\b~Y S' UD ANCHOP I _,.,, p _ _ .15" l /lN f LINER Du_ \\ / 20" l / j ,/ \\ l , 0" c. l-2%g s _,p j

o~..

l / ls I\\ \\ ,'/ N t / \\ / '\\ \\/ \\ \\ 4 l b .25" - I5" .20" .15" .10" 4 Spaces @ 1.873" = 7.5" LINE OF S' W_ T3 1 ] j CI~~ LINI? - S'M_IT?.IC ECL ING; 2'iSICN IN S 3 ~ FI.JP.I 2-1

4L 2 6 o.12.5 0.094 O.131 6 e o.175 = 14"

- ->w z :

A 's _ 1 - t-r l 6 O tD ? b-e O -t A 0 4 I e D .ha j "E G._ ~ t p O c v o n $ lI OJ f v v Doundary co'nditions at Nodes 1 tihrough 6, 113, 133 and 153 are from the parent ~ O model.(Calculation C-27.RC-06.373)

FIGURE 2-2 e

3 ll 0 dy N a, g w s-x l / Y' .EE P p 15'b 13 3 11 3 I y Stud s_ IINER/ STUD ANCllOR MODEL SYMMETRIC BUCKLING

NRC AUDIT FINDING i I. FINDINGS ^ A. CELL LINER DESIGN (3.8.3) 3. The corner of a cell where two perpendicular walls meet the floor needs to be analyzed, to determine its response when the floor is hot from a small pool spill and the walls a're cold. s s

RESPONSE

A typical tri-planar cell liner corner analysis will be performed on a model consisting of two walls and a floor panel. The model will include a 3/8 inch thick floor liner plate supported on an embedded beam and two' wall liner panels. A sketch of the model is shown in Figure 3-1. An elasto-plastic analysis will be performed on this tri-planar' corner region using the computer program ANSYS. The element types and sizes will be similar The model will be subjected to thosg of the models shown for the other areas. to 1000 F localized hot spot over the cell liner floor region while maintaining the two wall liners at operating temperature. The calculated results will be compared to the stress and strain. allowables of PSAR Section 3.8-B. 4 9 A M, h +~

[WBEMBEDDEDBEAM 45" 22Y' a . FACE OF WALL a h LINER FLOOR s ^ R SPAN e LINE OF SYMMETRY m i MODEL PLAN LINE OF SYMMETRY A Y i j, t y LINER PLATE I, - LINER STUD V j b. i --y c. 8 / CORNER ANCHORAGE f,,, i LINER FLOOR PLATE u / '% WB EMBEDMENT BEAM 4 jf

i. ' t/

y MODEL ISOMETRIC FIGURE 3-1

NRC AUDIT FINDINGS I.B. CONTAINMENT DESIGN (3.8.1) 2. Design calculations need to be provided that consider the tangential shear capability of the concrete walls sandwiching the steel containment below the operating floor.

RESPONSE

~ The design of the peripheral walls included the tangential shsar load. The reinforcing steel design was based on calculated maximum principal tensile stress using, concrete thin shell design theory. The design was checked for shear wall requirements and was found to be adequate. Specifically, the total reinforcement required was calculated as the total of reinforcement area required for axial load and bending plus the reinforcement area required for tangential shear. The results show that this total required reinforcement is less than the reinforcement provided under the current design. 4 e

NRC AUDIT FINDINGS I.C. SEISMIC ANALYSIS 3. The torsional moments in the steam generator building and reactor building predicted by the seismic model seem to be large compared to the other internal loads (that is, shear and overturning moment). This result needs to be further assessed by the NRC and the applicant. Unless the result can be justified, the overall adequacy of the seismic model needs to be questioned.

RESPONSE

NRC considers that the seismic torsional moments for CRBRP are large when compared with other seismic forces, presumably based on NRC experience with other plants. It should be noted that the CRBRP configuration with a single foundation mat and interconnected buildings is different from the configuration of most LWR plants. Smaller torsional moments are to be expected in individual and more regular structures that are founded on separate foundation mats. The CRBRP configuration results in a significant contribution from both mass eccentricities and the torsional mass moment of inertia particularly for the Steam Generator and Reactor Service Buildings. To verify the significance of this contribution the seismic analysis in the North-South direction was~ rerun with the torsional mass removed. As'can be seen from the attached results, the torsional momen't is significantly lower if the torsional mass moment of inertia is ignoied. l l I i .,.e ,,,.c---,

ELEV. AT ELEV. AT TORSIONAL MOMENT TOP OF BOTTOM OF BUILDlHG MEMBER MEMBER WITil TORSIONAL MASS NO TORSIONAL MASS RATIO

SGB 886 873 6.303E4 2.994E3-

.05 4.155E6 2.678E6 .64 857 846 787 -765 7.705E6 4.435E6 .58 746 733 7.517E6 2.453E6 .33 RSB 884 869 8.410E4 5.549E4 .66 840 816 3.776ES - 1.919ES .51 779 765 7.793E5 3.168E5 .41 755 733 1.819E6 9.873E5 .54 CONF. 992.5 969.6 1.210E4 0 0 910.8 884.0 5.409ES 0 0 843.0 816.0 1.742E6 9.780E5 .56 794.0 779.0 9.685ES 4.290E5 .44 755 733 8.823E5 3.876ES .44 RCD 816 800 1.308E6 6.460E5 .49 766 752.7 1.517E6 6.174E5 .41 752.7 733 1.584E6 6.500E5 .41 Units: Kip. ft. Torsional Moment Without Torsional Mass Torsional Moment With Torsional Ma,ss TABLE C.3 TORSIONAL MOMEN1S

NRC AUDIT FINDINGS I.C. SEISMIC ANALYSIS 4. The interface between the seismic analysis group and the structural design group in transmitting seismic loads needs to be reviewed by the applicant.

RESPONSE

The interface between the seismic analysis group and the structural design group was reviewed and found to be adequate. The following procedure was followed: Interface beismic loads were transmitted formally by the seismic group to the structural design group by CINDT (Control Information Data Transmittal Form). From 9/78 to date twenty-seven (27) CINDTs providing information for final design were transmitted to the building design grou~ps. This data included seismic forces on buildings, floor response spectra, maximum floor accelerations and displacements, maximum relative displacements between floors. The procedure regarding the use of seismic forces in the design is. included in WARD-D-0037, Seismic Design Criteria for CRBRP (PSAR Section 3.7A). CRBRP Project Procedure PC3.3, Technical Information Receipt and Control, specifies requirements for transmittal of technical information from one design group to another within Burns and Roe. In addition, the Quality Assurance Program for CRBRP requires that design reviews be conducted to assure design criteria, parameters and loadings have been properly incorporated into the design. Design reviews have been performed on the seismic design of structures completed to date. t 4 .n. ~ -

\\ ~ NRC AUDIT FINDINGS I.C. SEISMIC ANALYSIS (3.7.2, 3.7.3) 5. In some cases it appears that the applicant has used the square root of the sum of the squares (SRSS) rule to combine loads from the three earthquake directions and then used the SRSS again to combine stresses. This is generally nonconservative and the applicant needs to verify that the SRSS rule is used as specified in PSAR Section 3.7, Attachment B. The applicant's method for insuring that the procedure is followed should be provided.

RESPONSE

In the design of the Nuclear Island Buildings, the SRSS rule was used to combine the seismic effects (essentially the loads) from the three earthquake directions. This rule was not applied again to combine stresses except in the area described below. Moments, shears and axial forces were obtained for each earthquake direction independently and were combined by SRSS. The SRSS rule was used twice in the design of the area around the large penetrations in the Confinement. This is addressed i.n.the response to audit Item C.6. All other structure designs were re-examined to assure the SRSS rule was used only once. This has been confirmed. In addition the incorporation of seismic loads into the design followed the CINDT procedure and was subjected to design reviews in accordance with quality assurance requirements as discussed under Audit Item I.C.4. 9 1 i.-

NRC AUDIT FINDINGS I.C. SEISMIC ANALYSIS (3.7.2, 3.7:3) 6. In analyzing the confinement structure (Reference C-27.RC-6.105) the SRSS is used to include torsional and rntatipnal components and then used again to combine calculated stresses. This is nonconservative and the design should be evaluated again.

RESPONSE

In the analysis of the confinement structure in the area of the large penetra-tions, the torsional and rotational components of acceleration and displacement at the boundary were included by the SRSS method in accordance with the i requirements of PSAR Section 3.7-A. The design values of moments, shears and ~ axial loads obtained from the analysis were then combined by SRSS. This was based on the judgment that since the torsional and rotational. components of acceleration are small in the confinement and would not influence the design, use of the SRSS rule to combine the resultant loads will yield conservative results. In any event, this analysis has since been redone by separating the accelerations frcm each seismic direction, and then combining the effects by SRSS. The design values of the recalculated moments, shears and axial loads increased in some areas and decreased 1c ethers. In the areas where the design values increased, the combined stresses are within Code allowables and no increase in reinforcement is required at any section. l I

1 NRC AUDIT FINDINGS I.D. GENERAL FINDINGS (3.8.4) 1. In design book C-27.RC-6.105 for the confinement building the applicant miscopied the formuli for shear capacity of concrete in compression from ACI 349. A thorough check should be performed by the applicant to verify that the mistake was not carried on into any design calculations.

RESPONSE

In the Calculation Book C-27.RC-6.105, page 29, the formula for shear strength of concrete (formula 11-4 of ACI 349) was miscopied to show a factor of 200 instead of 2000. After a thorough review of the calculations, it was concluded that this was only a copying error. The actual design calculations used the correct value of 2000. Page 29 of the calculations has been revised to show the correct value. l t I l l i i l l

NRC AUDIT FINDINGS I.D. GENERAL FINDINGS (3.8.4) 2. The design of structures or structural elements in contact with com-pressible materials that experience large deformations r.eeds to be provided to'the NRC.

RESPONSE

Compressible material is provided at the following locations: Around penetrations located in the peripheral concrete wall of the a. Reactor Containment Building and the Containment Vessel b.etween elevations 733 ft. and 816 ft. The purpose of these gaps is to allow the Containment Vessel to move vertically independently of the concrete structures. b. Batween the exterior surface of the Reactor Vessel Support Ledge and the Reactor Cavity concrete. The width of the gap filler material will be large~enough to limit compressive forces on the structural elements. The structures will be checked for these forces and, if necessary, the compressive forces will be reduced further by increasing the width of the~ gap and filler. The material specified is cross-linked polyethylene manufacturered by W. R.. Grace and/or Progress Unlimited or approved equal. Attachment D.2.A describes the properties of the material manufacturered by W. R. Grace. Joints between the Nuclear Island Buildings and the Turbine Generator, Radwaste and Plant Service Buildings will be open with non-load bearing steel closure plates. 1

R6DOFOAM' H ' .u CELLULAR PLASTIC JOINT FILLER For SEISMIC MOVEMENT m TOTAL CAPABILITY MINIMUM LOAD TRANSFER xx MAXIMUM RADIATION RESISTANCE ENGINEERED @ JOINT FORMER TO BE LEFT IN PLACE DESCRIPTION DESIGN CRITERIA Cellular cross linked polyethylene in sheet form. SEISMIC MOVEMENT Sizes available: Thickness up to 3",in sheets 48" Su'ajected to seismic movement of 1.76 Hz 0.9 g by 48*. May be laminated f or greater thickness. acceleration. 31"/sec. velocity for 150 cycles with no damage to the Rodofcam or to the adjacent concrete. MINIMUM LOAD TRANSFER Under seismic testing, Icad transfer at 50% ,USE ccmpression was 17.8 psi, Designed especia!!y fo'r seismic separation joints RADIATION RESISTANCE in nuclear power plants, nuclear fuel recrocessing plants, fcssil fuel generating Exposure to 5 X los Rads before threshold damage and tc 1 X 107 Rads before 25% damage. plants, and pumped stcrage plants. With the Redofcam 11 adhered to a vertical concrete JOINT FORMING ' sur' ace with special adhesive, the seconc concrete At 2.8 psi lateral pressure of concrete (Pf aced at 2 ft. vertical lift per hour at 70*F.) deflection of g ' placement may be made against the Redefcam to Rodofcam 11 will be 2.5%. form the seismic separation joint. 2 O 2 RODOFOAM !! COMPRESSION DEFLECTION >0m 60 Oo I 50 E /' W E 40 i C E s0 o I o LJ 5 0 e 20 E l. I Z O l 2 b l c to m { l I -i 0 l m l l l l i to 20 30 40 50 60 70 80 >0 OEFLECTICN. . CE C AIGiNAL TMCKNEss g m O inO E A O CONSTRUCTION PRODUCTS DIVISION. W. R. GRACE & CO. NU '.% "A"",'",% Ei'OAl."' ?? $llllll.A,'.'.'N.,' 'l.. "N,'a.?l'l,0.".~,' $lA,".$,% C$' $l:!ai ;% ?

L.: ;'.:=,',::7,L':2:::::::,::::rll'::,~::".

.:::l."::;;:'::'.',';;'"lll",;, l,::; ns...,~,,,.. wta:avastiss: 6: w,etT*tv0tt avt CAv8 tit,;t. Mass. C:110

NRC AUDIT FINDINGS I.D. GENERAL FINDINGS (3.8.4) 3. All significant computer codes used for structural design need to be addressed in the PSAR along with verification documentation.

RESPONSE

The following computer programs will be added to Appendix A of the PSAR: B050R4 CODES EQUILIN FLUSH MODPROP MRI - STARDYNE (Revision) RESPECTPLOT STRUDL THAVSA 2DGENFRAME The description of the programs is attached. l l

BOSOR4 COMPUTER PROGRAM BOSOR4 is a computer program for stress, stability and vibration analysis of shells of revolution. The program was developed by D. Bushnell of Lockheed Missiles and Space Company (Reference 1). The computer code is based upon the linear, elastic, thin shell theory. The structure should be axisymmetric. The program can handle various kinds of wall materials and loadings. Both mechanical and thermal loads are permitted in the analysis. In cases involving stress analysis of a shell for non-axisymmetric loading, the program finds the Fourier series for the loads, calculates the shell response in each harmonic to the load components with that harmonic, and superposes.the results for all harmonics. The program has an option by which the stability analysis of a shell can be treated as a bifurcation buckling problem and mathematically it is treated as an eigenvalue problem. The program also handles shell vibration as an eigenvalue problem and finds mode 'hapes and s frequencies. BOSOR4 uses a finite-difference scheme as a numerical technique in the solution of shell problem. i Availability l This program is available through CDC - Cybernet.- Verificatio'n BOSOR4 is recognized and widely used in industry with a sufficient history of successful use to justify its validity. Application BOSOR4' has been used in the analysis of axisymmetrical structures. Reference (1)' Bush'nell D-Stress Stability and Vibration of Complsx Branched Shells of Revolution: Analysis and User's Manual for BOSOR4. NASA / Langley Research Center, Hampton, Virginia. Contrr:t-I NASI-10929.

COMPUTER PROGRAM CODES CODES is a Burns and Roe computer program that designs reinforced concrete wall and shell structures per the require-ments of ACI 318. The program calculates the reinforcement requirements due to axial load and bending, torsional moment, longitudinal and transverse shear. This is done for each of 14 load combinations and the maximum reinforcement areas for each group of elements are tabulated. In addition, the program contains an option wherein various loadings are combined and converted to principal forces. The components of the principal forces in the meridional and hoop directions are combined with the bending moments for the design of shell reinforcement. Availability ' CODES is developed by Burns and Roe. Verification The CODES verification has been done by hand calculation and is available in internal Burns and Roe documents. Application CODES is used in the design of reinforced concrete wall and shell structures. l t I l t t I

COMPUTER PROGRAM EQUILIN The computer program EQUILIN is designed to calculate an equivalent linear temperature distribution from a given non-linear distribution, according to the technique described in Reference (1). The computer program is provided with information of a concrete section subjected to a non-linear temperature distri-bution. The program determines mean temperature of the section from the non-linear temperature. Also, an equivalent linear temperature distribution is found such that it produces the same uncracked moment about the centerline of the section as does the non-linear temperature distribution. Availability EQUILIN is developed by Burns and Roe and is available in the CYBER 176 Computer of CDC-CYBERNET System. Verification EQUILIN has been verified against hand calculations. A verification problem is attached and the results are found .=atisfactory. Application EQUILIN has been used to determine equivalent linear tempera-ture distributions in thermal analysis of concrete structures. Reference ACI-349-76, Appendix A Commentary, Code Requirements.for l (1) Nuclear Safety Related Concrete Structures l 1 i f t { 1 I

EQUILIN VERIFICATION Attached figure shows a temperature profile through a concrete section. The following table provides a comparison between EQUILIN and hand calculations:

EQUILIN, HAND CALCULATIONS Mean Temperature 216.4 216.5 Difference in Temperature Through Wall Thickness 286.4 286.5 Inside Wall Temperature 359.6 359.7 73.2 73.2 Outside Wall Temperature Equivalent linear temperature profile is shown dotted in the figure.

5

A ?GO. 355 N \\ lL 320 \\ O .N 280 \\ 270 L11 \\ M s N 3 24 0~ 1-- <C M 200 N lu s 19 0 N Q_ N 2 IGC y tu s }-- t20 ' N A N N 60 x 55 4C o l* I l c s.5" 11.5" ss.s " J.o " = WALL THICKNESS., IN. I GIVEN TEM?ERATUF,5

EQ U I LI N TEMP.

_ QU I LIN K =.c u L i s c

COMPUTER PROGRAM FLUSH This is a computer program for approximate three dimensional analysis of soil-structure interaction problems. The program was developed by J. Lysmer, T. Udaka, C. F. Tsai, and H. B. Seed of the University of California, Berkeley. The program is a further development of the complex response finite element program LUSH. FLUSH includes additional features such as transmitting boundaries, beam elements, an approximate three dimensional capabilities, deconvolution within the program, etc. Availability This program is available through CDC-Cybernet.' ~ i verification It is recognized and widely used in industry with a sufficient l l history of successful applications to justify its validity. Application This program has been used for soil-structure interaction in seismic analysis. Reference Lysmer, J.; Udaka, T.; Tsai, C.; Seed, H. B.; FLUSH, A Computer Program for Approximate 3-D Analysis of Soil-Structure Interaction Report No. EERC-75-30, November 1975, University of Problems. California, Berkeley, California. 4 4

MODPROP This program calculates properties of a structure to be used in a " lumped mass" seismic analysis. The program calculates the masses and mass moments of inertia at specified elevations. It also calculates the member properties, such as area, shear area in two orthogonal axes (reference coordinates), area moment of inertia about these two axes, torsional rigidity, center of rigidity and centroid. Availability MODPROP has been produced by Burns and Roe and is available in the CYBE.R 176 computer of CDC - CYBERNET. Verification MODPROP has been verified against hand calculations. A verification problem is attached. Application MODPROP has been used to determine mass and stiffness properties of structures for seismic analysis. e 9 e +,

ATTACHMENT MODPROP VERIFICATION Figure shows a plan and elevation of a structure with concrete walls and slabs. Section properties of the' walls between Elevations 0.0 ft and 20.0 ft. were calculated. Also, mass properties (in weight units) were calculated between Elevation 10.0 ft and -2.0 ft. Both MODPROP and hand calculations were used. Results Units are feet and Kips. Moments of inertia are about centroidal axes. MODPROP HAND CAI.CULATIONS Shear Areas Ax = 224.43 224.4 Ay = 200.43 200.4 415.68 415.6 Total Area A = Area Mass of Inertia Ixx = 0.37012E5 37012.0 Iyy.= 0.37512E5 37512.0 8 Torsional Rigidity Kt = 0.7063E9 .7.063 x 10 953.9 953.9 Weight, W= Weight Moment of Inertia Iwx = 0.91827ES 91,829.0 Iwy = 0.96290E5 96,292.0 Iwz = 0.162192E6 162,195.0 Coordinates of Center of Ridigity, X = 14.53 14.53 10.36 10.36 y= Coordinates of Centroid: X= 12.74 12.735 Y= 9.96 9.957 l

f &_ OF CY LINDER w l'2' (% i l .d -4 b A INCL WAL A A FLOOR '/ J - & CF CYLINDER. t--- A A / SY2 _o y ELO'-O" s FLR 9 1 / 0 1 31, 10 _L N SX2 'l C - 1 ORIGIN OF .\\, to' COORDINATE =: SYSTE M CIRC WALL = l I 55.24 a t 'i j PLAN ,- E L OL O" t CI RC SLS g o h .9 a a a a a a h i k k, I 5Y2 o o i W W M _o = o o ,- EL O'- O' 3 l Ni C!RC SL51 SECTION A-A MODPROP VERIFICATICN PROELEM FIGUP2

COMPUTER PROGRAM RESPECTPLOT. This program computes response spectra from earthquake-accelerograms digitized at equal time intervals. The generated response spectra represent the maximum responses of a damped single degree of freedom system. The program is based upon the techniques described in Reference (1). Availability RESPECTPLOT is a Burns and Roe modified version of SPECEQ/SPECUQ It is available in the CYBER 176 Computer of CDC - (Reference 1). Cybernet System. Verification RESPECTPLOT results were verified against STARDYNE Program of Reference (2) and the results are found satisfactory. Application RESPECTPLOT has been used to develop spectra from the time-histories. References (1) N. C. Nigam, P. C. Jennings, Digital. Calculation of Response Earthquake Spectra frem Strong Motion Earthquake Forces, Engineering Research Laboratory, California Institute of Technology, Pasadena, California, June.1968 76079900 Cc.* trol Data Corproation, Publication No. (2)

STARDYNE,

e COMPUTER PROGRAM STRUDL STRUDL is a comprehensive structural static and dynamic analysis and design program. It stands for Structural Design Language and conceived, developed and initially released by the Deparment of Civil Engineering, Massachusetts Institute of Technology, Cambridge, Mass. Availability The computer program is available through Georgia Institute of Technology, Atlanta, Georgia (GTSTRUDL) and McDonell Douglas Automation Company ' (MCAUTC) of St. Louis, Missouri. Verification is recognized and widely used in the industry with It sufficient history of successful applications to justify 'its validity. Apolication It will be used for structural frame analysis of steel structures and supports. Reference (1) McDonell Douglas Automation Company, " ICES STRUDL User Manual" Georgia Institute of Technology, "GTSTRUDL User Information (2) Manual", Report No. SCEGIT-79-179, January 1979 e

o COMPUTER PROGRAM THAVSA This program calculates combined floor response spectra from spectra produced from independent seismic analyses for each of the earthquake directions (North-South, East"-West, veritical). It combines translational and rotational effects. The combination is based on equations 11, 12 and 13 of Appendix B of the Reference. Availability THAVSA was developed by Burns and Roe and is available in the CYBER 176 Computer of CDC - CYBERNET. Verification THAVSA was verified against hand calculations. The verifi-cation consisted of three steps: 1) Interpolation in a semilog spectrum plot (Figure 1) 2) Verification of the' calculations with equations 11, 12, 13 of Appendix B of WARD-D-0037. Results: THAVSA Hand Calculations Eq. 11 1.2630 1.2629 Eq. 12 1.3254 l.3254 Eq. 13 0.9084 0.9085 3) Plot Verification. Figure 2 shows a comparison of a spectral plot produced by TEAVSA and the results of hand calculations. A lication l l THAVSA has been used to produce Design Acceleratiod Response Spectra for Equipment Specifications and for the design of I structural components. Reference WARD-D-O'037 - CRBRP Seismic Design Criteria (PSAR Appendix 3.7-l A). l t I ( l

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c/ f A.61 MRI/STARDYNEf ~ The STARDYNE Analysis System is a series of compatible computer programs for analyzing the finite element method linear elastic structural models for a full range of static and dynamic input conditions. The static capa-bility includes the calculation of structural deformations and members loads / stresses caused by an arbitrary set of themal and/or applied loads. Prescribed displacement vectors can be used as input to compute resulting internal defomations, leads and stresses. The dynamic capability includes nomal mode response analyses for a wide range of loading conditions includ-ing transient, steady state hannonic, random and shock spectra.. Dynamic response results can, in general, be presented as structural deformations (displacements, velocities, or accelerations), and/or internal member loads / stresses. Availability STARDYNE(CDC-8400-2500) had been available on the CDC 7600 computer of Lawrence-Berkeley Laboratory. 9 r,H2 o v AB w ofse w oela6/>. :, '*n & COC lyA&& 6 Verification @ U"

Documentation of verification of the STARDYNE computer code per SRP' Section 3.9.1.II.2.c primary and intemediate can be found.in the reference.

Acolication The STARDYNE code is being used to perfom global dynamic analysis of the pumps including preliminary normal modes analysis and seismic and rotor inbalance response analyses. Reference &,f"s g#gpurd h $hbE M'rMme S ;r,yper of k-o Y MRI/STARDYN Finite Element Demonstration Problems, Document No. 84002500, Control Data Corporation, Minneapolis, Minn. e high A.-210

COMPUTER PROGRAM 2DGENFRAME 2DGENFRAME is a time-shared computer program which performs the bending analysis of two-dimensional frames. Frame members can be rigidly attached or pin connected. Loading condition include combinations of forces at the joints, concentrated forces on the members. Distributed loads on the members, concentrated moments on the members. Distorsions of the mebers, and temperature changes in the members. Data can be entered interactively or from data files. Availability 2DGENFRAME is commercially available to users through Control Data Corporation CDC KRONOS Time Sharing Computer Systems. Verifi~ cation Verification of this program was done by hand calculation and is available in internal Burns and Roe documents. Applicatibn 2DGENFRAME is used in the analysis of 2 dimensional frame structures. l + .g e e

T 4 + 3 i NRC AUDIT FINDING 's s,, s (s s. 3 s II. CONCERNS 3 1. The failure criterion for the cell liners is not appropriate. The strain limit criterion proposed is acceptable >1f the safe stress limit is not exceeded. More general failure criteria 'rieed to be developed that include the appropriate stress limits. (3.8.3)

RESPONSE

Failure criteria for cell liners of the CRBRP type are not provided explicitly in any of the current design codes and therefore special criteria were developed. The design limits useo for normal operating and faulted conditions, expressed in terms of strain, are corsidered appropriate for this type of structural system and sufficient to ensure structural integrity. The practice to express the criteria in terms of strain is consistent With the ASME B&PV Section III, Division 2, Subsection CC for containeent liners under service and factored ' load conditions. In accorda'r.ce with the NRC recuest to develop more general Erit'eria, design limits in terms of stresses have been added to provide a check for mechanical loads. The iriteria with strain limits have been retained to provide a check for the concination of mechanical a'nd thermal. loads. Cetails of the criteria in terms ci stress and strain limits are provided in the attachment. 1J y' ~ s x;\\ i s a

) ATTACHMENT CRITERIA'FOR CELL LINER STEEL The function of the cell liners is to provide a barrier to prevent sodium from contacting the concrete under design basis accidents. Design ~ criteria for such liners, to assure integrity under normal operating and faulted conditions, are not provided in any of the present codes and therefore special criteria were developed. In developing such criteria it is important to take into consideration not only the function of the liner system and the loading conditions, but also the nature of the loads and their potential to For this reason different design limits are provided for cause rupture. mechanical loads, i.e. directly applied, an'd thermal loads in combination with All loading combinations should be checked for both Cases I mechanical loads. ano 11 celow. I. Mechanical Loads In' addition to gravity loads, mechanical loads in this case include To ensure integrity under these loads, design pressure and seismic forces. This approach is consistent with the ASME limits are imposed on stresses. Code Case N-47 for Class I components; however, it should be noted that cell For mechanical loads the stress liners are not ASME Class I components. limits for the plate steel were adopted from Appendix F of ASME Code Section It should be noted that the Appendix F limits are applicable to primary III. The stresses and that verification for secondary stresses is not re conditions, are in accordance with ASME Section III, Division 2; Subsection CC The stress limits for anchors under faulted (large for containment liners. spill) conditions were adopted from Appendix F of the ASME Section III. The limits are given in Tables I and II. II. Thermal plus Mechanical Loads _ The loading conditions for the cell liner include thermal loads which Due to result in deformations much greater than those caused by other loads. the self limiting nature of thermal deformations, it is appropriate in this .It is, case to impose' design limits on the strains rather than stresses. recognized that in a structural system with discontinuities and elements with different stiffnesses certain areas may be subjected to strain concentrations. Fcr this reason it is essential that the system be analyzed with incremental elastic plastic analysis procedures and with eppecpriate discretization at the regions of expected strain concentrations. It should be pointed out that the imposition of strain limits for the cell liners is censistent with ASME Section III, Division 2, Subsection CC for containment liners under service and factored load conditions. The strain limits are given in Tables I and II.

APPLICABLITY OF CRITERIA TO REGIONS OF HIGH SHEAR The criteria for liner steel are expressed in terms of effective stress (for mechanical loads) or effective strain for thermal plus mechanical loads which include the shear stresses or strains. For this reason the criteria are considered appropriate for regions. where shear stresses or strains are predominant. It should be pointed out that the ASME Code Case N-47 recognizes the effective stress and strain as appropriate measures for inelastic analysis. Also the ASME B&PV Code, Section III, Division 1 (F-1321.1(C) accepts the von Mises yield criterion, which is equal to the effective strain at yield and makes no distinction for regions with predominant shear stresses. O s b e 1 I O l l

TABLE I CRITERIA FOR CELL LINERS PLATE DESIGN LIMIT (I) {l1) CONDIT10fi MECilANICAL LOADS TilERMAL + MECilANICAL LOADS CONTROL QUANTITY 'S CONTROL QUANTITY 'c ' e PRIMARY MEMBRANE OR MEMBRANE MEMBRANE MEMBRANE + BENDING PLUS IlENDING (a) Normal Operation and Upset c < 0.002 c < 0.004 sc ~ (l.oad Combinations (A) 1. st < 0.001 st < 0.003 c (11) in PSAR (b) Faulted, Small Spill , Illgher of: c < 0.005 c < 0.014 (Load Combination (C) in S + (S - S )/3 sc - sc - Y PSAR) 10.0lb 0.7 S C 1 0.003 c u st st c 1 0.5 c C 1 0.67 c (c) Faulted, large Spill e u e u (l.oad Combination (D) in PSAR) Ultimate (Nominal) Tensile Strength S = g Yield Stress 5 = y Uniform Elongation = c u Maximum Compressive Strain = r g Maximum Tensile Strain = 9 ~ I"2 - "3I

I"3- "1 I

j 2) Effective Stress = - f(o -0 S = e )2, IC Ef fective Strain = / (( _ g 3 l) l

3)

IC

~C = c -C 2 c j, "2 ' "3 are Principal Stresses o 03, cp c3 are Principal Strains i

TABLC 11 CRITERIA FOR CELL 1.lNERS STilii hfiCliGRS D E S I G N' LIM 1T CONulTION (1) (II) MECllANICAL LOADS TilERMAL t MECllANICAL LOADS CONIROL QUANTITY "F" OR "S" CONTROL QUANTITY "6 " OR "c " Lesser of-(a) floriiial. Operation and 0.25 6 0.67 f Ilpset (Load Combinations (A) & (D) in PSAR

0. 3'3 F ti (b) faulted, Small Spill Lesser of:

(Ioad Combinations (C) 0.9 F 0.50 6" in pSAR) 0.5FII (c)faulteUl.argeSpill liigher of: Membrane (l.oad Combination (D) S + (S" - S )/3 0.50 c" Y Y in PSAR) 0.7 5" 0.67 c" Membrane i Dending Liner Anchorage force Capacity l' = Liner Anchor Stress S = Liner Anchor Yield force Capacity l = y Liner Anchor Illtimate force Capacity l = tillimate (Nominal) Tensile Strength S = u Yield Strength S = y a tiltimate Displacement Capacity for Liner Anchors tiniform I.longation =

i. 9 Ihermal Loads Only

NRC AUDIT FINDINGS ITEM II. CONCERNS In addition to the Audit Findings the following ongoing concerns were addressed during the meeting. 3.b What effect does the angle of rock layering have on the excitation transmitted to the structure?

