ML19310A256
| ML19310A256 | |
| Person / Time | |
|---|---|
| Site: | Vallecitos File:GEH Hitachi icon.png |
| Issue date: | 05/01/1980 |
| From: | TERA CORP. |
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| References | |
| NUDOCS 8006060483 | |
| Download: ML19310A256 (93) | |
Text
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i 1
SEISMIC RUPTURE HAZARD AT THE
- l GENERAL ELECTRIC TEST REACTOR
1 A REVIEW AND ANALYSIS I
i s.orcea t:
1 I
Lawrence Livermore Laboratory P.O. Box 808 4
Livermore, California 94550 Attention: D. L. Bernreuter, L-90 i
May 1,1980 i
+
I TERA CORPORATION 1
2150 Shctfuck Avenue -
Berkefey. Cofifornic 94704 415 845 5200 Berkeiev Coafornic CCac$. IeuCS -
6etnesCO, VCrinCn3 WCSning*On. O C few York. Piew Y0rk j
Cei MCr. CCnforrtC SCtOn Rouge L0u$Cnc (7 80080864 6
TABLE OF CONTENTS Sec tico Pcce
1.0 INTRODUCTION
AND
SUMMARY
I-!
2.0 GEOLOGY AND SEISMOLOGY 2-i 2.1 Regional Geology.....................2-1 2.2 GETR Site Specific Ceology 2-5 2.3 Fault Pcrcmeters.....................2-6 2.4 Maximum Magnitudes - Verona Fault 2-15 2.5 Seismicity Model.....................
2-17 3.0 FAULT RUPTURE nAZARD 3-!
3.1 Ecrthqvcke Occurrerice Mocel 3-1 3.2 Fault Rupture Percmeters...
3-9 3.3 Fault Rupture Hc cro Mccel....
3-!3 3.4 Guciitotive Analysis of the B-2 cnc S-l/B-3 Snecrs 3-20 3.5 Extension of the Fault Mczcrc Mocel to GETR 3-20 4.0 R EF E R E N C E S..........................
I,- l APPEi4 DIX A Review of the ECAC Report APPENDIX B Review of the Benjcmin enc Associates Repcrt APPENDlX C Review of the Benjcmin enc Associates Report TERA CORPCRAilCN
1.0 INTRODUCTION
AND
SUMMARY
The seismic safety of the Genercl Electric Test Recctor (GETR) at the Vallecitos Nuclecr Center near Pleasanton, Californic has been under review since October 1977, when evidence of possible nearby fculting was discovered.
While there are conflicting and controversial views regcrding the location and capcbility of this postulated, necrby fault (called the Veronc Fault), General Electric has evaluated the GETR structural resistence to both seismic shcking cnd seismic rupture from possible ecrthquckes on the Verona Fcult. In pcrtici support for their re-analysis criteria, General Electric and their consultants performed a probcoilistic evaluation of earthquake suricce rupture under the GETR.
As port of the NRC review of this analysis, the NRC csked Lawrence Livermore Lcboratory (LLL) to review the rupture mocel and results. In order to provice several incependent reviews, the LLL subcontrccted with TERA Corporation to perform o sepcrate review.
Two reports representing the GE probcbility cnolysis have been thoroughly reviewed by TERA Corporction:
Probability Analysis of Surfcce Rupture Benecth Reactor e
Builcing, General Electric Test Reactor (GETR) by EDAC, Inc., April 12, 1979 Additional Probability Analyses of Surface Rupture Offset e
Beneath Recctor Building, Generci-Electric Test Reactor (GETR). by Benjcmin and Associates, March 12, 1980 In cddition, we have incorporated into our review cil of the related information docketed under Licence TR-1, Docket 50-70. These communications relate to the site geology and seismology as well as certain sensitivity analyses of the probabilistic model.
1-1 TERA CORPCRATION
i Our review is presented in detail in the following sections. Our specific review of the GE reports is summcrized in this section and presented in detail in the oppendices of this repor t.
The April 12, 1979. report wcs reviewed by Dr. Gregory Baecher of the Massachusetts Institute of Technology; his review results are presented in Appendix A.
The March 12, 1980 report wcs ciso reviewec by Dr. Baecher, and this review is presented in Appendix B.
For cdditionci diversity, we subjectec this report to cnother, indepencent review by our staff statistician. This detailed review is presented in Appendix C.
The overall conclusion from the reviews presented in the Appendices is as follows:
The presentation of the Benjamin models obscures the actual simplicity anu the properties of the models.
In Appendix C we dw e 'op in detail the model pro-perties and show that the two approaches in the 3/14/80 Benjamin report have substantially dif-ferent properties.
The first approach has pro-perties similar to the 4/12/73 model, namely:
a hazard function which decreases with time since the last c!! set, and an overall result which is indepen-dent of this time, as well as the number of o!! sets which have occurred. On the cther hand, the seccnd approach results in a hazard function which in-creases with time si:re the last offset, and an overall result which is dependent on this time and the number of of! sets which have occurred. While it is difficult to evaluate the correctness of one ap-prcach cver the other, it is reassuring that the results appear to be relatively insensitive to these model properties.
Furtheracre, the appendices identify certain as-sumptions that appear to be sensitive.
- n Appendix C we examine the sensitivity of certain assumptions and show that some assumptions are con-servative while others are nonconservative.
Thus, while we feel that the probability results are most likely correct, we have difficulty establishing a basis for this conclusicn without exploring alter-native approaches. Therefore, we have developed and applied a very di!!eren approach ta the problem.
This independent approach yields probabilities that are very similar to the Benjamin probabilities, and we therefore conclude that the Benjamin probability results are reasonable.
A detailed description is
.given in Section 3.0 1-2 TERACORPORATION
f Section 2.0 presents our cssestment of the site geology at the Vcliecitos Nuclect Center. This assessment is, of c.ourse, heavily weighted by the cvoilcble dotc but where dato are incomplete or absent we rely on our professional jucgment.
Summarily, we judge that the largest earthquake mag-nitude capable of being generated by the Verona Fault is 6.0.
Based upon site topography and its postulated relation to the Verona Fault, we estimate a slip rate for the fault of 0.02 cm/yr. We acknow-ledge the availability of certain age-dating infor-mation from site trenching operations that could also be used to estimate the slip rate, but we holi back these data to independently prcvide a quali-tative check on the model predictions. Finally, we use these fault parameters to develop a seismo-tectonic seismicity model for the Verona Fault.
This model predicts, for example, that the fault will generate 18 earthquakes of magnitude 3.5 every 100 years.
Section 3.0 decis with the rupture hazcrd. For completeness. significent model detail is presented. Section 3.1 summarizes the Poisson earthqucke occurrence mocel used in the enclysis. The next section (3.2) presents the fault rupture model cnd its percmeters. The model selected for tnis review is cicssicci and simple. The overall cpprocch is essentially to multiply the conditional possi-bilities, which correspond to the following questions, times the probability of on enrthquake occurrence (Section 3.1):
Given a certcin size and location of an earthque:, what is the rupture radius?
Given that the earthquake ruptures to the surfcce, how e
long is the rupture?
Given that the earthquake ruptures to the surface, how e
-large'is the displacement?
We have applied this model to an analysis cf the B2 and 31/B3 shears to test the validity of the mcdel.
The model predicts that the shears should experience one-meter displacements with a return period of 19,000 years.
Age-dating of the soils in these trenches indicates that one-meter displacements have occurred in the last 17,000 to 20,000 years.
1-3 TERA CORPORAiiON
Section 3.5 summcrizes hcw we extenced this fcult ru::ture model to cccount for the cbsence of f aulting between the shears. The overcil cpprocch is a Bayesien one in which the " prior" comes from cssuming that tne crec between the shears B-l/B-3 cnd B-2 hos the scme offset occurrence rate cs the Verona Fcult (derived from the slip rate model). The " posterior" comes from updating the prior with the information that offsets have not occurred between the shears in 40.000 years. In order to take into account the various uncertainties. we have intro-duced a parameter (k) which chorocterizes the relative confidence that we have in the mean offset rate of the prior cs compared to the offset rate implied by the upccting information (the smaller k is, the more confidence we have in the i
mean offset rate of the prior relctive to the updating information). We tr'en perform o sensitivity cnclysis on thct pcrcmeter.
We have used this model with the di!!erent sets of input. First, arbitrarily assuming that the age of geologic materials encountered in the ::enches is 40,000 years, we calculate that the probability of any size oHset occu m ng uder M ranges from
~g 7.6 X 10 to 1.4 X 10~.
This compares well with the results reported by GE. (obtained using the Benjamin model) of 1.7 X 10 ' to 1.5 X 10 '.
Sim-
~
~
ilarly, assuming that the age cf the geologic mate-rials in the trenches is 128,000 years, we calculate cc q esponding probabilities cf 3.5 X 10
- and 4.5 X 10 These numbers can be compared to the range of results jg the Ma:ch 12, 1980, Benjamin report of g
i 7.2 X 10 to 4 X 10 i
Tnere cre certain possible conservatisms contained in our results, presentec in more cetail in the text. Summcrily, however, these include the following:
The highest values in the range of probabilities. derived e
through our model and reported above (7.6 X 10-3 and 7.2 X 10-respectively) are obtained by assuming a high degree of confidence in the mecn offset rate of the prior.
Since this rate is set equoi to the offset rate of tne Verono Fault in the onclysis, the cssumption of a high cegree of confidence in this rate is a possibly substantial source of conservatism. Therefore. these higher probo-bilities cre most likely very conservative.
1-4 TERA CORPCRATION
1 Our cnclysis assumes that ecrthquckes of a given mcgni-e tude are uniformly distributed with depth over the fcult plcne. In fcct, Icrger earthquckes are probcbly preferen-tially locatec deeper.
e The onclysis uses a relation between magnitude and j
rupture dispiccement. The displacement values used in deriving this relation undoubtedly include some compo-nent of postseismic slip that resulted from one or more i
creep events following the ecrthquakes. The desired but uncvailable dato are for coseismic slip.
o The seismotectonic dato used as input to the hazcrd l
cnolyses cre believed to be conservative; however, there cre insufficient dato cvcilcole to support better, mecn-centered seismotectonic ccio, j
in our opinion, the uncertainty onc conservatism in the seismotectonic input
)
dominates any uncertainty associated with the model itself but nevertheless, we went to ccknowledge the following inccaquccies in our model:
e No f ault, pcrticulcrly tne Veronc Fcult, con ce classed as uniquely thrust or otherwise. Beccuse of this, we enose to include coto f rom mcny types of earthquakes in our model building.
e Our ecrthquake occurrence mocel for the Verona Fcult assumes that earthquakes of a given magnitude cre 1.
equcily likely to occur over the length cf the fault.
Although we consider this assumption to be reasoncble, it results in the conclusion that certain larger earthquakes
+
can rupture beyond the " ends" of the fcult. Other more complex but not more reasoncble models could result in i
sligntly greater predicted loads at the GETR.
i j
e The model for ecrthquckes is based on a Poisson occur-rence model which is time-independent. Although this is o conventional assumption, other models that include i
strain build-up and relaxation effects could possibly yield greater loads, depending on the time rycle.
.i i
I e
We have calculated the cccelerations at GETR using cn embeament model derived from actual cceelerotion data.
A statistical analysis of these date, different from-the one employed by us in this model, could yiela different 2
embedment models.
l i
.1-5 TERA CORPORATICN -
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2.0 GEOLOGY AND SEISMOLOGY 2.1 REGIONAL GEOLOGY 1
Geological Settina As shown in Figure. 2-1, Vallecitos Valley lies between the Vallecitos Hills on the north and east and the Sunol uplands to the south and west. This portion of the Centrol Coast Range of California is characterized by a relatively young straticrophic sequence of sedimentary rocks and a complex tectonic frcmework.
Bedrock, namely the Livermore formation that surrounds cnd underlies the.
valley, is of late Tertiary and Pleistocene age and has been folded and faulted between two major fault zones, the Calaveras and Greenville faults. These major faults are associated with the Son Andrecs system and constitute o portion of the boundary between the Pccific and North American Plates. Rignt lateral strike slip movement on the active Calaveros fault, located immediately west of the Vallecitos Valley, appears to have created east-west compressional forces a
resulting in the uplif t of the highlands surrounding the volley.
Geomorphically, the volley cnd uplands are of low to moderate relief with Icndforms depicting youthful stages of development. The uplands form rounded smooth moderate slopes and V-shaped intervening drainages that are actively eroding headword and downward. Ancient and modern landsliding is common in mcny of the more precipitous slopes. The degradation of the uplards is forming small alluvial fans and colluvial deposits on the low relief volley floor.
Predominantly southwest flowing drainages on the volley floor have incised the modern surface where modern soils are slowly being developed.
The study crea lies within the Alcmeda Creek Watershed that lies cbove Sunol Dem. Streams in the Vallecitos Valley are intermittent and flow southwest into Arroyo de la Laguna. Several perennial springs are located on the major stream floors within the Vollecitos Hills. The groundwate tcble beneath the volley floor varies from cn estimated 50 to 150 feet below tre ground surface.
2-1 TERA CORPORATICN
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,159 49 *'W TOPOGRAPHY OF THE m
SITE VICINITY
@g TERA CORPORATION
1 Bedrock The Vallecitos Volley is essentioily surrounded and undericin by interbedded gravel, sanc, silt, cnd clay becs and lenses belonging to the Livermore formation.