RESPONSE

The presence of shallow subgrade inclined layers of rock at the CRBRP site is not expected to result in a significant effect on the site response spectra. 1) Reference (1) shows the result from a series of problems addressing the effect of shallow basins on the free surface displacement amplitude variation for a variety of assumed incident shear wave angles and layer properties. These results show very good agreement between the more exact solution that includes the actual geometrical variation of the basin interface and a flat layer approximation with the depth of the flat layers set equal to the basin depth from a particular point of the surface. See attached Figure II-1. This shows that while within a local area below the free surface (local area being measured in the order of several kilometers) the layer has a certain slope and is inclined, the- ~ effect of this inclination on the response of the free surface above is i very close to the effect on a horizontal interface at that depth. Several examples are provided in Reference (1) which show that this is true for a variety of assumptions made regarding wave-length of incident wave, depth of shallow basin (which starts at a point of the free surface and dips downward to a depth of up to 5 kilometers), incident wave angle, I and property differences in the layer and the half space such as shear wave velocity and density. In all these cases, the flat layer approach shows good agreement with the more exact solution. The geologic configuration in the region of the CRBRP site is quite similar to Reference (1) study. The geologic profile of the CRBRP is discussed ( in PSAR Section 2.5 and PSAR Figures 2.5-1A and 2.5-17B are attached for reference. Attached Figure II-2 shows a cross-section of the rock profile underlying the Nuclear Island structures. For the CRBRP, a finite element seismic model was developed in which the very thin layers of alternating siltstone and limestone rock existing at the site were modeled as horizontal layers. See' attached Figure 11-4 for details. The response spectra generated using this approach were in , 8 - = ..,e - -. ~ - -,-..- - -

very good agreement with those generated by simulating the subgrade by ~ the equivalent impedances. This was presented to NRC in response 'to s Question 220.5. Attached Figure 11-5 shows a typical comparison of spectra calculated by each method. This is not surprising since (1) the design response spectra generated by simulating the subgrade by the equivalent impedances is an envelope of response spectra using the lower and upper bound properties of the layered rock and (2) the rock properties are quite similar. For example while Reference (1) used a shear wave velocity ratio of 5, the CRBRP rock layers have a maximum shear wave velocity ratio of about 1.5. See attached PSAR Table 3.7-5 for properties of the CRBRP rock layers. 2) An evaluation of wave paths under the Nuclear Island structures considering the inclined layers of siltstone and limestone was performed. Two cases of vertically propagating shear waves were considered using Snell's Law (Figure 11-3). Case I considered the waves propagating from the Basement (Shear Wave Velocity Vs = 8000 ft/sec). As the wave propagates towards the surface it refracts at the layer interfacgs. The resultant wave at the foundation level meets the structure at a 75 angle. Case II considered waves propagating vertically from the Knox Formation. The angle at the foundation g level is 80. At these angles there is no trapping of wave energy. The wave energy transmission is almost total for such closely similar materials (impedance ratio between limestone and siltstone is 1.43). Therefore, wave reflection is of minor significance due to minimal influence of inclined rock layers. In summary, since the flat layer approach is a good approximation to shallow inclined layers, and since the results of a finite element analysis are in good agreement with the results from the CRBRP seismic analysis which simulates the rock properties by equivalent impedances, it is concluded that the effect of inclined layers on the plant response is not significant. The influence of wave propagation and reflection on the ground motion at the N.I. foundation level also is of minor significance. Finally, the CRBRP site response spectra contains a much broader range of ground motion frequency content than would be expected for a specific geological profile. Its use in the design is therefore a very conservative approach. Reference (1): " Surface Motion of a Layered Medium Having an Irregular Inter-face due to Incident Plane SH Waves" by R. Aki and K. L. Larner of M.I.T., Journal of Geophysical Research, February 1970.. _ _ _

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Cumberla.nd Valley and Ridge Belt i. Unaka Delt i. Blue Ridge Belt i Plateou; c.a s a r e Aloif viLLE .re l e uNouvlL L E . -s. s ~ e ><.' . ' P.~ - _ ..r -w '.. . W.'"t V -;:iL=W:.<; :... &. - ,,, g ! .s

e

. ~ '... %... w ~.. w %. . ' >A%.'t v Qjs' - - -- u- - - : T: L..,- **--u.'. _ _ --- ~- -..m %n-y, ',q W::: : :.:.9 a.; * '.K. '. - o.. l*: x. .~. -4.k \\ ' ~r r -:.* ;' SE =--

pli,-"- ' - - =,%,

=..-- ..m. NW ~ -w m SCALE c io to so 4o so j 8 Ej i I e i .,,,,,,-'.*ll: %. l REGION AL GEOLOGIC CROSS-SECTION b t .on eese e.=m o.e w r*== THIN-SKINNED INTERPRETATION g,, 8 =d. :. -:::" :~;:,@ '.:.:ll'::::M:: ",.:= = ~ t4OTES: el ont.v nonveewesvenn convion or enoria.e snown at a.ocation or ensne save rnosecreo atona synene o y. 33 scouereso enone Meno ties 43 o :s et (D ~3 FIGURE 2.5-1A O. to' ' um

0) ' J

w 9 i r 2-3 p 4 W 3 .E .I .= g d 3 uountasu 0 P A u titsj g "1 M: s,4 coM A.auca 4,4 s e ......... x. it - , m% 3 ri.&,.. '*.-

M*-

g .. O.L E, tranLv caue n s AN. PREC AM.nt AN) ""'~ //////// A -~m //// /// // ~ ""' !amH ROME FORMATION not e.. g) T His PROFILE IE RASED ON REFEM.NCE. s s. i s....... r. a n o i.s JOY TEST WELL, PROJECTED rauty may as .n a LLow a. (SEE REFERENCE 13)

l. o.,t.s ac co n o.er rest no io ascen.nce e.n SCALE:

l' = 5000' Figure 2.5-17D Area Geologic Cross Section Schematic O e

TABLE: 3.7-5 ? ROCK AND SOIL PROPERTIES MODULUS OF ELASTICITY E (psi) UPPER LOWER UNIT P0ISS0ti'S HTL. BOUND AVERAGE COUND WEIG!!T (pfc) RATIO f(0. MATERIAL 6 6 6 Weathered 1.5x10 1.0x10 0.5x10 155.0 0.3 9 6 Unit B, Limestone 5 6 2.25x10 165.0-Sound 3.75x10 3.0x10 I 10 6 (Partially) 6 6 0.5x10 8 Weathered

1. 5x10 1.0x10 165.0 0.3 6

6 Unit A. Upper Siltstone 1.875x1)6 1.5x10 1.125x10 Sound o 11 6 6 6 Weathered 1.5x10 1.0x10

0. 5x10 155.0 0.3 4

6 6 6 Sound 3.75x10 3.0x10 2.25x10 165.0 Unit A, Limestone 7 6 6 h.0x106

0. 5x10 Weathered

1. 5x10 165.0 0.3

~ 3 6 Unit A, Lower Siltstone 6 6 1.125x10 Sound 1.875x10 1.5x10 6 0 6 6 Weathered 1.5x10 1.0x10 0.5x10 155.0 0.3 2 6 6 6 2.25x10 160.0 Knox Sound 3.75x10 3.0x10 I I 5 Wi Class A 37125.0 3125.0 37125.3 135.0 0.35 7@ 1 Fill j ~i .o. increased or The Modulus of Elasticity for Upper or Lower Bound Rock Properties has be 'fl,. NOTE: i sound rock. \\,,

/ f-8 NRC AUDIT FINDING ~ I. FINDINGS A. CELL LINER DESIGN (3.8.3) For at least four finite elemen't models of different regions of ,1. the cell liner system, improper boundary conditions were used that forced the liner into unlikely buckling modes. The models need to be changed so that the proper boundary conditions can be applied and the analyses should be redone. The reanalysis will involve at least the following four models: a. wall liner at a circular penetration b. wall liner at a rectangle penetration c. wall liner near embedded plates, and d. typical wall liner at a stud connection If the appligant has similar models that were not reviewed he should evaluate them in the light of this finding.

RESPONSE

A reanalysis of the finite element models noted in the Findings will be .pe rfo rmed. A description of each of the analyses to be performed is provided on the attached pages. Results of the analyses are provided for all ccmpleted analyses. ' Additionally, the finite element models not reviewed during the audit have-been reviewed for consistency with'the boundary conditions defined in the Findings. No additional analyses, beyond those identified in the Findings, recuire revision of the finite element model. w ,.n c--

( 1 of 10 sheets) NRC AUDIT FINDINGS ITEM II. CONCERNS In addition to the Audit Findings the following ongoing concerns were addressed during the meeting. 3.b h' hat effect does the angle of rock layering have en the excitation transmitted to the structure?

RESPONSE

The presence of shallow subgrade inclined layers of rock at the CRBRP site is not expected to result in a significant effect on the site response spectra. 1) Reference (1) shows the result from a series of problems addressing the effect of shallow basins on the free surface displacement amplitude variation for a variety of assumed incident shear wave anglis and. layer properties. These results show very good agreement between the more exact solution that includes the actual geometrical variation of the j basin interface and a flat layer approximation with the depth of the flat layers set equal to the basin depth from a particular point of th.e surface. See attached Figure II-1. This shows that while within'.a local area below the free surface (local area being measured in the order of several kilometers) the layer has a certain slope and is inclined, the effect of this inclination or the iesponse of the free surface above is very close to the effect of a horizontal interface at that depth. Several examples are provided in Reference (1) which show that this is true for a variety of assumptions made regarding wave-length of incident wave, depth of shallow basin (which starts at a point of the free surface and dips downward to a depth of up to 5 kilometers), incident wave angle, and property differences in the layer and the half space such as shear wave velocity and density. In all these cases, the flat layer approach s, hows good agreement with the more exact solution. The geologic configuration in the region of the CRBRP site is quite similar to Reference (1) study. The geologic profile of the CRBRP is discussed in PSAR Section 2.5 and PSAR Figures 2.5-1A and 2.5-17B are attached for reference. Attached Figure II-2 shows a cross-section of the rock profile underlying the Nuclear Island structures. For the CRBRP, a finite element seismic mode.1 was developed in which the L very thin layers of alternating siltstone and limestone. rock existing at j the site were modeled as horizontal layers. See attached Figure 11-4 for details. The response spectra generated using this approach were in

~ (2 of 10 sheets) very good agreement with those generated by simulating the subgrade by the equivalent impedances. This was presented to itRC in response to Question 220.5. (Attached Figure 11-5 shows a typical comparison of spectra calculated by each method.) This is not surprising since (1) the design response spectra generated by simulating the subgrade by the equivalent impedances is an envelope of response spectra using the lower and upper bound properties of the layered rock and (2) the rock properties a're quite similar. For example while Reference (1) used a shear wave velocity ratio of 5, the CRBRP rock layers have a maximum shear wave velocity ratio of about 1.5. See attached PSAR Table 3.7-5 for properties of the CRBRP rock layers. 2) An evaluation of wave paths under the f;uclear Island structures considering the inclined layers of siltstone and limestone was performed. Two cases of vertically propagating shear waves were considered using Snell's Law (Figure II-3). Case I considered the waves propagating from the Basement (Shear Wave Velocity Vs = 8000 ft/sec). As the wave propagates towards the surface it refracts at the layer interfacgs. The resultant wave at the foundation level meets the structure at a 75 angle. Case 11 considered waves propagating vertically from the Knox Formation. The angle at the foundation g level is 80. At these angles there is no trapping of wave energy. The wave energy transmission is almost total for such closely similar materials (impedance ratio between limestone and siltstone is 1.43). Therefore, wave reflection is of minor significance due to minimal influence of inclined rock layers and close compatability of rock material. ' In summary, since the flat layer approach is a good approximation to sharllow i.nclined layers, and since the results of a finite element analysis are in good agreement with the results from the CRBRP seismic analysis which simuiates the rock properties by equivalent impedances, it is concluded that the effect of inclined layers on the plant response is not significant. This was also confirmed by simple hand calculations which indicate minimal reflection due to the closeness of material properties of the layers. Finally, the CRBRP site response spectra contains a much broader range of ground motion frequency content than would be expected for a specific geological profile. Its use in the design is therefore a very conservative approach. Reference (1): " Surface Motion of a layered Medium Having an Irregular Inter-face due to Incident Plane SH Waves" by R. -Aki and K. L. Larner of M.I.T., Journal of Geophysical Research, February 1970.

,s '20-30 KM D I PLANT-FREE SURFACE 4 3 1-5 KM u k a6 ^w O 'X m HORIZONTAL g. o y e m. rt en BASIN PROFILE J FIGURE II-l 4

-2 g

(4 of 10 sheets) .A 2., 3 .I. I I .I II. a. l, 33 Ils

p I

= 8 g 11

2 i-8 I.

h ~ e. .a l3 f E I /U I2: I // //. 2 ,. '\\ / g m i l.i e s f 2 -(s su< Y J g i.

h.,i ye O

Zt m gI i k

\\g

/ t\\ / 2 ~ / O: \\\\, -H; I eA I O i t tu g

r

= .i. ( s s m I It

t M

a O i s; ~ f c: a-g = 0 2 / i / 5 Ii,; ) = / .g; a s. 3%$ I E'.I h.s. I. 2 s 3 (A3 343 mot ATA2'T3 S 9 e e I .o .c.-.

i ~,.

r NUCLEAR ISLAND

.* = l5.5* l \\ ~ o f @D 20.67* FOUNDATION LEVEL v.s-OC 14.5* / F. LIMESTONE VF= 5700 FT/SEC i N x.20.s- / / / p f Ob = 20.87* l CASE II. E-SILTSTONE WAVE PATH Ve-4000 FT/SE( OB=30 L D-W ESTONE i CASE I a Vo-5700 FT/Si i ^ /c? C-SILTSTONE / Vc= 4000 FT/SEC @A 50' B-KNOX FORMATION Ve,= 5700 FT/SEC G ~ R A-BASEMENT u VA= 8000 FT/SEC [ g WAVE PATil TilROUGI! INCLINED LAYERS FIGURE II-3

= = 4 + r.s em .c r,m p a .g = = s _= em e d d d d d d d d (6 of 10 sheets)

al a b u

C' Cd.g 2 5-a- 4-4 m C C em C C, ~~ $ .c 2 = 1

- m A

C U ~' ~ y ,8 O3 C'* C.

aI '

^

a: a

..: e a C 34 Z 4. Ze

i. 4 2m

= G J E OC CC' .ZC C@ b C, 3 l-=

i=
Ca

.= = 8 8 = b L. 4 I-i= J i-M

aa M to N e U C,

.8: 4 .E.m t= M M .- e.

e C*

E C -25

n.

2Cm =.C 4 3w

v. =

w ra. ~ ~ m8 = ~ it::::t iiii i t e i i n d -Ill!Ill' lIiL l l i 3 lill!!!' illI l l l l C = E N 3 a 4.. e i o w h l i if i l l l h i l- !!I< il i i ii i

i 1

I I E I 11 1 Ill il l I I l D u ltr il iI i l l l llll 11 1 I l = l j m i l ll e -= 1, L i fil' il i I lii l i ,t 1 Ill ll l l l l i li i i i~ in 11 i i l I h l I tmiilli I t l n l llll11111 i U l 1 l l 11ll1111ll l1 I i l I i 1111111 4 1 I l i . i .x i li 1 i 11ll111ll ll e L lilll6lIl Ii i i i g 1 lilililill i i i 1, I j l l littitilli ii i I L 1 i l i 0 11Illlllil l i i h l l-1 i ~ 4 8 liltit{ll l I i g C 11111llll ll i "Q j tillll!!i i i l v l llll111111 l l l t r tillililli il p lililllfil i i I ll

3 l

I Filitilll l l l l l I m 11llll!!! i 11 l l l l lillilill I I I i I, I l l 111111111 l l l I l il i i n.ninn ei i i, i l y g I i .a

n. nni llll l

5_ lli!!Ill! IlIl i I: ~ i l c. _' l l l O il lllli.. 1Iil t l 11 l l 4 _= d il'!;i, Ii!; ! i i l i I I l i i i i E li l; 'i8 i i l i i l I i i i i i I ~' 't d- .8.: i 1 1 I i i i i t I i 8 FLUSH MODEL FIGURE II-4 =, ..ne* I L.

~ (7 of 10 sheets) 1 .T

T.- ?.IS?OSSI S?ICTRA AT RCE O?IRATING FLOOR (IL. 816')

. ECR E-SOUTH TRANSI.ATION - SSI - 3% DAMPING .. =. i=- 't s ei s' CR3RP DISIGN S2ECTRA I / 2 e,, ~ 2 4 -s e it u .t.

/

V - s~+ t s z o-+ i- - = w + i .c y -.T w 3 .s. .w= ui C3 u 2 1 ~ t gt e 5 i 1 = E: Y I -.t+ { CR3RP,SIISMIC ANALYSIS .____ FLUSH ANALYSIS i c._g s, g. .2 .3 .s .a .s ) 's 5 .= s s ' io io

o o FREDUENCY IHZ)

FIGURE II-5 ( 5 l 4. g, I ^

i t i Cumberla.nd Valley and Ridge ' Belt i. Unaka Delt i. Blue Ridge Belt Plateau; C.a,7 E ..P s ENON VIL L E l A5H(WILg(, p' Q'.b'M.J'*.:..h* d!! f b% N, ...........u.... r....... g.~......._.... % q 7 %.' g. ~ ... 3. s, l SCALE 'l l l H" C 10 to M 40 W 8 l' E3

!'l]*

t

:::~ * '.

l,.T %*-**; I e 1 8 I ( { ,_ s . ~ - 1 ~- ~. REGIONAL GEOLOGIC CROSS.SECTION 1 THIN-SKINNED INTERPRETATION C. O. U**~,'l ~~ I o I Hi W~.~.W e W ~.. % * ***1 **

=..._.- - _

e-" m i

r j

(D (D ttm FAOTES: si ONLv NOniNw EsTERN PO R TION OF PMOPSLE SHOWN e 8 v alLOCAYlON OF CRSnP SITE PROJECTED ALONo BTRIME o r.

l MOOsrsEO rnc M nano (essel n

'.1 et tD

3 Q.

9 G* FIGURE 2.5-1A (n ss

.s u NW SE 9 E E 9-a 4 (W E I g h "b WHIT EO AM MOUNTalM g w F AU LTl.) x n.. CREEN FAULT un ,r-CO PPE R = s... -- .,+4+ 5::"-:: =- c .EA .E.E. ..A cowAmaUon 4 9 g sn* l A ,<,e n a M Au. A

l..

e 3... 4... 2.%'. MMOM MMON 4 88 4... e A... gggg 4... m=.".= ~.. n.. a.A..A r..m w y . O t.E 4... PAULT ...G ggggg .EotMENTARY ROCKS

gggg, lE A R LY C A M e R I A N.

PREC AM.Rf AN)' ""'~ //////// / a.E.M

/////// // ~ ' " "

Ow 3 ROHE FORHATION Q noT E.. tl THl.* PRt9 FILE t. WA.ED ON REFERENCE. i.. is am.a..r. Ano s.: J0Y TEST WELL, PROJECTED o" L PAuty May sE As .w a LLow as al . o. p.E G (SEE REFERENCE 13) rest AccoRoine To RErERaucE s.a m D" m SCALE: I" = 5000' e et Figure 2.5-17D Area Geologic Cross Section Schematic e 9 0 = 0

TABLE: 3.7-5 ROCK AND SOIL PROPERTIES MODULUS OF ELASTICITY E (psi) UPPER t.0HER UNIT POISSON'S NO. MATERIAL B0UND AVERAGE BOUND WEIGHT (pfc) RATIO MTL. 6 6 6 Weathered 1.5x10 1.0x10 0.5x10 155.0 0.3 9 6 Unit B, Limestone 5 6 2.25x10 165.0-Sound 3.75x10 3.0x10 10-8 (Partially) 6 6 0.5x10 6 Weathered

1. 5x10 1.0x10 165.0 ',

0.3 6 6 Unit A, Upper Siltstone 1.875x1)T 1.5x10 1.125x10 Sound 11 6 6 6 0.5x10 155.0 f 4 . Weathered 1.5x10 1.0x10 0.3 s Unit A, Limestone 6 6 2.25x10 165.0 Sound 3.75x10 3.0x10 7 6 6 6 Weathered 1.'5x10 1.0x10 0.5x10 N 0.3 3 6 Unit A. Lower Siltstone 6 6 1.125x10 So nd 1.875x10 1.5x10 I 6 6 1 Weathered 1.5x10 1.0x10 0.5x10 155.0 0.3 2 6 Knox 6 6 2.25x10 160.0 Sound 3.75x10 3.0x10 5 l 48 1 Fill class A 37125.0 3125.0 37125.3 135.0 0.35 E'E l The Modulus.of Elasticity for Upper or Lower Bound Rock Properties has been increased or g L decreased 50% of the average values for weathered rock and 25% cf the average values for NOTE: om e sound rock. 5

i....

E-n.

' 9..L y-~~-~~~~ 3 ,z

p. X.

f :/ ' h ,1 ~ hp ^.j5 , [ L' [ ^. ~ ' ' ~ -: n ?,: v

n..

m:. +.y 4 ~Y C+ h ' ~ } f[<. .a., j s v <' y ,4, ~, - .'s.

"r i

~ t t -u .g,'._ 4 * ? m_. W [, .!**'ema se Caesurssca Kaaaaam Voa 15. Not 1. Feassaar 10.197: .f[,

c

aAas.

'.1.)4[. ths NO nal q; Surface Motion of a Layered Mediurn Having rn M. N Irregular Interface Due to Incident Plane SH W ves $:1.f - czents in the Q.A

electrostatic

. J. G*ophys Emm Arz axn Kr.xxrra L. Lanura @w ~.2 W etrol An.stveu Department of Earth and Planetar; Scioness NU'hi den-Day, San Maseachusetts Institute of Technology, Cmbridoc, Massachuse:ta cesso W,Y < .' b.@0b. U. B. Potest A practical method is devised to calculate the elastie wave Seld i. a layer over-half-epace E.. medium with an irregular interface, when plane waves are incident from below. This method rn,cf Idoste may be used for studying the interface shape of the M discontinuity, for example, using the dp'M' - cativity ap-observed spectral amplitude and phase-delay anomalies due to telescimmie body waves. De &=Q I F- =- laitsd. Gw. method is also useful for the engineering eeismological etudy of earthquake motions of soft .W 1 F.%.OA super 5cial layns of various cross meetions. The scattered 6 eld is described sa a superposition 8-of plane waves, and appliestion of the continuity conditions at the interraec yields coupled 4 integral equations in the spectral coeScients. The equations are satis 5ed in the wave-number Gi. 0 domain when the interface shape is made periodie and the equations are Fourier transformed @/(f'44. and truncated. Frequency smoothing by using compler frequencies reduces lateral inter. C- ^ h.1[c. ferences suociated with the periodie interface share and permits coc2parison of computed results with those obtained from 6 nite bandwidth observations. Analyses of the residuals in W' ' '. the interface stress and displacement, performed for each computed solution, provided esti-DdM. mates of the errors. He relative root-mean equare residual errors were grnerally less thsa

n r$ !

MR 5% and often less than 1% for problems in which the amplitude of the interface irregularity hd,hn 5 and the abortest wavelength were comparable. The method is app!ied to several models of M P @T i

soft hamnn,' ' dented M discontinuity' and ' stepped M discontinuity.' The resu!:s are com-pared with those derived from the fat-Isyer theory and from the ray theory. In addition to

[f.$ 2 vertiesl Interference cEects famihar in the Est-Isyer theory, we observe the efects of lateral p, gj interference as well as those of ray geometry on the motion at the surface. y 3,s Inntent7erton cients are determined in such a way that the p- }! boundary condition is approximately satisSed. 1 - This paper describes a practical method of interpreting seismie observations on the surface There are several approximate ways to satisfy Lf+..gj. the boundary condition. Rayleigh used an stera-of a layer-over-half-epace with an. irregular.in-f'y g : terface, as shown m. Figure 1. We simph.fy the tive apptctimat. ion, expanding the bo'mdary condition in a power series of the ampb.tude of a problem by assuming that the med.ium is uni-form m. the y direct:en, and the depth { to the corrugat. ion. The same approximate method has s.s - s been applied to seismic problems by Sato i [1955]; Abubeker [1962s, b, c]; D:mkin and

g interface is a function of r alone. We are in-1 t

terested m the motion at the surface, whe Eringen [1962]; Asano (1960,1961,1966]; and place waves are tocident parallel to the :,a 3 M h U956] C'#- l place. The case of SH waves is discussed m this i ed a variational method in which the bound-hC4.\\ 3i paper, and those of P and SF waves will be arv condition is satisSed in the least-squares Q3 reported,m a subsequent paper by the present g g. g g, pg; authors (see abstract, Lcrner end Aki [1969]). 10'9$,; l 's but takes advantage of the fast l The prenple of our method ts not new, but g. g,3 g,, g gms b:ek to R yleigh [1907,1945), wh Tukev,1965] and satisSes the boundary condi-Mgp i studied scattering of plane waves by gratings ,x, (g ; tion m the wave-number doma.m. In our method, the teattered wave fielo.ts rep-re ented as a linear combination of plane wave, .\\n ther imique feature of our method is the D.'MMi t w.th dis rete hon.: ental wave numbers (' eIud-a cf complex frequeney. The t,me function of r v.3.. i m C"' E h " 3" b t h' I" '..., m, "y~ I in: tsho=o, creous waves), where the cd-in: : e frg:ency., a ecm;,e: ne=ber, we are f'- G,-*n;'.s C ltra t.y tie A:sence GdFin! M* Y 933 I s ~ . l _ ~

W$.,.S-WVINik.y5]9.,gz~y$.,WRUfiRTQQ@fQ ~K.G ,V Yh5$ Y%2.b r W '.G: a .,.....% 8E.M,.,::...p:y C;p; g,JX.4 y~5 g 9.g;o.,.t Qq. _ z T. .9 %'. g - 4.. p> .: y, -. w.yy -~.. ,h.,. w>...,.: n. y, 3.3.ihd.W.-$M-N[M Mk[ hN y 1 .,,o . p. g .,,yy3pg,.g,9 .d.h,-d f f -. y;g hfyg,j] 4 ~ a M ' 2,f 'if.y,-3f y,' , f /j J 5 t AKI AND LARNER ~, 934 "d Haskell method (Hoskell,1953]. It is impor-i )3 [5' y,, , r, w C II I' tant to know how the resonance conditions aro m D2 affected by the !steral variation of !syer thicky l i

" 'A mediu ness. We shsil show in the present paper some j results of the successful application of our ;

q) war

8 7,

,,?)i G d method to the above problems.

7. - O. I

, c]d g u w. . p. rs e, Fonut t mon or Pnostzu ,"aion.' i,l

==. 's Fig.1. Schematic cross section displaying the As shown in Figure 1, our medium consists of rgoro kms i ~, layered-medium configuration and coordinate axes a homogeneous, isotropic hyer with shest ve. g ?. .1 d locity # and density pi overlying a homogeneous !arly than t } amplitude and phase-deby enomalies smoothed isotropic half-space with shear velocity S. and F

i over a certain frequecey band, whose width density p.. The interface depth C b: function irregi

," y may be chosen appropriste for a given problem. of : alone. Our problem is to find the motion at In ma There are two msjor problems that have the free surfsee s = 0, when phne SH waves ptmg ( motivated us to develop the present method. polarized in the y direction with freqency. are l'e ne factor-One is the use of amplitude distribution and incident parallel to the -: plane at the angle i phase-delsy anomslies of telereismie body wave.

6. from the direction. This is a two-dimen.

re.eidu; {. would in the study of crustal structure.If the Afohero-sienal problem, and the solution is independent at the vicie discontinuity is not a horizontal phne, but of y. to asri l has an irregular shape, the amplitude of tele-Let the dispheement in the upper 1syer be tion t. u. ~ seismic body waves at the surface will show a vi (r, ) and that in the lower medium, u,(, ). to tes i'gQ. 3,' spatial variation owing to focusing and defocus-For e"*' time dependence, they may be ex. t i evalua p '1 ing. So far, only the rsy theoretical approach pressed as dition: has been undertaken [cf., MecAler and Rocard, er7',.y<l 1967; Mack,1969; Meres,1969]. The ray v,(x, ) - [-- [a,(4)e"'*"" The {($ g J. stress theory is inadequate in problems that invohe ^ of str 2.d wavelengths comparable to the linear dimen. + B,(k)e'"-""] dk (1) (gp.3,j.' ] The a l i sions of the interface irregularity. We are fj applying our method to the Afontana LASA u,(r, ) - e"***"** + [-. A,(k)/"*" dk tion 1 h.,OMi j data m order to determine a more unique pie-e;fy ture of the 5foho shspe under the array. The

% g'd, i j

result will be reported in a separate paper uhere k, and v. are the and a components of ,,(, b;.e i [Aki and Lorner,1969]. the wave number of the meident waves, respec-The other problem is the so-called ' ground tively, and The ir p$3 h, motion' problem. The spectrum of seismic mo-(,sjg,n,, gs)"' = w/S: ces 8 (3) [Mql tion at the earth's surface shows peaks and troughs owing to constructive and destructive ei - (w'/#,' - k') Qlh'sj,! e (O interferences within the surface layers. Earth- " (" /88 ~ Dy, f.:.1.. quake seismolegists are interested in this prob-h. lem because they can utilize the shape of spee-The signs of radicals v. and v, are chosen en (l2~ Ml.f '., eng:ceermg sets = ologists are concerned about term in the right hand side of (2) expres=e4 l l trum for studying the crustal layering. The that, when the frequency. is real, the first this problem beesuse the ground motion esn be the incident waves coming from = +:c, and c ;j ampliSed signiSesntly at the resonance frc-the second term expresses the regular waver {' g.jf g quencies. cattered back toward * = +co and inhomo-f ..E; The esse of SH waves in horizontally uni-geneous waves attentusting toward = +ce. forra byers has been extensively studied by (The extension to the esse of compler frequene) Kansi and his colleagues [Kanci,1952: Kcnci is discussed below.) et cl.,1050]. Cases invc! vin:: P nnd Sr une-Questiens have been raised [Uretsky,1905F g [ have been studied by #cstcli [1%0. IV.21. conecrning the vr.lidity of expressions such s r. r t. Phinney [199], and others, u.ing the Thomsen-(1) and (2) as denriptiens of the - sve field 1 ch I. t a..% : y 3 . c: n. 1.:.... m iUJ

~ j;., ,,, g? ' _ ;,,.c.. g.,.. e. -.. - - -., h.,; :,Y,6. % w u _hhYS ?. ,, Y o.l $'N;

.' * '- ?Y U

h5 ..w.;. ~ g k :i..'. ::.. ep. c. .w r + -- ary,d..,g A g.c..q q:.t 3<; r,. .4,d cc

y. - :

_l ~ . - ', m,. '..,:.,. n ,,;. y; ].;o.. ~ y ~ - ,pu.,- p 3..( r qs;

v

'll. ~~ L . '~ V - r s m ~ ',.'q'<. y .fl -l_ ~ 2 *;

- t
e

-m a Q, s_ J. m. ? m. ,..._.,__a

ffX.:. o., U t

r.. %.. '. 4: I V%g}

SURFACE MOTION OF LAYEltED MEDIUM 935 g, 3 7. ' "'r the interfsee. The essential dificulty is where p is the rigidity of the upper layer, p. .r.. 3r-bI ' N. 5.9* r:.at for ((r) < < (, where (, is the maxi-is that of the lower medium, and 3/dn repre-c.

- 3 1.

rr.um of ((:), the scattered field in the lower sents the rpsee derivative in the direction nor. m-dium may include waves locally propagating mal to the interface. 'k47 3 ne upward, as indicated in Figure 2. Therefore iR. g tra ur I U.$' g.M,!%y-i;) does not adeqt.ately represent the wave a !. eld near the interface, and the boundary con-g = uvad = n,(af,dr) -- n (8/de) W y,g, thrt are fEligf.%l suons at the interface cannot be satisfied er rc:orous!y. The discrepaney is greatest in prob. n, = -(df/ds)(1 + (df/d Y].'" (9) 10 1 {hih]$@k; krcs invohing istge interface s!cpes, partieu. n, = (1 + (df/ds)']" /* .e. }-- larly when the source wavelength is smaller I;fffg[7H us

i.. sine (2), (5), and (4 in to and (7), we get li 3d than the amplitude (C, - (.) of the interface 3dM':Ej' irregularity, where C. is the i-m mum of ((z).

an at In many problems, however, the upward-propa-f. ( A,(k)pn(k. :) + A,(k)pn(k, :)}e"' dk(($j'@.iN ,9 l 3.W! (33,. gating scattered waves near the interface may es .h $g0.N be nefgibly small, so that (2) may be satis- = h (:)e"** d re mit i .), factory for p.netical pt.rposes If so, small (10)

c. ;

j-d .~ g.,.M-Md. h residus!s or discontinuities left at the interface [ 3,g ),,,g,,) 4 g) g,,p... d n. u.uld generate only small observable motior.s ct

'l ', e st the free rurface. Therefore our approach i<

or = h,(s)e,,,, k gf 3e to assune that equation 2 is a good approxims- %,n6 jf.'y{ 7 .)' tion to the wave field near the interface, and where

G q. l T!r to test the adequacy of this assumption by x.

evaluating the residuals of the boundary con-Fu(4, :) = 2 cos rif(x) pla i d;tions in each practical problem. g,g,,),, _,o n,i l (, The boundary conditions are the vanishing 'f { ~g stress at the surface = 0, and the continuity p.,(k. :) = 2m[ika, cos y,f(x) L a-- t The strees-free require:nent at the surface is ~ 'in, sin rif(x)] of stress and displacement at the interface. 0 $: h wb. (ll) satisfied if we put B,(k) = A,(k) s.o that equa-gn(k,1) = n[ika, + inn,Jeart. -u.r m tion 1 becomes 2) A,(2) - e

y

,I u M: w(1, s) = 2 /- A,(k)e"* cos e : dk (5) h,(x) = n[in,k. - in,,.]e-"*" "%s'$ OI r:. ee c. lly s lving the above two integral equations uj;gi, The interface conditions are for A,(k) and A,(k), we can determine the r g4 e, a 3) w(1, f(x)] = u,[, f(:)) (6) wave field in any part of the medium. n(a /dn) = p,(Sw/dn) (7) APPRoIIMATE SOLUTION h(kg] 4) w f.@jE{$h 1 In order to solve the integral equations 10, we convert them into infinite-surn equations by { 8 aerw awr t,re,; assuming a periodicity in the interface depth fryi&3 'a'*** ((:); that is, M Y.d.l .Ins e'

- tt>>

O/ E

p. 61

('".4 in 1d 8 f( + r L) = f(x) m = al, 2,. - (12) bot -d. d( e' I-teg in the examplee that we consider in this paper, k .83 r-e. 2s h:y$ @y l the interface is plane exetpt for a loeslized e. M. interval in.the : coordinate. If L is taken long n r ]..g [ as comp:. red to the length of interface irregu-g. n o,. !arity, the c!ect of repeated irregular tres at y ! ic. 2., Sebemsta.c t!!ae.r tten of the recion datances cf mL can be cegieeted. As will be i~

9..!.

ir tt.e inter 4.ce displayi=c causes of the Ray-o e r s j .h r.m.:, rer, shown later. thu. cEcct can he easily d... g

m:ms
7g

.~ ;:T-}-h,} ', M' '.

~"' %

f[

.. ~.

.. :.. rn.. - .. ~ q m. _ w s. <,.. w ,., - -- -~e.s. .h d '. .A,1 i hjI ' ,.. y[ ' ~ ~ ffRf& pg 'ohN -f,j','f: WQ:'[: ?