The older, middle end younger members rcnge in age from an estimated 4.2 million to I million years. The formation may attain a thickness of 2,000 feet in the Vallecitos Valley and thickens eastwcrd to os much as 8,000 feet near the center of the adjoining Livermore Valley to the east.
Upper Miocene age rocks and Cierbo and Briones sandstones of unknown I
thickness probably underlie the Livermore formations within the valley. In turn, the Miocene rocks cre underlain by rocks of tne Great Valley Sequence (Cretaceous) and/or Franciscon essemblage (Jurassic).
The floor of the Vallecitos Valley has a thin mcntel of colluvial cnd paleosol deposits. Young fan deposits are built over older fans along the no theast side of the valley. Recent stream clluvium has accumulated in the larger draincges within the hills and on the volley floor. Slope wcsh cnd Icndslide debris cover most of the slope creas around the valley.
Structure The regional tectonic fabric (Figure 2-2) is complex and generally cons sts of northwest-southeast trending folds and faults, inclusive of the Calaveras, Liver-more and Greenville faults.
Within the Vallecitos Valley is the northwest trending Verona Fault which displays reverse dip slip movement cnd dips to the northeast. The fault trace, about 5 miles, troverses along the southwest front of the Vallecitos Hills from Highway 84 northward to the town of Pleasanton.
Associated with the Verona Fault are at least two minor subparallel shecrs located in the valley southwest of the main trace. These two shears (known as B-2 and H) can be measured in hundreds of feet in length and also display predominately reverse dip slip movement. The main Verona Fault and the two subsidiary shears, B-2 and H, are believed to be part of the same structure and are probobly connected.
2-3 TERA CCRPCRATION
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Normol/Strae Siio Fault A A A Reverse Siio Foutt FIGURE 2-2 MAJOR TECTONIC FEATURES LIVERMORE VALLEY REGION TERA COROC[AECN 24
There ore conflicting concepts for the origin of the shecrs mopoed within the Vallecitos Valley. The observed shears ccn be explained by both a landslide end c fault theory. The pros and cons for ecch have been addressed in detail through
^
previous investigations (ESA,1979). For this study the model for fculting was conservatively assumed.
Both the main trace of the Verono shears cnd the subsidiary shears (S-2 cnd H) have been well documented where observed by previous investigations (ESA, 1979). Trenching, boring and mcpping of the shears have provided a reasonable chcrocter'zation of the ages of last movement and the amounts of displacement.
Another major fault that was mcpped by Herd (1977) trends into the soutneastern portion of Vallecitos Valley. This fault is known cs the Los Positos Fault onc extends from the Greenville Fault clong the southern margin of the Livermore Valley across the Vallecitos Hills. At this time, field work including trenching and mcpping is being conducted to help determine the fcult's chcrccteristics.
2.2 GETR SITE SPECIFIC GEOLOGY The GETR facility is located adjacent to the northeastern side of the Vallecitos Valley opproximately 300 feet south of the hill front. The Vallecitos Hills rise obruptly cbout 700 feet above the volley floor to cn elevation of 1,286 feet. The southwest fccing slope is very irregular and has on overall slope ratio of 6 to I (horizontal to vertical).
The hills represent the southwestern ficnk of the no inwest-southeast trending Livermore syncline. The bedrock mapped on the entire slopes is the younger Livermore grovel of Plio-Pleistocene cge. Bedding dips on on overage 26 degrees to the northeast.
GETR is located near the cpex of a small alluvial fcn which was created by a drainage that exits the hill front immediately east of the facility. The fan is on older alluvial unit now incised by the some stream that created it. The apparent down-cutting could have been caused by either uplif t of the Vc!!ecitos Hills or sinking of the base level south of the hills.
2-5 TERA CORPORATICN
The fan surface is relatively smooth and slopes gently to the southwest. The fan is overlain by at least one or two soil units which are estimated to be relatively thin. Bedrock (Livermore Gravels) underlies the soil deposits. A younger fan located about 300 feet northwest of GETR is presently being developed by a youthful minor drainage flowing off the slope.
This younger fan is being deposited over the older fan surface in which GETR rests.
About 400 feet southwest of GETR a low smooth rounded elongated (northwest /
southeast trending) knoll interrupts the valley floor. Younger Livermore gravels presumably include a thin soil cover in the vicinity of the knoll. The Verona fault shears traverse the toe of the slope about 350 feet northeast of GETR. The B-2 shears trend along the southwestern fccing slope of the knoll a;>roximately 850 feet south of GETP,.
The Earth Science Associates report of February 1979, describes in detail the shears in trenches B-1, B-2 and B-3.
The trench logs describe the various materials observed, and fault history interpretations have been made using several dated layers.
2.3 F AULT PARAMETERS To determine the fault parameters of the Verono fault system, the geologic history during the Late Quaternary of the region had to be simplified. Several points are summarized below that help establish minimum and maximum ages during which it is estimated that faulting commenced in the Vallecitos Valley.
1.
There is general agreement that the Livermore gravels (youngest to oldest members) were deposited 4.2 to 1.0 million years before present (B.P.). It is further assumed that the Livermore gravels were deposited in a continental alluvial outwash environment and beds were deposited at low bedding dip ongles (e.g., = 5 degrees).
During filling of the central l
Livermore basin area, subsidence was initiated and continued as deposition l
progressed. As much as 8,000 feet of valley fill now exists in the central l
portion of Livermore Valley.
2-6 TERA CORPORATION r[
2.
Regional geologic mapping of the site crea indicates that all members of the Livermore grovels are tilted toward the east and apparently even the youngest gravels cre deformed as much as the older members.
This suggests that structural folding commenced af ter deposition of the gravels was complete, about 1.0 million years before present.
Some of the eastward tilting can be attributed to depositional subsidence; however, anticlines and synclines witnin the youngest gravels suggest that compres-sive forces have been active in the last 1.0 million years B.P.
3.
Structural folding continued up to the time of the development of the paleosols in the Vc!!ecitos Valley because the paleosols do not appecr to be deformed by folding.
The oldest paleosol in the Vallecitos Valley is believed to be about 400,000 to 500,003 years old.
(.
The faulting process could have commenced at the time compressional folding started since both types of deformation cre compatible and were produced by east-west compressional forces. If this is the cose, then a maximum conservative limit for the stort of faulting is about 1.0 million years B.P.
5.
A minimum limit cs to when faulting stcrted con be established if we assume that faulting commenced af ter the folding process stopped (e.g.,
400,000 to 500,000 years B.P.).
The compressional forces in the crea causing folding were transformed into o faulting mode that subsequently resulted in at least partial uplif t of the Vallecitos hills.
6.
It is possible that fculting could have started even later than the period of paleosol development, however, the slip rotes for the Veronc fault become excessive based on the assumed cmount of displacement.
The minimum and maximum dates for when faulting started M00,000 and 1,000,000 years B.P.) cre considered realistic, although conservative. The recson they are considered conservative is because cdditional geologic processes had to have taken place within the some time frame. A period of extensive erosion and 2-7 TERA CORPC' RATION
5 1
i i
perhaps mass wasting wcs necesscry to account for the stratigraphic section (Livermore grovels)- that is now missing west of the Veronc Fault.
The stratigraphic section through the Valecitos Hills is at least 5,000 feet thick. The 1
section west of the Verono Foult is estimated to be 2000+ feet thick, leaving about 3000 feet of gravel to be removed.
This projected section represents several cubic miles of gravels that were eroded and removed entirely from the Vollecitos Valley creo. The erosion process had to have occurred in the lost 1.0 million years since the lost gravels were deposited.
l With this in mind it is hard to believe that the paleosols could hcve survived inis vast erosion process on the Vallecitos Vcliey floor, it is difficult to believe that i
the paleosols are representative of the stable landscape periods between great 1
periods of erosion and removal. Because the paleosols cre well preserved and thinly stacked one upon the other the unstable periods of erosion between their j
development must have been minor. If this is correct then the great erosion i
period had to have occurred sometime between 1.0 million years B.P.
and the development of the ocleosols (400,000 to 500,000 years B.P.).
)
The cpparent discrepancy for the occurrence of the cbove stated processes is i
best explcined by on incorrect assumed age of the youngest Livermore Gravels (e.g., one million years). Two million years is a more reasoncble period for the o1carrence of the faulting, folding and erosion.
I in summary, vast erosion along with the folding cnd faulting, portions of which may have occurred simultaneously, occurred in the last 1.0 million years and more realistically in the lost two million years. It is our opinion that the slip rates described below cre conservative because it is very difficult to conceive l
that all of these processes took place in such a short (geologically) time span. It j
should be noted, however, that there is insufficient dato ovcilable to enable us to remove the conservatism in slip rates.
h 2-8 TERA CORPCRhTION
Slio Rates Verona Fcult Svstem As presented in Figure 2-3, the slip rate for the Verona Fault was established using a model based on the estimated cmount of throw clong the fault. A throw of 900 feet was calculated utilizing the elevation difference between the top of j
the Vallecitos Hills (~l300 feet) and the toe of the Hill (~600 feet). A near surface fault dip ongle of 26 degrees (bedding plane fault) that steepens to 60 degrees at depth was considered. The preceding section discussed ages of when faulting commenced on the Verona Fault. It is our best, although conservative, estimate that fculting started about 1.0 million yecrs B.P. cnd has continued up to at least the Middle Holocene period.
There is o realistic probability thct a portion of the Vallecitos Hills uplif t was a result of not only faulting, but folding and erosion (downcutting) of the vcIley floor. The following conservative slip rcte is realized not considering thct other processes contributed to the uplif t of the Vallecitos Hills:
900 feet of throw have occurred in 1.0 million years
= 0.03 cm/yr Assuming a reasonable 80 percent of uplift was due to faulting and 20 percent was due to other processes, the following more reasonable slip rate is estcb-lished:
650 feet of throw have occurred in l.0 mil! ion years
= 0.02 cm/yr For this analysis, it is our opinion that the.02 cm/yr slip rate is reasonable. An alternative slip rate model could be developed assum.ag that faulting com-menced two million years ago. This results in one-hcif of the above slip rates.
Fault Length The length of the Verona Fault is important in any specification of earthqucke capability. Actual fault lengths cannot readily be determined from the geologic dato provided by ES A (1979) and Herd (1977).
1 2-9 TERACORpORATION
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TERA CORPORATICN
In referring to Figure 2-4, o feature that could be interpreted as the Verona Fault can be well documented from field and air photo evidence, extending from locations B-C (2.4 km). On the bcsis of geomorphic and subsurface evidence, this I
represents the actual known total length of the postulated Verona fault.
Northwesterly from B toward Pleasanton, it is hypothetical that the fault is continuous.
However, cssuming the Vallecitos Hills were uplif ted along the Verona Fault by compressional forces, the fault could continue to Location A.
Position A is about the northern limit of the hills. Geologic data do not favor the Verona Fault continuing southeasterly past Highway 84 towcrd the Lcs Positos Fault. It is our opinion that the Verona Fault terminates near Hignway
- 84. The fault length from A to C is about 7.2 km. There is a remote possibility the fcult could continue northwesterly for on assumed distance of 1.5 km beneath the olluvium. Geomorphic data suggest that the Verona Fault might terminate near Point D.
A totcl Verona Fault length (A to D) of approximately 11.0 km is obtained cs the reasonable upper limit to the fault length.
Evidence to merge the Verona Fault with the postulated Pleasanton Fcult or the Cc!cveras Fault is lacking. f t is our opinion that the Verona Fault is a separate feature and is responsible, in pcrt, for the uplif t of the Vollecitos Hills clong their southwestern front. We further believe that much of the uplift of the hills from about points B to A is a result of folding.
As mentioned above it is our opinion the Verono Fault terminates at its southeastern end. No geologic evidence to date has been presented to suggest that the Verona and Los Positas faults merge into one through-going feature, cithough it cannot be ruled out.
The geometry and physics of merging a compressional thrust feature (Verona Foult) at en oblique engle with a tensioned normal fault (Los Positas Fault) are not easy to understand.
In summary, the range of fault lengths considered appropriate for the Verona Fault are os follows:
Verona Fault (odequate field evidence) 2.6 km Verona Fault (projected trace) 10.4 km 2-11 TERA CORPORATICN
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FIGURE 2-4 FAULT LENGTH 2-12 TERA CORCCRA'iCN
Fault Dio Ancle it hcs been suggested that the Verona Fault is a bedding plcne fault. This hcs not been confirmed; however, o model for bedding plane faulting fits the field evidence as well as a steeper dipping fault plane model. A bedding plcne fault con be a result of folding and not deep-seated crustal instability. For example, during the compressional folding of a syncline, the younger overlying beds tend to slip past one another in on apparent lengthening or thrustinc, cction. This is considered structural cleformation by faulting, but the faults move in response, perhaps slowly and periodically, to c slow folding cction or creep. It is doubtful that this type of fculting is ccpoble of producing on earthqucke. Since this cannot be easily confirmed, o steeper dipping fault plcne model was conserva-tively assumed.
I As shown in Figure 2-5 the dip cngle of the Verona Fcult plane wcs estimatec by using the location of known epicenters ecst of the fcult trace cnd estimated depths. The Livermore Fault was considered a limiting factor in that the Verona Fault probcbly does not penetrate the near vertical Livermore fault plane.
One roughly linear epicentral trend lies cbout 3.2 km east of the Verenc Fault and another 9 km to the east essentially located over the trend of the Livermore Fault. It is slightly possible that either of the linear epicentrol trends represent earthquckes on the Veronc Fault; therefore, we use these dctc in developing a fault dip model. Assumed depths of 8 km for shcIlow earthquake and 15 km for deep earthquckes were used to determine the dip ongle of the Verona fault plane.
Considering the above numbers, fault dip cngles vary from 41 to 79 degrees. The geometry of the inferred fault planes is shown on Figure 2-5.
Using a dip ongle of 26 degrees for the Verona Fault, os suggested in the slip rate calculations, would place the epicenter locations on the Verona Fcult approxi-motely 18 km to the ecst of GETR for shcIlow (eg. 8 km) earthquckes. This is not considered valid or realistic. 'in the ecse of the Verona, B-2 and H shears, the fault planes cre shallow, near the surface and increase in dip cngle wit!
2-13 Tei?A CORPORATICN n--~
i rautT ANCLE 9&
LIVERMORE FAULT vt.C {
VAcLECITOS POSSIBLE pOSSIBLE VERONA SILLS ~
j gVERONA EPICENTERS if EP! CENTERS 1
LIVERMOR.E VALLEY i;\\s')
1 i
FAOLT ANCLE n
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FIGURE 2-5 FAULT ANGLES (DEPTHS) ESTIMATED USING EPICENTER LOCATIONS (FIC 3 HERD 1942-19681.5 M+)
I 2-14 TERA CORCCRATON
depth. A 60 degree dip angle is considered reasonable for this study. This would place shallow earthquake (8 km) epicenters chout 4 km east of GETR and ceep earthquakes (15 km) about 9 miles to the east.
2.3 MAXIMUM MAGNITUDES - VERONA FAULT in this study, consideration was given to the length of the Verona Fault, amount of apparent single event displacements, the type of fault movement and the seismic history of the fault.
These data are needed to help deter.mine a maximum magnitude. Since there is a lack of a historic seismic history along the Verona Fault, geologic and empirical data were used to establish limits.
A hypothetical fault length was suggested earlier of about 11 km using a conserva-tive overrun at both ends of the fault. A more realistic fault length of 7.5 km is equal to about the length of the Vallecitos hills from Pleasanton to Highway 84.
This is the segment considered most likely to break since it corresponds to the topographic expression of the scarp and/or distal edge of the fault. At least one j
meter of displacement was recorded in trenches near GETR which is located
]
below the highest hill top in the Vallecitos Hills (e.g.,1,285 feet). The north end i
of the hills attains a maximum elevation of 800 feet or about 400 feet above the 1
J valley floor. There is appreciably less apparent maximum uplif t at the southeast 1
end near GETR. From this we rationalize that displacement may vary by 50
)
percent along the fault trace (e.g., 0.5 to 1.0 meters). This might further be l
explained by assuming that only portions of the fault move during a single event.
The following data summarize the various magnitudes, based on empirical relationships from fault lengths and amount of displacement of the Verona fault.
Fault Length lI km 7.5 km Slemmons (1977) 7.0 m 6.8 m Greensfelder (1972) 6.3 m 5.9 m Bonilla and Buchannon (1970) 5.8 m 5.4 m 2-15 TERA CCRDORATION
.im
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=
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Displacement I meter 0.5 meter Slemmons (1977) 7.0 m 6.8 m Bonillo cnd Buchennon (1970) 6.9 m 6.1 m Assuming that only one-half of the fault length was to rupture Juring a single event, we determined the following magnitudes:
Fault Length 0.5 x ! I km = 5.5 km Slemmons (1977) 6.75 Bonilla and Buchannan (1970) 5.0 Greensfelder (1972) 6.1 It is difficult to assign a single maximum magnitude to the Verono fault. It is our opinion that only a portion of the fcult will rupture the ground surface during on ecrthqucke equal to about one-half the total length of the fault. Our best estimate for on earthquake magnitude on the Verona Foult is therefore 6.0.
It is also our opinion that the B-2 and H shears are not ecpable of producing on I
earthquake independently. It is reasonable to assume that the two shecrs move sympcthically when the Verono fault moves.
2.4
SUMMARY
- FAULT PARAMETERS in summary, we judge that the following data represent either reasonoble or perhaps conservative best estimates and ranges of important input parameters to the calculations described in the remainder of this report.
Parameter Best Estimate Rcnce Units Slip Rote
.02
.01.03 cm/yr Fault Length lI.0 2.4 - 1I km Dip Angle 60 41 - 79 degrees Maximum Magnitude 6.0 5.5 - 6.5 ML 2-16 TERA CORPORATION
1 2.5
$EISMICITY MODEL 1
This section presents the development of a seismicity model for the Verona fault that will be used in the fault hazard assessment. Conventional and desirable practice is to base seismicity models on the historical seismic record, although l
such cs approach applied to the Verono Fault would be subject to unocceptable
~
uncertainty due to the sporce seismic records. - An citernative approach has been developed within the lost several years that somewhat avoids this problem by relating certain geologic data to the earthquake process. The starting point, cs I
outlined in Campbell (1977), is to relcte the geologic slip rate on a fault with the cccurrence of ecrthquakes. This model is developed below.
i Brune (1968) hos proposed on expression that con be used to estimate the total overage slip u on a fault from the sum of seismic moments of earthquckes on the fault. This expression is given by:
N*
I [M (2-1) pA o
where is the shear rigidity of the medium, A is the total crec of the fault zone given by its length L times its width 'N, cnd M is seismic moment g
o o
defined as AD,(where A is the crea of fault rupture for the event cnd D is the mean displacement).
Equation 2-1 ccn be generalized to establish a relationship between number of events having magnitudes m and greater, N(m), and the slip rate S. The total R
slip per year that ccn be expected to result from the occurrence of earthqvckes on the fault con be given by:
M i
u i
M,(m) ldN(m)l (2-2)
SR*
p o
I
.=
2-17 TERA CORPORATION
i 1
where M is the upperbound mcgnitude on the fcult, cnd cN(m) is the number u
of events per year having magnitudes equal to m. The cbsolute value is required because the number of events decrease with increasing m, giving a negative slope for the N(m) versus m relation. In Equction 2-2 only the cbsolute number of events is required.
in order to analyze Equation 2-2, relationships among M,(m), dN(m) and m are j
required. Let it. be assumed that, for the fault, the number of earthquake occurrences per yecr of a given magnitude or greater con be given by the magnitude-frequency law, 1
N(m) = 10c-bm; msM (2-3) i Taking the derivative of this expression gives, l
l dN(m)l = b Log,10 10 -bm dm (2-4)
Let it also be assumed that the seismic moment of on event con be related to its magnitude by a relationship of the form, C +C m M (m) = IO j 2
(2-5) 4 g
which has been suggested from both empirical and theoretical considerations.
An expression for slip rate is ootained by substituting Equations 2-4 and 2-5 into Equation 2-2 and integrating, thus obtaining, SR * (C -b pA o(' u
~
2 g
An expression for the number of events per year of magnitude greater than or equal to m may be obtained by solving this equation for 10 cnd multiplying each side by 10-bm,
5 p A (C -b) 10 (M -*)
(2-7) 2 b
N(m) =
u i
M (M )D o u 2-l8 TEDA CORPORATION
Similcr models have been recently proposed and applied 5y Anderson (1979) cnd Molncr (1979).
In terms of on application of this model, the dato required to estimate the seismicity of the Verona Fcult are listed in Table 2-1. The values for slip-rate, upper bound magnitude, and fcult length were taken cs the best estimate of these parameters os discussed cbove.
A lower band magnitude of 3.5 was selected to represent a threshold below which earthquake occurrences would have a negligible effect on the hazard results.
1 i
4 f
A relationship between seismic moment and magnitude wcs ceveloped from 167 recorded seismograms from earthquakes of 2.0 ( Mg ( 6.8 occuning in Southern California.
Seismic moments were estimated by several investigators (see
(.cmpbell,1977) who matched observed sourcc displacement spectra with specirc determined from dislocation theory.
A least-squares cnclysis of these dato l
re. salted in the following log-linear relationship:
i Log 10 M (m) = 16.3 1.4i rn (2-8) o where M,(m) is seism moment in dyne-cm. The fit resulted in c standard error of 0.42 and a correlation coefficient of 0.95. Based on this equation, the seismic 24 g.0 upper bcnd earthquake is estimated to be 5.8 x 10 dyne-6 moment of the M cm.
The fault width (W ) is the distance from the surfcce to the maximum depth of faulting as measured clong the dip of the fault plane. The maximum depth of l
fculting for the Livermore cree is probcbly less thcn or equal to cbout 15 km, based on the maximum depth at which earthqu'ckes are found to occur (Bolt and Miller, 1975). From this and the three-dimensional geometry of the Veronc Fault, the fault width was estimated to be 18 km and the crec of the fault plane 2
to be 193 km.
l i
s 2-19 TtRA CORPC-RATION
TABLE 2-1
SUMMARY
OF SEISMOTECTONIC DATA U5ED IN THE ESTIMATE OF SEISylCITY Parameter Symbol Estimates Slip Rate (cm/yr)
S 0.02 R
Upper Bound Magnitude M
6.0 24 Seismic Moment of M Id Y"*-C *)
M IM )
5.8 x 10 u
o u Lower Bound Magnitude M
3.5 Fault Length (km)
L II Fcult Width (km)
W 18 g
Fault Arec (km )
A
.93 g
s-Shear Rigidity (dyne /cm")
3 x 10 '
Seismic Moment Coef ficient C
I '* I 2
Richter b-value b
0.87 Seismicity (events /yr 2 M 4 )
N(M.7) 0.185 t
L 2-20 TERA CCiiPORATICN
A o-velue of 0.67 was ccoated from Compoell (1977) to be consistent with the seismicity of cctive regions within the western Unitec States.
The remcining seismotectonic data used in the estimate of seismicity cre presented in Table 2-1.
Based on equation 2-8, the recurrence curve for the Verona Fault was determined as follows:
Log 10 N(m) = 2.312 - 0.87 m (2-9)
This corresponds to 0.135 events per year of magnituce greater then or equal to 3.5, or rougnly one event every five and one-haif yects.
2-2i TERA COR:CRAT10N
1 3.0 FAULT RUPTURE HAZARD The proximity of severci shecrs to the GETR facility suggests a possible risk due to toult rupture.
A rigorous analysis of this risk would require o detciled knowledge of the fcult and frccture pattern in the immedicte vicinity of the facility, which is beyond the scope of this study. A general assessment may be made through a study of the fault rupture hazcrd associated with the adjacent postulated Verona Fault.
Fault rupture hazard is defined as the probcbility that the maximum surface displacement at a point on the icult exceeds a given value over the time period of interest. The general opprooch is cs follows. The fault system is divicec into o series of segments of equal seismicity. Earthquake occurrences within ecen segment cre treated as Poisson-Bernoulli processes. The random occurrence of all events from each fault is combined with a fault rupture model to develop a probobility distribution of surface disp!ccement for cny point on the fault. The calculation to this point essentially results in on assessment of the rupture hazard on one of the pre-existing shears. The next step in the cnolysis is to occount for the fact that trenches between the shears in the near vicinity of GETR have exposed undistrubed soils of c certcin age, thus indicating that the orec between the shears has not ruptured in this time.
3.1 EARTHOUAKE OCCURRENCE MODEL The basic input parcmeters of the earthquake occurrence model are the magnitude range (upper and lower bound magnitudes) cnd earthquake recurrence.
With respect to earthquake data, the magnitudes are discretized every 1/4 of c magnitude unit as is commonly done in dato recording. This representation permits the use of discrete models.
The development of the model involves three steps:
(1)
Assuming that earthquake occurrences form o Poisson process with mean rate of occurrence independent of 3-1 TtRA CORPORATION s
mcgnitude, o distribution is obtained on the number of occurrences for the time period considered.
(2)
Civen that on event has occurred, a distribution on the magnitude of events is determined. The process generot-ing model con be assumed to be Bernoulli. The orobcbility of success, pM)., corresponding to each trial, is defined a the probabilit that the event that has occurred is of magnitude M;. Thus, the probability of failure, qg,=1-p.g, at each trici is the probability that the event'is not of diagnitude M;. The proccoility of having r events of magnitude M;, given that a total of n events nove occurred, con therefore be obtained using the binomici distribution.
(3)
The distribution of the number of events of each magni-tude, independent of the number of trials, is obtained by combining steps one end two.
Earthoucke Occurrence (Poisson Model)
It is assumed, once the seismic sources have been located, that earthquake occurrences on each source form o Poisson process with mean rate of occurrence independent of magnitude. For earthqucke events to follow the Poisson model, the following ossumptions must be valid:
l.
Earthquakes are spatiolly independent 2.
Earthquakes cre temporally independent 1
3.
The probcbility that two seismic events will take place at the some time and at the some location approaches zero.
l l
l h
3-2 l
TERA CORPCRAT!ON
1 l
l The first cssumption implies that the occurrence or cbsence of a seismic event at one site does not of fect the occurrence or cbsence of another seismic event at some other site or the some site. The second cssumption implies that seismic events do not have memory. The assumptions of spatici and temporci indepen-dence have been verified by data when offershock sequences cre removed, and are commonly accepted. The degree of dependence between events, due to the duct mechanism of stress cceumulation cnd release, hos not yet been determined with ony amount of precision, but the ecrth's " memory" appears to fode quite rapidly with time (Corner and Knopoff,1974). The third assumption implies that, for a small time-interval, more than one seismic event connot occur on one source.