@;h (q.;l-;. W!Wl&*

h.l }, _ _ .y- '}

~

~< '.,,' ;c l v'h,. , 3sy.. e .x. gc.. ~ w. mv. .e : - -.,. v... m n .., 2 T. i' .j..x p ;C.- g,. ';,... ~y - 4 +fN- --] " Q; ?.,. C.. (;.f'g '

- v

.n ~ t 938 AICI AND L.ut.NER

voiding tl

-n ' O by making the frequency ecmplex in such a where complex. 'l way that the imaginary part of frequency is t t!.e set =ne sr;;e enough for waves to attenuate over the q,,o "..I. g,""(s),' "'* -"'"' dz 1 . ravel distance L. L* Phin"G' I f %M When f(x) is penodie, A,(:) c.nd pu(k, ) i e are also periodic because they depend on only G.."" = 7 fa F.""(s)e" "*- *'"' di (18) now looki: h bnIng t1 through t and df/d:. Ist us multiply equation 10 by e-'***. Then we have I penstialh ' ) H,. L h,(x):.....a t de s=plitudes d ) [ A (k)an(k, :) + A,(k)p,,(k.1)]e

dk The matrix G.."" should be well behaved as nary part

. i i i C, long as the variation of ((z) is small. In the the reasm ' w: ; 1 = h,(s) i=1'2 (13) case of constant f, for example, all G "" van-Nside ~ ~ where 1 = & ~ k Since the right hand side of ish except when n = m, and all H,. vanish varias pa (13) is periodic in 2, the left hand side likewise except when m = 0. Thus, all A."' vanieh ex-so aerve murt be. Since p.,(1, ) are also periodic, I can eept when n = 0, which corresponds to re-from a tri only tr.ke such vr. lues that satisfy Sected and refracted waves with the same wave ,g g,g,, number, k., as that of incident waves. In other the propt p .-'j g, " *g,,.g, words, there are no diferely scattered wave- ,g 3 7 8 l m this case. caused th '. l IL = 2rn n = 0, 1, 42,... Using the ' coefficients A.'" determined by ad th e {. l so!utan of equatans 17, we wn,te the approxi-turbance Thus, the integral equation 13 must be re-mate e lution for the d,splacement field as we take 1, i I..';%yiii placed by the idnite-eum equations I efect of 1

f. ' -

h. Y[:. } h f ( A.'" g,""(r) + A.'"g,""(1)]e'"*"'

"'.(1, :) = 2 {, A? cou?: e*' remond. The fre e - h (in layer) 6 f. = h,(s) j = 1, 2 (14) (19) ease is no tinuous s; M@ g where I u,(1, :) - e..... + [, A.,,e...,.,,,,... tional to e A.,,, = A,(k.) Ak. . = 1, 2 a--* -6Mf*M (in half. space) corresponc W NIM;j. g."" = p.,(k., 2) i, j = 1, 2 quency b . '%r

4..

(15) where frequency v N: 42i i

k. = k. + 2rn/L economise 3

,,rn $,.,9 6/.i i.l l A1, = gr/L

k. = k. + 7 transient t h,;g;[4}j'.

The sp Ni We now approximate the infinite-eum equa. oi., gijgs -- k,')"' y = 1, 2 phase cal t tions by the finite-sum equat,ons, be compa-i [v e e/ $ 3 by spectr Sd Suoertisso ar Taf t Unz or Court.tx

r. s Fnum S

'@l E ( A.'" g.""(s) + A.'" g,""(x)]e'""n. uiultiplied ] t.Qj,! The solution obtained above is the recpon* f(t) = e- = 4,(2) J = 1, 2 (16) to incident waves having the time function at each a G ~k ,p. e*'. Whm. is real, Ge dution corresponds of source t Instead of solving the above equstion directly t the steady-state case; that is, the respone When t in 2, we first take the Fourier transform of to sinusoidal oscillations 12 sting from the in-the length [3 'f. both sides by maltiplying (1/L)e "' and m-r te;;rsting over 0 < : < L. Then we have 4N + finite past. In this ease, the frequency spectrum ecmes cai [ ^ *%j - is a discrete line. In practical problems, we Ixity. Ho 2 simultaneous linear equations, .s - -d are always dealing with a signal of finite length, penents c which has a continuous spectrum over a cer-unique!y ( A, n G.,,, n + A.,, G. n,,] = H,. g.dn frequency ha.d. If we wanted to cover

udnatiss, the frequency range with the eclutions for line The cendi 1 = 1. 2.

m= -X. . o. +N spectra, we would need r.n inf. nite nu=ber of k= is th (17) rueh se:utions. A ri=p'e f.nd efective wr.y of mpHtud: l M_' web-' Ee e d A 4,. eye

W^- - ;y, ,;g. s,y, ; :.f~,j;'Qi.%m.-w.,,w .n. --e4 y,w. ] _...,,, N. 1 ~ -

K.G." j,[A "t

P,g, fg f " M ;j, y- ',.3 , _4 ^;..i-e g-: ..:q-p p J M y -( 7,3 g p;.. -s e.3 .g.

~

~J.

u..,
W a.

9, .~ . j' ..a

? y;, n,

..y. 's-r t. h* s y yy - * .? a, k .y.,.. s m-1 ~ i ,,. x

P -

SURFACE 6tOTION OF LAYERED SIEDIUM 937 Ii l s>oiding this proble:n is to mr.ks the frequency we request that not only the real part but also capkx. Re trefulnees of such a procedure in the imaginary part of phase kor -r.: be con- } ' ' d t!.e ceisnegrazn synthe:is was pointed out by stant along the wave frent.That is, b PMnney(19G5]. i, ' 8 i By mding the frequency complex, we are R e (ka): - Re (r ): e 'W '"#' dr (18) cow lookicg at a response to incident waves %p% + N I " (E'I' ~ I"'()#! " C "8t""" baving the nonstationary shape of an ex-

h. ' ~

ponentia!!y inerecaing ra metion, in which for : = x tan e. + constant smplitudes are ineter. sed by a factor e at every where e. is the m.eidence angle. It, then, fo!!ows

. y time interval v = 1/.,. Here,., is the imagi-that p

c3 behand a' m cary part of. and must be taken pontive for t-r cna!!. In the the reasons expla.med later. Re (k.) Im (k.) G- = tan e. (207 Q' Consider the contributions of disturbances at R e (n.) I m (n.) ~ **"I'[ F-various parts of the medit=n to the motion at .UM'- L',' vr.nc. h ex-an observation point at c = 0.The contribution Im (k ) I m (n.) w, rpends to m from a travel distance : must have originated Re (k.) " He (r.) " w] = e (21) g,5, g w$. - % ' the came wave at that source point at t = -r/#, where #. where e is rest neitise anel may be a function is h& the propsgataan velocity of the waves. If of., the real part of.. C'5.2.*C r/# :;> r, the amplitudes at the source that According to the periedicity argument sur. Y Y- "g *** caund the dicturbance were negligibly small, roundin,, quation 13, the integral variable k g and therefore the contribution from that dis-can take only the discrete values given by C

f turbinee may be neglected. For example, if p y?._ '

mt Ec!d as we tr.ke 1/., smaller than L/#, the undesirable

f. - k. - k.

h% k;$@R efect of the repeated interface shape can be (n = 0, s 1, 2...) - 2rn/L ', c ' * - removed. ne frequency spectrum for the complex = Thus the wave numbeni k. in the infinite-sum (19) case is no longer a discrete line, but is a con-equation 14 must all have imaginary parts equa.1 j 1 tinuous rpectrum with the bandwidth propor-to the imaginary part of k In the complex k tional to.,. Therefore, the solution abould plane, this means that, when a plane source F (in half-opart) correspond to a smoothed one over that fre-wave is given in terms of complex and k., L

/

quency band. By this procedure, we lose in periodicity imposes the requirement that the kg,,3k frequency resolution but gain in stability and summation in (14) be over discrete complex h.gf k3}W economize comptitation time for appliestion to values of k. that are equispaced along a line p t,. a trsndent waves. through k. parallel to the real k axis, The spatial distributions of amplitude and T_et us consider the complex k plane in more Mk j = 1, 2 phase calculated for complex frequency may <!etail, first discussing the branch point locations tN.$C be compared with those that we would obtain and remarking on the choice of signs of the -d.?K,[.b [%.N Cou rt.zx by spectral analysis of transient records pre-vertical components of wave number (equations multiplied by the time window of the shape 3 and 4). Since all the integrands in (10) are I tha response f(!) = e"", which should be properly delayed even functions of r, we need not worry about 59h tima function at each station according to the arrival time the sign of v. The sign of r, is so chosen that f.%ii n correrponda of sourco waves. the backscattered waves attenuate toward : = Q.{. jygg. , the response When the imaginary part of is introduced, +ce. In uther words, we use the top sheet from the in-the length of the wave. number vector /# be- [Laprood,1949] of the complex k plane where y eney spectrum ecmas ecmplex, where # is the medium ve-Im(r.) is positive. The branch cut is deSned by b/M problems, we teeity. However, with this introduction the com-Im(v,) = 0, which is a part of a hyperbo!a, fjrj3g f Enite length, pnnents of the wave-number vector are not as is well knows It is also well known that a;. Egg n ovtr a cer-i:nique'y determined. For their unique deter-Re(v,) is positive below the hyperbola and is .N K

ted to cover tiinatien, an additienal cenditice is required. negstive above in the Srst quad :nt of &

jf 'O. et!= for line TI.e e:nditien that is apptcpri:te to our prob-p!:ce for positive... .: c::.kr of 1.m is that the incident waves br.ve constant The locatiens cf branch lines and su=m:tien dra wcy of m;;!itude s.!:ng the wave frent. In other words, points k are shown in Figure 3. The e line, 7 t 1, nb GANSa _3RARY M _ESV __E, ?A. g m

+:Y);,q* -_ m c.,.~ n ~ we-o-n

q : 3*t -. K. -

. a: - -;* m&,3i 5: ~- (.. a.5, 12 2 s'.~'.n,..i s

. L d;

- %..s= =W..

e,.

g%.r y h-(, - r. .. :.. _, n.".h.l-

w..y.:e

g.,

t ,.,k'j.'l.t$Wski. gy;gsl.j/* f"&. n&.nQ ,e .c ~^ ., gD- : ., M in Q ::3' ll v

2. m,,%,...,

r .c x.e. .. y... ,. m.., j -_ d.' . { l~ .? ' ' s :l,,. sw g.[c's b ~.. _ . h.;.. "{. lkd .h-1 .w.w:4,;r~- ; .~ .,, e. e u,s r ;:. ~ - n - .. u .s. 4 f

.g;c e.Q

.x. a ;;.. d Qi, .., 3.f.! ,[ .'l .;.l ny, t .{ g3g AEI AND !.ARNER In t al T..e varisbl

1. mdary S, I.

'8"" M' the outwars

~ r 3

I '21eo t;. n G sati!

- cf., ! 6.in Im te,)

O sO

a. ( 2 < 0 pa + pa 1

W != (e,1

o

.) P e ('2 )

0 throughout og /

Sir) is the e '- , ly[Jl /u N Now, let --- k.,y a a nre:ular in a a a a x x-x x.- x x x x x x x x x 3 y o,ma* ***

a h

n,, m n necting the . '., !. i y

e. The e
eiy

/ e I). The exa fid the ses -e .,ii taver as d . L4 Fig. 3. Locations of the summation wave numbers k. on the top aheet of the cornples k Ti>ere, fore, ? plane for the esse w =..(1 + in). The solid square is the branch point, the open circle w. g g;g the source wave number, and the crosses are the locatior.s k amount of we can ne which satisfes equation 21, is' also indicated. characteristic lengths of the interface irregu- ./ The plane waves corresponding to the points larity relative to the wavelength, and propsgs. ' " '['l

I,

% } lying cn this line have constant amplitude along tion direction of the incident plane wave. i he that solu "INM%] [. ! ?"J. t a wave front and have a real unambiguous in. Since no exset solutions exist for this problem, r san shes sh cidence angle 6. as defined in equation 20. The we have used relative methods to estimate the

c b. %DD I incidence waves satisfy this condition, but the accuracy of our solutions. The various error

'M"I* '"' h hi$.k 3f 'I.! up a both scattered waves do not unless e = 0. Thus criteria considered are based upon evaluation of t ns to a ' W JJ.1. 3 f whenever is ecmplex, the ecmputed wave the displacement and stress discontinuities de-f[ amplitudes A.** and A." pertain to waves termined at the interisee. A relation that fo!- diference ((I7@[hl0 a W 'Mh! whose propagstion directions are uncertain. In lows from the conservation of energy provides I"."' U "' ** KirchhoE : fact, for k. to the right of the dashed hyperbola ancther check of accuracy. However, this error motion can ',g'? ii in Figure 3, Re(r.) < 0 so that those waves measure is a comparatively insensitive one that the errors i J. ' j.u.!'t attenuste downward toward a = +co while is equivalent to a weighted integrs! of the l "'* I f.p'7 appearing to propagste upward. Thus it is boundary condition residuals. We obtain a more "'pg-;;g%}i t difficult to attach simple physical meaning to meaningful estimste of the accuracy of the com-Aun(r, 0) l ".yrdQ the wavts that contribute to our frequency puted rurface motion by using a representation ' %]y"%'{ Q: 3;, smoothed solutions. theorem in which the residuals at the interface [ are taken as sources. " J**"* yy Axazrsta or Elutofta aND Rr. sot.t: Tron Residuale et the interface. To estimate the N C MXf'{ Sources of errors. In a previous section we errors in the wave field associated with the [.f.dfj{ ?; cited the existence, in our method, of an intrinsic interface discontinuities, let us first consider the di 1 l ,.;, -l.,. z. error attributable to the incomplete description Kirchhoff integral solution to the scalar Helm-l "h"' J of the wave feld near the interface (Ray-holta equation. If u is a rolution to the equation .,y. leigh ansatz error, Uretsky [1965]; Meecham that is valid at all points of a region R enclosed I [A, d (1956]). Other errors thst can arise include by the boundary S, then it may be expressed arg( ', j - ' l'd tho*e attnbutable to truncatmg the infinite-as the integral. ,. _ ;,.?. '. - sum equations, to introducing the periodie in-

t. the errot

, ' " ~ terface shape, and to Emoothing with the u<e '2 D e '; *,' ',) aufz', s') the interfac ( ef complex frequecey. The list ef errors can "N' d ", " an The Gre l Wo include those occurnng in the numerica! < e rurfaev c;.!cu!stiens. The sizes of thne errors depend ~ upon critical p rsmeters rueh r.s tha shape and - *(2,e ',) aafr, :: z', s') e n eer. sir dS (22,' 6n . ;iven by s. e,, T .Aa. ., L,. Y. 'usw

9-...a '. r5.. >.. s. ~ --,,.m; a s'p*F'?.TS

"? ~~ 5'WN..

<-c _ ~ ..b ?- h*

d.I'.

~~ . b,. 3;- ' :..v.. f.". ya. - : -Q, if.q.g.,y,.,x- ,.r..,.' v.. . t p;,, !: ~ < '^ry .n ' ~;... " :3 r -: y:: '.. .\\.a. K.* 1,;,..: ; * * -[.h.: %..'. Q, j.'._ .. n. j r w. l w;.. s ~: z 1 k SURFACE MOTION OF LAYERED MEDIUM 939 E... 1' I' The variables t and / are coerd'estes on the 3*I le:ndary S, and 3/an is the spatial derivative in " E _N'o,/wQ N) + N'n,D~ N,\\.[ (25) s, - i \\ /w .?. the outward direction normal to S. The fune. 8 t.cn O satisSes the inhomogeneous wave equa. ^ 'i n where i!" ' ) B'O + pu'G = - 3(r - /) 3( - s') (23) R = [(z - t), + (r - s')']in gr. ; J.(.7lj throughout R. Here, p is the mass density and N, " N* ~ #,), Y (# Y ),3in atz) is the Dirac delta function. Now, let S consist of the free surface, the II.'" is the zero order Hankel function of the kJ.?, irregular interface, and vertical surfaces con. first kind. Aleng the free surface, s = 0; this D.C5 [p{grM hecomes necting the free surface and interface at x =

c. The enclosed region is the layer (medium

[M t). The exact solution w to our problem satis-c(i,o; t,p). _.8._ y," .7 m 2m fies the scalar Helmholta equation within the iQ y < laver as does our approximate solution u,,. where 59' p '**N*I T!$ere, fore, the diference Aw, = w, - u, r = [( - t), + t']in <;U;t. s-pen circle la must satisfy (22). By assuming that a sms!! amount of attenuation exists within the layer, By usir:g (24) we could compute the exact W we e3n negIcet the contributions to motion free-surface motion, provided that we knew $(, nterface irregu. from sources along the Vertical surface at in. the errors,m stress and di!pheement at the

,: ,g finity. We now take the Green's function G to interisce. However, we do not know these i' ':.i h, end propsgs.

be that solution to (23) whose normal derivative errors; instead, se know only the diferences j one ' wave, sanishes sloeg the free surface. The stress-free (residuals) between the approximate solutions t fcr this problem, 1 to estimste the requirement at the free surface was impo=ed' in the layer and in the half-space. These q e various error up n both the exact and the approximate solu. residuals are just the diferences between the on evaluation of ti ns to our problem, and, hence, upan the real errors (at the interface) in the layer and { ieontinuities de-diference Aw,. With our choice of the Green's in the half-spsee. That is, et the interface

Isti:n that fol, function, the free-surface contribution to the d "
v ~~ "$ ' " dW" - dW" energy provides Kirchbos integral is zero; the error in surface (26) i rever, this error motion can therefore be expressed in terms of

. 3 n:itive one that the errors in dispiseement and stree alone the f = ri, - r.., = Arin - Ar., integral of the interface. We have where d and f are the displacement and stress MS b'O.@ e obtain a nacre and Arw residuals, respectively; and Au n sey of the com- "4,0) are the real errors in the half epace at the inter-b @' % a representation fr.ce. In order to make a low order estunate of nt the interisee r the error in free-surface motion, we shall assume = }'""'"* G(r, 0; /, f) Aro,(/, f) s that the residuals are comparable to the actual R,. 4 to estimate the errors at the interface. That is, at the interface, 13M

AWr(2,, f) ag(,, o: f, 7)"

isted with the dS (24) d = Aum, f = Arw. ng rrt consider the h M[$'@ i ie scalar Helm. Using d and f as sourcos in (24) yields the to tha equation ni ere estimated error in displacement at the free S.~f

surface, Guidr', f) _ du,(r5 f pfh don R eneIceed y be expremi ar,(f, g).

C(r 0) = h H e" ($g$ 0" 0* i e the error in the y component of stress along w u dS (27) M [g __)/ cn, ("I _,) g' th interface. 4, \\ 3, B, _ l. 04 The Green's function that s:tisses C3), the ic 1 d.- i e surface ecndit:en, and the radiation eendi. } when-i n consistent with the e-'" time dependence dS ph - dip aagte of the interface [' I given by a I(w i W. 3

f. %K..,,.

s a4%WE.f.QM w.r.w

.N, f."w..;.'.g? f wllQ %:.2.%.n ^sh. ; - 7 >,,a.- 4 ~ = - - r y P 1 l . s'i9 : W{Q ' E. MQ:Q.g.,s.QQyyp:,. ll fu,, ',

, p ;

_ *9.f 3 ' ~ Yp. Ql:V.%QQW ' ' ' 3 : D.. %.h?:,%% J ~ .m.., g.;.-1

p.

~ -?Y.; l?:dh$'$$ INS

a. :.; )l % ' ~~.b, '

(. ' 'y. a ++.:.:.mW.y 1 . u.~, - 4 . :.c _. [ n r - ' .y, a

c r..;.

+~ l ,3 -s t. g) AE1 AND LARNER the truncadon is TABLE L Eoowncan-equan (rtr.s) Errors and Conservation of Energy Errors (4) for .. - f-tLe En= pies Shown la the Figures for the esses pl 3 3 - r in Table 1. l ^ -: 0.', Comervatio's Fiscre 2N + 1 deg c/A. s, 4/F rme Error a 4 g og p ru::sted bounda 4 E3 0 0.1 0.5 0.3 5 0.112 ' O [' 79 0 0.1 0.5 0.3 5 0.0053 (1957] caed a .Y 5 79 40 0.1 0.67 2.0 32 0.(US ecnservation of j (1 40 0.1 0.67 2.0 32 0.054 macy of thir - t 25 40 0.1 0.67 2.0 32 0.074 5 6 65 0 0.0 0.67 0.3 5 0.0019 5 X 10-. of energ state .y 65 0 0.01 0.67 0.3 0.00039 volving real es, ./-"-- a; 65 0 0.1 0.67 0.3 5 0.00034 energy aerces a 't 7 65 0 0.1 0.2 0.05 5 0.012 spee mW g,8 8 65 0 0.1 0.5 0.05 2.5 0.062 8 9 79 0 0.01 0.25 0.15 13 0.017 the mterface, t I S. 10 65 0 0.0 0.25 0.3 5 0.00055 7 X 10-* pletely represes i. 'I 65 51 0.0 0.25 0.3 5 0.0013 3 X 10-' waves' I 11 53 42 0.1 0.67 0.3 5 0.0040 .l ! 53 43 0.1 0.67 0.3 5 0.0031 .2 53 55 0.1 0.67 0.3 5 0.0041 g,(y,,) - c"" 53 61 0.1 0.67 0.3 5 0.0075 53 78 0.1 0.67 0.3 5 0.C 41 83.9 0.1 0.67 0.3 5' O.0.l05 g 12 53 42 0.1 0.67 0.3 5 0.0185 .. p. 53 43 0.1 0.67 0.3 5 0.0091 De mathemati ".T.Mg $3 55 0.1 0.67 0.3 5 0.0053 tion of energy rt 'r.1: ./ - 53 64 0.1 0.67 0.3 5 0.0078 i

%g. t_ f 53 78 0.1 0.67 0.3 5

0.0092 {lC l M.+F*? 4 f W 41 89.9 0.1 0.67 0.3 5 0.0061 J%'.;y M(ir:I.h..k 79 40 0.1 0.67 2.0 32 0.038 when A. the 13 79 50 0.1 0.67 2.0 32 0.048 is P(u t. %;; sp?.:

79 32 0.1 0.67 2.0 32 0.027 the direction of 79 18 0.1 0.67 2.0 32 0.012 scattered plane C,%7.7) ii I

79 9 0.1 0.67 2.0 32 0.019 sH nguhr pla-

e.

79 0 0.1 0.G7 2.0 32 0.027 ZY.3{6 8 } 79 -9 0.1 0.67 2.0 32 0.(U2 genswam . '#47:4 79 - 18 0.1 0.67 2.0 32 0.(n5 a!!y away from gif;[]lI]: s ' f'.M i 79 - 32 0.1 0.67 2.0 32 0.052 hrge depth). 79 - 40 0.1 0.67 2.0 32 0.049 When the ca 8 4 %. ~~. h 79 - 50 0.1 0.67 2.0 32 0.075 our approx 2:nati ~h ' Positive nlum imply sourm wave incoming fro:n the lower left. place the exact 7 q.M* : left side of (30) U.hQ s In the appendix ' ?'5Y. ^-JF- ' '*" [z - z') (R5fSE)' 8 can be expre Eik't \\ ( i the interface r Hius = fint order Hankel functics of the E +

"II **#

1 ...M'-l fint kind. (2A) rutely evaluated u . E~D The rehtire importanee of the dimensionless E

'*1 D"

+ tu ples myoh $0 Q raiduah f/ui and tb/Bi, in influencing surface m:Pi.4 ;, motion, is dispbyed in (27). For enmple, in the where d, and f, are the displacement and stns> [,?j; case of a shallow depth of interface (wf/S:K 1), residuals, rerpectively, at position f alog the O 4 the eurface motion is determined primarily by interface; um, and um, are the ec=peted vr.!ues ..i "j the displacementa at the interface because of disp!seement at podtion j sle: ;; the interface g R o'(:) dominates H.o)(:) for small values cf :. in the layer r.nd hr.!f.nsee; and rm, and ri.v, 3,,'bb l i As a measure of residu.ls at the interfr.ce, we are the computed rtreexs at t!.ose podtior.s. de'.ne rchtive roct-:nean-squ.* e (rms) e ror r.s The postbus cre equi.paccd ab.; the = di-th'ICIb*it:: "E"" rection, and u is reveral time.s hrc.: than N. T

, :1 L a.:.N C...' h

^

--,,e.w v

',n'.-,- .. '. ?. ". ~ ..,, e neef? 5 ". ; -.. ,T'... ] '. 'k A l* ,e a t ~ ' jf'E.

&*f h c'.

f ~ .r. ~ ..,N k ry a. L- ?~ ~ ,) 'a m_ -~uL } 3 .. 1

-F SURFACE h!OTION OF !.AYEltED SIEDIUh!

HI l i von (8) fer the truncation index in (19). The rms enor values that ue truncate the m5 nite-sum equations to [' ' fer the enses presented in this paper are listed include an upper limit of 79 scatter orders (cor- .a in Table 1. responding to the 2N + 1 wave numbers).This . R Cor.servcdon of energy. In dis,ussmg the truncation impo=es the principal limitation on 2 Crror a re!!ection of plane neourtle waves from cor. the accuracy of o.tr solutions. For a given M~ rucated boundaries Meechen (1956] and Heaps medium confguration (characterized by the f.ne L2 G (1957] need a relationship derived from the layer thickness. the medium parameters, and f( [ conservation of energy as a check on the ac-the shape of the interface irregularity), the M ". g%. ' 4 cursey of their computations. The conservation truncation error is dependent upon the fe!!ow-H9 5 X 10% of energy statement is that, for problems in-ing quantities: C volving rea!, the time averaged net flux of C' - 1. c/A, the ratios of the amplitude of the Q 034 energy across a place at large depth in the half-interface irregubrity to the wavelengths m-M- 2 space must be zero. Below the deepest point en volved (e = [(, - {.l}* d 7 the interface, the exact solution can be com-b'M"iF s,== %*/dt!,, the maximum slope of the .Q 055 7 X 10-* pletely represented by a superposition of plane interface. 13 3 X 10' <o 3. e., the angle of.meidence. 'Ms, - waves'

4. L/W, the ratio el the periodicity length N~i 31 g(y, g) - e"" * "" + { C,e"" * "* "

for the interface shape to the width of the 41 anomalous tone. C5 > fu (29) 5. e, the rr.tio cf the imaginary to res! parts Pfl; g $5 of frequency. N.L gg na mathematical statement of the conserva-p 53 tion of energy requirement is that For a fixed number of scatter orders, the gj T8 i truncation errer increases with incressmg valum y { lC,j' cos S./cos S. - 1 (30) of c/A, s,, P., and L/W. Since the intrinsic 22 f (Rayleigh ansatz) error also increases with e/A g where S. is the acute angle between vertical and and s,, we cannot always distingmah between T the direction of propagation of the nth order the two errors. The quantities e/A and s, in-4-/ I scattered plane wave. The summation is over fluence the rates of decrease for the amplitudes 3 a!! regular plane wave orders (the inhomo-of the higher order wave-number terms. To our I geneous waves, i.e, those that decay exponenti-surprise, the intrinsic error does not depend on J aHy away from the ' terface, are hible at the angle of incidence e., and the truncation M g m W[@M I large depth). error depends only elightly on S., as wiIl be l When the coeScients A.", determined using shown later. We find that, with 79 scatter I ~ our approximate ms. hod (see equation 19), re-orders, the r=s error is generaDy less than 1% MA place the exact solution coe5eients C. In the when c/A and s, are less than unity. Thus we LN left side of (30), the right side becomes I + 3. are able to study problems involving reasonably y In the appendix, we demonstrate that the error irregular interface shapes and wavelengths com-g H $%j 8 can be expressed as a weighted integral of parable to and larger than the sise of the irregu-the interface residuals. This conservation of larity. A$g energy error measure is more easily and accu-One means for studying the truncation error, rately evaluated than is the rms error. for a given scattering problem, is to compare g m ) (gg) ne error 3 is included in Table 1 for those solutions obtained using different truncation g n r n m/ examples invoking res!.. The small values numbers. Figure 4 displays one such example. g.4 T of 8 confrm the accurney of the numerical The problem configuration is shown at the bot-hg@f '*~~*** " d **#*** co=putations as well as the fact that the wave tom of the figure. A wave of wavelength 50 km .on f alcag the equations are indeed satisSed. is incident vertically upon a basin 5 km deep gg

P N ** "*"

Exemple of the trunection error. Our method by 50 km wide. The interface periodicity length h.; Mb* for solving these wave-scattering prob! cms is is 056 km in all the examples deeeribed in this l - ;-

d % ""k.**

m.de fea ible by the rpeed and large core paper (the use cf an explicit unit of distance i U. W* ** = emery ef modern digital ecmruters. Even so, is ci= ply for convenience; of course the eclu- [ j , f.' ec sputer ti=e and rterage een:tminte require tiens are unchanged when all lengths t.re seahd m hy i

h @r. m i.3M

_ M ma

.;. ?;:~;- M' 4,.g Y, L 's; &~n,S:.. w a r.;,r. s..4. Q .i'l." W :.f,~;Q [^% g g &Z Q. %m w-j Q f aw"~~**** ; ~r n-;- Q-QW.QRh ,. ~ ^ u . :'? ,p.,_ ~.w, 5._. A..y e,p. ;.,,. y. c :;p g.. ~... y. . >. n pyp m 4., ..g . g.-w.. q.. m- ,. s , g',h if y. g.. ..e.qf ~;., v me. - ,. Lg . r.. g;;;.7g..u.y-Q -g ( V . g q. .7 n. - . g ,.4 .,m. 4 .. m.3 -0 1; ;..q q g v. , ~, -....q y m., , 3., WT

m j y g..e p

~ ~ y r. = -- -4

s mm I

LI }l2 AEl AND LARNER pertain to scattered waves covering a broader m (' g span of directions than do the wave amplitude.- 3}'T i -n-~.- in the layer. However, the wave amplitudes arr-lh more basically functions of le. - 0.! than of '/a' by ~ lnl. hence the half epace wave amplitudes de-tre E s k .n 4 4 8 hn eay more rapidly with increasing lnj than do 8**u== *wa** *

t

.r,.I ---i -$5 g ~ % U 'L = those in the layer, and thus may explain the se '~ l *i interesting finding that when an insuf!!cient thrt j r, ); hr 'l %g number (53) of scatter orders ts used, the error el } 6 1 i in the half-space remains cnall, i.e., the solu-t her ,4 .a tion in the half-epsee suffers little error due to rw i~..,. s-+- a -

p truncation of the wave-number spectrum.

rede l;'.W/"'. The normalized amplitudes of displacement at a!th W h::: the free surface are shown at the top of Figure iyi r s.***

4. The normalir.ation is made with respect to erro the amp tude that would be observed in the in t li i.

.I same problem if the layer had uniform thick. Fig. 4. Spatial distribution of the normalized ness given by the thickness away from the inter-amplitudes of free-curface displacement and the (see tre;ularity (0.01 km in this esse). Note e amplitudes of interface dispiscement (arbitrary that the surface mot. ion computed using 53 units), displaying the efect of truncation of the ll ' -i infinite sum equations for a soft-basin problem. scatter orders departs very little from the more The normalization is made with respect to the accurate solution. Thus, in this case, the rme o .nl3r.,, j t ; '- displacement that would be obtained for the f.at. error is a pessimistic measure of the accurney j M.P.M :] layer problem in the absence of the interface of tbe computed motion at tbe free surface. The .vp-9 i irregularity. p-lA ki errors in the layer decay rap.dly away from i l the interface because these errors occur pre-kg. { p} }1: .g. w g,: equally). The interface shape has the form of dominantly in the higher order wave-number g g [: j a single cycle cosine. Away from the basin, the terms, which are attenuated due to our smooth- { h,ij, I layer thickness is 0.01 km. Solutions were ob-ing with the use of complex frequency. g&.;.d !.@i. ] tained using 53 and 79 scatter orders, respee- 'lle dots in Figure 4 are values of surface tively.The curves in the center of the 6gure are displacement computed using the Thomson- ?sMT i spatial distributions of interface displacement IIaskell approximation assuming that the layer h; amplitude. When 53 coefficients are used, the has uniform thickness equal to the local thick- ((r thG' A' l-solution in the layer (daabed curve) osci!!stes ness. Further cemparisons with the flat-layer O.(3)[ about the one in the half-space (solid curve). theory are described in a later section. wg[h- 'When 79 coefficients are used, the interface diu Example of the inherent error. In some

n. gig placement distributions are indistinguishable, in problems, we encounter relatively large inter-My the fgure, from the solid curve. The rms error face residuals that cannot be redueed by in-i gly n i for these two cases (Table 1) are 11.2% and creasing the number of scattered wave orders.

2? 0.5%, respectively. Obviously, since the rms In fact, these residual errors increase as more 2[Q 3 error is so small for the 79 coefficient example, scatter orders are included. This type of error the Rayleigh ansatz error is insigniScant in this is probably the intrinsic Rayleigh ansatz error r [ O f. i problem. Cretzky suggests that the intrinsic error i-i, N W Note that dividmg k. = k. + 2-n/L by k, manifested by asymptotic tchavior of the serie< C yields sin 6. = sin e. + nA/L, n = 0, 1, representation in equations like (19). That is. [N.1;; m2 where e,is the acute angle from vertical as N increases from small values, the serie-associated with the nth crder scattered wave approximation (19) first approaches to, then (the en;ie 6. may be ec=p ex), and A = h/le, diverges from, the true so!: tion. This is the b<. is wave:e:gth. Becau.e the wavelen:th m the havict exhibited in the example shown in Fi;;ur - half-space is five timee larger than that in the 5. layer for t!e example shown in Figure 4. the in this pnblem, a 10-km wavelength was - wave amplitudes in the half-space sclution is inellent at e. = 40' frcm vertical upon. [ 4 b ?).%%.