Thus, considering all the events of magnitude greater then on arbitrcry lower bound, o distribution is obtained for the number of occurrences in a given period of time, t. The lower bound is chosen so that ecrthquckes of mcgnitude smcIler than the one specified, which have o negligible demoge potential, con thus be disregcrded. This distribution is obtained for each seismic source.
In its general form, the Poisson law con be written as P (n/ \\ ) = e " ( \\ t)"
t 0 ; n integer 0,
(3-1)
N n!
where P (n/ ) = pr bability of having n events in time period t, given N
n = number of events
\\ = mean rate of occurrence per unit of time Thus, if the mecn rate of occurrence \\ is known, the probability distribution function con be defined completely.
The pcrometer \\ is obtained from the dato and con be modified subjectively. In the present case, it is expressed as the mean rate of occurrence, per year, of ecrthqvckes !crger then magnitude 3.5.
3-3 TEiiA CORPORAilON
Estimate of A If one assumes that the number of seismic events for o future time t follows o Poisson probability law, there is still uncertainty cbout the parameter
, the mean rote of occurrence (Equation 3-1). Therefore, A is treated as a random variable. The probabilistic information on A con be obtained-through historical dato or from the subjective knowledge of the analyst. The subjective probcbility distribution on A is colled the " prior distribution."
The concept of conjugate prior is used for analytical simplicity (Raiffo and Schloifer,1961). Therefore, the prior distribution for the rcndom varicole is chosen as the gcmma distribution with pcrameters N' cnd v'.
Since the gemme distribution con fit a !crge variety of shapes, this choice does not introduce any major limitations in the model.
Using the historical information, one con obtain the sample likelihood function for A. The posterior distribution for A con be obtained by combining the prior distribution and the sample likelihood function by means of Boyes' theorem.
Let f',g ( A ) be the prior probability distribution function for A, and L( A ) be the sample likelihood function for A (given by the Poisson low for n observations),
then the posterior distribution f",g( A ) is obtained as f",g ( A ) = N, L( A )f',1( A ),
(3-2) where N; is o normalizing constant. The posterior distribution of A is also gomma type with parameters A" and u ",
in Equation 3-1, the conditional probability on the number of events n is based on I
A. The unconditional er the marginal distribution on n can be obta:ned by using Equation 3-1 together with Equation 3-2 and integrating over all A 's. Thus, 1
l m
P I"I
- N(n/ A )f'(( A )dA (3-3)
N 0
3-4 TERA CORPORATION
4 which leads to
\\"
P (n) =
(3-4) n! ( v ")
( t + A ")"'
I4 i
for n integer a 0, y " > 0, \\ " > 0, t > 0.
i 1
Equation 3-4 is called the " marginal Bayesicn distribution of n." This distribu-tion, af ter the uncertainties on the mean rate of occurrence are considered, I
gives the probcbility of the number of events cbove a predetermined lower bound M, in time period t.
j l
Distribution of Moanitudes (Bernoulli Model)
A Bernoulli trict is used to model information on magnitudes. Given that on event hos occurred, the probability that it is of cny given Richter magnitude con I
be represented in terms of a Bernoulli trial. If the seismic event that has occurred is of the M; under consideration, then the outcome of the Bernoulli trici is a success. Conversely, failure at ecch trial implies that the seismic event that has occurred is of some other mcgnitude.
t If pM. = pr bobility of success at each trial corresponding to M; s
cnd qM.
- i-PM.
I I
probability of failure at ecch trial,
=
then-using the binomici law,
.i r
r
"~ 'M.
M.
M.
P I'M.I"'PM.) = C
'PM.II-PM.)
(3-5)
R n
i a
i for n integer >0, rM. integer; 0 s rM. s n, nd 0 s pM.
- I*
4 1
I i
3-5 TERA CCROGRATlCN
)
j where P (rR g}n, pg,) is recd cs the probcbility that rM. events of mcgnitude M; will occur out of a t'otal of n events, given that the prcbability of occurrence of M; is pg, at ecch trici, cnd i
4 I
i M.
1 C
n M. !(n - rM.}I r
i i
A A different probability pM. is obtained for each M; considered in the model.
similcr equation is thus 'obtained for each of the other magnitudes.
The probabilities pM. re mutu lly exclusive within the range of selected magnitudes, i
- hence, pM. = 1.0.
all M; As an example, for M; = 6 and n = 5, i
P (0 events of M = 6 given 5 earthquakes) = (1 - p6)
P (1 events of M = 6 given 5 earthquakes) = 5 x p6
- II - P) q j
P (3 events of M = 6 given 5 ecrthquakes) = p6 l
It should be noted that i
n[
p(r/n) = 1.0.
r=0 r
To be consistent with the Gutenberg-Richter ecrthqvcke recurrence relationship f
the probability pM. should be consistent with on exponentici distribution of l
magnitudes. Cornell (1971) has proposed such a distribution which incorporates both lower cnd upper bound limits on magnitude:
3 4
3-6 TERA CORDORATION
(
'. f P(M s mt = K I - Exp
- J(m - M,)
(3-6)
- where, P(M s m) = probcbility of M s m j
f "I K=
1 - Exp -d(M -M) 6 = 2.3b M = upper bound magnitude u
j = lower bound mcgnitude M
1 The probability that on earthquake has a magnitude within the rer.ge m;; m/2, equivalent to the Bernoulli parcmeter pM., then becomes I
g, = P(M s m; + a m/2) - P(M s m; - a m/2)
(3-7) p l
l Equation 3-5 represents the generating process for the ni. nber of events M;.
However, this information is conditional on the knowledge cbout pg,, the probability of success corresponding to M;.
]
Estimate of o,
g 1
The conjugate prior distribution on pg,, f'p (pg,), is assumed to be beta type I
with parameters
'l' and E'.
Since the ' normalize'd beto distribution is bounded I
between 0 and I, and fits a large variety of shapes, this choice does not introduce any major limitations in the model. A prior distribution of a similar l
form has to be assumed for each of the mcgnitudes considered.
l l
The usual format of the availcble data indicates that, cmong the n ecrthquckes observed for a given source, rM. were f M;. This information is used in the construction of the sample likelihood function. The sample likelihood function 1
L(pg /n,rM.), my be obtained from the generating process on p g,,
I (Equation 3-5). '
The posterior distribution f"p(pM.) is given by I
f"p(pM.) = N;L(pg /n,rg,) f'p(pg,)
(3-8) 4 i
i i
3-7 TERA CORpORAT!ON
where N; is a normalizing constant. The posterior cistribution on p.
is also -
beta type with parameters 7" cnd (".
in Equation 3-5, the conditional probcbility on the nt,mber of successes, rM.' IS based on pg, and n.
The condition on pM. con be removed using Equation'3-8 and integrati'ng over all the values of pM. s follows:
4 i
(3-9)
P ('M./n) =
R P
R i
0 e
i i
i ff F( M.}
("M.}
( M.- "M.}
rM.
i i
i i
=C
(" )
r (dy) n f (!" ) f( n" M.
M.
M.
i I
8 i
for n integer > 0, rM. integer; 0 s ry, s n, i
4 E
and i g, =
M.*'M.
1 I
i 99 M.
- 7 M.*"
1 I
H g, + ( M. * "M.
r I
1 i
ff
"M.*d.
M 1
I The cbove expression is the distribution on the number of ecrthquckes of magnitude M; given that n earthquakes have occurred.
There is a similar distribution for each M; considered.
Marginal Distribution on the Number of Mconitudes The distribution of the number of events of each mognitude independent of the number of trials, is obtained from Equations 3-4 and 3-9, thus j
3-8 itRA CORPORATIC'N
eo P I'M ")P (n)
(3-10)
P (' M.)
I R
N R
i n=0 r( n" )
i
'M.
M.
= {
C ' r( g " ) r( n"
- (" )
n n=0 M;
M; M;
n y, + t,) r(n + 73,-ry, - ! y,)
r(r y
i i
i i
i
=
n (n + 7M.)
1 r(n + e ")t" \\" u n! P( v ") (t + \\ ")"
This distribution describes tcrolly the seismicity of the source considered in terms of the two porometers mcgnitude (M;) and number of occurrences (n).
The Bernoulli model has the advantage that the probability of occurrence of cn ecrthqucke of any given mcgnitude (pg,) con be established and updated independently of other magnitudes. It also offers greater flexibility in the use of historical seismicity dato and in combining it with subjective information through a Boyesion approach.
3.2 FAULT RUPTURE PARAMETERS The three fault rupture parameters required in the development of the hazard model are fault rupture length (L), fault rupture dispiccement !D), and fault rupture radius (R). Models used to estimate these parameters from earthquake magnitude (M) were developed from regression onclyses using the method of least squares. A statistical summary of these onclyses is presented in Table 3-I, cnd tabulated values of the percmeters cre provided in Tcble 3-2.
i 1
3-9 TERA CORPORATION
1 ABLE 3-1
SUMMARY
OF REGilESSIOt t ANALYSES FOlt FAULl Li ilGIlf, FAULT DISPLACEMENI AtID SOUltCli RADIUS Ln Y = AyBy M Coef ficients Ston<l rd Correlation A
il Pororneier Y
Y Er or Coefficient H
i Foult Length, L(kin)
-4.670 1.185 0.83 0.76 73 i
Foult Displacernent, D(cm)
-3.797 1.273 0.1M 0.76 73 Source Radius, R(ken)
-3.391 0.843 0.63 0.73 163 y
E I
t I
l O
-O m
O Z
4 l
l
TABLE 3-2 T ABULATED VALUES OF FAULT LENGTH, FAULT DISPLACEucNT, AND SOURCE RADIUS Magnitude Fault Length Source Radius Fault Dispiccement (M )
L(km)
R(km)
D(em)
L 3.5 0.6 0.6 1.9 4.0 1.1 l.0 3.7 4.5 2.0 1.5 6.9 5.0
.6 2.3 13.0 5.5 6.3 3.5 25.0 6.0 12.0 5.3 47.0 6.5 21.0 8.0 88.0 7.0 38.0 12.0 166.0 3-11 TERA CORPCRAI!CN
i 1
1 i
i Fcult Lengtn (L) i Slemmons (1977) has recently tabulcted fault length onc cisplacement date for 87 worldwide eqrthquakes occurring since 1819. He used these dato to develop relationships between mcgnitude and fault length, where fault length was the l
independent variable. His equations are thus valid for estimating magnitude j
from fault lengtr..
1 i
In our model, what is required is on estimate of length when mcgnitude is known.
I Therefore, the regression cnolyses were repeated using M cs the independent variable. The limited amount of dato availcble on normal-colique slip faults precluded their exclusive use in the onelysis. Therefore, it was decided to use dato from cil fault types in the regression.
i Based en the 73 observations for 4.0 s M s 8.7 in Slemmons (1977), the log-lineer relationship between L c, A was found to be Ln L(M) = 4.67 + 1.19 M (3-1I) l for fault length in kilometers.
i
~
Fault Disolacement (D)
A similar enclysis was r ;n for maximum fault disp!ccement. Agcin using the Slemmons (1977) dato for di fcult types, the log-linear relationship between D and M was determined as I
Ln D(M) = -3.80 + 1.27 M (3-12) for displacement in centimeters.
i f
3-12 TERA CORPCRAiiON
Source Radius (R)
Source rcdius is a measure of the true rupture dimensions, not just those observed on the surface of ter the earthquake. It requires that theoretical source spectrum shapes be fitted to observed spectra, and thct the source pcrometers be estimated from theoretical dislocation models. Source radius thus represents the radius of a circular rupture surface whose crea is equivalent to that of the cotual rupture surface.
From Brune's (1970) dislocation model, source radius may be computed from the relationship R = f-(3-13) where J is the sheor-wave velocity of the medium, and f is comer frequency g
(the point where the high frequency decay begins on the source displacement spectrum).
Using this model, Thatcher cnd Hanks (1973) estimated the source radius of many Southern Californio earthquakes. Ccmpbell (1977) expandea this set to incluce 163 earthquakes of 3.0 s M 5 6.8.
Using these data, the following log-linecr g
relaticnship between R (in kilometers) and M was established:
Ln R(M) = -3.39 + 0.84 M (3-14) 3.3 FAULT RUPTURE HAZARD MODEL For en ecrthquake of given magnitude and location, the probability of observing a displacement greater then a particular level at a point on the fault located at a distance, x;, is comprised of a joint probability of three events. The first is the probcbility that surface rupture occurs, which we will designate os " event E."
3 The second is the probcbility that surface rupture extends at least cs for os the point of interest (designated as " event E g"). The last is the probcbility that the displacement exceeds the specified value (designated as " event E ")*
d I
3-13 TERA CORPORAilCN l
i i
f 1
Mathemcticclly, this joint probcbility may be expressed as i
P(D >d l M;, x;) =
El'EI P(Ed' s
I I
where M; is the magnitude under consideration cnd x; is the distence clong the fault from segment i (the location of the ecrthquake) to the point in question (Figure 3-1).
Simplif ying, by mecns of conditional probability theory,
= P(E l EgE )
- P(EjnE )
P(D > d l M;, x;)
d 3
s j
= P(E I E
f'E
I
( ~IN d
/
s s
s i
with o!! events being contingent upon the occurrence of a given ecrthqvcke.