-,n-y' " , aer** ~v'W' "~=%y'v X ~ 7 ,,7 t bFJG p ~, _W ~ 'k -. c. n.. .. n n; '.p, m & u -: n x,0.l*ll~l h l.l &,;. -

?: *
J 1

x.' ?- ~.. ,., w + ~ y% ~ _ ; ;;'* '., .,. i U,: " t:;.,.. - ' ra . s., - ,,a y^ * ~.- ',$ e, ,,s es4m ~- ,-e f., s .?- ..t

g

, s Q '.l p ~ s -m e

z. -

r si SURFACE MOTION OF LAYERED MEDIUM 943 i e.sverin a broeder ner 25 km thick, with a severe irregularity

4. As N increases, the oscilLtions in the layer i

tl.: wave emplituden otep) in which the thickness varies 5 km as diminish as does the stress rms error slightly. trave amptitudes are i half-cycle cosine wave over 4 km, and thus As N increases further, however, the stress rms j, ,f l6 - 0.] tb n of ie.s a '" mum gradie::t of almost 2. The error a; sin increases while the spatial oscilla-N ,;.:ve amp!!tudes de- ,:rcues at the interface (rolid curve for the tions in the hs!f-space become more rapid and ., 8 'f - ere.tsmg lnj than do I..ver and dashed curve for the half-epace) the o<eillation smplitudes become larger and "h. n:s may explain the are dirplayed in the center of the 6gure for more cor.centrated near the step. The number p chen an insu5eient three solutions obtained using 2N + 1 = 25,41, of coemeients beyond which the displacement N-ters is used, the error and 79 coeSeients. The rms rtress residuals for residuals become divergent is genera!!y dif-i. c=all, i.e, the solu-the*e solutions are 0.061, 0.000, and 0.064, re-ferent from that beyond which the stress re-(Y rrs little error due to

eetively. The sizes of these rms errors do not siduals diverge. In these cases involving 25, sber spectrum.

rHleet the large residual localized at the step 41, and 79 coemeients the rms displacement re-h.L tes of di:p!acement at although they are determined predominantly sidusts declined from 0.0816 to 0.04SO and h at the top of Figure by it. In the 25-coemeients case, the trunestion 0.0145, respectively. .Q(i^ tade with respect to error is manifested as an oscillation of stresses It is discult to assess the efect of the irre- / be cbeerved in the in the layer as in the example shown in Figure movable localized residuals in these step prob-

(t.

had uniferm thick. O away from the inter. f; in this ca2e). Note { h(,. ecmputed i: sing 53 _i, ,,,,,,,n.o a.,,,,,,,,, --- e,u,es litt!s from the more q io [g ~ n thb case, the rm. rtere n!.he accurney {" ~ w {g the free surface. The h,wy rapidly away from ca errors occur pre-y crder wave-number .i

==...r. h .1 d due t3 our smooth- . f-eney. m icing the nomson- -}, Q_ .re values of surface k c:nin: that the layer I'- a with the fat-layer i*- h;{5 e t. 57 al to the local thick-Ik-

as

-s ter section. } yf mi error. In oo:ne am.a, as a C l ueed D ~~~ -= nitered Wave orders. d; 3 %[ crs increase as more L Dis type of error -*-e -e e ro o to 9 e a ' " *

m

'.zy!eigh ansats error ,t,,, ..... a, o o.. n u, 7-k[.o is intriraic error i- - + ^ ' ' ' ' * "

behnior of the serie-

' f" Q s like (19). That it. 8 ',' M * *,' *". dl values, the serie' ,,. 'f'

.1 i,...

approach-s to, thes ,p ;q Fic. 5. SpatarJ distnbutions of the normalized amphtude, of free-aurface delseement and y ~j CE int 4rface strem dispisyi::g the stress residuals (cr thrr+ solutions computed using 2N +

25. 41 r.d M coescients. The normalirstion is made with rcerwet to the, disp!see:nce g

m wr.s elen-h wa' ' acuL oe obtained for the f:st-layer Troblcm, nn.1 stresee. s.c expressed in equive.! cot una ram v(rtie:! upon a of dglace:nent by multiplying by (1/h.si ). j i .J. -..- .Q.24] .bn. ;w <;r we,m u

Y~""'" S - "* %,

&,,, t.W, c,.. s.3,f, dY.pl.S.. me@ C;?/C. v.,.. m... g.,*Ehr'.Wi[* Ss

.YJ . m. m' i.. fb. h ~-, g /.t?.d.4M~,y q,se.W. -e...s ~,.,.y. 7 .A ~ ..t f -b. c. -., m-:t.W-z.,w.. --t 3- , p, - ..c *. -. Q.p,Y.fy f '! . m - s-k )N. .".l. fi.f5

f.

c - k..c-%. ..q .,...,...I <.. i, sv. . ; ;* y. .,-; p,q.; ..x.,. ,-.. Ki. y }* ...s...., /.. .e.. qu ' p.L *; ;.,*:, .. d. , y,~. -v .. ~ : .m s,M.. a. .m.g, x' .g.,. . ~ ; ~.)t. 3,

.J.

~., .c.. c. ~, , 7,,7p s. . 9.rg,g.. -. v:.. y,m;. .,.,.;.. 3, J ..:. %.9 7 zw... ,j . f,..c p, _

m. m.

y.... c I ! l AKI AND LARNER -a[ pg g. os: ills ' f' .A I es-s r } seteril s gp t t 1 I t t t t f s o ao .o ao so oo iso.ao eso ao roo aro e.os. In the i ?.~ ,, e s.re sharp l

@t**r5 inner 6.,

.,,.so si t f are at . i use o t v.. i gide lg s ,i a ' - -i ,,. ss rm I owe g. fog n. so te Q. s a.) retent J Icl/l Tl o.- wa1.i j:, t est eases d' _ 4,, ; - io -- d:,stral = we ... ; i T.. flat-b . a.an beeon '. / .e, .m , N.- efeet ..p, st tbe mam , u

..,o.

.r.. < '.,.. - ( l introc [ -* 0" 9 M.~/jll QYiN* narfai ' i e (j o -M ro -C o e ao 3o . g v =2av ,J i t d h I Se.ee.' or..e mome.e Ial D" hph Fig. 6. (Upper portson) Spatial distributions of the normah-d amplitude of free. surface focusi displacement for a dented M discontinuity problem, displaying the smoothing effects of the reveri W E D,j#; use of complex frequency. The normalisation is the same as that described in Figure 4. P'MMA j } g;[j Weer portion) Warecumber spectrum of spectral amplitude rat.io versas scatter. order Th. p.g number; e is the ratio of the imaginary to real parts of frequency. wave ,'@M - cases. M;.,1 %. lems upon the computed displacements at the large inherent errors. We find that if the the r free surface. The stepped 31 discontinuity ex-truncation errors are small when e. = 0, they sus 11 Q( c.Q.M 5r amples presented below must be considered in remm small when 6. increases even to grazing plex: (.j.Wy'b, j 1 that light. The fact that the rtresses in the layer insidence (6. = v/2). The inherent error ap-19), M' (41 ceefficient case) roughly reflect the step pears to be independent of 6 2-n/. ,'h,! shape of the interface suggests that the errors Example of the smoothing efect of compter snnn ...q : in the layer may not be so severe as the re-frequency. Figure 6 demonstrates the smooth-tudes GC.y siduals ruggest. ing of the spatial distributics of ecmputed free- <= ,y, & The irremovable errors are larger only in surfsee displacements efected by the use of a= M. ;.-h problems involving steeply sloping interfaces, complex frequeney. The problem configuration deca) jj and in these problems particularly when the consists of a half-epnec over!ain by a layer 6% order E,1*l {f, j wavelength is rmall as compared with the wavelengths thielc with a 5-kn depresion in alrea. g . (.i. amplitudes of the interface anomalies. For less the interface. The depression has a co irr Alt [(i Tz, j severe interface chapes rueh as those in the shape for one cycle (50 km). A 10-km wave-si:nif length wave is incident vertically. The top (d:ft ir d'i -; dented hi discontinuity problems presented be- ' [ 4 ' ', ' low, the rms residuals are small and we!! dis-curves are the computed free-curface displace-ray t tributed over the length of the interface. In ment amplitude (ecch normalired to the in th those cases, the errors centinually deeresse as re pective Est-hyer eclution es deteribed presi-nye 2 25* + 1 increases to the mar. mum value cf ~9 ously) for three value: cf <, the ratio ef i=ap Erge Surpridngly, we End that the existence of nary to re ! pirt of frequeney. The amplitu - e aus-r shi. dew renes in our prebiems does not imply teste applies to the < = 0.1 eurve; the othc-i er r tm l ..g.e .. J. L'. '

? 'h{QQW.f1*OR*'.TSW".".yKt2:,M:'(. *.'*>-'.g:am.m,n.;z:*:3*f-v.wy c* ' ' J 5 ~ 4 ; ;,.. y,*. .V _,, ~ ' ' g .y;.

..~

,qyy

y.

.y W W r 1 } s. .y ; w ,n<.t i .y _ y . ~ _7 r_ _. _ { , } s Ip[.d 5 SURFACE MOTION OF I.AYERED MEDIUM 945

gg curses are displaced upward. The use of com-tions (not shown here) for problems involving VM pin frequency severely damps the oscillations senttering at the irreguhr interface between two h" '.

alcag the limbs of the < = 0 distribution.Those half-spaces. In those problems, hyering effects ,p ovillations have an S-km wavelength, char-are absent. We find that the spectral amplitude h [DW ; seteristic of the cutoff wavenumber (N = 32 ratios decay more slowly with increasing k/k, .pga n' in these namples), and are thus caused by the for proble=s involving longer wavelengths rel-p sharp cutof of the warenumber spectrum. The ative to the dimensions of the interface irregu-pb Q - iver two-or three-eide lobes in the top curve tarity. are actual lateral wave interference efects. The 1.^if:+ use of complex frequency smooths out these Sorr BAsm Pacer.sw g' 4.+ M side lobes. I4t us consider ground motien at the surface U M.. For real frequency, the byer vibrates in a of a soft medium basin when plane SH waves resonance condition where the thickness is 6% are incident from below. This problem has been NTN6ii [E'e.)-ih wave!engths (see the discussion of the soft basin studied by other investigators primarily under !'Y { esses below). Over the depression, the nmplitude the assumptica that the basin structure con-PC-h'{ distribution displays these characteristics of sists of horizontally flat layers having the rame f'at-layer interference. When the frequency stratification as the one directly beneath the becomes complex, these vertical-interference observation point. In other words, a problem E' f" I M' i < ',j j e"eets deteriorate with the results that the of three dimensiens has been treated as if it main side lobes are less deep and tie amplitude were one-dimensional. We wish to test the .1 at the center of the anomaly is increased. The validity of this assumptien. In a horizontally h M.! introduction of complex frequency siters the flat-layered medium, the surface motion, due I's?gd rurface displacement distribution from one that to an incident plane wave, is determined by i is dnminted by f!at-layer interference efects the interference between upgoing and down-i to one domimted by wave focusing and d'- going waves that have the same phase velocity focusmg efects (the later-arriving multiple in the horizontal direction. This phenomenon 1 Figure 4. reverberations are de-emphasized), of vertical interference can be completely de-i y atter. order The bottom portion of Figure 6 contains scribed by the Thomson-Haskell method. If the L 1 wave-number epectra for the c = 0 and < = 0.1 interface is not plane, scattered waves with l 8 esses. Rese are plots (on a decibel scale) of horizontal phase velocities diferent from that i that if the the spectral amplitude ratios lA.*1/lA.* l ver-of the incident wave are generated, and the i 0. = 0, they sus the scatter order number n. A.** is the com-lateral interference can become as important 'td O JA,[pj Sea to gra:ing p:ex amplitude of waves in the layer (equation as the vertical one. - [ e,r 19), with horizontal wave number given by It will ba shown that, so long as the inEr.fsee sent error ap-Sn/L. For normal incidence, the spectra are slope is small, the flat-laver theory using the f. t erJ cf compter spnmetric about n = 0; thus, the < = 0 ampli-slFatiEcation directly_beneath_each obervation pi ' j es the smooth. tudes are plotted only to the right and the point gives a natisfactory result. As the inter-f sn=puted free- < = 0.1 amplitudes only to the left of the (su becomes more irregular, the efect of hteral %frg 27 the use of n = 0 line. For e = 0, the amplitudes do not interference becomes important. We shall show Mrfg i 2 eenfiguration decay sufficiently rapidly out to the cutof an extreme example, where the hteral inter-pjhyr by a layer 6% order number, thus causing the S-km oscillations ference of pseudo-Love waves gives nse to a gg d,precian in already mentioned. brge amplitude variation across a basin that g-g yfp'8: y has a cosine Although these amplitude ratios are small, a has a uniform inickness over a finite area. We M i 10-km wave. si:nificant degree of wave-number coupling shall also observe that the hteral interference [,g}g(gQ cry ne top (diffraction) is indicated. On the basis of simple becomes mere s:gninennt when the direction of risce dbplace. rey theory refraction. neglecting reverberatiens incident waves tecomes e!oser to the horisontah (:4 to tb in the byer, we expect contributions only frem The first example is illustrated m Figure 7. ggy c:r 5-d presi. nves v ho e lr.l < 3. However, we require the The velocity and der.sity of_ ha!f.cpaee are 5.;,- 7 ~id 1.4_ tunes.those ofTthe raft bann, reepeo- --ti3 cf i ne;~ 1 ger wave nu=ber (hrger n) waves to ad- .. - r.: p'itud - cate!y satirfy the boun6:y conditiens. We tie!y, cs shoc: in the finire; therefore the t',e othe i-terpret the wave-cumter ecup'ing as diftne-impehnee ratio is 7. The mdth cf the bw.,n a,s e 'n pri=arily on the basis of si=ihr computa-50 km, and the depth in 1 km at the deepest u, l =2+ t g-. f '2"I'*?M"2-. )

.m,. ....v...%....,,,s.7..- - u ; ,r. u.s M..,,,g $;i~: %, W{'.g2.,,? W(... .g ..,,,. m,, % w.,.. ,.4..... ..f ~ ~?<.,. x.y * ", ~l ':.. U 3.. ':v@)S.-GQ q O:; ...o. c,..m[s[.59)h 2, hy [ h.';, .h!hlk( f W

r r,

W%t qu. ,b. j -v.w/ .;.. 71[..;.7g.ggg"; gg;g..g[ w,,., c., 7.W~p 7t.. i .. :~ y;p..W.n%.j % Q...-U. l b \\ n . %..'J 3.:E + ~:.p,:.. ; b j , f,_ - " f jf '* '; ~ ~'? . M. t -y \\ j I 946 AKI AND LA!!NER , -t spectral amplitudes decay, rly with increasinr section }to lnl indicating signiSer.nt wave-number couplin:

s. etion

.... n.o.v.,,u- " g attributable both to diffraction and multiple interfas ~ } reverberations in the layer. cycle ov In the second mmple, the incident wave-wavelen; . sej length in the layer is chosen as enetly twice t:mes tl ,~* "r the greatest depth of the basin, as shown in satisfyin 3: so oo iro ao (' ^ Figure S. The Hat-layer theory predicts a before.1 S.,,eee p.e. a tam) resonance condition at depths that are (2n - partof ~ 1)/4 times the wavelength and an anti-The f u,, 8**'"'" a"-**'= resonance condition at depths that are n/2 distribut i, ,,.ro r, e .e.i e g s,.$ J. l ia times the wavelength, where n is an mteger. the top. @h 8a. a s w/" In this example, antirescnance occurs where gives s: "~ md ^***"'", the basin depth is manmum, and resonance above s "d 8-'"* occurs where the depth is half the manmum. Bat-laye la'.'1/la';1 Figure S showa that the. solution _obtained by at the N'' g s ". l 3 our method agrees well with the predictien by horizont [ /' k M } the caTLIyEtheory ~although the wave. number wave 12 -so" \\ rpectrum foTtTis case indicates larger amplitude Love w

I

~ -l / ..... k E no. In the third example, the basin structure between \\ scattTred Traves than in the previous example. cludes t .E.E.o o A 5""*'""**"' . p,, censists of a long section of uniform thicknes frequen ydf.5d [ Fig. 7. (Upper portion) Normalised displace-bounded by short sections of rapidly changirc in the la ,; 4 ti] y ment amplitude at the free surface of a soft basin thickness. As shown in Figure 9, the uniform We int ?N. (mmmum depth la % wavelength); the solid yE .5 line is the solution computed by the present seross t I@M'T)3 ^ Q-method: thi dots are amplitudes computed by the between (r"-[d'.-{ local Domar.3.Rankell approximation using local 1.[ scatterei layer parametcu. (latoer portion) Wave-number hI demonst bg;g[.i p{ $

JJ I

spectmm of spectral.emplitude ratio versus scat. j3y te% order number. 2.o.- r.h..e:H ? $f 2, g 'o U point. Tha shape of the interfacedr_garit_v do do ' Jo do do [Q ' N.' (ih, is a cosine form for one eYC!C ~Ws:555tf~ curve in the top of Figure 7 T 8 I,, . <j! shows the amplitude of surface motion, cal. ,,.m.,- Nr. k@JC(h. ,8 culated by our method, when plane waves ^ * * * * ' '

I "'"

h '*. h whose wavelength is 5 km in the layer (25 km ,,. u m,- in the half-spsee) are incident vertically. The p 2..,M., imaginary part of frequency is set at 10c'c of 8-a* 'o' w us.va [.'a/y:1 R. 9 :. the real part. He dots m the same fgure are P -Ea amplitudes calculated b,y_ the_ fat-}sI_er_th.e.ory .k, 9,,,,,, M./ % h assuming, at each point that_the, basin structure e u r-s,.jf' is a horizontal layer having a constant thick-YF"" . W " nees equal to that directly beneath the point. /"" q h'. ; U - na s'o.o 4-Agreement between the deta and the smooth .*o.m.ao.o o no e ,ui ..%[ cune a exQh demonstrating that the fat-Fig. S. (Upper portion) Normalized displace-c layer asr_.tmp. hon.,s_ a gooTene m calculaticg rnent amplitude at the free surface of a soft basin i t. p e - the surface motion for this case. (rnr.zimum depth ts % wavelength in the layer). -The'waTeiYropagatic; vertical!v (n = 0) The dots are amp'itud-s co=puted by the local do=inate the scattered wases in this case, as Tho=xnaas'gl aproximation uing local layer F dicated in the wr.te-number spectrum then par.neters. (Iot-e portion) Weve-nu=ber eiw-a:ft m tru= cf rpeet.1-e.=;'itude rr. tis vc rus a:atter-by at the bottorn of Figure 7 However, the crder nu=ber. l Ws /'N _h{- .; r. .;L

WEQ@Q@;'Q'{-[ONMENW'M '~~?MG' &: M,M'w'y@!W-mW..w'.' ':...k NlS v.y *m-.: -'i M N D 'W.h. }@.bh$$n[%': ' ~ h 4 M};*N P '? yv@&': 9 %. q x } ; =-s - Q 1l~ i,M ;.' f. N ?'s::q :y;\\ 3,? W'%"$%y? QfE.i'Q"-i'Q' ' ' ::.." " % M;. :.;'. w 7:.k d * .. t ~. 7-s'- 3 w :f.w +r,.g.e;.. ' .x c- ; w 2 w ~i ].} } - - P' ! N,'.; -

~

e l

'V r. a ". f. ",. - ~., - : c_ ,.-r u w,, s - I . s. g."_.,,, 4 -- s,,,, e - ..g: i . E %~ c t, e r

g s

,-?w, r b SURFACE MOTION OF !.AYERED MEDIUM 947 . ~ ~ g... # iincreasin;. section is I km deep and 80 km wide. This observation on a two-dimensional structure by W i W. rr coupline s etion is &nted on each side by a sloping a theory appropriate for a one-dimensional Nes' # $ CCItiple interface, having the shape of cocine for a half structure, the observed and theoretical ampli- " 6, % 3 $5,$@fs.. - eyete over a horizontal distance of 10 km. The tudes can di5er by a factor of 2. p lent wave-wavelen.-th of waves in the layer is exactly 4 The last example in this section (Figure 10) pn@L' @f ictly twiec t;mes the thickness of the uniform section, compares the motions of a soft basin for waves id- .V., shown in satisfying the resonance conditien mentioned of long wavelength (20 km in the layer) inci-predicts a before. Note that in this example the imaginary dent from two diferent directions. For the e >J i [f'1;/ } ~ [. its (23 - part of frequency is 1% of the real part. sero degree incidence angle case, we see again . T.9 1 an enti-The f!st-layer theory predicts the amplitude that the Hat-Isyer theory predicts the result h:..dh.d,

are n/2 d: stribution as shown by the dashed line at calculated by our method. For the 51' incidence r

,Wh P.W' 7.4@ .n integer. the top of Figure 9.The solution by our method angle case, however, we see a signiSeant dis-ars where gnes amplitudes that fluctuate considerably crepancy between the theoretical and calculated recocance above and below the curve predicted by the solutions. The flat Isyer theory predicts a con-4:1..$ & 3 M(...-C,-gh ) f(# 9 maximum. f!st-Isyer theory. The wave-number spectrum siderable decrease of amplitude with increase of Nqi [ M dy*y.C . tuned by at the bottom shows secondary peaks at the incidence angle. On the other hand, our solution .) diction by horizontal wave number corresponding to Love shows a nestly equal or even slightly greater 'e-cumber waves in the uniform portion of the basin. The amplitude for the larger incidence angle. This W/ *N'M 43.'f 4M. ! Q:t# cmplitude Love wave region indiested in Figure 9 in-result is intuitively seceptable because, for e n m pia. c!udes those horizo=tal wave numbers that are wave!cagths comparable to the width of the structure between /p. and./#i, where is the angular basin, the motion of a soft basin is perhaps h,gg-gd. thickness frequency and B, and B, are the shear velocities determined primarily by the vibration of the gg(eg changinc in the layer and in the half-space, respectively. basement, relatively independent of the inci-d2 g i uniform We interpret the large amplitude variation dence direction for the wave that forces that ~ scross the basin as due to lateral interferences vibration. We may interpret this result as j between the primary waves (n = 0) and implying that relatively more energy is trapped .2 scattered pseudo-Iove waves. This example in the basin as the propsgation direction of 9 demonstrates that if we try to explain the incident waves becomes closer to horizontal. N.$o,d N'a. p. z 3., M !g[!.j w -W..- m.e 3l! } eof,. i gdriF## - d I lgl p I --- ac w-s== ,y.m E ey ff 1, s.>- l I'D' a @gIM; N 1 *" y ~ l AJsgeh &y.x>ih s x b y. %,,Y. a ,,..u.. gl a r. p. 3 a e='.=, e l k'l/K1 l .l

,uw-4;, md....%.

'q, e- "9** e.a =.g.. W.n M v. s .o o. ,l, .q m .a..<. . ng,,.. Wh.A.1 l i t m e ... so ?.. - ~. j'% M %y n i displace-ofL bitin .y j t,,, -- . ;-%.%,k j, .,4 % -r p? _ " 9 a layer). m. c. the local c 1 kyer Fi: 9. (Upper pardon) Normalir-d dis;1:ccment a=phtude at the free surfsee of a fat F/ p - h R 8.' soft bada (traximum depth is % u nselength in the layer).The data are amplitudes computed /*., y by the local Tho= son Hanell approximation usi:g locrJ 1syer psrumeters. (Loirer P>rded ,1

C*f Wr.re-oucher specteum caf rpectrs!-e=plitude ratio vern;s scatter-order number.

c{,; j i i .:g _ ' f:~ f. ~.z. :.. ma 'N

T . ",,;m

- - fym K~~~r.g* - W M.... Cny:,.'.*.M"w_~.,.y['y. a.f;.a.w*.e.y.g ry:.~mp.. M.p.g.~;g c.p;g-M

~*

i:~~

!r.~g. .m. :- v L..>.4 g%g -.. - g n u, a

f. i

.~...s.w

- ', g,z. y.

+ .._. :' u .m .. ~ ;, m., ~. - p _ 'w. v ~p, .'y ~. Se m:, rj L _ ;,.;. s ? . W,. ;;:.. y.gf?%y.7 ... r - y ~. \\., ;;.......,,..; u-

q.3.pa. 4..)

_.; x,. ; pgy

w: zsr9

, z.w ..,.y'. a :; e -; ;V

v.,. m,a yo.

.. u* ~ au;,,. 'g m-4 ; w

t. @ ggt;: .; n m.x. K r.- x9-m..

..g. s n,; u.- N l~ _. _ D i.l T ' y~ W'U '% WNM h

.w

~ .x....

-[

-l[* ]_ the free surface.1 M3 AKI AND LARNER 170 nro artificialm of the ray-theory / ' 7.o-j *a# -- - N tive of interface i '6 [. e e e tw ierar eno aate. 09 senable, smoother a.e 50- ... eu w,=, te.. m so!utions. When t .I g duced to 5 km 12 , ~.N z ~ p f... ! .1 so'o im'o i'o oso a' PCsk at 149 km

/

s,<=. o m at 164 km moves 54

a. : 1 a

theory lobe. Also, W 3,. gy c. t 8" re a8 cestly coincides 1 'if [ a:cureer ausgest. These consistent a s.u y yl

e. se-

e. o-4.u

. xp esen to graz:ng u J ""..'t o w 89.9' amplitude e a 'g im:ty away fras Fig.10. Normalized dirplseement amplitudes at the free suHace of a soft basin for two dIrcetio:s of i=eidence cf lo:g wavel-ngth (100 km in the half 4 pace) wsves. The dots are amplitudes cot:puted by the local Thomson-Harkell approximation czing local layer pa-e, rametars. ] $5 I' e g )0 DaTen an Str.Prc> M Drscorrrnurry with wavelength 10 km in the half-space (7.5 3, ? 00b If the M discontinuity and the interfaces in km in the layer) are incident from below at { the earth's crust are horizontal planes, the various incidence angles. The top traces are % N [b = amplitude observed at the surface due to tele. free-surface displacement amplitudes, for each l I'.N} h l M seismic body waves will not show rpatial varia. incidence angle, normahed to the respective 2 ' [%'f h tion within a small area in which the incident amplitudes obtained for the plane layer case 1,'. $h U 6 waves may be regarded as plane waves. We without the dent. The phase delays are likewise ( '.W.c p H are interested, in this section, in possible ampli. relative to the phases cateulated in the plane !y.% E tude ae,n6 that may be caused by irregular layer case. The amplitude and phase-delay i l q ';h - .U shapes of these interfaces. This problem is scales are shown for the uppermost curve in 5 on I somewhat diferent from the previous problem each case. I ai N.M g i.' of a aoft basin because the distance from the The solid circles are projections of the trough 688 JQ[; ' observation poirt to the irregular interface is of the interface depression along geometric ray 5 fQ: creater. In this problem, the ray-gecmetrical paths. The maxima of f!ux density calculated by E l t ' V. E efects, such as focuing and defocusing, start ray theory occur at nearly these same positions .).MQ : playing an apparently significant role. in this case (these locations are indicated by the M'Nh-Nd In the ezample presanted below, the imminnary arrows). The double arrows for the e. = 64' part of frequency ut is sufficiently large so that curves denote the intersections of caustics with ,.h the exponential window s r* is down to 1/e the free surface. No ray theory arrows are ' M.%g. 1 i in 3.98 seconds. This time is short compared shown for the e. = 73* and 89.9' cases be- ' ' M f[* with the travel time through the layer (10 cause the ray. theory relutions have shadow i 3.-$,$, seconds for one-way vertical path); therefore, none gsps. The rms errors (Table 1) are Emsll l l 7.m. the efects of multiples are nearly absent in even for the case S. = S19*. Note that the these exampics. amplitude varistions and the breadths of the Figure 11 shows results for a case in which phase-delay ano=alies ineresse with incresi.ng "i! the interisee, located at a nominal depth of incidence angle and that the qualitative shapes 25 k=:, has a depression with depth 5 km and of the anomalies are consistent with one another Fig.11. 5,, l width 50 km. The shape of de::t is a cosine whib 0. chang s. surfs-e 4!:c "M I h*O l for ene eyele. The ve!ocities end dentities of The dashed curves for the e = 55' are the byer end Er.!f4psee reu-b!y correspcnd to :=plitudes and phat: d:kys e:mputed uint pts fe$. tho e of the crust tnd upper mantie, re pec-the r:y theory. The a=p!!tude is ve.y Isrge a e ray tie:.- tively, as shown in the fi;ure. The piene waves

r. car 149 km beesme the foes! region is neu i

.'-e +.. -..

%'??.*t."ffW'"K.*Lf;**)[*'"'NQQ'H:&' ' W Rhf$&5.;:,$% 6; '%Qf ~~ -[,;M,y*

* Y,.i #

? "," ' &y *NW!Qd;Y} ' ; .'( W & K & : q? W Q - & ? -h ."Y ? .&..... Tk.s & d.j. Q Q i r ? W W ?. "" M

' ?

's

,7.;'.?.),.b W -}?.. - 1l %,:Q % .r Q.'C

.yrK,:

^ }} .,3, s - -N.# ,9 .r. ~. .__ ~ =.._ - { r SURFACE MOTION OF !.AYEHED MEDIUM M9 1.. ~.. n j the free surfsee. The discontinuities at 120 and is that waves at nearly grazing incidence have f 1*0 erc artiScial manifestations of the sensitivity 's en' the repeated depressicas rsGer th:n a q. 3 of the ray theory to!ution to the second deriva. 25-km thick Sat layer. f r. tive of interft.ee shape. Our solutions are rea. The next ext =ple is the case of a rise as se tsb!s, smoother venions of the ray-theory shown in Figure 12. The media parameters for 1 fa so!utions. When the incident wavelength is re-this case are identical to those for the precedmg " A ' ;. f. l duced to 5 km in our method, the amplitude one. The arrows indicate the pocitions where p(If,%-j peak at 149 km increases, and the depression the minima of flux intensity and rearma of f y-( at 164 km mova to the left toward the ray-phase advance calculated by ray theory occur. ,2 U/+ : ~ y theory lobe. Also, the phase-delay curve then The solid cize!es indicate the projected posi-O nearly coincides with the ray-theory solution. tions of the erest of the rise afecg the geometrie h ',%- These consistent suites of curves and the good ray paths. The residual errers are again sms!!. L. 7. a accurney suggest that the solutions are valid The amplitude variations are less ri=pfe than (/Q P." e.en to grazing incidence. Note that the 6. = those in the previous case. However, the e. = fM. 4,'-f r. 89.9' amplitude curve does not normalize to 55' curve co= pans well with the ray-theory imity array from the anomaly. The reason solution except at the artiEcial discontinuities in for two

'A'h to dots are

'69 Yi';:'.L[ layer pa- @,yy,' 23

. s";

%,o .,4 ~~ ,o s' . :r dp'@^f hlf-spsee (73 3,, 8.** from below at 8

N e.....

cp trates are ,e so ?- 1' A. 5 My udes, for. each l Ii / 13,,,,pective p .. /ihy- [ io m layer case 8...e* l ys r.re likewi e -W G to gg.W. 1 in the plane 'd '8, y .,,.s.f, so i phsse-delay M H, - nos+, curve in g o rs j. g%$ 0 s..,.- M iof tha trough (s ' A !NP L.9: oo

e.... -

gaometric rsy 5

a.. s s

A If!1 oo 4

er.lculated by E 8.**** A c, o .R d

=e paitiens I

*2' O Q-d i oo diented by the f a'o.'o a'o i.' o e s' o aco am'o a.'o a.'o .' v W g,4:gj o .o .co iso iso th2 6. = 64, casrsnee a.s $Q.$yv$ .v ' eaustics with y arrows are t i e,. 5 o. u ypj'a ),gs e3.es be. caust so n. 2s== ,,. e e..u. ' p?[I'U.$ .n-g O have shadow 1 I E. i 1) are ess!1 u,, ru '",,. - ei.'

s *a a**

o s-so.- 5 ,, a ..u e pfy. Tote that the ".;G%p*, tadths of the r,. so n. r & ss./

' M.,%.

,, o., (th incressm.g c r-g. g me . r* itative abspe. g,gy h cne another Fig.11. Spatial distributions cf nor=atized amplitudes and phase delays for the free- );Y ~ *'.'2. surface disp!sec=ents in the downward. dented M discenti:uity p oblem as functioco of the ,..., c a=gle of incidesee s.. Tt,e dots show the treiceted positions of the trough of the deprescon g l re. ah-- the geometric rsy paths. The r.rrows show the poeirions of per.k fus intensities and - :!ed u% rhrE delt.y: prefeted frcm rsy theory (ex:!uhng mul:ip!c re cetions). The dar.hed curr s a i v:.y 1:rre are rey theoreGrr.1 solutions. [ l <-!ca b new i 'l -pe.,, j .i n. .m. -5 !=. ' Lw.... e.-

va p ej,.pg -y * ..-o.z.,.... - v,c e,9..gya .,c e--vfg.-m,--.q.w.,7,g..e neg. m..-- ;g:'r;g.m. s n , j.. , ;.apg 3_ q-'g:gggypg4.g,g jg4

gg h: ~.j&

g.QV g.g .g,.p. ,q!4 .x .,,. f. g(l:i: g ,y e. ..a _.,. ' ?.;.y y~ ; g,,,t;,,-; :.2 W : K.y

~g,"'

T. Q,jf! Qg r ~. - . :,.J. 33,, A.., ; X i. 2 '.,. .as> m..- , ^. e - 3 v.. ..e. 4 4 - :,m, ^ ^ it.I T -@ &- ; 2.$ .s .y T g,'O AKI AND LARNER {- perturb: tien '2 ~s on9 i n ; ihrrer o m ,o '8 ,,.rF-~--~--~----~ putatiemi! sc ~ 9 c.s ,o c:dly epl:e* g e..s

io e.. s'

.o (1902]. t Als } e.. *** y eo method is bt .(p { C tion for the

9 applied to sr

.'.l. cf arbitrary oss 8.****. t u = o.o P.syleigh ans: E h$ 8** V comput.ition o.o il co to problern g J W g in tha paper. t ..e. v than = rima -c; 9 I W '112e metho 'i l 3M 63 at the a, i i i i e i i 3 o 49 eo 6o to soo IZo

- o a6e seo 2co 22o 24o too oisrAmet (tal p M 1on W.

.I ..' i scattering ord "Il t given prob I n economize s'e m. 8 80** a,s a,.",.'u..,'."- caust a souree ya ee:';;puter tuni 1 1 If AfirLE 89 3* Cg.4 o k./Sas e.. rc - ,,. n r..a mu:n number f i (;' @J ik .c y/ 5,. 4 n. ict the first (p '~tW.@g e...c. .c#.7 .. o. i , secoods for er if 4e Although 5 .7. Y. <,>y i '.1 --d sc.: paper, the co 5,9 7 C. A

1, Ug.12. Spatial distributions of normalized amplitudes and phase delays for the free-surface

).h 1 displacements in the upward dented M discontinuity problem as functions of the angle of @Mems invc f."c.i'& y M( incidence s The dots show the projected positions of the minimal Sur intensities and phase 1 74M J p - delaya. respectively, predicted frons the ray theory (excluding multiple resections). The ( i- 'V; j. L'j f

dashed curves are rey theoretical solutions.

Q, i... l I F- -d D ta the ray-theory solution. The phase delays ment between the projected points of center s '..; % )d l 4 '. compare very well with ray-theory predictions. of step and the maximum of amphtude caleu. ) L .~!! Again the anomalies increase with incidence lated by our method. .I y' angle. $uch amplitude and phase-delay anoma-The above three examples illustrate that the

.m

.'.~ lies for near-grazing incidence msy be useful as amplitude-distribution and phase-delay ancmnly y-7 y -$-+ interpretation tools in refraction seismology. observed at the surface are sensitive functions r I The last era.mple is the esse of ' stepped'.TI of the interface shape. The change of amplitude f f,,- ~1 ' discontinuity as shown in Figure 13. The height distribution with the change of incidence angk h' ]; of s'tep is 5 k=, the depth to the center of step is especia!!y diagnostic of the depth of the ir-i l g. y is 25 km. The shape of step is a cosine for a regular interface. We believe that our method l half cycie (with wave!ength 8 km) connected can serve as a new useful tool in the stud > . s'j. i to half wavele::gth cosines on both s: des. The of regionalgeophysics. [.Nj ;, solid circles again ind:este the projected position Conclusion. Our method provides a practie.C r-I v,; ,j-of the center of step along geo netriest ray means for the study of aspects of the wan e raths. Incident waves base the nme waveier.g*h fields peculiar to the rimple two-di=ensionn I and the same imninary to res! part rat.o as models discussed. The emplitudes and s!ge those in the preceding two cu:np!cs. When the of the interfsee irre;ularities that we study at j waves are incidcnt from the ude of thinner crust Istger th:n those :1!oveed in the iterative al % U-l to th:t of thicker erm. v.e !!nd 1 general arce. proximatien method ef Pay!2i;h or in vr.riou [*~ i \\ \\

so

m~i 7. r, ser=w rs-re:r*7 s. !,.:

m yo=* ~* y. - v = ~~. v s ~ ': ~-* v.W ~.

e.

r

G M.~ n,g. M.c? s n u n-r,L.. ~ -.~ > $'.

~.. n:.N 2. Y*%.;;rq " C'.M.,., ."m _... (..-lN e h'c- "? @. ~.e p c y m.. w v [ . ~. n.o.., s ' M..,.s.$. $ .? .1 L,., :.<c :,l. $ :' ' ?.

M

~ ,t- . r.:o n ~.. - I f a'. '-7 ' &. * ' *,,N.. '; ?- ? q- . 7,. - .a e. ..s. ~,. t .T - s .a< > m.s.*~.. g, : : ,a.-=. ( s \\ SURFACE MOTION OF LAYEHED MEDIUM 951 M* r. prturb:6on methods (Gilbert and Knopoj, isotropie) media where all liut one of the inter-j 3 1p;0; ncrrera, IM4; Meltor,10CS). The com-faces are plane and parallel. This includes the f.. in p:.stier.:t scheme is straightforward and more problem of variable topography. The method is y, so e:-27 applied than u' the method of Ber.augh readily extended to ine!ude problems involving

. y -

68 [172]. (Also, see Sharrna (1967]). Banaugh's additional irregular interfaces; however, we M. '8 z.-thod is baeed upon an integral representa-must sserifce either accursey or roughness of f.6's. J. l [h'i' t;cn for the solution and, in princip!e, can be the irregular interfsee in the problems that i .M app!ied to scattering of wates from obstacles can be solved under the present computational el arbittsry shape without su6ering from the constraints, f,cq E:yleigh ansats error. However, for comparable c.f - c4 8Lo ec=putation efort, when his method is applied A N DIX [D;. 40 to problem geon2ctries similar to those treated The error in censertation of energy. When .Q. C8 L is this paper, errors arise that are more severe frequency is real, if u is a solution to the y,Q;;,,' than our trunestion errors. homogeneous sealst Helmholtz equation through-E@' Tc'e method wu programmed for the IBM out a region R, then the real and imaginary 3 y"r,'p. f. f,YA5 at the M.I.T. computation center. Com-parts are individually solutions, throughout R, D'/:gs i/ pf.stion time is controlled by the number of to the rame equation. A censequence of Green's c:tteri::g orders representing our solution. For theorem is that a given problem confguration and frequency, (. d ',' ra ecosemize by hand!!ng cases involving vari-Im f s' 0 I. 'C dS = 0 (AI) y c: cource-wave dire:tions concurrently. Typical ,f 8 ec=puter time for problems involving the maxi-where Im denotes the imaginary part of, the c,:m number of scattering orders is 4 minutes asterisic denotes compZer cor,fincie, and S is the n; v !ar the fint incident. wave direction and 20 surface enclosing R. ";s secends for each additional direction. Let us apply (A1) to the approximate solution W; y g Although we showed no examples in this ute in the re;; ion of the half-space bounded by u paper, the computer program is applicable to the interface, the plane at large constant s, 7. p fremrface prr,blems involving multilayered (homogeneous, and vertical surfaces at = 0 and z = L (L is p ,o ooe a s and phase n M-, me,;, i:na). The W e y v V p;e:e e.- p .o '* & 'l T WS A n:f@ n l w N 9. !.i 3 l etrate that the 'Q- %{h so to l ay -omsix i dtive functions a 88 ^ h@.D

e cf amplitude j,o. o* @ in

- M '8 0* h@f., J ,.nendence acgle t 4 h'; i.-

- ~

?pth of the ir- ,,, a g y % .g.g.co o ao e so m it ar method ao ,,.w..,, pgr-p' in tha study 22] = ,,. t e.... s t* W w

" ~s As.
o.=,.=

.D. i ,,,a. I ides a practies! 8 :

H '* '",

1 -2. r

c. ;-

l t of the waV h

?

i.e.,.8 ".*... M '3' /l. h,..,/ i ! ro-di=ansiona l Ve d end fI0pc j [L" 7."$.'.',."

h.

j I '3 I ng. 13. Spati:1 distributions of the normahacd amphtmh > of free-+urface displseement ~2 # *.. 6 f - the stepped M disco:tinuity problem as functior.s of the acgle of incidence s.. The dota f t w de poMied poition of the center of the step along the geometric ray paths. f. l cr ta v:.r; u - t e t l 1 .u. J7:U. -N,,.., 'l iWQ.,o.bs.f6f.a. " 1 g x .A--

.m-- * ~.-- --e r pv. re ; ;' m v,'e. v. .., j.yy2 - e p,. w < - .v., ' i.e., g,....+ - q t m c.,r... ig.g h i ,f ~ 5 ..a .. s_... -. -.w.c. , w.- n : ? m.c

,w;lx.y.

f.,.. g._;~.:4 q w f.,S& Q N ?' ~ 9.,. ;4 .~ .w. fS.S,.,...X. :w; " p. '$ PM ' ~.n ,,, y :.' 2f,,W@ t m. ~., ;. %,,.. w n..,-ur g. 7.,y.y ';,z .-+ .w. .y, 7 ... g \\ 't.. ',. [ . E' .= } ~ ' .~ ~N W;;? v:n+:w ;.2.M *?.9.J %.5.d.6%@5 N: _. e. - '.$1.z, .. e :_ .J.f.; .y i.:. M: ~ ,...-z, % < h. ?:+ ~ 9'2 AEI AND LARNER

P the pe:iod of the interface shape). For con-

'\\ i ves.lenes, we make the horisontal component of e dx = 0 when m#n g 6 w:ve nu:nber le (equation 2) of the source paper w2re equal to 2r(/L where (, an inte;;cr, is Recalling that the inhomogeneous waves do not p [ the nurnber of wavelen;ths across L. With this contribute when :is large, we 6nd thus choice, both the exact and approximate solutions f"." h. y! have the period L in the horisontal direction; { e'ce' 6. [ A."' j' - i ,9,, .f a! consequently, the inte;;rals over the verties! = g;, 1 portlans of S at = = 0 and : = L cance!. Equation "g' Natio At becomem + uL cos 6. 's 1 + 3 (A5) so::3, ,,s y, I , in Im u *rw de The quantity 8 is the departure from the con-n servation of energy requirement ascribsble to e the residuals at the interface. The error 8 is F .' s. more easily and accurately evaluated, by means = 1m u'rn dS se m (A2) of summation over the sesttered wave orden on 7'. ..-i s . l .r. the left ride of (A5), than is the r=s error. The Een Equatico A2 deSnes the quantity 9 crror 8 is not a sensitive one for two reasons. Abubs dj Similarly, let us apply (AI) to the approxi-First, the amplitudes and phases of the residuals SH mate solution w in ti.e region of the layer ::ecerally osci!!ste alo:g the interface, tending fer' ' produce small values of 9; second, this error Ab I 22. bounded by the free surface, the interface, and { )dk[ A'H the vertical surfaces at : = 0 and = = 4. measur'= dces not have preferential weighting of ( .'M Again, the integrals along the vertical surfaces the larger residusis (localized anomalies) as does g, b.l ' cancel. Also, the integrsi along the free surface the rms error entenon. ,p,, D vanishes because of the stress-free condition. We n te that the conservata,an of energy [7 yi *M anoi hd We have, then criterion provides a measure of the error in the stras y MN,3 <obtion for the half-space only, and states 2 54 itsWf 3 t -' nothing about the solution in the layer. To see An'no g* :g/j tm { u,.,*ri,ds - 0 (A3) this, we replace w in (A3) by the exset solu-y b 9 ,C* 1, j suu r. tion 14, and note that w = Au + 14 at the g l - I IJsing (26) and (A3), we rewrite the integral interfsee. We 6ad that g,,y,, ~ ' Q j over the interface in ( A2) as way i= f...g.,tu.* A r + a = r. 'e jtg..Q i... <. a n - im 7a., - u,.., J... w.Q,c.,.,,j g. Q + au.'ary]dS j ~~'-- NkIi. Y - a* rw) dS (A4) This integrs! for n is independent of the ap-Banau ,N.D.i8 F.3.. proximate solution in the Isyer; it vanishes with = l That is,9.dus!s weighted by the approximate is an integral of stress and disp! ace-1 r-.a; ch the errors do and Ar m half-space solution. g&~s.4. i u ment resi pp, i I k 4' . When frequency.is complex, the rest and solution in the Isycr. Since 9 vanishes with the imsgmary parts of u no longer mdividually go,- ( j .i residuals, i't is a measure of the accuracy of I

p. - 9 :[i the approximate solution.

satisfy the homogeneous sen!st wave equation. Fou. The. tegral along the p!sne at Istge :. Appliestion of Green s theorem yields a volume Dusti

v... -

9, t I m is integral on the right side of (AI). In that cax. of c a. readily evaluated when we replsee w.r., by an

r...

4 expresson obtamed from the computed solution. we could attsch no physical mencing to an error half-fi ~ tin y [ -T. equation 19. Beesuse the solution is periodic for mese based upon de m.tegrs!. 9 E' O^ our choice of k., we eso use the ortho;cnslitt Achowledenents. An outline of the raethod Gike enndition, described in the preeent peper wts conceived by f-n 6 i the penior author (Keiiti Aki), when he wr.s E. s working at tl.e Nr.tienal Center for Earthetske j..t. fe c ' * " " * * * ' di - /. when n =,, thereh. US. Geo ogical Survey, Menlo Park. i, Californis. under a Tapcr arpoint:nent. IIe owe-r a., I l l . - - ~ ? 6 . 7.. t .- r t

$ $ f? Y'E IYM"fi?'EW .Q W W ^.V;,5gk?.M; [<W ~Mm.*,,.w. b.., w.- ?.5$- v D.' '.k F1[.;.;NM3*.jft@dk.,. FY: $h. m.,: ~.? .nlh. N? ENiWM: 2 ".q/h W. P. N.[

f. %

v r m n.. c... \\.?Q.m.,;A;q. n.~ m.,'d n.. 4 s :, Q,-1 ;;,. ?, ng.:; .Ty

-ll.?-

w ~w -v ,%:4n :n n.: 4.-y; F;.3, y . x. v.; p. J.g' " q. ; n v m .,,p., ^ N.:y.v..gm. w.,, g.. 1..g.t. - - ,y,.

,, w,,x.j y

. c. .w. . A.-- .w.....

n. L, n,, y

^9+ = :3 9394: ' T',_,. 1.'.;n' ..Q

m....:.r;%

c L -

n.. c. g.

,y. '.. n t '.

. ~.., ;,-;y :, _. u.. u ..;.c? r v. uc 3.; ,'l. ' : ; ,j v. L;..:.r. :;.e ~ y V SURFACE MOTION OF LAYERED A!EDIUM 953 y,;.- a pie:r.st and prcStable summer to L. C. Pakiner Haskell, N. A, Crustal ref!ection of plane SH I'.. b when a pd n and 1.4 colleagues at Menlo Park. The fast waves, J. Geophp. Res., 65, 4147, 1000. Fouric transform program used in the present Ifaskell. N. A, Crustal re!!ection of plane P and ,i f p2per was wnt en by Ralph Uggins of the SV waves, /. Geophp. Res., 67, 4751, 1962. 1 cou. waves do not Yaar,athusetts Institute of Technology. The au-Hesps H. S, Refection of plane waves of sound (, fDd thors tt snk Dr. Widna and Dr. David doore from a minusoidal surface, J. Appt Phys., 23, y,., i hr sugesting valuable improvements cor the D' N-L -* original manuscript. A part of the comp tation ras done at the computttion center, M.I.T. Herrers. I., A perturbation method for elastic R..' This research was supported partly by the wave propsgation,1, Nonparallel boundaries, J. g'V Natio=al Science Foundation under grant CA. Geophys. Res., 62, %t5,10Gl. y s!+3 (A5) c:2, and partly by the Advanced Researth Proj-Kanai, K Relation between the nature of sur-p/ set Agency; it was monitored by the Air Force face layer and the arnplitude of earthquake L-ure frc a the con. 05ee of ScientiSe Research under contract AF motions, Bult Earthquake Res. Inst., Tokyo Q niv, 3, 31, 2. Q1 tent ascribsble to E*"'I' K, T. Tanaka, and S. Yoshisswa, Com-kg',

r. The error 3 is R27:nnrczs parative studies of earthquake motions on the L

,ju:ted, by mesns Ahabakar, I, ReCection and refractien of plane ground and underground, Bult Earthquake Res. P {C ed rave orders on SH waves at irregular interfaces,1, J. Php. Inst., Tokyo Univ.,37,53,1959. hs r:n1 errer. The Ecrth (Tokyo),10,1,19C4. Lapwood, E. R, The disturbance due to a line i fer two reasons. Abubakar, I, Refection and refrnetion of plane source in a semi-inScite elastie medium, Phit ( c3 of the re:idus!s SH waves at irre:r. dst interfaces, 2, J. Php. Tr:ns. Roy. Soc. London, A, !?!, 63, 1949. h,, interfcce, tending E:rth (Tokyo),10,15, It026. Larner, K. L, and K. Aki, Response of a layered I ttr:nd, this error Abubakar, I, Buried comaremional line source in half 4 pace with as irregu!st interface due to p a half-space with an irres dar boundary, l. incident plane P and SV waves (abstract), Eos, a .id we ;htinE of Php. Earth (Tokyo), 10,21,10Cc. Trans. Amer. Geophys. Union, 50, 400, 1969. t' ch. t Aki, K, and E. L. Larner Interpretation of Levy, A, and H. Deresiewicz, Ref!ection and b spectral amplitude distribution and phase-delay trensmision of elastic waves in a system of M istion of energy anomaly of P waves observed at LASA (ab-corrugated layers, BufL Seismol Soc. Amer.,57,

Mb

.f the error in the street), Kos, Trans. Amer. Geophys. Union, 60, 393, 1967. D enly, and rtstes 244,19es. Sfack, H, The deconvolution of LASA short. N'D the layer. To see Asano, S, Resect. ion and refraction of elastic period seismograms (abstract), Eos, Trans. t ,y the erset relu-wans a,t a,dence of SH wave Bull Earthquake Amer. Geophys. Union,60,245,1969. n$ corrugsted boundary surface,1. The D case of inci I E* Seattering of plane elastic warm ." y AICIV0 face irnperfectiens. paper presented at N + n at the Res. tr.st., Tokyo Univ.,33,177,1060. by sur g sp As:no, S, Resection and refraction of elastic the fall meeting of the Eastern Section of the b~F waves at a corrugated boundary surface,2, BufL Seismologies! Society of America, Cambridge, f,.S sthquake Res. Inst. Tokyo Unny., 39, 3G7, Stass.,0ctober 10G8. ?, 43 5fech!er, P, and Y. Rocard, Geologie limitations jy Asano, S, Refleetion and refraction of elastie of the use of an array of seismometers (ab- -M bg M waves at a e trugated boundary surface, 3. straet). Abstracts of Papen, Int. Au. Sctamot A%* Ar,) dS BufL Seunot Soc. Amer., do, 210, 1066. Phys. Earth's Intenor,123, Zurich, Switserland, s f,e$hi 2 dent cf the ap. Sansugh, R. P, Seattering of scoustic and elast.ie 1967.

it vanishes with waves by surfaces of arbitrary shape, Univ.

Afeechsm W. C, Variational method for the Cahl. Radiataon leb. Rep. UCRL 6770, 174 calculation of the distribution of energy re-E.% dfpe ation. W' f!ceted from a periodic surface,1, J. Appl 74 x, the rest and Cooley, J. W, and J. W. Tukey, An algon.thm Phys., 27, 361,1956. kt :- yer ind..d g;y for the machine computation of complex Stereu. R. F., Effect of Mohorovie topogrsphy gg' m r me equation. Fourier series, Math. Comput., 10, 207,10G5. on the amplitudes of sets =ie P waves, J. Geo-n.- h'.rY t yieIds a yo!ume Duntin, J. W, and A. Eringen, The ref!ection phys. Res., 74,4371,1959. y.gg'5 61). In that case, of t!astie waves from the wavy boundarv of a Phinney, R. A, Structure of the earth's enist m fr m spectral behavsor of long-penod body p:g aning to an error half 4 pace, in Proceedings of the 4th U.S. No. q a timl Contreu on Applied Mechanics, Uni. waves. J. Ceophp. Res., CO, 2007, 1964. TcNty of California Press, Berkeley, 143. 1902. e cf the m thod Ci!b t, F, and L. Knopoff, Seismie scattering Phinney R. A. Theoretical calculation of the

p..e '

n. e:necived by fr^ m topog ephic imgularities, J. Geophp. rpectrum of fint arrivals in twem! clastic r-mediums J. Geophys. Res., 70, 510,,, 100,5.

c

). then he was R. y cy, ;g37, in), r fct,,M: nab t Cut 11, N. A The dimerrion of curface w:ves Ra31eich. Lord (J. W. Strutt), On the dynamica! ,... ~ _J r g,..er!) Puk [

i..nulti-!s>cred medis, Bull Scumot S,e.

thiory cf gr::ings. Proc. Roy. Soc. London, a, s. .j h r snt ITe oro

g, 399,1907, a.c7,,43,37,gg33, I

g" ~ l, l f X. -,. ^^ < g 4.1 *. %.d.w.t [

...,'??:.:9.;.;,. %q.,r.gs.

. y. .g_..., p.gy;fy.(,:g.pM.,g,,..........?;;.1 ~ . n.gt,sgg%g] kE h %gyxy,1hNi

y

$ "'. h [. N,s uty;gh.?? k h D8.h g+th g $ q >u._.x [h 5.,.'[c,N.$5-M;-(h'+:h.z

,-9. L MI}M7';M g.

4,a e'in,%. 5,utSIa-:. Qq.,,Q;.g~ :. ~ n.7,y,.%.,z.,.%z,m.c,ymerz ,,;g%yg_~.,y;., p,q.,;. 1 um p~..=... j: m. . ~ m....m < 3.+ ,,,.. a ]. f;q.4 y .=m. g 'g : t,.,..$-y. a.-. ;,':.g: y~w_q .+q-- t. ,s u . ), .y... , ~. - ..., e,fi's e,e.*.g ;. u. . + .w

"- ~;t y Mh, * ;, ~ ;n.f.._.

.."hfl5 N k v,:: l. Q h h. k. y.;.[;, M [m.dDN

\\ '. ;'

g.;:.- g.: + q., .U,

_ s.,

..m c.,...s. ,;'_. W l m ',.; s _3. r J.h,,3 7,%.'*" y-,W,s;J G' b C..;

x_

q m q

.se:

._;f. a 954 AKI AND !.ARNER M"" # Raylei-h, Ierd (J. W. Strutt), TAs Theory of by surfaces of arbitrary shape, Bull Seismal dotand, vol 2, SM pp, Dover, New York, IN5. Soc. Amer.,57, 795, 1967. Sato, R, Reflection of elastie waves at a cor. Uretsky, J. L, The scattering of plane waves froni ru;;sted free surface (in Japanese), J. Seismot periodie surfaces, Ann. Phys.,.*,3, 400,1965. Soc. Japan (Jidin), Ser. 2,8,121,1955. Sharma, D. I, Scattering of steady elt.stie waves (Received June 23, 1969.) . ) . 1 i I The for j km.M = time durs pandent.. Network shown to wave fort 3~ 25 km, tl ob:erved with a sta e i s I-The domin: i. f.;.['".4.: j acteristics in a 5 '.; 'I long-period F p 1,"M, investigators. .,v -- transfer funct i

d.y-n a

~ "Tri g structures ha' l 24 - ferences exist t

1 - L.

b.. '.g e 10 see. These t ;. ;.J.& simi!srity thz t -0.f3 E. period P.way ,.U $ Stations of th j ~ 'j earthquake. 'I k,(gjf to inquire ints s' di! "3 ter=c;rspha i *,- T3rious time I. ~ -k'.j.. ; j r e q gstin;;-source ,y...t pitneh, the t! '.. c, for a time-de C ,3-tics of eartl [' - J; velocity from -:t sented in a ] 4 paivr 1). T1 azinsuthal mt [ 3 fault theory ~ t=itted radia ..x lyzed in pape y tu!u indientec an i interpret. [ { e*CT! So Chtr.i C trf.t C H j i I .a... ' k, .*4 6 $.e ". 9 g se.. I e h

% ;:e.. J..c.,: +. ~ @... . V ; ?.' M S, T,..,2 f. p* *T.M @ a!a%.~w,.w,w..in W.w.+. +%. +: m c-sw4-Wy m m :+. W e. &...-,:..xf..w....M.y5'E4}6 9 W $'M m:..~...,..q. .. ; ~... m..a.!m.t c :.#.., c.>, y.w..!.g.s!vOSUM. w.v v.D..;,r .r.M.... _..,,..". 3 erm m, v . <x. >.. c.m x%...w.,,: M W-FD.. .v$@ T n' F. Q. >..n.i. ..,==.% ..s..., :?.. .-.. w.. s.. ~ .w --e -. 2.. _ -e n - . m. ; e - -. ,,-m,.s.. 's... :v. A %.. -- ,7.. m. %.';C.%py:::2;;.M,f m. 3 .I n.s.y :.. H:m :.m.:s:~-.4w 7. w,;c

f. e. m..p n.iu p.c..e..

v.,...f. d.~.... - 5.J.eun cf i.be Seanmoiorci Sonety of Amenca. Vol 70. No. 4. pp.1060-12% August 1%0 M.9Q.MH.C3X yH3 U$5 % 1 c + THE SEISMIC RESPONSE OF SEDIMENT-FILLED VALLEYS. PART 1. .t W... W,. ~N ' W;.,. @, g,,%,, THE CASE OF INCIDENT SH WAVES - I'M MY.W 4MfW C F.n@,.v>.;y' f i 2 BY PrzRRE.YVES BARD AND MICHEL BoUCHON . l% p. W tfN w .. g.m g ww:- { ABSTRACT E'* N.N '- (N, Y.5 ,m. ~~.T. N+.M.:<., 3 .s www in this study is presented the extension to time domain calculations of the e a Aki-Larner method (Aki and Lamer,1970), developed to investigate the scatter. [, M sg.' @" * *m nik ? sng of plane waves at irregular interface. Seismograms computed at the surface t.o H

MW u.,.7 W,.p t. M.:.C yr Z l M..

e,- i ,8 of a soft basin for SH waves vertically incident are compared with results 4 ,y "&.Pffg.g.%:W.' 3 obtained by finite difference, finite element, and asymptotic ray theory methods. j .} . The metho,d is then applied to a study of the seismic response of' sediment-filled [, D'MQ valleys to incident,SH waves. Various, geometries and theological parameters t* g * 'Lp g y.g. i, tM...,*u. . w,e..x..%"'6 are considered. The study shows the important role played by the nonplanar aM v interface, which~when the inc. dent wavelengths are comparable to the depth of j"Mk"MI.Cf.h.ih.% i pW :=gyQ i the val ley,* results in the generation of Love waves which may have much larger, fb#M E.v.wMbe amplitude t.han the disturbance associated with the direct incident signal. In th.e Q.'M L. r. presence of a high-velocity contrast between the sediments and the under!ying ~,, W r.e yjjp g,.,:. r.% ..edrock. these. local surf.acd waves c.an. be reflected seve.ral times. at the edges : 7 p b 1.. t, m m.a* e

m..

.m n p gg,.: . of the va!Iey, resulting in a long duration of the ground shaking in the basin.ln S ~.

p the case cf a lower impedance contrast,these waves may produce disturbances.

c kgghs'$y(g:< rgy$ y h.,..g@..f gg. pn the outer sides of,the valley. - e3# .j *- ,,.n, ~ ). dw.,,.,wc..p4. o.,...i.- w. m...,,. p Q pgT;,.yMi@g. k... y. '. _ v -m. arw w g.:. IN ........ T.,RODUCTION .,,... c. s ~ a. - - ., [.,- ' E Mi Wh'en looking ai' intensity ma'psir either great-or middle. size earthquakes, the t T stretching of griat'cFintensity coniours'along alluvial valleys or sedimentary basins.^. 4 ' ~ [ - %.:.or.w.dady ' is often a striking fqathre, and becaush. of.the greater concentration of' population in'..~ 3 s d > tl.ese area',it has bro. -.h. ~. b......ug t ear er seismologists and engineers to study this particular -- s agg / ,~. j gg R eifect [for a-completi review, see Gufenberg (1957)]. For a long time, such studies;- j,{y e d ". Q U: have been essentially qualitative,.resulting from field observations of great earth " ".^ i, j quake damage, or.from do'mparison bet 4 den records of the same events on varioust pI.f4 yp eological sites..From these vadous works,it is well established that alluvial vallevs < : B % ems, s>-e-4 bd sedimentary basins are generalg exposed to surface motion' amplification, andg...h- '~y ([k.6,3 {hq.}g;p{dtgy that signal records are longer and more, complex on such sites than.on crystalhne." m ip 1 SQW ~ rocks. These 'studiss, hInvever, also lead.to the conclusion that the scatterin; and. t [s.h3.My$j;%T rrpeated reflections ofseismic rays af giologicalinterfaces induce a very complicated :- 9 'f M y['j$ky&:. pattern, so that the jnfluenbe of irr,egttlar geological structure or' topography may ove shadow the.eff'et.of. local site conditions. This was. pointed out by IIudson g ggpjg (1972) for the Sa2 Femando earthqdake strong motion records obtained in the., _ y.g;%g-g*g, P:dadena aies whifn ' exhibit quite different characteristics at very near sites and ' - T .dr-ihr m:mmum displacements on crys'alline rock and on several hundred meters S-wi,pym, m%.G i.g rf.gA.z- .r-... .~.r.-..w,

n. -

\\ cf.Aluvw.m.

:..M......

g such effects, one must keep in mmd fo...;s.fferent phenomena.e.... M.,.P%.%.pl. n x% p .di UtgwMg-d,, y'2-Thus, for studym. ur g-jd. Q$, 'Ihe first is the'dislacem'ent amplificiti5n which occurs when a' seismic ray travels. ' g f M y@ g g

.hrongh an interface'ffom lit high rigidity. medium'to}a lover one. The second is the~

WpC mechnnical ~risonanb4[of; plane layers at critical' frequencies. Asl pointed out by gggg i l 'iaskell (1960), for a'SH~waie verycally incident on h plane layer ~having a shear pW7MKQ.! i n M V..WS.c 4. vebcity Si and a thickness 2, these frequencies are Q.??: ;. L..:. .;~ g. ~ f,, - (2n - 1) g$3 6- . W

-+

~ ( ~ R.; *,

. l.7L ' *--

f-f: v 'l oy 3#.t W. t..iM;..i.f-45 y .. N=r wC .D D.U*t$ I . c..R. _ w.(( ;,,, ' F ~~~ED 7' g 4., .J.. ~. .s.h.....w. .. m.u.. :.= -... +.zw. n.;. ' -m.a-- N.ovs n '. ~." a t*kM-z.=2. .. - :,u Ai ^ c n

- - ~~'y;. + = y.

. m.y.;.m. y3.~,,R

.$ %n 3.,.fy ctQ, %.MjW;d.., W,.m$?.,i[,, Q.4i., 2.Y!$$,.n.Q.cr... - ~.. ..r. ,m M qy no q n c_ G,t... g

,. 9 n,.

.i.u g. y :+.y y; m p xy.. 4.a., z.... w ., +,.g. 4.. g,:.~g%; g-

.y.;-_,: s.s...:.

q.15;xr.m m...u ;n v. w.w.~,p~Q. g q. K. -3, g. 4y. .w.

M.-.qg,,J.Q.,.rd.. y-%,.,

~. U: 9 M: s... p win

2. +.y

.-:;,.:gr.. .n d g.ge u -1 .i; -I 1264 PIERRE-YVES BARD AND MICHEL Bot?CHoN SEISMIC the resulting$tMcAdisplacemrM ampJification being pronortional to the imt edance considered, is smr contraTsIIe.. density x velocity ratio) between the two media. The third corr.es outside T (Boucht from the nEnlineanty of the ground response, the importance of which increases As incident sign when the soil rigidity decreases, and its main effect is the attenuation of high frequencies. Finally, the fourth phenomenon is the influence of geological lateral heterogeneities and topography, which may lead, among other effects, to the focusing /U of seismic rays or to surface wave generation. It is because of the scarcity of appropriate data which are only starting to become available [e.g., Tucker and King i The shape of this: (19~9)] that attempts have been made in the last decade to cornpute theoretically ~ such effects. One such first study of the effect of a soft two-dimensional basin on a i plane-incident' signal.is due to Aki and Larner (1970), who developed a new semi- ? q analvtical, semi. numerical techruque for their calculation.Trifunac (1971) and Wong } I iifunac (1974) brought in' analytical methods to ' valuate surface motion of i e . perfectly elastic serhi-elliptical' valleys for incident plane SH waves. Boore et al. 1 ,(1971),' Smith (1975),'and H6ng ind Kosloff (in preparation) used finite difference .j M and finite element techniques to investigate irregular underground interface effects. _.J These methods ' generally remnin valid as long as the incident wavelength remains i v of the order of the sediment thickness and become inappropriate for high frequen. .u , _cies ' Ori th' centrary,'.Hong'and:Helmberger (1978) proposed a high-frequency M 9~. ; <,'9'. e ' -'.me:thodi. referred..to as Glorified Optics, based on the first motion approrimation d

which.they applied to incident SH waves.:-

3 . (, 7., - - X.