After developing models for the cbove probcbilities and unconditionalizing with a
respect to M; cnd x;, Equation 3-15 may then represent the hazard from cli possible earthquakes on the fault.
P(E_)~
J Surface rupture during o given ecrthquake is assumed to occur if the source radius (R) is greater than the distance along the fault plane from the center of the rupture to the ground surface, designated as "w;" (Figure 3-1). Since, for a given ecrthquake, the horizontal extent of faulting tends to be greater than the vertical extent, this definition of surface rupture is considered conservctive.
For o given mcgnitude M;, we may compute source radius from Equation (3-14)
M (3-16)
Ln Rj = AR*ER j Because of uncertainty in this expression, Rj may be considered a random l
variable. Let us assume Rj to be lognormally distributed with a medicn of Ln RJ l-onc a standard deviation a i
R ( n Ln R) equal to the standard error of estimatc,.
j.
TERA CORPORAilCN
i I
1 F AULT TRACE SUP.F ACE SITE V,'
RUCTURE
.o N\\
DO
\\
V1
,m
.f\\ \\ '
AULTo_Ar:E
/Er:Ti I
E CIRCUL AR RUPTURE I
SUPFACE i
y -
I "O
l z
l v
FIGURE 3-1
-SCHEMATIC CF THE FAULT RUPTURE HAZAPD MODEL 3-15 TERA CORPORATCN
1-1 f
1 Table 3-1.
Then, the probcbility that Rj is greater then w; (the event E )
j s
s given by i
P(E )
P(Rj > w;)
=
3 l
l - P(Rj s w.)
=
Ln w;
~
I-I v - Ln Ri Exp
-l/2
'RM2' R /
l
/ Ln w. - Ln Rj h (3-17)
I - c l\\
=
'a
/
j where ? (*) represents the stondcrd-normal cumulative distribution function.
P(E[z E) 1
-t-Given that a surface rupture occurs, let it be assumed that its length is equally distributed in both directions along the fcult trcce (i.e., bilateral rupture). The of this rupture is directly cbove the assumed point of initiation midpoint (Figure 3-1). For rupture to occur at the site of interest, it must proceed at least cs for as that distance from the midpoint of the surface expression to the site, x..
Therefore, rupture occurs at the site if L./2 > X., where L. is the J
i j
i
- surface rupture length associated with a magnitude M. event.
J i
L. is computed from the data in Table 3-2, thus i
j Ln L; = Ag + a M; (3-18) g i
l To account for uncertainty in the regression, L. may be considered lognormally j
j distributed, with median Ln L; and standard deviation (on Ln L) of a L*
3-16 F
TERA CORPORAiiCN
4
.s 1
I The probooility of the event Eg occurring (given E ), becomes r
P(E l E )
P(L;/2 > x;)
=
j r
I P(L. > 2x;)
=
J i
l - P(L. s 2x.)
=
J 8
( Ln L - 2xi \\
(3-19) j I
= l - 41
(
L
/
P(E l E, n Ej g
1
)
Given that st'rfcce rupture occurs and that it proceeds at least as for os the site, the surfcce displacement associated with the ecrthquake M. mcy be estimated J
from Equation 3-12 as follows:
(-
Ln D; = AD*OD j Accounting for uncertainty in this estimate, D. is considered legnormally J
distributed, with median Ln D; and standard deviation (on Ln D) of The D.
probcoility that the displacement exceeds some specified value d then becomes P(E E O E) = P(D. > d) d f
s j
l - P(D.6 d)
=
)
/ Ln D. - Ln d h l-+l I
(3-21)
=
(
'D
/
Disolacement Hazard The surface displacement hazard for a point on a fault cssocicted with on ecrthqvcke on segment i of mcgnitude M.
is obtained by substituting J
Equations 3-17,3-19 and 3-21 into Equation 3-15.
3-17 T~RA CORPCRATiCN
A 1
LnD. - Lnd\\
LnL. - Lnx.
'[
P(D > d M., x.)
1 -o 1 -o
=
O L
[LnR. - Ln w. (_
~
I l-o:
k 1
The total hazard at the point in question requires combining the hazorcs associated with all possible earthquakes hypothesized to occur on the fault. For this purpose, the two-dimensioncl fault plcne of crea A is divided into cn equal o
i number of segments of equal seismicity.
t Contribution of One Seament The contribution to disp!ccement greater than or eqvci to d; of all events M; occurring on the some segment is computed as:
P(D > d;) = pP(t4;) +
1 - (1 - p)
P(2ta;)....
1 - (1 - p)"
P(nM.)
(3-22) i
+
J 1
i where P(D > d;)-
probability of obtaining displacement greater
=
than or equal to d; at least once i
P(kM.)
probability of k occurrences of event M. with
=
j j
k = 1, 2... n l
l P(D 2 d;/M;), probability of obtaining a dis-p
=
l placement grecter or equal to c; given on event M..
J
+
i 3-18 l
_ TERA CCRPCRATION l
l-
=
L 4
j Setting a = 1 - p, the cbove expression con be rewritten:
n k
P(kM;)
(3-23)
P(D 5 d;) = P(no M;).
q k=1 with 'n chosen so that q P(nM.) con be neglected.
l 1
I The chove discussion assumes independence cmong events. Hence, the contribu-tion of all possible events can be combined cs follows:
P(D > d.)
=1 U
(3-24) i - P(D > d.)
all
' M.
one segment l'
M.
J
]
l The whole rcnge of magnitudes is included, from the largest one down to tne smallest one that generates a noticecble effect at the site (M; 2 Mmin sa function of distence). This eliminates the consideration of a large number of events.
Contribution of One or Several Sources Because the events cre assumed to be independent from segment to segment, the contribution of each segment of a source is combined as in Equation 3-24.
I P(D > d;)
=1-n 1 - P(D > d;)
(3-25) one source cll one segments segment -
t
.i When several sources are considered, the same principle is cpplied for each source. Thus, P(D > d.) = 1-G 1 - P(D > d.)
(3-25) i I
all one sources
- source, a
3-19 TERA CORPORATICN 4
. ~.
J l
q This expression gives the probability that, at least once during tne period of interest, d; will be exceeded.
3.4 QUALITATIVE ANALYSIS OF THE B-2 cnd B-l/B-3 SHEARS 1
The model described cbove permits analysis of a specific fault at a particular i
location, and therefore, relative to the Vallecitos Nuclear Center, permits en ossessment of the known shears on the site. We have excercized this mocel using the seismicity model presented in Section 2.0 to determine the return period of various magnitudes of surfcce displacement..The results con be qualitatively compared with the data from the trenches. As con be seen from Figure 3-2, tne model predicts ruptures of one meter with a return period of roughly 19,000 years.
Age dcting soils in both B-2 and B-l/B-3 trenches indicate that one meter displacements have occurred within the Icst 20,000 years.
We ccknowledge the uncertainties associated with these cge dates and therefore, conclude that this agreement does not validate the model, but instead indepen-dently supports the model.
In the next section, we extend this model to predict rupture displacements between the shecrs, cnd particularly at the GETR.
3.5 EXTENSION OF THE FAULT HAZARD MODEL TO GETR As indicated previously, the model described above is designed for on essessment of a specitic fcult. This section describes how we have extended the model for i
opplication to the GETR site to account for the obsence of surface faulting between the existing shears.
To cpply this model to the problem of cciculating the probability of an offset under the GETR, we used the following opproach:
l 3-20 4
TERA CORPCRATiCN
a l
1.000,0C0, l
i i
l 6
i I
t i
(
i l
l/
i 5
l l
l i
l i
a t
100,000 1
1 s
I i
/
/
t c
)
i
/
i G
i i
i 4
{
r I
i D
I i
i g
l
+
10.000 r
1 I
.=.,
6/
I 3
/,j
/ :
l 1
I i
i l
l 1
I,000-1 I
I I
{
i I
+
notation I
I (c.b)
I
)
c : sliprote b = mcximum magnitude a
i i
i l
1 100 0
50 100 150 200 250 DISPLACEMENT (CM)
. FIGURE 3-2
.1 PREDICTED RUPTURE DISPLACEMENT V.S.
RETURN PERIOD FOR THE SITE SHEARS TERA CORPCRAT.CN
~
.,-m..
,w,-.
,e,
,--,.--a
(1)
Assume initially that the entire creo between shecrs B-l/B-3 cnd B-2 hcs the some leve! of seismic cetivity as the Verona Fault.
(2)
Calculate, using the model given in this report, the mecn rate of offset occurrence between the shears, given this assumption.
(3)
Assume that offsets between the shears form o Poisson process (with parameter \\).
(4)
Tcke the value obtained in (2) to be the mean of a gamma prior distribution (with parameters \\' and e') for \\, and let the standard deviction of this distribution ( a) be o constant times the mean. (This constant will be vcried.)
(5)
Assume that there have been 0 offsets over some period of geologic history and obtain the resulting posterior distribution for \\ and its mean.
(6)
Calculate the ratio of the mean vclue obtained in (2) and the mecn value calculated in (5) to obtcin the factor by which knowledge of 0 offsets in the given period de-creases the mean rote of offsets between the shears.
(Coll this the " reduction factor".)
l (7)
Divide the mecn rate of earthquake occurrence on the Verono Fault (as obtained in Section 2.0) by the reduction fccior to obtain the mean rate of earthqvcke occurrence in the crea between the existing shears (taking into account knowledge of 0 offsets in the given period).
(8)
Use this value, and the model described in this report to calculate the probability of having an offset between the shears with a displacement greater than various magni-tudes.
l (9)
Multiply this probability by.058 (which is the probability
~
i of an offset under GETR given an offset between the l
shears, assuming a uniform spatial distribution of offsets RB/BS n the Benjamin Report)
- between the shears, i.e., P i
to get the probability of fMe occurrence of offsets of various magnitudes under GETR.
l l
The justification of (7) is that the mean rate of offsets is directly proportional to the mecn rate of earthquakes, and thus, o reduction in the mean rate of offsets l-by a given factor implies a corresponding reduction in the mean rate of l
earthquakes. Thus, we have bypassed our lack of knowledge about the occur-i 3-22 TERACORPCRATION
rence of ecrthquakes on faults which may exist under the crea in between the existing shears by using our knowledge about offsets in the crec (i.e., O in a certain geologic period) and the prcportional relationship between the mecn rate of offsets and of ecrthqvckes to obtcin cn estimate of the mecn rate of earthquckes in the creo. This then cllows us to calculate, using the model given in this report, the probcbility of the occurrence of offset between the shears.
(Note that assuming initially that the crea between the shecrs has the same level of seismic activity as the Verona Fault con reasonably be judged to be conservative since one would not normally expect en orbitrary crec, even one between two known shears, to be os seismically active os a known fault.)
(1) and (2) yieid a (prior) estimate of the mecn rate of offset of I = 2.754 x 10-3. (The model actually eclculated this to be the yearly probcbility of getting on offset of greater than.005 cm, but this differs insignificantly from the probability of getting cn offset.)
(3) and (4) imply, using the formulos for the mecn cnd standard deviction of c gamma distribution. that I=
\\'
- and, a=keA=
where k is some constant.
Solving for v'ond \\' yields
=( ; )2
(;
=
I i
1
\\, -
a2-k I
3-23 TERACORPORATION
a i
For (5), tne posterior distribution for \\ is also a gamma cistributien with 1
i parcmeters l
v"= v' +0 J
(" = A' + 40,000 4
Thus the mean of the posterior distribution is given by
}
v
=
a
~~. 40,000 l
y, e
W (T}I 2
l
. 40,000
\\
2 i
I. 40,000 ~. k i
For (6), the reduction factor, RF(k), is given by 4
\\
i RF(k) = \\ f
\\
2
\\ l + 40,000i' k
/
2
= 1 + 40,000I = k 2
= 1 + 110.16 k i
i For (7) cnd (8), the mean rate of ecrthqucke occurrence on the Verona Fault wcs calculated to be.185.
Tcble 3-3 below presents the probability of the occurrence offsets of various mcgnitudes under GETR, assuming that there have been no offsets in the Icst 1
40,000 years, while Table 3-4 similarily presents the probabilities assuming no l
offsets in the Icst 128,000 years.
1 l
3-24
- TERA CORPORATION l
T ABLE 3-3 PRCBABILITY OF OFFSETS UNDER CETR (WITH 0 OFFSETS IN 40,000 YEARS)
Probc. ility Probcbility Probability of of of K
Offset Offset im Offset 2.5m,
.I 7.6 x 10-5 1.4 x 10-6 1.0 x 10-7
.25 2.0 x 10-5 3,8 x 10-7 2.8 x 10-8
.5 5.6 x 10-6 1.0 x 10-7 7.7 x 10-9
.75 2.5 x 10-0 4.7 x 10-8 3.5 x 10-1.0 1.4 x 10-6 2.7 x 10-8 2.0 x 10-9 TER/ CORPORATICN
TABLE 3-4 PROBABILITY OF OFFSETS UNDER GETR (WITH 0 OFFSETS IN 128,000 YE AF,5)
Probability Probability Probability of of of K
Offset Offset im Offset 2.5m
.I 3.5 x 10-5 6.6 x 10-7 4.8 x 10-8
.25 6.9 x 10-6 1.3 x 10-7 9.5 x 10-9
.5 1.9 x 10-6 3.3 x 10-8 2.5 x 10~
.75 8.0 x 10-7 1.5 x 10-8 1.1 x 10-9 I.0 4.5 x 10-7 8.4 x 10-9 6.2 x 10-10 3-26 TERA CORDCRAT!CN
4.0 REFERENCES
Anderson, J. G. (1979). "Estimcting the Seismicity from Geological Structure for Seismic Risk Studies," Bull. Seism. Soc. Am., 69(1):135-158.