The purpose of the present study is first to test the applicability of the method. 9 24

'idev'eloried by Aki and Larner (Aki an_d Larner,1970; Larner,1970) to time domain M .c.. caleplations by comparing'it with other' methods, and secondly, to investigate y 3 2 systemitically, by.rneans of this powerful technique, the elastic response of two-1 5" x dimensiorial alluvial valleys to plan'e-incident SH waves /~ - )3 ...w.x.& m.s D i u.rn. : .. -.. ~.... ... SHoRT PREsEN. TAT.I.oN A.ND ACCURACY OF THE AEI.LARNER AfETHOD .[ _ 'a . c *.

s. :

.; : w ^ Thep'roblem'and the.rderhod. The' target is 'to evalua'te the elastic response of a ' tw'o. dimensional 6asinif arbitrarylshape. to a plane. incident SH signal of arbitrary ' ~'i form. In this aim, the Aki.Larner method is first used to compute the surface,.,'! displacenient produce,d by a plane stationary incident SH wave at a number of Eites f* inside and outside the valley, and then the FFT is used to obtain the displacement

3-i 0

. kin the time domnin'.:;m.c <. w:b 'c - . - = .i V The Aki Larner me'thod relies in part on Rayleigh's representation of the scattered wave field produced by a plane wave incident on a corn 2 gated periodic surface. The 9 t ' scattered elastic.fielli is described as.a~ linear combination of plane waves having discrete horizontalDvenumberE The' application of the stress and displacement 3 ^ continuity conditions at the interface leads to a system of linear equations which A are' solved after being Fourier transformed in space and truncated. The method i Ric$rN2[No"p* ( requires a periodicity L:in the interfaci shape, L being set much greater than the

- - site 5 **d an horizontal size of tlie interface anoinaly.In ordei to save on computational time and M

~? l to gain;in stabilfty, the method also 2nakes use of complex frequency, the effect of N! l~ ihich is later remove.d from the time ~ domain solution. Afost calculations presented j "E l here are made for 32 frequencies /e' ually space'd between Oand feo, fm, being,take.p_ .j

e. 'equ'al to 2e/t). T q

l ' <sual to gih h d freXdejotes theNn@nuntdepth.of_the%IisThTimaginary { osci!!ations due to t , part of the frequency is chosen as er = r/md32, so that the frequency spectrum is because of the use j continuous and quite smoothly varying without ripples over the whole interval these cutoff freque: [0 /mo). The time domain results, however, are quite insensitive to the particular incident signal rathe choice of er, as long as eV, where T denotes the length of the time window more high.frequene: l .:a C...N,9 *....r.. u.s.... g. . a. u - a e..c....;. - .;uu w..m, .K u na w n

' p;jfW %v.M y.g.7g.Lp Q gy q' g-.(

- +:- ;-:~r ~-- - ...., - 9 _.. . m_,_?; $_; E... d d M % &.. t '?-: . '. u G G v. ' y. _ ; y - ~ F J.:

W n\\; Q.i. ;}.?. & y ~, Q. 6. ?t*L' j.3;. X [n.,rh' %. $. Q.:..i & Q W

%&.ydfp T :.:. % % W m R C~.?.1.f._S~.Q.tW%ggyf.ey;. q% ? E*d i$. T.Q %... Q-n r ,J ,m, @m...s n. :W g ".2y: g.. ;..~.,g.plge.;g. ~'g(ggg.q4g ew 1 };^.x : r.,.j>& y _._ Wcp g 3-A q '-~ 1266 P!ERRE YVEs BARD AND MICHEL BOUCHON SEISMIC RE Internal consistency of the Aki.Larner method. The Aki Larner technique con-tains three sources of error. The ftrst and most important one comes from the incomplete description of the scattered wave field. From Huygens' principle, a f stress complete desen tion of the wave field in the region within the range of variation of a the interface depth should include up-and.down going plane waves of nonconstant i-amplitude. The representation used here, however, only provides an approximate description of the wave field in this region. The resulting error, named Rayleigh C ansatz error, becomes important when the mean interface slope is large or when the g'. f incident wavelength becomes comparable to the depth of the basin. This Rayleigh ,r; g ansatz error is the main cause of limitation of the method in the high-frequency J. displacement; I td ' ' ~ 9-domain. The two other sources of error result from approximations made during the .l. calculation and can be kept at ari arbitrary low level by choosing adequate compu- .J tational parameters. The assumed interface periodicity, which allows the discreti-y zation in the horizontal wavenumber space, may introduce parasitic effects due to 3-disturbances coming from adjacerit basins. Such effects were proved negligible by i k varying ths periodicity len'gthL and c'omparing the results, as is shown in Figure 1 3( ,ws, e -- ['~ for one of the configurations studied. & The' shattered wave field is represented as a linear combination of plane waves -l ,.) B withiliserete horizontal wavenumbers k. = 2:n/L (see Aki and Larner,1970). To perform nume'ica1' calculations, this representation'must be limited to values of k. -d r such th'at'k. < ky - 2:N/L. In all of our computations, the truncation number N N -.Z was cho'sen so that the amplitudes of the corresponding scattered waves inside the { 'i dispiacement .. valley converge toward 0 when k tends toward k.v. This requires k.v to be chosen, y for each frequency, fairly larger than the corresponding. fundamental Love mode 5 ~

I E avenumb.er. This fact appears clearly in Figures Ea and 9a.

Although these error sources may be controlled the best test of reliability of the j me'thod is,to estimate the error. This can be achieved by comparing stresses and y displacements on eacit side of the interface either in the frequency domain or in the l _, O time domnin^as is do6e in Figure 2. This figure shows the good matching of the. boundary conditions for one of the configurations studied. '[ g Comparison of the present method with other techniques. As previously men '. t tioned,several other techniques have been developed to compute the elastic response ' A of two-dimensional geological structures. Recently, Hong and Helmberger (1978) compared their results,with the, results obtained by Boore et al. (1971) and Hong y a ~ ~ and Kosloff (in preparation), for a inodel consisting of a 50-km-wide,6-km-deep

- 1. 8****${

~ valley surrounded by a 14nn-thick layer overlying a half space. The theological '.1 i" parameten used were p, denoting the density and B,the shear velocity, l ~ ~ h; Bi= 0.7 km sec-' d. ..pW2.0 gm cm-8; =. '] ~ -:.for the. layer"...:... : - .s u. [. ,for t$e half' spacer ;.p:Q3, gm cm-'; Bi y3.5 km sec-1 ~. c... e with a R' icker wavelh as. incident sighal, having a characteristic. time t, of IS 3 sec. d y In order to test the Aki-Larner meth~od,it was applied to the same configuration cd u ~~ using a cutoff freg'uency of 0.137L Hz and the results compared with the finite ' di'ference, finite element, and GloriEed Optics results. This compa:ison is depicted in Fi;;ure 3 and shows the good agreement of our method with other techniques. [QNhy charactensua or the u especially the finite element method. Moreover, the Aki-Lamer method allows very er.sily a study over lon;er time intervals which is quite impcriant. as the signal durrion proves to be greatly increased by the pres-nce of local heterogeneities.

....,.. m....;,..u.m~,;,w,.,,. m n. n..u ~ n.a...~..m... a. m.... ;.. ~:,.,.... ... ~.. .u. - ... a. m;mg ,g..gy

..g s

we.ev.w m- . u~.. 3 ;;.n ra,Eme e,.,. . c. ! p.a t;+m.~.._. w.. b,. .e V*g. N,l:.* p* 4~.C..,.. c.. e.a .st.;.. t .4 .r ,$ ?. '&s j-Q* D5..'; '..7 i+.,' $$,Ql.Ue'..*.. M.i. .i.. SE!**.!!C P.ESPONSE OF SED!.NIENT.T11!.ED YAI.I.EY3 TO SH WAVES 1267 M, s

iMdf '.Wi.,

&.. f. -la tr,, '--.t.. :*P.* >*,W.. ~l

UC h 1 i +d.,. M.

9...,/ : a.y... .j.4 ;? r 'I.... 4. ti 9g J MnieY.4ir.;- r L.G, t:.~.<Wr,1 rO/,7Qe.;yFl.,%?/L i i. '. - e ..a...,"m.w %u,.n, x.~. e +.1 i stress t,. r.;.v.w.Is'M;.,I.3a. ws,.v;,::..r s:

1

..c+. t . v. rm - y. v-.. i.. p:.~~t s :M W. R.* -rtD{% "$sp,,)q,,,.;'; < e r,1:**%.*v. 1s.e .ae r. g sk. 9. -+ - .oa %.B %. A.*mr,.rs b-Ab ' e-e y. wy...,: :.. ( l A f\\ sn .n, n p*E,V. f.g ) -y - ..u #s.2:c. f. g.s. a l ( n

- 14.3 fj.N:,.o., '+;,@ F#

displa cemefit ' -w n. .v.e

wx.ti1QW..

l A /\\ c:w a \\ ~f~ fiS ?hbq?bh.v'l~N. 'Y - W.ui. w..&.,$. $.r h.: **.h ~ 4 .a ~' k$$h$?') ' k$..?;$$I-h: N.$TY 'M s D ^ _^ On _n^- n n C,..,c. Q:p'.h ).g. 547 ev .: m _. m -V.y v.y g-W & p..g, g g(g, %. fey, I y - .V V,.,. 9 "2 % jq W/ u- 'n y . t t-stress M w a.4%*Cy O G., Dh _'O A - _. n. y v ijf nn hn p r% :-c. _v y - y y -..v y v ~ [GJ,Q.-M*.',*-3ffp %.f:..',. . t..o we a u.. w,,.r'%,,.... f V: p. w, An *..,&rt. s ~ B. .% t f.Os

t.,,,ie -N. i.%k..

~ ~ > s.*r-o ace.- w lq vg W.":* a.^ 4.t efe.-., A f ** /\\. [ ' ' W[. h. ;'>,,-@-,.v,so.. m

6

' y.'

[ F d k r*K ; ' i displapemeritj, ' s' ., P A.rr'.g~hxwwws.a. n~ ~ P.*%.. 1 !, 7.,te @,w*rn...',& M- *.- A /\\ . ~ _ ' ~ r S r s..,q;n.e.v. 4 ne -- ~ m y 4 @g,Av.v g.,peggd g A.... ,@****dK4'

p. p

.c. ,e ~... G L y.

}

.., a t .n v.. s. . n f\\ ; _ ps ',, _ .a . f\\ .... sy. (*. f.. %,try fg.t/; . ~ -. _n n , n 1 v. w2 a-v '. y q v, T,4 ~ s n . gg,;.g V stress - J^-A

n. v/.\\ t _ ~y s.-

g.t, g mes 7g: n m g4 u.. ,y y n-h A [ b L.x h.4 s.s sp.< %:n w. L ,. m ;i. y x @.y w% Q 4 u.,.,,,.,; n ,-f ' pf rg .n. n g5ae P n ~. v,:p z m -f. Yy h' :5 L

VA~9w

a. %.. ye,os.5;M.h.;p',.ysa.M* '

.%.,* l,.)., : a. a displac.ement 9n o. s o.- +W:

q.m A ~ v._

m A i:;*- -; ?;.n v w r . ~ 1 4. t n ::::Q:.t."nt9 y< 2 ty s . ~~- y. a5

M;; -' bw.ac'a.

, d#- -Y% vW ',-.

r. .,=t' 7 4

dI hbv

.h. > h n %w,e:.f n.;.~ Wl g.:-m w s' - rw e m. e 0.*.;<.k.i.y,ew.,.,QM.w.,%'. l .-s;tpng s., w-C .l 'h:10.re '. .i'[g B , C a. :y. ' u.g -.i<.. %. w~.h, m . D2 50. a.. c. m ~ 'C "g M. C: 8.25 - %= m s.. ~~,w-e l , p. f.,. s ., w... ..., ~ . W.' f.,%,4,'&. *[*$.~.!!.$g,,@ ' &~ .~4- %q#,,:y Dc! 2. Cocma:ison of s ressa and diplaceInents obtained at three couples of pomts on each side of i s Se m:erface. Tne loue trace is inside the valley. and the upper one is m the half. space. The

c. $. g,. 4,4 m;f i$.,, -

7., 8.. d.aranerutacs cf the valley and of the incident signal are speeded on the figure. ..v.. s-S' O, ~..,.;.. .7, c: T. - ] :.I.',,. c. ;.,... ' . ;c. a.. w. '.* n I <. M. j . 3.:...r.... n. ep V '. c .?s. ; ,-m, _,n- .,s... _ - ~. ~ - .. ~ s. w- ,,... 2 c,:,...,,g 3,,;,,;; g. .,. o. rk., r E',. , A..:.. ....-e ...... ;s /. ;.. f.,.,., s., s_at,.2.v*~*:,,,.,,,,,....... ~.g.,......,. ...,;.. ;.f.,.. , s r, % s,m.,, 3... - ...e c. .p.s.. 3,s... . g. o. ; 3.e:.. i.,.,,, . f... :... ;. - W; j..,,u., .u. - _ v ..,...a*...w y. 4

a,r,

..n. ..n.. ,j .y;,;.-

s...,. ; -:

. g g;sp.; _; g...,.

p.

~-

.L';i'.CQ. ;.?., ;...?%&L?:]f._ B:-l9L. :.%> y?ls;.w;:.%M. Mna:$y 4.;%,.l.%.'.ip%lP.Q;, ?.'W x f:.~; &

Q: W p :.(. 5 g;.3. y.. g =tg g.g. y..g.g.,n t
.e n.

rk GmA,;._. 2 ~2 -- Aw .p. t., ;..m?a.. =' :,;&s..a. mm.-.% >.:. :. r;~x~.v: ~.r:r..y. p %:..,m:, ~w.:u a c- ._.u. g.. M W v.:.1. u .,.:;.r.,w.n &,.: .e.. w ~ xg< re; _ ~ 4.: v SEISMIC RES: 1268 PIERRE.YVES BARD AND MICHEL BOUCHON On the other hand, in From these various tests,it is concluded that the Aki-Larner method is remarkably lateral variations, the c well suited to investigate the elastic response of laterally heterogeneous geological shaped interface is cor structures. GENERAL FEATURES OF THE SH SEISwie RESPONSE OF g(g), g SEDU. TENT.FII.t.ED BASINS h. Samples considered. A valley is defined through two classes of parameters: f(x) = 3 geometrical and rheological ones, and the problem is completely set when the y incident sirnal characteristics are specified. f f(x) = 0 g w ~ ~. c'j-These two types of ba .;.1 without any change, f. .... -. _.. ~. .f. width of the valley,2 s - M- _. - w s

$c vertical, c2dence cas4 m.

N 1 -- w v 3 g. ..,.., p.J .16 _ l$., .....r _.. ~... .2. -- s .n q n -(, 1a--- f.,,s.. g,.- /\\ .,. m,. m,,,,.i ..~.... vy 7 -.. c /p.. ~ ^ . g. G "{ -t. Ws~W_- -. s

f. % p t

Frc. 4. Geological con

s.,

/ x. 2. h "Ti mediu.m. its density shea: , y' j 32 -- - ~ e: o, 1 --- Gw.ea ocies W. W ' ~ ~~C - Fate e"eeeace ~ f b" ' %,,,,v%"'s. ' g ^l - and in the low cor.trast c: O .7 -l V 200 2ao 4 .,4 . o.. _ -ao.. oo..... uo t w i h = 200 m,500 m. a = 0.20 for type 1 an ' Fic: 3: Compari. son between the results of finite difference doore et al.19711. finite element (Hong and Kosleff. m preparanon). GlortSed Optics (Hong and Helmberger.1976), and present discrete Generall' IheoIo h g wavenumber method.The traces are the surface displacement at six sites located at a horiwntal d25tance of 0 to :0 km from the valley center.The valley structure and the meident signal characteristics are given impedance contrast ~ tion), and Hong an- ,. "." " '. *.. -. 1 1'.. P - 4, (which gives a p ane ><.r m. . Two ma,m types ofgeometncal structures referred to as type 1 and type 2 valleys somewhat unusual. (Figure -1) are ;constdered. These two geometries serve different purposes. To lying directly on cr.s investigate the focusing of seismic rays, a one cycle cosine-shaped interface of width main charactensm 2D and mwi-num depth h. is defined by geomett,j (type 2 v. 42 = 1.75) was also f(x) = g 1 + cos - for l:1 < D ,1. .".3.w 3 x. i a case. both for ver h c 4 3 O O The incident sig: its peak frequency i()-O for l l > D. . =n g 5 j) I

' T~-~* ~" * -"'~% 7; <,M, :..-3.~;~~+,.;g.m... ;,. ;,g...x-vrrg,..,1 7, [,*,..'? ..1,, :p.~.. t.~.

p.:p. :..