Bonillo, M. G.,
and Jane M. Buchanon (1970).
" Interim Report Worldwice Historic Surface Faulting," USGS Open-File Report.
Bolt, B. A. and R. D. Miller (1975).
Catalogue of Earthquakes in Northern California and Adjoining Arecs, January 1,1910 to December 31, 1972, Seismographic Stations, Univ. of California, Berkeley, California, I-567.
Brune, J. N. (1968). " Seismic Moment, Seismicity and Slip Rate Along Mcjor Fault Znnes," J. GeochYs. Res., 73:777-784.
Campbell, K. W. (1977). "The Use of Seismotectonics in the Bayesion Estimation of Seismic Risk," School of Engineering and Applied Science, University of California, Los Angeles, UCLA-ENG-7744.
Cornell, C. A. (1971). "Probabilistic Anclysis of Damage to Structures Under Seismic Locds," Dvncmic Waves in Civil Encineerina, edited by D. A.
Howells et al., John Wiley and Sons, Loncon.
Earth Sciences Associates (1978). Geologic Evoluctions of GETR Structural Design Criteria, Reports I: Effects of Earthquake Source and Trcnsmission Path Geology on Strong Ground Motion at GETR Site, Report 2: Ground Motion and Displacement on a Hypothetical Verona Fcult, Report 3:
Geologic and Seismologic Pcrameters of Near Field Strong Motion Records, for General Electric Company.
Vallecitos Nuclear Center, Pleosanton, California.
(1978 cnd 1979). Geologic investigation, General Electric Test Reactor Site, Vallecitos, California.
Prepared for General Electric Company, Phase I and 11.
1 Gcrdner, J. F., and L. Knopoff (1974).
"Is the Sequence of Ecrthquakes in Southern California, with Af tershocks Removed, Poissonion?," Bull. Seism.
Soc. Am., 64:1363-1367.
I Greensfelder, R. W.
(1972).
" Maximum Credible Rock Accelerations in Californic: Map Sheet 23," California Division of Mines and Geology.
Herd,.D. H. (1977). Geologic Map of the Las Positas, Greenville end Verona Foults, Eastern Alcmeda Co., Cclifornia. USGS Open File Report 77-689.
Molnar, P. (1979). "Ecrthquake Recurrence Intervals and Plate Tectonics," Bull.
1 Seism. Soc. Am., 69(l):115-133.
l l
I 1
h 4-1 TERA CORPCRATION l
Raif fo, H., and R. Schcifer (1962).
Acolied Statisticci Decision Thearv.
Mcsscenusetts institute of Technology Press, Ccmoricge.
Slemmons, D. B. (1977). "Fcults and Ecrthqucke Mcgnitude," in State-of-the Art for Assessire; Earthaucke Hazcrds in the United States.
U.S.
Army Engineers Waterways :ixperiment Station, vicksourg, iviiss., Misc. Paper S-73-1,Rept.6.
- Thatcher, W.,
and T. C. Hanks (1973).
" Source Parameters of Southern California Earthquckes," J. Geochvs. Res., 78:3547-8576.
4-2 TERA CCPRCRAiiCN
APPENDlX A REVIEW OF THE EDAC REPORT "Probcbility Anclysis of Surface Rupture Beneath Recctor Building, GETR" Reviewed by Dr. Gregory Bcecr.er TERA CCRPCRATiCN
]a L J MEMORANDUM TO Larry Wight DATE-January 24, 1980 Rm Gregory Baecher CORES TO project 4011 SAECT EDAC, INC., Report, " Probability Analysis of Surface Rupture Beneath Reactor Buildiag, General Electric Test Reactor (GETR)."
This review concentrates on methodological aspects of the proposed proba-bilistic model and the numerical results to which the model leads.
No attemot is made here to assess the geological data upon which the analysis rests, or to questien associated estimates of geclogical age. The conclu-sicn of this review is t* tat the probabilistic nodel is flawed, and that tne probabilities of surf ace rupturing estimated from it are difficult to su:::rt.
Gec1ccical Data Ary probabilistic nodeling of seismological or geological phenomer.c nust be cased firmly on the geological record.
In the present case the data include oeservations of offsets on presumably three major shears measured in trenche at the site. These are reported in Table Al of the EDAC re-port, and (omprise 18 locations and the inferred motion on each during four (4) g(ological time periods, dating to 128,000 to 197,000 years bp.
As noted in the report, the field evidence is for total offset during a presumed time period, and not of numbers of discrete events leading to offset.
An important field observation for the latter probabilistic modeling is that all detected offsets in the past 128,000 to 195,000 years have occurred on one of three major shears (B2, Bl/3, H).
Presumably there would be other major shears at the site were this not.the case. The sediments upon which the 4
GETR is founded are estimated to be at least 195,000 years old, and probably older.
In essence, these are the field data upon which the modeling rests.
The model does not introduce mechanical concepts or other first principles to enhance the data.
January 24, 19B0 MEC To:
Larry Wight From: Gregory Baecher Page 2 "Bavesian" Model The basic probability model is straight-forward and easy to interpret. This is a considerable advantage.
It decomposes the event that surface rupturing occurs beneath the reactor building into two parts:
the occurrance of sur-face offsets anywhere between the major shears, and the conditional occurance of an offset under the reactor building given it occurs somewhere between the major shears. The data are then used to estimate the probability of the first of these (P ), and intuitive reasoning combined with sensitivity studies 3
are v;ed to estimate the probability of the second (P )*
2 The main objection concerns the estimate of P. Tne occurrance of offsets l
between the major shears is itself decomposed into temporal and spatial event occurrance models. The temocral model is simply a Poisson process in which the probability of more than one offset in At = 1.0 year is assumed negligible.
This is as supportable as any other similar assumption, and given the data base, probably as good an assumption as can be made.
Furthermore, it is a widely used model in seismic hazard analysis, even if certain workers in the field object to it.
The spatial model is a simple Bernoulli process with some fixed (conditional) probability of an offset occurring betweer, the major.
shears, and the complimentary probability of it occurring)on one of the major shears. Thus, the concosed model becomes P1 = $1 exp(- A, where $
is the Bernoulli probability and A the Poisson density (year -1).
To statistically estimate $ and A, a historical record of ever.ts rather than e:ca!ctive offse: is necessary.
This, is not available in the data. Therefore, some assumption must be made. The authors circumvent this problem by taking advantage of a most peculiar mthematicc! property of the model: Neither the number of historical events nor their total offset influences the statisti-cal estimate of P.l Compared with the imoortance of this result, procedural questions like the choice of prior on (' A,& ) whether uniform or non-infomative distributions better reflect ignorance, etc., seem trivial and are just as well ignored.
The surprising conclusion results from an oddity of the decomposition of P),
and not simply from an error as might have been thought.
The likelihood function of ( A, & ) is given in the report by equation 5.7.
Simplifying only a little (by considering history as continuous rather than divided into four periods),
A" t (),
)n L ( A,$l data)
=
e where n is the total number of offsets (i.e., events) in time t.
Uncertainties on A and 4 are independent in the likelihood, thus if they are independent
January 24, 1980
)
MEMO To: Larry Wight From: Gregory Baecher Pa;e 3 in the prior they are also independent in the posterior. Taking a unifor-prior (tne argument is identical for otner independent priors),
-nt f'
( Al data) =A" e
(1-4)"
f'
( &l data)
=
Looking at the " predictive" probability for P) (called E) in the report),
F) = /g N, (1-n)" t d t f"
N A"e -nt(le- )dA y
(n+1) / (n+2)t
=
-I
= t Where N and N are normalizing constants.
y Thus, for moderately large n, only the temporal length of record influences P) (and thus hazard to the reactor building).
Two interesting -- and clearly incorrect -- consequences would be the following:
- 1) If in addition to the period of record of, say, 200,000 years one added an hypothetical earlier 200,000 years during which no offsets occurred, F) would be halved.... even though nothing changed in thq geology.
- 2) For two sites, one of which experienced an offset event each year over the past 200,000 years, the other of which experienced only one each 10,000 years, F) would be the same.
While this answer is correct mathematically, it makes no sense. What is happening is that, for given t, higher n leads to higher Poisson frequencies ( A) but correspondingly lower Bernoulli probabilities ( 4). The probability of an event in the coming year goes up as (n+1), but the probability that it will be between rather than on a major shear goes down as (n+2)-l.
Thus, the inferrences cancel. One can make the probability of surface rupture as small as one wants; it has nothing to do with the field data.
" Alternate Probability Analysis" This analysis concentrates on estimates of 4, and in principle is not different from that above. Conceptually the argument is, in 128,000 to 195,000 years no offsets have occurred between (rather than on) the major shears. Assuming each year to be an independent Bernoulli trial, the probability of such an offset next year is something like 1/t.
January 24, 1930 Y.EH3 To: Larry Wight From: Gregory Baecher Page 4 This is a familiar problem in the analysis of rare events, and not solved to almost anyone's satisfaction. The authors allude to one-sided chi-squared distributiors and other statistical concepts, but could describe their procedure in more specific detail. Given zero occurrences in k trials (here t),
in any future trial the maximum likelihood estimation of the probability l data) as a probability is 5 = o, not very useful result. Renomalizing ' (p distribution, the expected value of p is E(p) = (k+2)-I, a result much like the naive estimate 1/k. These are back of the envelope calculations anc difficult to cefend very far.
Alternative probabilistic Model In the nature of speculation, one might propose an alternative model for tne occurrance of surface offsets. From the data themselves one has no way of knowing the nuccer of individual events leacing to the accumlative offsets.
However, one does know that those offsets have occurred on at least three
^
shears (i.e., 3-2, B-1/3, and H).
One mignt assume then, that instead of seeing many offsets one is seeing only three, but in different spatial loca-tions.
So one has seen three shears in about 150,000 years in different locations. Therefore, the probabilitj of a new shear feming is something on the order of 1 in 50,000 per year. Making the assumption that these shears fom, say, subparallel to the existing shears and within a band about 2,500 feet wide, the probability of a new surface offset intersecting tne reactor building can be calculated as per equation 5-14 (page 5-8) of the original report. Assuming the width of the reactor building to be 72 feet and the width of the offset at the ground surface to be four feet, the annual probability becomes something like 6 x 10-7.
3 shears 72 ft + 4 ft
= 6 x 10-7 per yr.
150,000 years 2500 ft + 4 ft One wouldn't necessarily cnoose this number in preference to others; it is only derived to show that there are many ways to interpret the data and to build a probabilistic model. The question is very much conceptual, not mathematical; and it is on this conceptual level that the regulatory issue must be resolved.
l Conclusion Essentially the issue resolves to this:
in 128,000 to 195,000 years no surface offsets have been observed between rather than on the major shears.
Given this observation, how does one estimate the probability of an offset occurring between the major shears? The problem is conceptual and not l
mathematical.
The EDAC report essentially reasons that when no events have been obsarved in k trials, the probability on the next trial is 1/k.
The mathematica: model proposed in the report to some extent merely confuses j
the issue.
APPENDIX B REVIEW OF THE BENJAMIN AND ASSOCIATES REPORT
" Additional Probability Analyses of Surface Rupture Offset Beneath Recctor Building, GETR" Reviewed by Dr. Gregory Sce:her TERACORPCRATICN
Gregory 3. Baecher c/o von 31s=arck Elektrastrasse 73 50C0 Munich 51 7 April 1950 Mr. Lawrence Wight TERA Corporation 2150 Shattuck Avenue 3erkeley, CA 94707 t'.o.a.
a a
v
Dear Larry:
As you asked, I have reviewed the March 12, 1950 report by Jack R. Benjamin and Associates titled. " Additional Probability Analysis of Surface Rupture Offset Beneath Reactor Building GETR," and herewith effer =y co==ents.
These are divided into three parts:
general co= cents on the conceptual =cdel, para =eter esti=ation, and =iner =ethodol:gical suggestions.
As in =y review Of the earlier EDAO repert (Reference 1), I find the present analysis a ce=petent atte=pt to deal with a difficult problem.
There are very few local data available and no established
=: del within which to interpret those that are.
An atte=pt has been
=ade :: structure a reasonable =odel and deduce hazard fro = the data.
D1 spite this, my criticism of the results of the April 12, 1979 report applies to the present as well.
A co= plicated model has been deve10 ped, but there are too few data to estimate its parameters.
Therefore values must be chosen by arbitrary criteria, and what we are left with is precisely an argument of for=,"no offsets have been observed in t years, therefore the probability next year is se=ething like 1/t."
There is nothing i==ediately wrong with this simple argument, for at least its logic and li=itations are clear.
I fear that in the bDAC and J.R.3enja=in and Associcates reports too much is being =ade of the analysis behind a result actually of this type, and that the 33 pages of mathe=atics obfuscate this fact.
This is different from saying that I find the results
" wrong," they are si= ply difficult to defend.