. -s -

~~~

s.--* n~,~,., A;m ...g:.s; ;.;.:2 :n,..gh. . p.q 2.. ,n,.. . :f '; 4.X::.,~ 'f. ... ;; y.g. 9.,*..W. ','r.Q+w.'.W* C.Q.* *. y,.'.[:yh,' e,:v.Q;v r.rp'"..~,... ;A.a;;.*,.:. :,* R. 9,. ;;, g: zu~,a,.. i. p4'?y1. & i.71r '%~Q h s w?. ~., .. W

n. ',.v, s:;

ryl-u ..g.s..m.. -.c...a,' 7.e, a.,,,,. ..,. s. : ,, ; y- (. ^c~ ,.,.... 7.. e% % : 1 -. < ;, 4. 3..,.,j',9, t-- q,. .y s. s. ..,3,' ' p... w c D.

f _

-- ^ ^ ~ ~ ~ ' * .+. .' ;A ,:,. ;..-tm; c ;t;. .,.s; 3. . y;%.,;,;.r.k.ia.?,?c.*w f>y?) '., '. cl... p.;.e.o.m[h{w,.b.by r-t

,,, &.['9;[ih$WN'O i'

,,' &j 4 sEtsuic RESPONSE OF SEDIMENT Fn. LED VALLEYS TO SN WAVES 1269 IG 1 l':.O N g'$ %%y g g.- 'n the other hand, in order to study the surface wave generation on geometrica! <g.w.df v, rr Q jfl. .cera.1 var:stions. the case of a plane layer closed on each side by a half. cycle cosine-

f. ! [$w#w.9g.42. ?

k{j?Q. d.2 ped interface is considered ? ,7. W. e E

u-
.2ygg.g W.g.g..,.

P ' \\ p. y for l x l < di i(x) - h }. N..:..~,vu.@%w. g'. .q,.w wm s. h ~ (x - di) T g.. } ynt a,. n g.. w = w u.

u. j.tyy.,

for di < l x l < di + di i(x) = 2.1 a-cos ds s h.. v.w,-. s.

g'.$.y'..%r E

i(x) = 0 for l x l > di + d2 - D. M..,..

f pp. idy w

r {p;y. @g}s g These two types of basins happen to be symmetrical, but the method remains valid, wi6out anv change, for arbitrarily shaped interfaces. In all the samples stuc2ed, the q ggl.gg wid h of the valley, 2D, was equal to 10 km, and ds/di was equal to 0.25. For the ,s Q verticalincidence case, three shape ratios were studied h/D: 0.04,0.10, and 0.20 (i.e., i ~ .#v.%y*di!Wi'.n$'sM M.w.wt.-r w n } hT 8'. 2: j d'~ di h +4. W.r..g m :wyJ,4 l..e p.... t ;w W'.gte;p. fi a e s. pl,* ~t Y5l[{b."R.h.g.gg s f Type 2 5, ***'i l ~.. tSc Qf,% S'%*??.*/j, ;4y

AS.

,.n -g p, p, p, h f.:. :.y %.-[EM,.A s y.et M pFC e i=.

r, F, n

.\\ S* *.t?". 7@fd *.s.sb *,."tD*dl1 i s t st; %* h. r-*5.'.%9. M W., A.

D * *3 i

eQ : s%t {

* ~

M. Q~ sug %..gr 1 E.s u,,) D WW N. Lv l g.- g&t.#.!!,. a$ V 4 u.6e-"M.Q .t t L-. l p

>=%9. ~

4-p-~A .y TY P' ' TG '.S ~ .s{y .cc ,r. u n e e,.,

- ea3 h

p P es T 238'$..> 8M 3)"$@RQfg? 's i u- ./ p$jQI ? L ZL ' ~ 'Ffgd, y [ ti l f r each y 4 4g g; rme.c.n. i:s aensiry, shear velocity, and rigidity. Is the high contrast case, there.respec ve y, o rio. 4 ceological cor4uration of vallev types studied. p p, a represent. .1:3.tg.g is g.g,...,M%g.sh I h. 4.Hi4M pi = 2.0 gm em'8; Si = 0." km sec~' lb m KJf ( W **g pi = 3.3 sm em:' B: = 3.5 km *+e" q a.rd in the low con:rsst case 2;y y3} 9 r.%,.4 1.;5.W J,Q gg.,s.,$. +;.(-?y,,.cy, c QP Si = 2.0lun sec-'. ~ - p, = 2.3 gm em:. S: 3.6 km sec". i. r~ p,4Mi.p@ g<- pa = 2.8 gm em'*; ~ g, -p.gy. g - -g,k.. w h - 200 m,500 m, and 1 km), and for oblique incidence, one ratio was studied: h/D 1;

5@f.qd"

= 0.20 for type 1 and h/D = 0.10 for type 2 basins. 0 ggg,y,q..g;tg$ TT(.pg Generally, rheological parameters were chosen to represent a strong mechanical i pedance contrast. Following Boore et cl (1971), Hong and Kosloff (in prepala- ',1 Lg gf 39.; g) .g.gjg?,j v5 tice), and Hong and Helmberger (1978), in most cases, pdpi - 1.65'and BdSi = 5 m Prhich gives a plane layer'ampli5 cation C of 8.25 at resonance frequencies). Kithougn y ,,<;g gq q,g g ggg;$cy,*gg.. 4jfs - mmewhat unusual in nature, since it corresponds to poorly consolidated sediments 6 $g;g[.g;;y,.g.,g g gg Wing directly on crystsIline' rock, such a contrast allowed us to clearly point out the inain. characteristic features of a valley response. Ho,w.yver,- for qrte_partici21ar_ g v,p....3.;.gg g l 3.m. 4 c. .teometry (t>~Pe 2 valley with h/D = 0.10),' a much lower contrast (pd. pi = 1.22, #d ?;w r.75Ga~s also-~5:nichWd~to see w'oether si=:.lar ef:.ects were to occur m sucy c,.. g. w ,3., g. g<.q%.n. c .s- ." g. 7 4,; pg-. ..g..;; que meidence.

case, both for ver:Ical and obb.

The incident si:nalis a Ricker wavelet whose main parameter t,is c,nosen so that ,.g., g, ia pak frequency roughly corresponds to a third of the maximE-n frequen , ', g. _.., __, .. r ,,.; _. c;... q p s g L'. m' g. _....s.s.. _ _. d'. E e..

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9 'M. 7..w.;: p- @, M,. %.+..(..g y M,m. i ._._ m.,._2. ; yhQ p - - - vu. v s :...n n 1270 PIERRE-YVES BARD AND MICHEL BOUCHON SElsMIC RESPc considered for the given problem. This maximum frequency is generally about four times the equivalent plane layer fundamental resonance frequency. In two cases >b, specified in the Sgure captions, however, f o was taken equal to 15 #i/h in order to A study higher frequency Ricker wavelets. ('M First, the verticalincidence result.s are presented, both for high. and low velocity contrast, and then the changes brought about by an oblique incidence are shown. h[ Our results are scaled for a valley half-width of 5 km (which corresponds in all the Sgures to D - 50 because the length unit is 100 m), but they can obviously be [ ogi*og O extended to other geometrical or rheological values, provided that dimensionless rj hmf ratios h/D, tSi/h, and B #i remain unchanged. (_ J .,.:."- p. : py 2-My ), . VERTICAL INCIDENCE ? I considered, h/D - 0.N, corresponds to a madmum slope of less than 4*. For such jqk N@ Type i valleys; high impedance contrast case. The lowest shape ratio that is M@ y a smooth valley, one might expect the focusing and lateral effects to be negligible c j ~, and the. seismic response to be quite similar to the one of a 11at layer of similar 3 . thickness. The actual dis' placement response, as was computed, is shown in Figure 9, 5a. The flat layer apprEimation is valid only in the valley center during the first 4 i see of thd disturbance.The reasons for the inadequacy of the FLA in this case can 4. t ' be clearly seen in Figure 5b. This illustration shows the generatio,n of a wave i ~ . disturbance at the edge of the valley and its subsequent' lateral propagation toward *[ the,6ther. edge. This prop'agation is characterized by a very distinct dispersion. I where higher velocities are clearly associated with lower interface depths and lower J frequencies. Phase, velocity values vary from 0.S7 to 2.6 km sec-' and agree quite .[ well with fundamental Love wave phase velocities for' a flat layer having a thickness ? ~ 0 equal to thi mnemum depth of our, valley (it will be referred to as the " equivalent plane layer"). Moreover, the meeting of waves generated at each edge produces in i.1 the. center of the basin a vsry stiong ampli5 cation equal to more than 5.5 time > the surface displacement which would occur xithout the valley. t r Simnn features can be observed in Figures 6 and 7 for valleys which are, respectively,2.5 and 5 times deeper. Furthermore, owing to the lower maximum frequency considered, p /h, the signals were computed for a longer duration and. ( j thereby, the behavior of these scattered' lateral waves was observed when they j 2*i T,, arrived at the opposite edge of the basin.These waves are reflected, at least partially, r g g g on the raising interface. A comparison between the seismograms obtained at the ~ 2 valley center in both cases (Figures 6A and 7a) shows that the amplitude ratio for .t' [ [ two successive lateral wave arrivals is much higher in the case of a deeper basin, 4 i I. [ p

r therefore, indicating that the amount of energy reflected on the interface increases

[ 7 y e with de interface mean slope. (, P { T The dependence of signal duration on valley depth, which can be represented as [ [ i the ratio of the total, signal ~ duration to the time length of the incident signal is L .? dif6eult to establish' A comparison of Figurss Sa, 6a, and 7a indicates that two ~ opposite phe'nomena ihtirfere. For shallow valleys, the incident signal will be shorter f. y p.f h h (otherwise there would be no lateral waves generated), but the surface wave

a

g amplitude will decrease rapidly with each reflection. For deeper valleys, the incident 7

L signal islonger, but'the surface wave amplitude decreases much more slowly. This - t ' may help us to understand the great dispersion in data connecting signal duration and local sediment thickness (e g., Gutenberg,1957; Hudsen 1972). From a comparison between Fi;ures 5b,6b and 7b. we can a so infer that the dispersior. panern depends on the shape ratio. For shallow va' leys, several lobes ~ ~- .w_ = ._. _ ~.3

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pr E.<-Q 7.p-,y.g :q$-1jf[ k hfk [ 4,-..w_ whhyjpMMM me, _a v ~~ ~' ' T ~ % s W,.,..,e _ __qgg; w - f,kT,*h h h.Nf'k.. g -., %.,a.,e... e aj. g4q;, .g m y ,4 a. .~ - - :[h. hdb y --- SEISMIC REsPO.' PIERRE-YVEs BARD AND MICHEt. BOUCHON 1274 were observed propagating over a short distance with a distinct but constant phaw velocity (Figure 5b), while for deeper valleys only one pulse was observed propagat-ing over a longer distance with a varying phase velocity (Figure 7b). The seismograms obtained for shallow valleys are characterize'd by the presence ( of very strong. but very localized (in space and time) pulses, which willinduce strong s torsional stresses. For deeper valleys, the peak amplitude is much lower, and the 7 affected area is wider. Also in this latter case, the mean amplitude is somewhat greater and stays almost constant over the whole zone within which the lateral t waves propagate. To further investigate the nature of these waves requires a look at the evolution of displacement: with depth at the center of the basin. The amplitude versus depth k curve,. shown in Figure 6c, clearly exhibits the well-known cosine dependence of fundamental Love modes. Another confirmation that the elastic' wave field induced y 5 in the basin is mostly comprised of Love waves is obtained by looking at the amplitude of the up-and-down-going wYves usidWhe valley in the frequency- [> horizontal wavenumber domain. For the three shape ratios h/D considered, a pattern.was obtained similar to the one shown in Figure Sa. Although not easily [ g O interpretable, this pattern. exhibits two characteristic features. For each frequency, j (7 qCY (c0 the. highest scatter-order excited waves correspond to the fundamental Love mode of the equivalent plane layer. Furthermore, they are excited as soon as the frequency A g '- exceeds the' fundamental Haskell resonance frequency, fo - #i/4h. Otherwise, the r n (,'_ g N # .. complexity.of such a pattern comes from the continuous depth valiation in the 3 1 val!6y, and also at higi.er frequencies from the excitation of higher mode Love waves. An attempt was done to point out the presence of such higher modes by e cornputing the valley response for higher frequency signals, but the dominant feature; always remains the fundamental mode. Nevertheless, this higher frequency study4 allowed us'to point out two features. First, the peak displacement amplitude at each point decreases when'the peak frequency departs from tne fundamental equivalent ') plarm layer resonance _ trecuency. Second, the zone within which Love waves prop-j agate widens' when the frequency increases. This observation is particularly clear in -r Figure 8b where a given site becomes excited as soon as the incident wave reaches the fundamental resonance frequency corresponding to the sediment thickness at this site. This latter property may.not be valid for deeper valleys, however, because .i of the larger deflection of the rays due to 'the greater slope. Finally,in the sediment. thickness range s'tudied, the focusing effect that we expected to find with cosine-3 shaped valleys does not affect the direct arrival of the incident signal, but acts essentially on the Love waves generated at each edge by trapping and entertauung 4 them inside the central part of the vallev. .t Type 2 Valleys (a) High. impedance contrast case. The sediment thickness is now uniform over four. fifths of the vallev. In this geometry, quite pure Love waves may be expected ~.f 4 t'o develop at the' edges and to propagate over the whole width of the basin. This is i 4 v'ery well confirmed in the frequency domain by Figure 9a, where are clearly pointing out the first three resonance frequencies of the equivalent plane layer and simulta-d CI 'neously, the excitation of the fir' t three Love modes. These modes develop, respec-s tively, as soon as 'the frequency exceeds the corresponding plane layer resonance frequency. .l This is also quite clear in the time domain. as is shown in Figures 10 and 11: Love waves are generated at each edge of the valley because of the deflection ofincident m. ~ w. .D'

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}..l:<p'.. q.m-5'?$5ffh.'g&.. b2?&';w p-$i.}[$S&s.4.,m..,Qy% ~ Q-W5 . _. _. g2 & %in T5fk, W MQ t- .d. .x. ..w -.7. g. g.p ,pg g,qq.g. gg w ,y:.j;n,..g--3,py.g gy p.,...,1q. g pp p.g_ - - - .9 c.. e.g + ___ _ SEISMIC RESPO PIERRE-YVES BARD AND MICHEI. BOUCHON 12 % vertical rays toward the propagate back and fortl Like the results for ty layer is shallower. How which causes the amplit Q valley (Figure ob),the p 'i measurement of phase indicate that, even for : propagating surface was long time, Figure 10 rev depth and greater phase 2 tinnfh h, modes. Their presence is N that all the Love modes !!!ss!!!N!! 3, enough. unmn mun $$.g$s!!!';d"Lg[A e p' # # gh N A comparison betwee d!! "'- i t i seismograms become m, ..mmmg#*$$f1h us~naL = due to the increased m, : y' "hmmu "!!!!# i:L"," -i 3 %kh.ikh { g quite clear in Figure 1 j j causes the mean and pe = 4,ynggdmy c, 2
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~~t for deep basins, wheres + nmn s.t'"+ /fi, i.. I shallow valleys. The i m u s<f 2 investigated, and it w unj:, k significantly decreases- ' 'rs;;M,-~g ^ 7 i q the most " dangerous" a . ~pmyE!;- 14 u::# A ~ . c. of increasing the freque 2 ~ t = 3 ment pulses, at a given g% ( 'W it W A comparison betwe 8 kd, hsft 'I o t. shows that, for the sam 4.- 1 N:$$j@3 g } ~ = '7 - ~ MEEE:; j 2 in type 1 than in type 2 ? 5 t feature is particularly e 1 TOWN;$ %sk.Q.'4
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90l$5M %.g.vz.h.r87=W' sE!sstic RESPONSE OF SED 15 TENT-FtLLED VALLEYS TO SH waves 1279 j ngigQFh,JEd).Ef,.. M.u.a m.w w:gn Mk,k"NMNb = urticrJ : y: tov;ard the horizontal direction by the oblique interface. They then hk propagate back and forth within the valley, reflecting on the edges. 4..g?.!yC / i - p ue the resuhs for type 1 vaj, eys, more dispersion lobes are observed when the ,i.r, W*4g/y.Rg..e-fy
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!ayer is shallower. However, owing to the constancy of the sediment thickness. -F.w.w 'f Ne - ' qip'wTo%(.gf,i'gM. ,N M w hich causes the amplitude frequency response to remain quite similar all over the ,U d E C % @ M ii M7 g valley Wigure 9b), the phase velocity is practically the same for all the lobes. The S-mpr 2 measurement of phase and group velocities and the observation of Figure 10e -qr.aw.; M % g n %an'*M;N.M/0Z' indicate that, even for the higher frequency signal that was chosen in Figure 10, Y $.NTW T NMTM[@4 propagating surface waves are mainly fundamental Love modes. Nevertheless, at ^ M'r long time. Figure 10 reveals some high-frequency oscillations whose variation with S.' N Y N ? k r3. depth and greater phase velocity lead us to interpret as the first higher Love wave $7[hdd $hMi M. g'b DS(%% modes. Their presence is predicted in Figure 9a, the extrapolation of which indicates,. that all the Love modes may be excited provided that the incident frequency is high d V p$i g g.m w w,;sxg:.c vg n enough. l Whdg 4 A. comparison between Figures 10a and lla shows that, like for type 1 valleys, $k"J.i@amg%._=1-seismograms become more and more complex as the vallev depth increases. This is I due to the increased interference.between the up.and.down reflections (which are WNN4"751-dNN {n. @h '* ~$[ ; quite clear in Figure 10) and the lateral Love waves. This interference pattern + . f 4 causes the.mean~ and peak amplitudes to be greater in the central part of the valley ~ @[Q% gig y.i-for deep bna, whereas edges are exposed to slightly higher, peak amplitudes for 3 shallow valleys. The-influence of the incident signal peak frequency was also j g}gggg,$y.yEyJf(. MET 4f-Rg investigated.-and it was found, as for type 1 valleys, that the peak amplitude . the most " dangerous".of which appears to be the fundamental one. Another effect gpm@p9g. sigm5cantl'y. decreases when peak frequency departs frotn a resonance frequency, 1;v c.cd7 - M h:D t of increasing the frequency is to shorten the lateral dimension of surface displace. ~ J l s I ment pulses, at a given time, and thereby enhance torsional stresses. .j; ' 7'Q ui. Cs e 3 [ A comparison between the responses of_the two valley geometries considered W ' E 8
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%gyggg@s shows that, for the same shape ratio h/D, the Love wave amplitude is much greater in type 1 than in type 2 valleys, over all the zorie within which they propagate. This ggggh%gg.} feamre is particularly enhanced in the valley center where symmetrical Love wave 1 Orj qggg?" f p$w.v%" trains meet each other. In the former type, continuously increasing depth reinforces M .Y h 6 (P11 f.% ~ Love waves all over their travel toward the center. On the opposite side, the longer 4, 4 2 -14 E lateral travel path of Love waves and their better trapping due to the higher slope t % r c y s A._gg. +-w-7.w ~ of the edges, must cause signal duratiott to be greater in the type 2 valley. .h (b) Low. impedance contrast dasc. These high. contrast cases have drawn attention N.56,f 6 ~~~iU-M N s. . to the importance of Love wave generation and reflection on nonplanar interfaces.
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, FM 3h1M!Nmd.Mll.th Ficure 12, where the rheological parameters used are pi = EI"g~m cm-8, p2 - 2.8 gm 2-f.hiM5$@SMk c'm, Si - 2.0 km sec'*, and S2 - 3.6 km sec-*, shows that these phenomena, though v-lJ FEM M p.M.y ghanritativbly much less emphasized, remnin cualitativelv_ the sam'e when the s wxtw 97p:-- MM4-W[F impedance contrast is reduced to a low value. Althouch the rank amolitude at each k -hk-I$ ' point is n5Iaf{ays recorded at the direct.arivn1 n' +e incident signal, the Love' ' contrast, Love waves are not efficiently reflected at the edges. Later propagations _ h h?.; 9.i7 N., DWA.W 9 A wave' generation and propag'ation are quite dlear. However, because of the lower J. ~ ~ U.. w. n p...J.? m" m r g,0 w w. observed in' higher contrast studies are no. w. 'ahard!v perceptible. As a corollary, the e.r...i ay.&..- cdMC live wave energy now travels through the r ising interface and slightly affects the, ,i,,,a [ Ns t. p % a. w WW*h.. t ...m.s.? B y ground displacement outidde of the valley. Therefore, one must keep in mind, when 67 msp cna'y:ing seismogrm in order to identify the various arrivals, the whole geological N.--..Zwd.- Qjr.i..c .-..w mueture of quite a wide area around the recordm.; site. 2 w p.9n. x..y. . r. > r... w- . ','s '}} ~3...i *, .;.c - *.. n....cu. e x. , c y @= %
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14) than in the case of a type 1 valley (Figure,13). An inte f r f undation des,gnm t
~ becomes very striking. a the resuhs show tne 22 i vallef, The disproportion between the two Love wave tra ns h d n as anincident sign for'a type 2 valley with low. impedance contrast (Ficure 15). In thislatter case, t e s p signi5 candy different ri i l direct IAve wavd is as important as the. incident signal fh incident signal for each synunetrical wave train. This produces, on thethan in the ve We thmk Guy Perrier for edge of the valley, a peak amplitude about 30 per cent greater 'i na p i h u Pon* O incidence case.' As'inother consequence of this direct i f to efficiently J side of'the valley because of the inability of a low-contrast nter acei s 7 for 9 Mi. E and FA Lener m reflect Love waves.,.4 f~ Another outcome 'of oblique incidence studies, w f ce displace-3 to incident plane cH w, Boore D.M K LLarner. microzonation purposes,is the shifting of the zone where maximum sur a 71 ~ m'ent occurs.'As the' peak amplitude-liefst the meeting zone o t e t4 y *ringProM fh wo Love wave ' trains, and as the sh'iftingisproportional to the length of the Love wave prop N Bouchon. M. d9t9). Dis zone, this shiftmg will be' greater intype 2, han in type 1 valleys. t b. sicns..i. Geophys. Res.
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j Gutenber:. s. (195-). Efree- ? .K. Coscwstoss p. A Haskett N. (IMOL Crus:4L These various results lead to'se'veral co'n'chisions. Fir'st, th'e Aki-Lamer w Hong. T. L and D. V. Hein s. (L ,p Buu. Sew.u Soc. Am. L Hudson D.E. n9 21. Locn l . proves to be a powerful and reliable too to stu y ain comparable to the h the time domain at least when the incicent wavelengt s rem + ne.gs. l Luner. K cetot Near te vertical characteristic dimensions of the structural anoma y.Second, hanne ineplu in:en* ~ induced by boundary sediment-fdled valleys, it was shown that the main effect E' T.
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Mr h(@

hc;:$1 of a high4 elocity contrast between the sediments and the underlying bedrock, these f
2i M M Jr local surface waves can be reflected several times at the edges of the basin. resulting k~N.. k, i. %@. _ _.N~t.sdf in a long duration of the ground shaking inside the basin. The number of reflections } pfrf+3 v Yp: is related to the impedance contrast and to the mean slope of the valley boundaries. Mni@$. h 1T 'T In the case cf a lower contrast, these Love waves produce parasitic signals on the lht-% M 6$ outer sides of the valley. The amplitude of the Love wave surface displacement may GA.N k M !.#i be much more important than the one of the direct disturbance. The resulting s Wdi&CPNP*#'.' amolification, which is sometimes up to one order of magnitude, stronciv depends $(hMk-h4 O on klie site location inside the valley. The lateral dimensions of the zone dere these high amplitudes occur is related to trie period of the Love waves. When due to $$(.%'fhb g[ I d 9d!pF N N.Su h fusdamental Love modes, this zone extends over a length of the order of the lQ.%[ghyygM,5 sediment thici ness and is, thus,.y_e v narrow for'shallTTallevs, lT@g.W.gOL-ic[gf@M ~ lthough the. assumed planarity c?th:-incidentrave field restricts our results to pM 2.kM.M3.E A the teleseismic domain, one may quite reasonably expect similar phenomena to take ffY.$gh_,p.9;,2w I N $.'% place in the near. field. The outcomes of the present work can t. hen shed usefullight ap pg. kg.a. Q..pf.:R.@ghg m earthquake engineering and microzonation stuc2es. For instance, the flat layer 9 -theory appears strongly inadequate to compute reliable ground motions. Thus, the dds^'J.NTpWW.E sc.me can be said of all engineering computational methods which consider the i3M LYN 6b bedrock as.infinitelv. rigid because the interface displacement has been shown h?hM~@M k (Figure 2) to lastlo5g after the passage of the incident signal. Also shown (Figure ,h y %; yp.pppe ~ ' 9) was the possibl,e development of higher Love modes which may be very important ..nd~@.m.an.%@wyL s-M i t for foundation designing. Finally, from a more purely seismological point of view the results show the.importance of local surface waves wh2ch seem to develop as pu ty.gs.,-ig.3% soon as an incident signal encounters a curved boundary between two media having githi D Q R W "g@4 FDM.My ' _.: d 'YEd,;h7 b> ~(..j si;nificantly different rigidities (e.g., at least 50 per cent differences). D,%3 2 wn. T'9 WF ', $V %.:g:S ACENOWLEDGMENTs f >:~:. % %Q'PA:'Wf'lM R.I'Z%;&khN. k? ItF We thank Guy Perrier for dirussions and comments Tha werk u as supported by the French Ministry ofIndustry and by the Institut National d'Astronomie %[@*+.-g~!-Q3"ges et ee ccophyuque. ~ REFERENCES y M W racr'. - p w.r,e+ gfG b M m -.ut4Wy ~~ .Gi. K. and K.L Lamer (19 0). Surface motion of a layered medium harmg an irregdar interface due 4yhy@hhMM't to me: dent plane SH waves, J. Geophys. Res. 75,933-954. I. hth
Boore. D' M., K.L larner,and K.Aki ( 971). Comparison of two independent methods for the solution of e se tienng probl response of a sedimentary basin to incident SH waves. J. Geophys.Res.
y ,k - c. M d -$. y h. $n s Ni Bouchon, M. (1979). Discrete. wave. ' number representation of elastic wave fielEsin three space dimen-i 4. o w..: >,fq.t,2Q1 uons. J. Geophys. Res. 84, 3 % 3614: ~ Ej [$.jQgg:p% y. gr. Getenberg. B. (1957). E5ecti of ground on earthquake motion. Bd Seism. Soc. Am. 47,001-250. sig M.- ~gs.g yyy@ H,ng.T. L and D. V. Hel=berger (19'7M~GiTn5ed Opnts and wa$e prdu,414r 4450. ,inketcN (1%o) CrustaDeficcdon of plane SH waves. J. Geophss. Res 65 SN** 5,i n.<.$E8.q9t.gg k M e_ 7.A Q. 2 i on in nonpHnar structure. is ru + pic~s. Ss, scu-:. Sve. Ar es, 1313-1330. T*ff M e,--.C ' f-).[-i*, N N~C 4 l Hucn. D. E. C9701. tocal distribunon of strong earthquake ground motions. Ed Seism. Soc. Am. 61 h E.3.'.t.'- ~~ M; U.." '. l J. ~ Un . ~.. - 17f.~.-!?M W K CFN. Scar receiver scattenng of teleseis=ie body waves m layered Crust. Mantle modeh hmr.: cregdar mierfaws, Ph D. Thesu. M.I.T., Cambridge, MA. h;n'#'.ykijl:M ~~ 'f. ^N...'. ,.c ~ . fl.. W Wh,l r s q...".,'?. Q .3.,3. .g e n f_je.
2....- ; y i? ) W
r.$.
fE5h'?~. & .. d;;.4.u.S@,.S, ;r Ws p . 2 "^%%h::*;p..%.rcM.-Lr;;t .... m m
. m w[5.,d,'f?,,N'IMM "*.. ~,Y-7,L,DJ?.:,Nj'$Q,hh,*/,;L 5%$,,f[M $i1'?[ ' 4Y.. IM7.. m.-.rm.-. E
... G
-f *8. -r .. -.'[. _ ', . ^;.' - ,%f. -. 0.E w.t.'t'$.g.JW@iegg;dy"?MrWd^ M "M,Ay,;;, MWsIPSu 'p..,-* '. e - , -- ; v., ;.. ;-.,., -- m.c; y,.'.q>,.n :.y.t,.-~p.'m,?b.fWijb f a:p "tg ?"~.',4%fer$!.Mh.*.?;g&?i='vk .:. >.... hr a -~ : ', <. ncr<- n qv;r . ~u.,..z #. %.s.,'. :'7.2^'y '%.** W&. p;.i~.v.,,s.p:,~'s ; .r r.: -~WFp
.gs.w.m 3:

w. -:-
.- t M V.9.:.G sW. . : n iQ,.&. Y * "%...:
- = '.
. 4 pr?. 7. 3.-e. n ::::w..m
= ;Qs.:w R.;:..
n./-. - y ;;.
;.. r. f.y.n.;.y:,L.e... :u-y..:. : --. g a,..,,...
s...... ;r.y.:g'. m.p c.. x =. :.q p - ....,.:cn:;g, ,.-::4.*..;;,.. a. . - - - >, n.. p..~. f n.y.,. .n r .t Jd; 16.56 PIERRE-YVEs BARD AND MICHEL BOUCHON Bu;'et:n ef the Set Sm:th. W. D. (19751. The appbeaticn of finite element analys:s to body wace,propagatiori problems. Geophys J. 42.747 76o. Tnfunac. M. D. (1971L Surface motion of a semi-cylindncal alluvia.1 valley for incident plane SH w aves, .\\TTENUATION OF Ed Setsm. Soc. Am. 61. 1755-1770. Tu:Ler, E. and J. King (1979). Some expenmental observauor.s of the t esponse of sediment-filled valleys ,'u, to weak and strong earthquake mocon (abs.ract). Earth.;ucke Notes 49,73.~ b -[h Won;;. T. I. and M. D. Tnfunae (1974). Surface motion of a semi.elbpucal alluvial valley for incident plane SH waves, Bd Seism. Soc. Am. 64, 1389-1408. W,..
gy*:
With data from 8 LsaoRAmraz oz GioraTsatz IsTrasz, AssocIt At C.N.RS. (ERA 603) tion equation is cer
".LG.M.
y L.stvtasrTt Sc:tsTIrigt.z zT MtoicA1.z oz Garsost.z ' bi{ This compares ver) BP 53X 3K 41 GazNost.z. FRANCE
fg San Andreas regior
~7' Manusenpt received December 27,1979 - .-n s+ 4 A knowledge of hor ..,c distance is essential tc .. M.+ absence of instrument - c:n i may be obtained indi: 2, attenuates with eP cer i felt-intensity with acc first step. t e: .,a: ' -5 Eighty-tluee eanhq p been found suitable to Z (1895) isoseismal map ,2 3 from Minoza et al(19 ~ from various UNESC ~,. t. ~ t. the years 1960 to 1976 +; The intensity scale t T.:.- scale. Intensit.y data .J' correspondence of thi ~ 3 1. .;I The 83 eanhquakes .a... 2, ...a.: 'Io G r: no. Following the well4.: ..,...c... ~... Richter (1942,1956), t .q . r:: c. ^ ~ l ? .i* f.- n.;. .E A standt.rd procede 1.[ I,(R ), a least-squares - %-? - ~ ^.- =? r~ s...;.,, The coefficient b. l associated with the g least-squares conditio of the fluctuations of
Attachment #3 MEETING NOTICE
Subject:
Seismic / Structures / Cell Liners Analysis Working Meeting Location: WLLCO, Landow Building, Rm. 1111 7910 Woodmont Ave. Bethesda, Maryland Time: 8:30 a.m. Date: November 17, 1982 Pu r po se : To resolve the outstanding seismic / structures cell liners issues. Scope: This meeting will address: c) The October 5, 1982 NRC questions b) The effects of inclined layers c) The NRC Audit Report findings d) Identification of Closed Issues Seismic / Structures / Cell Liners Analysis Meeting Agenda f I. Introduction DOE /PO II. The effects of inclined layers B&R III. Project's responses to the B&R October 5, 1982 NRC question ) IV. Cell liner criteria B&R i V. Responses to Audit Report B&R findings VI. Discussion and resolution DOE /PO, B&R VII. Identification of Closed Issues DOE /FO, B&R t I
Attachment #4 Attendance Sheet 11-17-82 Name Organization Phone # 1. W. E. Manrod CRBR-P0 615-574-2266 2. J. E. Beavers CRBR-P0 615-574-3117 3. J. R. Ellwanger B&R 201-265-2000 4. R. E. Palm B&R 201-265-2000 5. T. A. Butler Los Alamos 505-667-5171 6. J. G. Bennett Los Alamos 505-667-7888 7. P. R. Washer CRBR-P0 615-576-6179 8. G. B. Sigal B&R 201-265-2000 9. N. N. Kaushal CRBR-P0 615-576-6020 10. A. Morrone W-ARD 412-722-5498 11. P. T. Kuo NRC 301-492-8442 12. C. P. Tan NRC 301-492-8424 13. A. T. Dajani B&R 201-265-2000 14 R. L. Rothman NRC 301-492-8217 15. J. R. Pearring NRC 301-492-9563 16. G. N. Freskakis B&R 201-265-2000 17. R. Stark NRC 18. Pat Docherty WLLC0 I l l s t I l l
N ATTACHMENT II I I l t I l t I
RESP 0llSES TO NRC DESIGN AUDIT ITEf1S ON CRBRP STRUCTURAL AND SEISMIC DESIGN I
NRC AUDIT FINDINGS 1 1. A. CELL LINER DESIGN - SilMMARY OF FINDINGS 1. REVISE AT LEAST 4 LINER MODELS AND BUCKLING ANALYSES e INCLUDE OTHER ANALYSE'S NOT REVIEWED 2. REFINE THE FINITE ELEMENT MESH FOR LINER AND STUD ANCHOR MODEL AND ANALYSIS 3. ANALYZE TRI-PLANAR CORNER FOR HOT FLOOR AND COLD WALLS CONDITION 11.1. (UNDER ' CONCERNS') MORE GENERAL LINER STRESS CRITERIA IS NEEDED.
NRC AUDIT FINDINGS ITEM II: CONCERNS 1. MORE GENERAL LINER FAILURE CRITERIA IS NEEDED TilAT INCLUDES APPROPRIATE STRESS LIMITS.
RESPONSE
e FAILURE CRITERIA FOR CELL LINERS NOT COVERED BY CURRENT DESIGN CODES. e DESIGN LIMITS IN TERMS OF STRAIN ARE CONSIDERED APPROPRIATE FOR THIS TYPE OF STRUCTURAL SYSTEM. CONSISTENT WITH CONTAINMENT LINER CRITERIA - ASME III, DIVISION 2 1 e STRESS LIMITS HAVE BEEN ADDED TO CRBRP CRITERIA FOR MECHANICAL LOADS. e STRAIN LIMITS RETAINED FOR COMBINED MECHANICAL AND THERMAL LOADS.
NRC AUDIT FINDINGS ITEM II.1: CONCERNS
RESPONSE