Short of collecting more site specific or collateral data it is not clear what else can be done.
MODEL As I understand the model equation 2-1, the probability of surface rupture is divided into two parts. One part is the contribution by new or undetected shears; the other is the contribution by shears 3-1/3-3 and 3-2.
In the first part, the ter= Pgg see=s to =e to be I
r t
i 1
W:ght/2.
a du==y variable.
Its value is set as 1.0. and it has no effect 7
the on the analysis. The real actor in the first par is P
/C..,
prebability of new or undiscovered shearing between the ex cting and discovered shears.
This is taken as a Bernculli process probability and estimated frc the datur that nc effsets have been observed in t years.
How exactly to get a number cut of this datur is problematic, but a widely kncwn questien.
The "n Ive" estimator wculd be 1/t".
The =az =u likelihood esti= ate is obvicusly
- ero.
A 3ayesian distribution could be cbtained by assuming a diffuse prior and rencr:alizing the likelihood (1-p)t.
The authors have chosen equation 2-2, which depending on C can vary frc= cero te one.
The cr:terien for chcosing C is no: discussed--or at least I may have m'.ssei it--in the report.
See the ce==ents under section 3 below on 3ayesian analysis.
I would suggest that, nc matter what estimator we use, conceptually there is no such difference; they all reduce to something like 1/t".
The differences I take no issue with the esti= ate of P-,/iner, type distribution nz
-a between using uniform, nor=al, beta, or any c between the shears are =incr.
Therefore the probability of a new cr undetected chear leadine to surface rupture beneath the reactor building is driven by the time since the last displacement and nothing else. As the authors note, this is the same analysis as in the 1979 report and my opinion of its strengths and weaknesses is unchanged frc: cy earlier review.
Ecwever, the authors have greatly simplified the mathematical logic leading to this result, and I would support this approach as perhaps all we can do.
I should also say that one of my central objections to the 1979 results has been changed.
In the present results P U
3S/O" is estimated from the ti=e since the last offset (or period of no known offset), rather than from the time during..which the known effsets occurred and continuing to the present.
The second part of the analysis leaves me =cre troubled, and given the data base I am not sure that it is necessary.
It seems dcuble counting to me.
If we assume that there is only one geological process taking place, then the probability of part one describes the ha:ard perhaps as well as we can.
Part two introduces a second geological process leading to a probability of also something like the reciprocal of some time since the last event. Continuing the logic, why couldn't one hypothesize n processes--none of which have been observed in t years--and arrive at a hazard of n/t? Since n is arbitrary, the ha:Ord could be anything between 0 and 1.
l i
t
- u. s-u... n.
n
.i s e.. v.r. v:t r e, r.y.r.,7.0 -
. n n.
The main thing that disturbs me tbout part two is the proble: Of estimating six paracter: fre three data and two relationships at ng them...in other w:rds, with n: re=aining degrees of freed::.
As I see the
- iel the f:llowing cust be estimated in part two:
N nuster of effsets in some time t, r
rate of slip on shears averaged over t, I
average time between effsets, T,
total offset in time t, x
average size o. c:.rsets, c
f standard deviation of times between Offsets.
k To de this there are tree data:
A Sieh's data on the San Andreas move =ents (Reference 5) leading to a ' direct' estimate of (?T, total offset, T, inferred fr:m various site dets, average rate, r7 by assuming T te have cccurred in ti=e t.
And there are als: two relationships:
N = T_/T (4-7) 1 (4-a) x = rt Therefore, sc=ething cust be arbitrarily assumed.
In ANALYSIS APPROACH 1, this is t.
In ANALYSIS APPROACH 2, this is N.
There is no getting out of this trap.
In some ways, as far as the model of part 2 is concerned the way one arbitrarily choosed T and N doesn't matter.
They seem to cancel anyway, which was my major objection to the 1979 results.
If the averace time between offsets is increased, by 4-4 and 4-7 the number of i
offse:c is decreased, and vice versa. Higher T's mean higher P,7 's but correspondingly lower P s, and vice versa.
In the end, the only thing that matters is the /0Nti=e since the onset of faulting, BS which seems intuitively unappealing.
MINOR SUGGESTIONS ON METHODOLOGY The following are short suggestions on individual aspects of the report, and are less important in ny overall impression of the reasonableness of the results than the comments above.
< -u.
- n.,..i First, there seems c:nfusion throughout the report en whether er net to use 3ayesian =ethed: of inference.
By this, I do not mean sub.ie:tive probability, but rather distribution functions on model para =eters rather than poin* estimates.
Given the t ese y.a -
e+.a..
e.e*4
-*es,
- d. '.
k
,.. e-.
s..
4.
4.. 1.....
4..*-
- 4. -
au
.... su...,
is my cpinion thst carrying through that uncertainty with density functions wculd give -a better idea of how the data were being used than poin* esti=ates do.
This is particularly true of the esti=ates of 2-2 and 2-3 which can be arbitrarily varied by changing C.
- Granted, such an approach w:uld not help in dealing with f(:) and therefore the predictive distributien on
- after considering 2-4 since u w.ve.,
.u.e e-.4
...e 4.e.4..,
u..-.
4.n.-
,. v.
w.
.ia w. --....~
m.
.g deceribei :n pages 4-3, 4-4 and en 4-5, 4-5 illustrate what I e...a..... o A o., a,,.,- 4 p.c.<se.
>.e.
4 r,-
..e-.u..---...4.u..4..-
a4
~,;
s..
p(N/T ) using 3 aves' Theore=, a "=axi=u= likelihood" esti= ate of N is ai3pted. This is =azi=um likelih: d only because a flat prior was used, and if one wants a =azi=um liklehood esti= ate whv go through Sayes' Theore= to begin with. Cn the ether hand, if one has the ilstribution f(N/...), why net use it in the =edel se that this part cf the c:stistical uncertainty, at least, is reflected in the e_-..s.
Second, the =edel does not recognize that there are at leas:
- hree shears at the site which may not all have occurred at the same ::=e.
I noted this in my earlier review, and I don't think ths it is i=pertant enough to redevelop here.
Thiri, *he argu=ent at the bottc= of page 3-1 ec= paring the strength of the shears with that of concrete speci= ens seems
=isdirected. The streng*h variaations of geological =aterials depends very =uch on the area or volume tha+ is being averaged over.
Therefore the co=parisen seems invalid.
estimated fr== topographic relief takes Fourth, the =azi=a= T,d erosion.
ne account of weathering an yifth, also on page 3-2, there is no support except supposition that shears have some characteristic offset rather than an exponential distribution. One could hold that the average of an exponential distribution is itself a " Characteristic offset," and there are =any j
geological phenc=ena that are exponentially distributed (e.g., joint spacings).
Sixth, where does the characteristic offset, x, of Table 3-2 c. -. < -...,,
b
"n. y... / :
4
-.p.
s a..
- a..d..- a..=. e c
'*-a,...e.
- a.
S e v e..*g*.., w"v s h. o ".'.'.-... " ".,, ~. e, a.
.'.a*.
...a haVL s '.i'sa-. d 's s - a. -e '....
- o s a.
.' - "e 4.- s u- '. i e k.e s- -?.
y C 0 v.C ' t.M. T.
'..'c o
s In concept, the present results are based on -uch the same codel as these of 1079 Therefere. they cannot te used to indeper.-
d-*
tk..c s e e u- -14. e. -e e..'.*..e.
Gi ve..
- .h. a.
e -.
d.*.~;
^.
d.e n. *. '.v va.. ' ',-
that the present analyses may be as much as one can do, and frc w
... s
. w.....
.w._.
A -.....
o.
,.,4 e y I w..,_ve 14. 1e <.4..+.4..
c
-..,. e. 4. e.
,.4...
w-
- 2
.. e e-p..r. e.
u.
..u..
-a -. s. 4.
.a 2,
. L s.,,... o,, 2 n
.e..
e-
.w..
- 4..
.,..s...
..,. +
4e.
.w,
- v..
w, e e. 4.,.,.
'..c. e a. d. -.- =-
o".d.*."..'.,-
w'.. a... '..'e.+.
'. *. ' e.
..e
.#a y
- d a.
'.,.e.'.e.'..-.
.e..d a.
- k..a*. *. h e ma *.he m * *. i.a s c '.#" s *. e
- a*.b..a..
ve s-s'-,
a.
. -. -.. +., a.. d.
In =aking recommendations based on this clarify the basic logic.
w...- *. '. k. e k.a.e 4 - 1 n' ~ 'a.
..o
. '. "..e e *. e 4...
w k.ea, ia
..'..d.
e.
a e,c.4.e y.
- 1.,/ +...e 7..se.
neVe ne.es,e.
C...
4 1.
1 v
... -... e
... e
- e.,
- 4. w.,o m
+w...e e..we..,.
. ".. e. =.
- 4. s g- ".s *.
- .C
.d.- *. a-w'*.*.'a*...".'.-
0...
4....c.... y-c e...,..
m.. -
eg....,b-w 4 14.. - c e 1.
. 1..
4.
.-..s
- c. ho.%n A. A.e.o
.e....% na.
r ssess..e..
in n.A..
- s. 0 n,'.r.e
- s. %..+...
w w.4 e
,.e d...
0 be u..
... e...
nem.o.
a.-
O.r.se.
e.
e
.e
..l1e+e., in.ec
..,,n
.... o.w..e.
v.
c.
m w.
w o..r..e,, s s e.u.. e c.e d _, +. a-5.u.A.
1
$"..b...-s h ve
...e,*.ed w ' *. ".
w e v
e "s *. a a
<, u '. *. o v a. - e.*. '. a.
- k a.
.. s-4.
D.e #. -a...c e ".
W*.. e*.*..a..-.'"....k.a..-. '.. s' *.. d..-.- - c l'. e c *. d. -..- 3 e a - e..-..'.. -.
s p
s.
w c.
1./s.s seems.....<.....,
4
.w
. 2.n.o sul a....a w
2 3
eae
.o4e.24.g wou.a_
although I am sure there would be proponents for such a course.
0; if If you need clarification of anything I have said here, I :sy 'ce of further assistance, please let me knew, ei..e..,..,ou..,
e..... c J (3
a i
A I
(
LLI Gre cry -3. Stecher
APPENDIX C REVIEW OF TFE BENJAMIN AND ASSOCIATES REPORT
" Additional Probability Analyses of Surface Rupture Off set Beneath Reccter Building, GETR" Reviewed by Mr. Srien Davis TERA CORPORADON
i MEMORANDUM ic L. H. Wignt pqE April 14,1980
- 7CM B. J. Davis CODES TO Review of the 3/14/80 Benjamin and Associates Report:
3;3,ge.
"Acditional Probability Analyses of Surface Ruoture Offset Beneath Reactor Cuilding General Electric Test Reactor" INTRODUCTION This meno oresents an analysis of the accroach taken by the Benjamin anc Associates Report to obtaining an estimate of the probability "of tne occurrence of a future surface rupture offset ceneath :ne (General Electric) reactor building (at the Vallecitos Nuclear Center)".
Section 2.1 of this memo presents the general accroach of.the recort (i.e., a description of the equation which the report uses to express this probability). Section 2.2 discusses several statements in the report whicn seem to go against the approach which is actually taken.
Sections 2.3 through 2.5 oresent analyses of tne various terms of the equation for tnis probability. Section 2.7 presents a summary of the results of this analysis.
4 Section 2.1 General Accroach In the report, the yearly probability of a future surface rupture beneath the reactor building is given by the equation:
2 i
i
.P
+ : P
.P
.P 6.P P=P BSTi RB l25 i=1 ON BS ',0N RB!BS (where the terms are as defined in the report).
I Two different approaches are used to calculate several of these tems.
j I
l These terms are set equal to the following quantities in the report:
Pg=1 i
E Sl D?i = 1 - (1-C) i
+U for Approach el 1-
)
CN = <
=
i (1 - Q(to)0(to + 1)) P(- l t, t, c,N)dt for Approach s2 I
o s
,o
-(-
P BSION i
the mode of the P(N!Tx) distribution for Approach el where N =)
I a fixed (assumed) value for Approach #2' t
i I+b p
RBlBS L-D where C = a " confidence level" i
t* = age of soil beneath reactor building For the existing shears:
l t = time since last offset g
Q(t) = probability that the time between offsets is greater than t I
l t= mean of. the time between offsets a = standard deviation of the time between offsets l
t = time period for offsets s
N = number of offsets (for the past t years) i s
i I = width of the reactor building L = distance between the two shears b = width of an offset at the ground surface l
i l The combined terms P g. PBS jM,PON
- PBS!ON using Approach al, and P;.PBSl0:1 using Approach =2 are discussed in sections 2.3, 2.4, and 2.5, g7 respectively.
P is discussed in section 2.o.
RBlBS Section 2.2 Preliminarv Discussion i
j On page 2-1 of the report it is stateo that "If an offset occurs away from j
the existing shears, it is conservative to assume that it will occur between the two shears", and "it is conservative to assume that a future offset will occur in the area bounded by and including the two existing shears". This is certainly true, but the approach which is given in the report seems to go against this assumotion.
T,u assumption would seem to imply (trivially) that PBSIM, the prob bility of an offset between the existing shears, civen an offset on unknown, undiscovered shears in the region, should be 1 (since an I
offset on an unknown shear in the region is obviously not an offset occurring on tne existing (known) shears, and according to this assumption an offset i
can only occur on these two shears or between them).