(CONT'D) e MECHANICAL LOADS GRAVITY, PRESSURE AND SEISMIC LOADS LINER PLATE STRESS LIMITS IN ACCORDANCE WITH ASME III, APPENDIX F STUD ANCHOR STRESS LIMITS FOR NORMAL, UPSET AND FAULTED (SMALL SPILL) CONDITIONS IN ACCORDANCE WITH ASME III, DIVISION 2. STUD ANC110R STRESS LIMITS UNDER FAULTED (LARGE SPILL) CONDITIONS ADOPTED FROM APPENDIX f 0F ASME Ill. e THERMAL PLuS MECHANICAL LOADS STRAIN LIMITS IMPOSED BASED ON SELF LIMITING THERMAL DEFORMATION. CONSISTENT WITH ASME III, DIVISION 2 FOR LINERS.
i 1ABLE I CRITERIA FOR CELL LlflERS PLAll'~ DISIGN IIMil (I) .(lI) CONDI T10fl HECilANICAL LOADS TilERMAL
MECllANICAL LOADS CONTROL QUANTITY 'S CONTROL QllANTITY 't.
e PRIMARY MEMBRANE OR MEMBRANE MEMBRANE MEMBRAllE
BEllDING PLUS BENDING 1 0.002 s.sc 1 0.00<1 (a) Normal Operption and Upset i-sc (Load Combinations (A) A 0.003 st < 0.001 t.s t r
(B) in PSAR (b) Faulted, Small Spill liigher of: (Load Combination (C) in S + (S - S }/3 r.sc < 0.005
'sc < 0.014 Y
Y PSAR) 5 0.010 0.7 S 'st 1 0.003 37 c u s 0.67 c 10.5 t c (c) faulted, large Spill c u e e (toad Combination (D) in PSAR) UItimate (Nominal) Tensile Strength S = u Yield Stress S = y Uniform Elongation = e "d* *"*
E"S S "
^" e sc Maximum Tensile Strain = c 3t I"3- "l} I
2),' I"2 ~ "3)
Effective Stress = ((oj S -0 = e e (( )2, { )2, 4C Effective Strain =. l} I = r ~C 3 "l' "2' "3 are Principal Stresses cj, c2' '3 are Principal Strains
TABLE II CRIIERIA FOR CELL LINERS STliD fNCHORS ~ / DESIGN LIMIT l (11) CONDITION (1) MECHANICAL LOADS TilERMAL + MECilANICAL LOADS CONTROL QUANTITY "F" OR "S" CONTROL QUANTITY "6 " OR "t (a) Nonnal Operation and Lesser of: 0.25 6 0.67 F Upset (Load Combinations (A) & (B) in PSAR 0.33 F i u Lesser of: (b) Faulted, Small Spill 0.50 6 0.9 F (Load Combinations (C) in PSAR) 0.5 F 'l (c) Faulted, large Spill liigher of: S + (S - b )/3 0.50 c Membrane (Load Combination (D) u u y I in PSAR) 0.7 S" 0.67 c" Membrane + Bending Liner Anchorage Force Capacity F = Liner Anchor Stress I S = Liner Anchor Yield Force Capacity F = y Liner Anchor Ultimate Force Capacity F = u Ultimate (Nominal) Tensile Strength S = u Yield Strength S = Ultimate Displacement Capacity for Liner Anchors 6 = u Unifonn Elongation f = cu
Thermal Loads Only
ilRC AUDIT FINDINGS I.A. CELL LINER DESIGN l.A. PROVIDE REANALYSIS FOR WALL LINER AT A CIRCULAR PENETRATION. RESP 0tlSE: REANALYSIS TO BE PERFORMED BASED ON 1/fl SEGMENT MODEL VS. ORIGINAL 1/8 (115 e fl0 DEL COMPLETED e NON-SYMMETRICAL BUCKLING TO BE EXAMINED e CALCULATED RESULTS WILL BE COMPARED WITH CRITERIA e DESIGN WILL BE MODIFIED IF REQUIRED
1' s / a L li N/\\/N/\\/\\/\\/\\/' /N/N/N/\\/N/ \\/N N/\\/N /N/N/\\ /N/ /\\/\\/N/N/\\ /\\/\\ N/\\/N/\\/\\/\\/N/ / N/'N/ N/N/\\ / \\/N VV\\'NN /\\/\\/\\/\\/=m N! N/N/N /N/N s'x/\\/ 'N /'N/\\/ \\/N/1 \\ \\ N/NvN/N/N~ ' /x i'S 7 37 hx'\\,/NTN/ \\/X / / L N/NJ/N = a=:c / v N IN/N / N/' l
M J/
I//A Ai/N !m 253 13' ~ k \\ 13 -, i, ;,y g g l ~ l l ' g' c7'_py g I 2 ; 5 ' O, c, EA. W AY l Q = l\\.39 l}.htl_$ ' _ 3 $ $ ^ $'L 323.75 .x : .:::r R A : ; ::::aA:::::s FIGURr 1.A-1
NRC AUDIT FINDINGS I.A. CELL LINER DESIGN 1.B. PROVIDE REANALYSIS FOR WALL LINER AT A RECTANGULAR PENETRATION. RESP 0flSE: fl0 DEL AND REANALYSIS WILL BE SIMILAR TO CIRCULAR PENETRATION (flNDING I. A.l. A) 9 e
NRC AUDIT FINDINGS I.A. CELL LINER DESIGN 1.C. PROVIDE REANALYSIS FOR WALL LINER NEAR EMBEDDED PLATES. 4
RESPONSE
e ORIGINAL STRIP MODEL REVISED TO A 1/4 SEGMENT MODEL EMBEDDED PLATES SPACED AT 45 INCHES APART IS CRITICAL REPRESENTATION e ANALYSIS COMPLETED 2.4% MAXIMUM STRAIN = vs. 13.4% ALLOWABLE STRAIN = i f s. y 9' N 4 l 1 p t v
,v 3 - k N s FIOURE 1.C-1 v LINER AT EMBEDDED PLATE. Allevable Strain i LINER ETRAISS (IN/IN) \\- s' 0.1000 M = Z. ~ o,1340 'J~ h I p3 8 MAME: ~ '\\' STRAIN P 0239 (P B)-. 0235 (P B) .0205 (M+3) .0231 (M+B) I.0060 (M) f.0069(M) [.0072(M) [.0071(M) j N ]/'N/ /h i \\ -(, l /\\ /' .0205 (M+3) / S A[ s i l=} kb \\j m .0071 (M) N N l.x) t 1 1 ( I N/ ' 7\\; / / = ' 'O,dl-l -k /\\ \\/ I [ [ N / N gs N ,f / \\ '6 \\ g ~; " h \\ ~ / .0235 (M-3) i -/ .0239 (M+3) .f \\/ j \\ \\ j .0069 (M) . ? > 'h i. _, __ M ~ ' ! 'sl/ \\ ,o237 <3 3) g; j / I ^ .c060 (M) I _ b ! /O / ['\\c N,py X'y! l \\l =,_ i -m \\7 g j t X;j .- ] 0 - 7 X / / O r = --. s, a a o ,n <<n.- _ _ ;,_ j
e. -
,~ - -. s = i ~ -
- n :s t=
n_, .N-- y , _ M O -' L.. _ _:. ,O C% ' ~.t =, i. i
_A :
c'9 .' C 5 ^ s-E j:: E7M5N- ___,.,s c
d. N s,. a s a --r w.,
.N%,=u '. 1 g R. 4 0 R-s 5 L \\ f -f?- g
I FIGURE 1.C-2 LINER AT EM3EDDED PLA!Z STUD STMINS (IN/IN) Z. ManMcM STMIN Allevable Strain 0.1000 M = 0.1340 43 = .082 (M+3) .099 (M+3) .100 (M+3) .041 (M) / .049 (M) / .051 (M) / / / 5 k ? / \\/ / \\/ m N U N W! i ~/N~ .099 (M*3) D M /.049 (M) / n iY 1 -4 WKvN / N W 1 - T <P l Ni b t
\\
m:,/\\ ,l . 082 -(M--3; /.0.t on j!gl! N_/ N 9 g 1
yl O c 5.75 4, 27. 5.'
i u M:__. _s C / S e e. m I, l m l
==e--g i L 'e 's '. s Il 4 ' b.M b U I /.~ 3 3 * '_ N 3 k m - 5 u s E s.t s v- = ; - = o - T C 5-' i -NC.-CR _ _ _, m -. _ . v :. [ _~ 2*w w ~ .'s. 'w I. w* ~ ! A ~ % ("I V. '.. \\ g -,_.__m-, -
NRC AUDIT FINDINGS I.A. CELL LINER DESIGN 1.D. PROVIDE REANALYSIS FOR TYPICAL WALL LINER AT A STUD CONNECTION.
RESPONSE
e f10 DEL REVISED TO A 1/2 SEGMENT MODEL VS. 1/8 MODEL (AT IIS F) 6 flAXIMUM ASYMMETRICAL BUCKLING CONSIDERED BUCKLING INDUCED AWAY FROM WALL e ANALYSIS COMPLETED AND MAXIMUM CALCULATED STRAIN IS WITil!N ALLOWABLE
E } _. A r=
* **27 STM 9
.207 .221 .242 .250 .250
i s
S JL 1 5 .138 .157 .207 .243 .251 gR! 8 ' 3/8 Liner (Typ.) .056 - .015 .089 .195 .237 I, I. N .q - 323 .258 .104 3.67 .152 II N ii q srw ,[N .611 .536 .352 .144 .044 37 41 Q = 1/2" dia. O Stud j .901 .822 .642 .448 .355 l, I. N sN -1.163 -1.102 .951 .803 .741 lt g q ii l l l -1.366 -1.321 -1.225 -1.132 -1.095 ii N Y q y - 1. 4!.4 -1.410 -I.340 pl.280 g ~ 73 7 ="u 7 l SYM SOTE: Posi:ive displace-cents represent displace =ents/ deflection tcwards l the concrete. FIGL~ rte 1.D-1 DISPLACEMOiTS
(MEMBRANE, MEMBRANE + BENDING) T 1 (0.0094, 0.012) \\ \\5 (M) (WB) !I N 2 3 y ~ / r N (0.0056, 0.0113)- (M) (M+3) iI N .i q lI N ( (0.006h,0.0195) (M) (M+3) -(0.012, 0.065) 'N deud 41 (M)h (M'3) (0.0051, 0.0137)- / Q (M) (M*3)__O, ,e -(0.0054, 0.0207) LQ k (M) (M+B) ll N j Q (0.0050, 0.0146) - (M) (M'3) ii Nq l \\ Yi II Nq I l kl --(0.0100, 0.0243) f I (M) (M*3) ik/ I 'i N, Y [ X l 77 73 l -9 I, \\- - syM - i - - (0.0062, 0.0162) (M) (PS) Allevable S trai:1 FIGURE 1.D-2 0.0703 NON-SYMMETRIC SUCIC ING M = 0.0942 STRAINS M*3 = m
- - _ _ =. NRC AUDIT FINDINGS j 1.A. CELL LINER DESIGN l PROVIDE I 2. FINITE ELEMENT MESil IS TOO COARSE AT DISCONTINUITIES SUCil AS AT ANCHOR STUD. REANALYSIS USING REFINED MESH OR OTHER METHOD. RESP 0flSE: o REANALYSIS WILL BE PERFORMED USING REFINED MESHES CONSIDERING SYMMETRICAL AND N0t!-SYMMETRICAL BUCKLING e SYMMET,RICAL ANALYSIS COMPLETED STRESSES AND STRAINS ARE WITHIN ALLOWABLES i-0 NON-SYMMETRIC ANALYSIS WILL BE PERFORMED l 3-DIMENS10NAL fl0 DEL fl0ST SEVERE LOAD CONDITIONS IMPOSED PER OTHER ANALYSES RESULTS
l' F as 2e o.12.5 0 074 0.131 6 e o.17 5 1.4 " = w 5 1 5 4 E e 1 d 'sch l~ t-9 O tD 0 b e O N 6 i l N io ~J i t 7 L .-- q O c c o a a l q v e M 3 g 2 o u Boundary conditions at tiodes 1 throngli 6, 113, 133 hnd'153 are from the parent Ou model.(Calculation C-27.RC-06.373) < i FIGURE 2-2 l s I o .. A> UY [ + -X Y' .E7 4 y Jf 153 13 3 11 3 i g Stud - 3 LINER / STUD ANCllOR t10 DEL SYMMETRIC BUCKLING
NRC AUDIT FINDINGS I.A. CELL LINER DESIGN 3. ANALYSIS NEEDED FOR TRI-PLANAR CORNER RESPONSE WHEN FLOOR IS HOT FROM A SMALL SPILL AND WALLS ARE COLD.
RESPONSE
e ANALYSIS WILL BE PERFORMED e LOCALIZED HOT SPOT AT 1000 F ON FLOOR WITH WALLS AT NORMAL OPERATING TEMPERATURE e CALCULATED VALUES WILL BE COMPARED AGAINST CRITERIA. DESIGN WILL BE MODIFIED IF REQUIRED I l
[WBEMBEDDEDBEAM 22h" 45" a . FACE OF WALL dV LINER FLOOR I 3 ^ R SPAN e LINE OF SYMMETRY l m MODEL PLAN LINE OF SYMMETRY A ) i l - g LINER PLATE l l, LINER STUD l j {y b .-.- -- -- g / ! b CORNER ANCHORAGE ,/ #,, i N, LINER FLOOR PLATE / J j j % W{ EMBEEMENT BEAM
j y
o MODEL ISOMETRIC-l FIGURE 3-1 L
4 DESIGN AUDIT ITEF.S ITEM 1.B.2: CONTAINMENT DESIGN PROVIDE DESIGN CALCULATIONS THAT CONSIDER TANGENTAll SHEAR CAPABILITY OF THE OU WALLS THAT SANDWICH THE CONTAINMENT VESSEL.
RESPONSE
e TANGENTIAL SHEAR LOAD INCLUDED IN DESIGN e CONCRETE THIN SHELL THEORY USED IN ORIGINAL DESIGN e DESIGN CHECKED FOR SHEAR WALL REQUIREMENTS e REQUIRED REINFORCEMENT STEEL IS LESS THAN STEEL PROVIDED IN DESIGN
(. NRC AUDIT FINDINGS ITEM [.C.3: SEISMIC ANALYSIS TORSIONAL MOMENTS IN STEAM GENERATOR AND REACTOR SERVICE BUILDINGS SEEM LARGE COMPARED TO OTHER SEISMIC LOADS. RESULTS NEED TO BE ASSESSED FURTHER.
RESPONSE
NRC PRESUMABLY IS COMPARING CRBRP TORSIONAL EFFECTS WITH OTHER NUCLEAR PLANTS e CRBRP CONFIGURATION IS DIFFERENT FROM LWR'S e SINGLE FOUNDATION MAT AND INTERCONNECTED STRUCTURES LWR STRUCTURES ARE SMALLER AND REGULAR AND SUPPORTED ON SEPARATE MATS SMALLER TORSIONAL MOMENTS EXPECTED IN TYPICAL LWR STRUCTURES e O
ITEM I.C.3: SEISMIC ANALYSIS RESPONSE (CONTINUED) e LARGER TORSIONAL MOMENTS IN CRBRP DUE TO CONFIGURATION FROM MASS ECCENTRICITIES FROM TORSIONAL MASS MOMENT OF INERTIA SEISMIC ANALYSIS RERUN IN NORTH-SOUTH DIRECTION WITH TORSIONAL MASS REtiOVED e IORSIONAL MOMENT SIGNIFICANTLY LOWER CALCULATED TORSIONAL MOMENTS FOR CRBRP DESIGN ARE REASONABLE e
ELEV. AT ELEV. AT TORSIONAL MOMENT TOP Of B0TTOM OF BUILDING MEMBER MEMBER Will! TORSIONAL MASS NO TORSIONAL MASS RATIO
SGB 886 873 6.303E4 2.994E3
.05 857 846 4.155E6 2.678E6 .64 786 765 6.704E6 4.435E6 .58 746 733 7.517E6 2.453E6 .33 RSB 884 869 8.410E4 5.549E4 .66 840 816 3.776ES 1.919E5 .51 779 765 7.793E5 3.168E5 .41 755 733 1.819E6 9.873E5 .54 1 CONF. 992.5 969.6 1.210E4 0 0 910.8 884.0 5.409E5 0 0 843.0 816.0 1.742E6 9.780E5 .56 704.0 779.0 9.685ES 4.290E5 .44 755 733 8.823E5 3.876ES .44 RCB 816 800 1.308E6 6.460E5 .49 766 752.7 1.469E6 6.174E5 .42 752.7 733 1.584E6 6.500E5 .41 Units: Kip. ft. Torsional Moment Without Torsional Mass
Ratio =
Torsional Moment With Torsional Mass TABLE C.3 TORSIONAL MOMENTS
NRC AUDIT FINDINGS ITEM I.C.4: SEISMIC ANALYSIS REVIEW INTERFACE BETWEEN SEISMIC ANALYSIS AND STRUCTURAL DESIGN GROUPS IN TRANSMITTING OF SEISMIC LOADS.
RESPONSE
INTERFACE LOADS TRANSMITTED BY CINDT FROM SEISMIC GROUP TO STRUCTURES GROUP 0 G IMPLEMENTATION OF SEISMIC FORCES INTO DESIGN IS INCLUDED IN SEISMIC CRITERIA DOCUMENT CINDT PROCESSING IS IN ACCORDANCE WITH PROJECT PROCEDURE PC 3.3 e e 0.A. PROGRAM REQUIRES DESIGN REVIEWS REVIEWS HAVE BEEN PERFORMED ON COMPLETED SEISMIC DESIGNS 9 INTERFACE BETWEEN SEISMIC AND STRUCTURES GROUPS IS CONTROLLED AND IS ADEQUATE
NRC AUDIT FINDINGS ITEM I.C.6: SEISMIC ANALYSIS FOR CONFINEMENT ANALYSIS THE SRSS RULE IS USED TO COMBINE TORSIONAL AND ROTATIONAL LOAD COMPONENTS AND USED AGAIN TO COMBINE STRESSES. THIS IS UNCONSERVATIVE AND DESIGN SHOULD BE RE-EVALUATED.
RESPONSE
THE LOAD COMPONENTS AND STRESSES WERE NOT BOTH COMBINED BY SRSS e COMBINING OF TORSIONAL AND ROTATIONAL COMPONENTS OF ACCELERATION AND COMBINING e OF DESIGN LOADING WERE BOTH CALCULRTED BY SRSS. i INFLUENCE OF THESE INPUTS ON THE DESIGN WAS JUDGED AS MINOR RE-ANALYSIS PERFORMED BY SEPARATION OF EARTHQUAKE ACCELERATION AND COMBINING LOAD S ONLY BY SRSS S DESIGN VALUES INCREASED IN SOME AREAS BUT REINFORCEMENT IS ADEQUATE
NRC AUDIT FINDINGS ITEM I.D.1: GENERAL t Tile FORMULA FOR SHEAR CAPACITY OF CONCRETE WAS MISCOPIED FROM ACI 3119 IN Tile DESIGN BOOK. VERIFY THAT THIS ERROR WAS NOT CARRIED INTO THE CALCULATIONS.
RESPONSE
1 A THOROUGH REVIEW WAS CONDUCTED ON ALL CALCULATIONS UTILIZING ACI 3119 SifEAR e CAPACITY FORMULA e THE SHEAR STRENGTH FORMULA WAS MISCOPIED CALCULATION PAGE REVISED THE ACTUAL DESIGN CALCULATIONS USED THE CORRECT ACI VALUE e 1 l
HRC AUDIT FINDlHGS ITEM I.D.2: GENERAL PROVIDE DESIGN OF STRUCTURES OR ELEMENTS IN CONTACT W!TH COMPRESSIBLE MATERIALS THAT EXPERIENCE LARGE DEFORMATIONS. RESP 0ilSE: e COMPRESSIBLE MATERIAL PROVIDED AT FOLLOWING LOCATIONS: AROUND CONTAINMENT VESSEL PENETRATIONS BELOW ELEVATION 816 FT. LOCATED IN THE PERIPHERAL CONCRETE WALL BETWEEN OUTSIDE FACE OF REACTOR VESSEL SUPPORT LEDGE AND REACTOR CAVITY CONCRETE e GAP FILLER MATERIAL WILL BE LARGE ENOUGH TO LIMIT COMPRESSIVE FORCES ON STRUCTURAL e ELEMENTS STRESSES ON STRUCTURES WILL BE CHECKED e NO COMPRESSIBLE MATERIAL IN JOINTS BETWEEN N.I. BUILDINGS AND ADJACENT BUILDINGS USE OF OPEN JOINTS WITH NON-LOAD BEARING STEEL CLOSURES l
NRC AUDIT FINDil1GS ITEf1 I.D.3: GENERAL ADDRESS IN THE PSAR ALL SIGNIFICANT COMPUTER CODES USED FOR STRUCTURAL DESIGN INCLUDINC-VERIFICATION DOCUMENTATION. RESP 0flSE: e APPENDIX A 0F THE PSAR WILL BE UPDATED ACCORDINGLY e LIST AND DOCUMENTATION OF ADDITIONAL CODES IS PROVIDED
SUMMARY
OF PRESENTAT10'1 CRBRP MEETING WITH NRC ON SEISMIC AND STRUCTURAL DESIGN NOVEMBER 17, 1982 RESPONSES TO UNRESOLVED ITEMS ON SEISMIC DESIGN FROM NRC NEETING OF OCTOBER 5, 1982 e e RESPONSES TO NRC DESIGN AUDIT ITEMS ON STRUCTURAL AND SEISMIC DESIGN 6 INFLUENCE OF INCLINED ROCK LAYERS ON SEISMIC MOTION TRANSMITTED TO THE STRUCTURES /
RESPONSES TO UNRESOLVED ITEf1S FROM MEETING OF OCTOBER 5, 1982 ON CRBRP SEISMIC ANALYSIS e TEN ITEMS IDENTIFIED BY NRC ON SEI'SMIC DESIGN EIGHT ITEMS RESPONDED BY LETTER ll0:S:82:116 0F OCTOBER 28, 1982 FROM e J. LONGENECKER TO P. CHECK e RESPONSE TO TWO REMAINING ITEMS TO BE PRESENTED AT THIS MEETING l SEISMIC RESPONSE AT 20-35 llZ RANGE USING FLUSil INFLUENCE OF INCLINED ROCK PLANES ON SEISMIC EXCITATION
MEETING OF OCTOBER 5, 1982 ITEM I.A: THE FLUSH ANALYSIS HAD A FREQUENCY CUT 0FF OF 20 liz. PROVIDE SPECTRA FOR A FREQUENCY CUTOFF EXTENDED TO ABOVE 30 liz.
RESPONSE
e FLUSil ANALYSIS CUTOFF FREQUENCY EXTENDED TO 35 liz. e RESPONSE SPECTRA AT SEVERAL LOCATIONS INDICATE MINOR CHANGE IN RESPONSE ABOVE 20 liz.
u. u. -. ~,. 4 f l I .a .= k.= ng **en. e. k U..
v..A s J.g,e-n..w. M eg A t ww.* wA d

e e pps. e =L e
i ..w. w n J 3,
r... -: v.. n
... L.. 10..s - Sa.c. - 3 e., D, A,.,m p.m... v e a m 9,
Om.
f
.n.
o l 4 y l I i 2,- .s. el. .f .= 3 = T 7* 3 a ,.1 j 4 i 3 i. J\\. v e a. % bq.,a r p.L P f rM -e 1 u ,T. V, ~- 4 2. s u.g c...... L' 3 i t d 'Y 8 g+
2. *--
.==. i ~ 'Y - Average Properties (35 Hz) M3 c.:. i - Average Properties (20 H ) e.:. l i 1 .= ~ 8 l '.2 .2 4 .5 .3 4 2 3 4 5 t LO 4. ip f. n f.IC. N ' "s ( W. ' l u I l i i I g-, - w- ,7,- .-u, y .n -<~---m.-----.----

~,---n.-,-.--m-,,~v

-- --~ -- .,--,e-..- -,,,-----n

m eww

I 1, t ) ?.ESPONSE SPECTP.A AT ?.CE OPE?aTI:iG FLOOR (EL. S16') NORTH-SCU H TF i';SLATION - SSE - 3*. *JAMP!'IG 4 f 2, 3 .j +
+
.2 '. ~, - a I 3 a.=. s
a.
6 ~ y A 1 .- + . 1. 'l I 2 a c '1 \\. I+ A E. 5,. i.\\.._ W s i -. t w- = u E c I, i 35 4 4 ? g= .T_ r = r. - Average Properties (35 H:) i e 'j a.= - Average Properties (20 Hz) '5 + i i .4 i 2 2 = s a : 2:
4 4
....t.*,t*: NL" v. ( G..T. )
.u 1
4 4 I l 1 J -.--ee.e ,,m.c. ..,_----...,w_ ,.---...w-..,. .--m---
RESP 0flSE SPECTRA AT RSB Efl0 (EL. 816') NORTH-SOUTH TRANSLATI0t! - SSE - 3% DAMPING I -+t
t 7
T. 't e al. 3 5 ~ 4 u. z -- s C R-i-t s -1 w \\N e ,I-C e W Wt U
u I C a,**:
i 3 +': - Average Propertie ; 8-35 Hz E' - Average Propertie i p T 20 Hz i g: .r s l 91 5 2 31 T I 2 .2 .4 f t, e' 2 3 frequency iHZ
.I RESP 0ftSE SPECTPA AT EL. 899 (MID-POIlT) tiORTH-SOUTH TRAtlSLATI0t; - SEE - 3". DAMPIi4G sv r .t ~ t -r 'T* el.i l 2 3.:. z. I: 4 aJ .u z -r og - s -t c: c: h w ' a.wT U
u i Cn**:*
U - Average Properties 35 ilz - Average Properties p' ~ 20 Hz t 3.y: n M. I l ,,+2 l f ~ 1- ~ ~ l '.2 .2 .4 .s . "i 1 3 V a s to 20
D 53 FREQUENCY (NZ) l l
l l l I
RESPONSE SPECTRA AT SGB END (El 816') NORTH - SOUTH TRANSLATI0fl - SSE - 3" DAfiPING s,- 1 - =- - T. 't!"t!9 9.i z-C i.- ~.L s e '; xwawI uv : C9 I - AVERAGE ~I PROPERTIES I 35 Hz - AVERAGE g, PROPERTIES T. 20 Hz T 5:T at i 3 Eh I ? i
I
'2 .1 .4 .h .9 't '2 ~ 3 h 8 8 ~ 10 2C 10 44 ~ ~' FREQUENCY (HZ1
t i v, v_: Ae.+. e. w.. C =.
e.
.r_ _ :-. e. n...e e.. spe.r...a.4. :-. ;o v ..s,._...,v.......,,. - e:.. - a,, D.s.1, .,.G a .u.... .r... ..,,:..n-:vu.a C.'a'.'a*.** %~.~..*.'."..*.*v- 1 =. e."..e. s .. =. = =":" ,. =. i E. a.=.
=
3: f% C". / 'h ,. 4.v -= .\\o* g' v

C*.L.
I
= .=. V r. c: ' N. .=. ' t, \\ .a..=. ~..m-n ra - u= : a n u..,. r,.J.. _ =.
a. w.
C" ". M... ". 0D 7.' 3 (.-..st.. v_r tS.v-e..v.) s .-'x 92 (.5..:) US y0Dr.z 2 a c.. = + ya "" W ...N FIUSH MODEL (20H:) -*+ i FFIE FIELD f .i
c..
[ t t CASE I l .2 .4 1 3 e s a to I: 3 t . =. r... i N. e (.a. l J l l l c- _m g-3 s_--..g--. 7 s -W
MEETING OF OCTOBER 5, 1982 ITEM 1.B: PROVIDE FREE-FIELD VERIFICATION BACKUP INFORMATION FOR FLUSil.
RESPONSE
COMPARISON OF RESPONSE SPECTRA PROVIDED FOR 2 MODELS ~ e FROM THE FLUSH PROGRAMS MANUAL e ONE MODEL WITH EXTENDED BOUNDARY' S ONE MODEL WITH TRANSMITTING BOUNDARY e RESULTS SHOW CLOSE AGREEMENT BETWEEN RESPONSES AT TWO LOCATIONS / h i / r h ,\\ s ,/ 8- / b a + 6 ? \\
-f 1 .(, 7 4 ~ i> 10 ~ ,s s 9 Ee .t g I e ,-s \\ p' r} a K,',c t . 3. ' w', - O 4 -1 -P. I : 7 ,f .a ,,e 9a 3ot .I' y .= ,- ~~. - e i - '.. -z _l \\s a ,,, o ' i n- ~
\\
t e 5 ldth.,!.! s QfK slE, tic R 4 t ~ Lh kli INiliRiirl!I!l'_ 4 ' ~ i i< s e t h lili l l.. l l l'. ~< =,, i i 4 s \\ i- _'l..lij l t i l ; l l l l l - 5 1 5 = l', .~ f i g l t l i i. i i i ig l g j e, j ,u-l49
1. 9{
j ' { C E g a :c .s 1 e nin i.iiiii s, s_a 1 s II!!l l ll8 l' .! */ I 'I. s ( E o =1 g i-i m: m, s - =. .in,, t i n.n...i iii. < j, w w -i E llill l ! !' ll l '!. 1-a n.su ii.i i.,
a. -
-,. _'2 1 'i. - wz w a c, In 6 _e n un. i,- in 1 ii. t. i t ' s li i jnnnitisi iiiii.T = s. . i s i i i, g 'i l liiillilllil lll!l,= j 'ii!!i t i I (!! i, J
I L
I tl . r .e i innnnin ii,. m --.T i i eni.. <i i In i
r r a n n o n i t i iti i..
y eu i viina ~ ff I W!!!!ritil! -' t i l l ! ! _I i,, _! I i si,siit (si ilt isti li, l .g = l = t = _.i e. ,c _ . -e e. e e / c e l. W '- E .,J s
== x, oc -n e T T' e c-m r e s e s a i e 1 e 3 W d i m. a c .i e 2 ~ e g ~ e, ~,. - 5 ] a 3 2 n s _FE i I-22 e' g 5 4- ~ e ~ C 7, ' / s. 5 l ', ! E E 1 4 t T, b, t s ., c. 3 1 i,.. ~ = sw = .s. .e ,s ..t a .3' = -t. p= . 1 i e. c,. ,4% c c =- .~ i 1 t i. i a. z ':' i : o u g e ;< s ; - e.s c. - ~ o m:3.:. m _ -~ _ r
c,-
1( i i 4 4 o e e 1 B ,3 t, : :, n
$.! N-v% g
_ i. s e c,' $? 5 5 l
n., e c N
s, ~ 1 e g C
n
== g. {m l u
5 w
s-M 5.'
o l
I i e t ^) i t e i c. O c c o O 'e e o. C o o e C C C ~ ~ e 4. 4 A -. 4. g e -. u.ime ion =es e.....m,.q, imaos D p p e i' s wl A q** w s 1 ^ i - . s* O OO- '"M N es g ,g g,,,,
MEETING OF OCTOBER 5, 1982 ITEM I.D.: SUBMIT RESPONSE SPECTRA FROM FLUSH AT DIFFERENT LOCATIONS ON THE FOUNDATION MAT. RESP 0flSE: e RESPONSE SPECTRA COMPARED AT 3 MAT LOCATIONS e FOR LOWER BOUND, AVERAGE AND UPPER BOUND ROCK PROPERTIES 9 M.;NOR DIFFERENCES IN RESPONSE BETWEEN 3 LOCATIONS e CONFIRMS RIGIDITY OF FOUNDATION MAT AND REPRESENTATION OF MAT BY SINGLE LUMPED MASS ON CRBRP SEISMIC MODEL IS APPROPRIATE e IHIS RESPONSE ALSO SATISFIES NRC-DESIGN AUDIT ITEM ll.3.A
O i COMPARISON OF RESPONSE SPECTRA AT MAT ~ NORTH-SOUTH TRANSLATION - SSE - 3% DAMPING l I u a.s. =+ 3
+
= L ~ 't n. "+
== ~= I I N-g C .T + v z .N'., z? C9 .M w = C 3 A C ".T 'D = y H -I- '\\- -= u = u 3 C' = 1 .i .c ~ MID POINT OF MAT a gl _ _ _,_ _ SOUTH END OF MAT s .?

NORTH END OF MAT c.+

=i UPPER BOUND PROPERTIES .-a.:. E -=*+ = e
t-
.'i i 3 i 's e : 2: 3: s: .2 .4 .a FREQUENCY (HZ1 FIGURE 13.1 .EE
i COMPARISON OF RESPONSE SPECTRA AT MAT NORTH-SOUTH TRANSLATION - SSE,- 3% DAMPING %-+ 'Y +=
+
=i. wa "+ 5
== ee 1 t I -x e.. ~ ..a_ = j A i .y 5 -= w i - = O.: . 3% \\ 9. ,i (A.',, uT w l N u= u : r =~z \\ ~ 1.- t -s 'I E.-I-1
?
=.5 MID POINT OF MAT l ae f - - - - SOUTH END OF MAT .= a: NORTH END OF MAT 'l = S AVERAGE PROPERTIES + j r i g i. ~. 2 .2 .h .8 .5 i D 3 'e 6 e it 20 30 C FREQUENCY (N Z ) FIGURE I3.2 i m sD
s COMPARISON OF RESPONSE SPECTRA AT MAT NORTH-SOUTH TRANSLATION - SSE - 3: DAMPING 2. .z a.i. a+ W
+
5=- = 't a. "+ 5 i + + u4
E N A
.1. N ~ = C .E. s - = ' 't w + c: s DA. p c '.i A .: v w e J. =. s w3 y = U E c ~:. ./ 1 94 41 ... =. =* MID POINT OF MAT - - - - SOU~d END 07 MAT ..I NCRE END OF MAT
i-
.= .3. LO'n'IR 30UND PROPERT*3 g* '.2 .s 1 2 i s a ic to
s.:
FREQUENCY (HI) FIGURE I3.3 e 23
MEETING OF OCTOBER 5, 1982 ITEM l.E: PROVIDE FINAL SHEAR MODULUS VALUES FOR S0ll BY FLUSil AFTER ITERATIONS.
RESPONSE
e FillAL SHEAR MODULUS VALUES SHOWN IN THE FINITE ELEMENT FIGURES e llIGH AND LOW EXTREME VALUES FALL WITHIN TABULAR VALUES PRESENTED AT OCTOBER STn MEETING e OAMPING VALUES ARE REASONABLE FINAL VALUES SHEAR STRAIN SHEAR Il0DULuS DAMPlNG (IN/IN) (PSI) 2 x 10-4 15,000 5.7 2.5 x 10-3 6,000 12.9
i 1 i DISTANCES FitOH IJI CENTEHl.lHE DISTAllCl:S FttOP NI CICHT10ltl.I NE (TOWAltDS HORTil IN FT.) lTOWAltliS SollTIl IN FT.) u. e in. m. m. en. .o. in. m. m. .s. a. en. m. .o. in ws m. m. en. m. m. u.ii m ,e m e. u s. n e-e4 e-r- a m m m e-a m m e m e.,4 O.e N.e e.s e.e .v EI.EVATION to to y M et M O G3 O M e4 (D se en n erg n m n n n n n n m n n n w w e m o (FT.) .. J1h 09. L' "I* 'I ' * 'Y'i W 8IO 89 8t & a l's t $ slie ele 1 not seie l l T(;n FotillDATION MAT
g. '
61'4 9 toe q lis t. esia atua s's& s lys I fin el s ! fe(5 se'e l s.91 _j gy s a lg uti no .. n nis ini !g g z e.9 viss rico mis t 23, uer tus oss t.3 i._ plo ess9 eni saw unn.67 n o t. an d in s it es se si is s' ass sino pn in n. sino sios i6ei sii6 in9 esai gr.a es t o eie ir a n-sisi 795.% k "M '-1!_ " #4_. '2P. " 11 8 ' 'li* 'm "96 'I'5 "85 "d8 'oa t ss u,. 790.44 g sosc ..in
i. e.,
suo it.s i e s9 3. ..n un nce ois n'a s n9
1. n
..al si9 : in o e.a e... 1ai o " ~ ~ ~ 7as.n N Es4 EMll@~ N' NEEil WillW~ ~~ M i s ', 'VM sa h.
ael sied y sity eles n's e soi6 sola pa iii 780.22
'p q 4 3: ee 02 et ea nyt esti SII.TSTONE st Cd 82 Li =43 isai 158 81L 77s.11 su <p,4 soin nos oo6 sua ia n nas e.g. Sec q.c s ii, 77o.n0 7' -~753 ll1713 ~~ ~3113fR AllF ~4)))fkMJllb ~ AllllK Mllijk ~' .I Sil.TSTONE l UNITS: ksf T---- ..e ~ t l" t I 1 -W i FINAL. SiiEAlt MODtil.IIS FGH SOll. IN Fl.IISil ANAL.YSIS l CASF. II - AVICHAGE l'ItOl'EltT1 Eti SilEAR MODULUS VALUES: "i " ~
1) Siltstone: 83077 ksf rlcune i.4-1 ii) Limestone: 166154 ksf

m. l..
. - e.1 1-1 - e u.. <. = = =
TABLE 05.2 STRAIN DEPENDENT PROPERTIES FOR CLASS 'A' STRUCTURAL BACKFILL SHEAR STRAIN SHEAR MODULUS DAMPING (IN/IN) (psi) (%) -6 5 x 10 22,600 5.6 1 x 10~ 20,850 5.6 2 x 10-5 20,000 5.6 5 x 10-5 19,650 5.6 1 x 10-4 18,000 5.6 2 x 10~# 15,000 5.7 5 x 10-4 11,200 6.7 1 x 10-3 8,000 8.0 4 x 10-3 5,000 14.8 1 x 10-2 3,500 17.2 3.67 x 10-2 2,000 19.0 1 x lo-I 1,500 20.0 i ( , [" _ _
MEETING OF OCTOBER S, 1982 ITEM 1.F: PROVIDE NUCLEAR ISLAND MASS MOMENT OF INERTIA AND COORDINATES OF C.G. FOR BOTH FLUSil AND CRBRP SEISMIC MODetS
RESPONSE
e IABULATION OF MODEL VALUES DEVELOPED e CORRELATION IS CLOSE FOR MASS MOMENT OF INERTIA VALUES e OVERALL CENTERS OF MASS IN'BOTH MODELS ARE CLOSE
NUCLEAR ISLAND BUILDINGS EQUIVALENT MULTIMASS MODEL FINITE ELEMENT i MODEL f1 ASS (K-SEC /FT) 34934 34934 8 MASS Il0 MENT OF INERTIA (K-FT-SEC ) 7.94 x 10" 8.50 x 10 ABOUT CENTER OF BASE COORDINATES OF CENTER OF GRAVITY (FT.) ELEVATION 781.4 768.5 NORTH-SOUTH 10.30 0 EAST-WEST 3.73 0
MEETING OF OCTOBER 5, 1982 ITEM I.G: PROVIDE FOUNDATION SPRING VALUES FOR LOWER AND UPPER BOUND ROCK PROPERTIES FOR CPBRP MODEL.
RESPONSE
e REQUESTED INFORMATION TRANSMITTED BY LETTER ll0:S:82:116 0F OCTOBER 28Tii.
l MEETING OF OCTOBER 5, 1982 1 ITEM I.11: PROVIDE MODEL FREQUENCIES AND PARTICIPATION FACTORS FOR AVERAGE ROCK PROPERTIES FOR CRBRP MODEL. 4
RESPONSE
i e REQUESTED INFORMATION TRANSMITTED BY LETTER HQ:S:82:116 0F OCTOBER 28TH. f l l t i l
MEETING OF OCTOBER 5, 1982 ITEM I.I: RESTRUCTURE RESPONSE TO PREVIOUS HRC QUESTIONS REGARDING JUSTIFICATION OF RADIATION DAMPING.
RESPONSE
e FROM STATIC FINITE ELEMENT ANALYSIS OF ROCK /Soll PROFILE: SHEAR MODULUS DETERMINED FOR AN EQUIVALENT HOMOGENEOUS IIALF-SPACE e FROM EQUIVALENT SHEAR MODULUS: IMPEDANCE FUNCTIONS OF THE HALF-SPACE CALCULATED (STIFFNESS AND DAMPING) 8 IMPEDANCE FUNCTIONS CHECKED AGAINST VALUE DETERMINED BY HALF-SPACE THEORY USING UPPER SILTSTONE PROPERTIES GOOD AGREEMENT
i l ITEM I.I, RESPONSE (CONTINUED) 1 e ASSUMPTIONS AND APPROACH USED IS JUSTIFIED: UPPER SILTSTONE LAYER IS DOMINANT l DIFFERENCE IN SILTSTONE AND LIMESTONE PROPERTIES IS SMALL e SEISMIC RESPONSES OBTAINED BY TWO DIFFERENT ANALYSES ARE IN GOOD AGREEMENT CRBRP METHOD VALIDATED e
FLEETING OF OCTOBER 5, 1982 ITEM I.J: PROVIDE FORMAL SUBMITTAL OF FIXED BASE ANALYSIS RESULTS TO NRC THAT INCLUDES NORTil-SOUTH AND EAST-WEST RESPONSE SPECTRA FOR EACil BUILDING. I
RESPONSE
e RESPONSE SPECTRA TRANSMITTED TO NRC BY LETTER HQ:5:82:116 OF OCTOBER 281H e TilERE IS CONSISTENCY BETWEEN " FIXED BASE" RESULTS AND CRBRP ANALYSIS RESULTS e llIGilER LOCAL PEAKS FROM FIXED BASE ANALYSIS ARE NOT APPROPRIATE FOR CRBRP NON-RIGID ROCK FOUNDATION 80ll-STRUCTURE INTERACTION MODEL IS MORE REPRESENTATIVE OF CRBRP SITE CONDITIONS 4 RESULTS FROM TWO INDEPENDENT ANALYSES CONFIRM ADEQUACY OF SolL-STRUCTURE INTERACTION MODEL - RESPONSES ARE VERY CONSISTENT e CRBRP DESIGN SPECTRA EXCEEDS FIXED BASE RESPONSES IN MANY REGIONS
ITEM 1.J., RESPONSE (CONTINUED) e SEVERAL CONSERVATISMS ARE INCLUDED IN CALCULATED BUILDING RESPONSES LARGEST EARTHQUAKE IN TECTONIC PROVINCE ASSUMED TO OCCUR AT CRBRP SITE THE SITE DESIGN RESPONSE SPECTRA IS A SM00TilENED WIDE-BAND AND ENVELOPED SPECTRA BASED ON STATISTICAL STUDIES OF MANY EARTHOUAKE RECORDS RESPONSE SPECTRA PRODUCED BY THE ARTIFICIAL ACCELERATION TIME-HISTORIES ENVELOPE AND FOR MOST FREQUENCIES ARE ABOVE THE SITE DESIGN SPECTRA CONTROL MOTION INPUT AT THE FOUNDATION LEVEL IS THE SAME MOTION AT GRADE i LEVEL l e CRBRP SEISMIC ANALYSIS IS APPROPRIATE AND CONSERVATIVE FOR THE CRBRP SITE AND THE SEISMIC DESIGN )
NRC AUDIT FINDINGS ITEM II: CONCERNS TREATf1ENT OF S0ll-STRUCTURE INTERACTION IS INADEQUATE. 3.B. WHAT EFFECT DOES THE ANGLE OF ROCK LAYERING HAVE ON THE EXCITATION TRANSMITTED TO THE STRUCTURE?
RESPONSE
1. LITERATURE SHALLOW SUBGRADE WITli INCLINED LAYERS OF ROCK AT CRBRP SITE IS MINOR INFLUENCE e ON GROUND MOTION INPUT TO STRUCTURES, PAPER BY AKI AND LARNER OF M.I.T., JOURNAL OF GEOPHYSICAL RESEARCll, e FEBRUARY 1970, " SURFACE MOTION OF A LAYERED MEDIUM HAVING AN IRREGULAR INTERFACE Due TO INCIDENT PLANE SH WAVES" e ADDRESSES EFFECT OF LATERAL VARIATION OF LAYER THICKNESS ON GROUND SURFACE MOTIONe A VARIETY OF CONDITIONS ARE CONSIDERED. VARIATION IN INCIDENT SHEAR WAVE ANGLES. VARIATION IN LAYER PROPERTIEC. l l v'
NRC AllDIT FINDINGS ITEf1 II.3.B: CONCERNS
RESPONSE
(CONT'D) e VERY GOOD AGREEMENT IN FREE SURFACE RESPONSE WHEN COMPARING: GEOMETRICAL VARIATION OF BASIN. INTERFACE. A FLAT LAYER APPROXIMATION. e FOR A LOCAL AREA BELOW Tile FREE SURFACE (SEVERAL KMS) THE LAYER HAS A CERTAIN SLOPE OR INCLINATION. e EFFECT OF INCLINATION ON RESPONSE IS VERY CLOSE TO EFFECT OF HORIZONTAL INTERFACE ASSUMED AT SAME DEPTil. e RESULTS ARE NOT INFLUENCED BY VARIATIONS IN ASSUMPTIONS. f! AVE-LENGTH OF INCIDENT WAVE DEPTH OF BASIN INCIDENT HAVE ANGLE l l l
NRC AUDIT FINDINGS ITEM II.3.B CONTINUED i ADDITIONAL LITERATURE PAPER BY BARD AND B0uCHON, BULLETIN OF THE SEISMOLOGICAL SOCIETY OF AMERICA, VOL. 70, NO. 4, AUGUST 1980, "THE SEISMIC RESPONSE OF SEDIMENT-FILLED VALLEYS. PART I. THE I CASE OF INCIDENT SH WAVES PES 0 NANCE ZONES ON THE SUFFACE OF A DIPPING LAYER DUE TO PLANE Sll SEISMIC INPUT BY G. L. Il0JCIK, JANUARY 1979; PREPARED FOR NATIONAL SCIENCE FOUNDATION 6 RANT NO. PFR 78-15049 l t
NRC AUDIT FINDINGS ITEM II.3.B: CONCERNS 2. CRBRP CONDITIONS e CRBRP CONDITIONS AND EVALUATIONS ARE ANALOGOUS TO REFERENCE. VERYGbODAGREEMENTINRESPONSESPECTRAFROMHORIZONTALLAYERVS. EQUIVALENT IMPEDANCE REPRESENTATION OF SUBGRADE. VARIATION IN ROCK PROPERTIES (IMPEDANCE RATIOS) IS SMALL: CRBRP = 1.5 REFERENCE = 5 0 e FLAT LAYER APPROACH IS GOOD APPROXIMATION OF SHALLOW INCLINED LAYERS. e FINITE ELEMENT ANALYSIS RESULTS ARE IN VERY CLOSE AGREEMENT WITH CRBRP ANALYSIS RESULTS WHERE ROCK IS SIMULATED BY EQUIVALENT IMPEDANCES. e CRBRP SITE RESPONSE SPECTRA CONTAINS A VERY BROAD FREQUENCY CONTENT.
NRC AUDIT FINDINGS ITEM II.3.B: CONCERNS
RESPONSE
(CONT'D) 3. WAVE ENERGY IRANSMISSION e EVALUATION OF WAVE PATHS UNDER NUCLEAR ISLAND STRUCTURES CONSIDERING SILTSTONE AND LIMESTONE LAYERING AND INCLINATION. e VERTICAL PROP 0GATION FROM BASEMENT. WAVE ANGLE AT 75 AT MAT. NO TRAPPING OF WAVE ENERGY. ALMOST TOTAL TRANSMISSION BETWEEN LAYERS. e VERTICAL PROP 0GATION FROM KN0X FORMATION SAME AS ABOVE EXCEPT ANGLE AT 80. e WAVE REFLECTION IS OF MINOR SIGNIFICANCE DUE TO MINIMAL INFLUENCE OF INCLINED ROCK LAYERS.
NRC AUDIT FINDINGS ITEM ll.3.B: CONCERNS
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(CONT'D) 3. CONCLUSIONS 0 INFLUENCE OF INCLINED ROCK LAYERS ON SEISMIC RESPONSE AT NUCLEAR ISLAND STRUCTURES IS VERY MINOR. 1.10RIZONTAL LAYER ASSUMPTION.IS REASONABLE FOR SHALLOW, LOCAL INCLINATIONS OF ROCK. VARIATION IN CRBRP ROCK PROPERTIES IS SMALL. IMPEDANCE MISMATCH NOT SIGNIFICANT. CRBRP RESPONSES ARE VERY CLOSE FOR TWO DIFFERENT APPROACHES. MINOR INFLUENCE ON SUBSURFACE WAVE PROPAGATION. CRBRP SITE RESPONSE SPECTRA flAS A MUCH BROADER FREQUENCY CONTENT THAN FOR A SITE WITH A SPECIFIC GEOLOGIC PROFILE.
O 2 i l d) ~ DJ o4 LL L' D 1 (D I i tow M tt 2 g O O _ l l t ? N ~ a = 9 2 a [ k f
m ..s (~ / Cumberland Valley and Ridge Belt i Unaka Belt i Blue Ridge Belt Plateau; c.n. n r ive A5Hf veLLE k { ; $ff[.- NW SE SCALr nw 't Ui sr j C_ 40 20 30 go go 'l !!E] 9 lil Ein== 'o - ' =. E i< 'l 1 l = 2.2-C -:::= *.- REGION AL GEOLOGIC CROSS-SECTION g ] I THIN-SKINNED INTERPRETATION =% 1*~ ' g
=..:. :w..w.
~ L.
== a m.* ~ ~ " ,y-. _., m. e f10TES: il onLv nonvenwestenn consson or enoreLE snown al Loc AYBoM OF Cn.nr SITE rnOJECTED ALoNG SYnjME O> 31 uoner seu rnoM Ming {teg4) oa et tu 3 ct flGURE 2.5-1A $,y cn y
O u e .L E e_ g E I >W h WE 8 WnITEDAM MOUNTAIN h g d PA u tilst x an. po ' f C O """ " C"" " " U 'T a... '": :c 1i M : ."'L...L ~ .... - s,%,%4 % ..A L. CoH A. A UG A [ "a 4 aa n.C AMAU.A q ...A !') (19 ~' I888 8 7 88 CONASAUGA '??{,*g, ~ O ,g, ?+ ' M'i ~ y o, SOLE FAULT ~' 9.... SEDIM EN T A n Y nOCM. g.... (E A paLY C A M e nI A N, en EC AM.nt AN) ""'~ /// / //// /8 - n- ///////// ~ '"" not e s. ROME FORMATION si vues enor Le as masco on nereneNCES is,is........r.Ano i.s JOY TEST WELL, PROJECTED L FAULT MAY BE AS .n ALLOW A. al
s. O. o.E (SEE REFERENCE 13) rumf ACComoinG To nerenewCe e.s SCALE:
l' = 5000' Figure 2,5-1711 Area Geologic Cross Section Schematic D N. y
{ REACTOA Cont. St... IIIf tnest Af t. AF - t al.T.WI. .AOUNG UAFACE Coset AOL .L.. 00- . 00 e i.P.P et Atutag. .Pt A.N.f.. 0 0P..As-. rt..v. 0 5 .. A,N T ' ' ' e A a,, 0 r- .00 .a - , 0,,0,, 0, ,s x\\ -t. A... s s E usus e t empt.t Dast 700 g 700 3 en ss NsN Weslt A WPP.R 8Lt.10NE I NN Wulf A testa
DNN Uest? A \\\\%g j
\\ N tv.15 seg u... 0 qs I 00- .00 FMON io M..A,f 40.N.. F00 0 s ? a .a t ,.00 FIGURE 6 CROSS SECTION - NUCLEAR ISLAND .}}

ML20077K543

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