However, P g3 g s most i
}
definitely not set equal to 1 in the analysis, and so thi conservative assumption is not consistent with the approach which is actually given in the report.
]
Section 2.3 Analysis of Pg.Pg3 5g In both approaches in the report, Pg is set equal to 1 and PBSl M s set i
equal to 1 - (1 - C) /t*
It is stated that setting Pg equal to 1 is con-1 servative, and at first glance, it seems that doing this could be considered -
to be grossly conservative, since it is very reasonable to assume that Pg is in fact fairly small (at least relative to the extreme value of 1).
However, when one looks at_ the way in which this assumption (that Pg = 1) has been incorporated into the expression for PBSlM, it becomes clear that the i
-w
a
..t-issue is not quite so simple.
The report comes to the conclusion that Pg3;g should equal 1 - (1 - C) /t* by assuming that there have been "zero offset events in t* years" (between the known shears) and that " offsets, which occur due to unknown undiscovered shears, are independent with respect to time and fit a binomial distribution". (See Appendix A for a discussion of the expression 1 - (1 - C) IU*.) But PBSlg s the probability of an offset between the i
existing shears, oiven an offset on unknown undiscovered shears in the recion.
Thus, the number of years during which there have been no offsets between the shears is irrelavant to the calculation of PBSlM -- the only relevant statistic is the number of offsets which have occurred on unknown, undiscovered shears in the region (of which zero have been between the shears). Thus, if, for example, there have been 100 offsets on unknown undiscovered shears in the i
region (with none having occurred between the shears) the correct estimate BS!M s 1 - (1 - C) / 3 0, i
2 whether it has taken 100,1000, or 100,000 -
of P years for these 100 offsets to have occurred. Thus the only scenario which gggg = 1 - (1 - C) /t* is one in which tnere have been justifies setting P t* offsets on unknown undiscovered shears in the region (with none having occurred between the existing shears).
Thus, in addition to assuming P g = 1 in any future year, the report has in effect assumed that P g has. equaled 1 for each of the past t* years -- i.e., that an offset has occurred on an unknown, undiscovered shear in the region every year for the past t* years. This is clearly an extremely unrealistic assumption (considering that the report takes t* to be at least 128,000 years). The effect of this is to greatly decrease the value of PBSig, since if in fact there have only been n(<t*) offsets on unknown shears in the region in the last t* years, then PBSl0iiwouldequal 1 - (1 - C) I"
- I rather than the smaller value 1 - (1 - C) /t* + 1, (see
~
Appendix A for an explanation of why the exponents /n + 1 and 1/t* + 1 are used).
l I
(
i For example, if n=20, t*=128,000 and C=.9, the report would calculate Pg3;g =
2e:c:
1 - (1
.9) f:
1.8 x 10-5, whereas the correct value would be Pg3;g77 =
=
- /::
1.0 x 10-).
And if in fact there are no undiscovered 1 - (1
.9)
=
shears in the region (which is certainly a possibility), PBSlEi would equal BSlDII
- I - ( -
}"3 whatever choice of C was made (i.a., P above example). Thus, while setting Pg = 1 is almost certainly grossly 1
BSl0ii = 1 - (1 - C) /*** is almost certainly grossly conservative, setting P non-conservative (a fact wnicn the report does not mention) and thus further analysis is necessary to determine whether the combined tenn Pg BS!DIT
- P is conservative or non-censervative.
Such an analysis is not obvious since by assuming Pg = 1, the report has avoided stating a probability function to use in computing Pg.
However, the fact that they take Pg = 1 and having no offsets between the known shears j
,in t* years to imply that there have been t' offsets on unknown, undiscovered shears (and not more than t*), suggests that they may view each year as a Bernoulli trial w.r.t. occurrence of an offset on an unknown shear -- i.e.,
l either 0 or 1 offsets can occur on an unknown shear in the region each year.
If this is correct, then Pg would clearly equal ti.e parameter of this Bernoulli distribution.
If we somehow knew that there had been n offsets on unknown shears in the region in the past t* years, then it is easy to sho'<
that the maximum likelihcod estimator (MLE) of this Bernoulli parameter (and g3lg = 1 - (1 - C) I" + I thusofPg)isy..
Also, we would have that P Thus, we see that the approach described in the report is equivalent to taking l
n = t* in the above model.
Thus, assuming this model, the question of whether as described in the report is conservative is equivalent to the Pg*PBSl0N
~ -
~
se n,
. question of whether takino n = t* is conservative compared to taking n < t* in the abm 9 model, i.e., is 1
1
, (1 - (1 - C)"
- I) > {. (1 - (1 - C)" ' l) for n = 0, 1, 2,..., t* - 1
?
Appendix B shows that this inequality is true for n = 0,1,..., t* - 1 and g
BS g as described in the report is, in fact, conservative thus the term P P
in the sense described above.
However, Appendix B also shows a peculiarity of the model being considered -- namely the insensitivity of the probability to n.
For n between 215 and 128,000 there is less than a 1% difference in thevalueofh(1-(1-C)"*I).
For n = 50 there is only a 4.3% difference, and compared to the n = 1 case, the n = 128,000 result is only 3.4 times as large -- not a huge difference considering the difference in scenarios between having an offset on an unknown shear each year for 128,000 years and having one offset in 128,000 years.
Thus, the value given to Pg g3lg (which is
- P the product of a grossly conservative term and a grossly non-conservative term) is more conservative than the product of terms which more and 1 - (1 - C) I", respectively, accurately estimate Pg and PBSlM (I'.
for some n < t*), but only by a slight amount.
Section 2.4 Analysis of P
- PBSION -- Approach #1 ON WithrespecttoPf3 andPfSON(i=1,2),moreofanattemptismadeto and estimate each of these quantities accurately than in the case of Pg BSlM (where only Pg BS!M as a semblence of accuracy -- not Pg and
- P h
P is set equal to 1 - (1 - C) /N where N = the g3lg separately).
BSl0N P
P number 1of offsets on the known shear being considered.
In Approach 1 of c
i the report, to estimate N, estimates of Ty, r, and j are obtained from the i
i
data, and the density of T, giv n a particular value of N is taken to be:
x P(T,jN) = f (W)
I where:
fg(W) = standardizea normal density function T
U"E
~
and x
W=
r c (N N
t (the mean time between offsets on either of the known shears assuming that his time is normally distributed) is taken to be t (which is the MLE of t),
g g (since $ is assumed to be.5).
Thus, using and so a =.5 t t
T - Nrt Approach I in the report, W = 2 x e in f (W) (as above).
g i
rt IN g
The density of N, given a particular value of T,is given by x
T lN)
- P(N)
P(NlT)=
x P(T )
x where P(T )
- i 1
P(Tli)p(i) x x
(note the = upper limit rather than the incorrect N which appears in the report.)
p(i) is taken to be a diffuse distribution -- i.e., we can let p(i) = h for thatP(NlT)=P(TlN)~h-
, P(T lN) for N = 1, 2,..., M.
i = 1
..., M, so x
x x
M M
E P(T !i)*
(x I}
x i=1 1=l ThenwecanletM-=togetP(N!T)=P(TlN) for N = 1, 2,..., =.
x x
,$ P(T li) x 1=1 The N which maximizes P(NlT ) (i.e., the mode of the P(NlT ) distribution) is x
x chosen as the estimate of N to use in PBS ON = 1 - (1 - )
However, since n
n r
I
, l
=
,I P(T li) is a just constant (for a fixed T ), it is obvious that the N which x
1=1 maximizes P(NjT ), also maximizes P(T lti), and vice versa.
But what value of x
x 2
W I
N maximizes P(T lN) =
eT x
Q2:
2 (T ~ N#U) x I
- 2(r II)
=
Q2 2(T - Nrt )2 x
g Nt
'r-g5"e
=
Clearly, the values of N which are close to N = T x
are the only candidates rt g since any other choices make (T - Hrt )2 large and thus P(T lN) small.
x g
x The table below gives the value of P(T lN) for different values of N x
C/
l (assuming the "best estimate" values for t, t, T and r which are used o
x in the report):
I l.
For shears Bl/B3 N
P(T!N) x T = 210 x
100 1.3 x 10-r = 1.34 x 10-4 180
.060 190
.694 194
.964 i
t = 8000 195
.992 g
196 1.000 197
.988 T
l x = 195.9 198
.956 I
rt 210
.150 32 l
300 4.2 x 10'n l
400 3.5 x 10-I max P(T li) =.3989 x
i Table 1:
Relative values of P(T lN) for i = 196 X
(Shears B-1/B-3)
, l l
2.
For shear B2 T = 140 x
100 7.9 x 10-
=.89 x 10-4 180
.046 r
190
.631 t = 8000 195
.975 0
196
.997 197 1.000 T
198
.983 I = 196.6 199
.946
- 'o 210
.182 n
300 1.2 x 10,u 400 1.5 x 10-
- j*P(Tli)=.3984 Table 2:
Relative values of P(T jft) for i = 197 (Shear B-2) r t )2 (P(T ii) =
e-2 (a
-1
'I
)
Q2 :
i (r t )2 g
Thus, we see that the procedure given in Approach I for computing N (a procedure which uses normal distributions, conditional probabilities, prior distributions, and maximum likelihood estimates) essentially boils down
'x to taking fl = r t (or, if fl is required to be an integer, taking N to be o
l T
the integer closest to
).
Note that this is not necessarily unreasonable, rt g since the total ground displacement (T ) divided by the average ground dis-x placement per offset (r t) should approximately equal the total number of offsets (N) (assuming that the values for T
,.r, and t which are used are x
correct). However, it is also true that the complexity of the analysis.in the report obscures the simplicity of what is actually being done.
^
- m-w y_
i i i
- I - (I - }
To analyze PBSiON,wecanwritePBSION I
in (1 - C)
=1-e in (1 - C)
=1-e N
in (1 - C)
Using Taylor's expansion for e N
and simplifying we obtain (1)
PBSj04 "
.4 9*
Since in the report the smallest value N takes on is about 30 (for case 7 in Table 4-1 with T = 52, r = 2.18 x 10-4, and t = 8000), we have that x
g 4
I 1
1 I"l-C 1
(For the "best estimate" case of N = 196,
= I n 1
. 9,TJ
~
30 f4 I
I" (I C)
- In (1 - C)
. ) Thus we are justified in writing PBS ! ON' N
4 1
ro in (1 - C) -- thus we have that PBSIOfi f
is inversely proportional to N, l
f
'x and proportional to tg (approximately).
O(I ) - Q(t + 1), 3. Q(t + 1) wnere Q(t) = P(time P,, is set equal.to o
o o
0" '
Q(t )
Q(t )
g o
between offsets (on whichever known fault is being considered) is at least t).
Thus P is the probability of an offset next year given that it has been t 0f1 g
years since the last offset.
In Approach 1 it is assumed that the time between offsets either has a Weibull distribution (F (t) = 1 - e (
) or a g
Normal distribution -
(f(t) = [3
- (t - t)
EG:
).
e a
4 f
v Assuming the ncrmal distribution t -1 o
x 1
p l
e It - t)-
7 _.
3
- (t - tP I..'
n:
-e dt -
e s.
2 _.
e 2:*
at CN J % 2- -
J gi2 t
t t -1 o
o o
=
1 e- (t - i)2
('. - t)2 2 :-
dt e
dt
\\
7 t
o Tre rer rt uses the maxinum likelihood estimator of t, which is to, and assu-'esf=.5,so:
0/
Suostituting these values in, we CDtain
=
2 1
- -l
! -l
- o o
o f.
..n,-
-2v dV
. ?f2. l ' dt Ct e
I e
e
.g
, 7_
n t
t
.C
_- C
_ o wnere V =;- 6 "n.; _
i.
+o s.
x x
"C'
. ?ti. 1):
2v dv
~
e e
't dt e
d.
o u
s,"
.t t
o 0
o inus. P.,. is a mcnotonica lly decreasing function of t.
Further nore, since n.
n v
-2V I
wnich
.999999 e
-. 1.0 for V: [0,,,0nn] (4000 is the srallest value of t g u
, sa the re; ort considers), and since
= %@r-F 5 -2v 2
, we can write
.o
= '--
e o
1r-
'c -
1 dv
/---
0
_ \\ 8/-
1.6 So P,,is inversely p.g.2 t
c
~
o
'o
'5-v proportional to t (approximately).
g
Assaming the Weibull distribution, ft *I)r fTt it it +1;K o
Q(t +1)
-[u 0(tb P
- I~
- I-I-*
3 ON it;_2 lK 0
-l u ;
e-If t = the mean of the Weibull distribution (the report's use of t is confusing since it is used to represent the mean of both the Weibull and Normal distributions even thougn these quantities are different), then fA
' (1+2v) - 1 and
$ =.5 (tne report's estimate) imply
!9
=
't-
'(1-2/K) that K is appproximately equal to 2 (the report does not state what value of K it used, but tables of garana function values yield this value).
Thus 2t 1
v U
P
=1-e(
u:
) and taking the MLE for u (=t )--as the reocr does--
ON g
we obtain:
2+
1+-)
P
= 1 - e' 'o "o
ON
- (- - +.)
Calculating the Taylor expansion for e "o
"o and simolifying we obtain 2
1 the expression PON " T ~ t, + higher order tems.
Since the lowest value o
o of t which is considered is t = 4000, we see that PON =
a'nd thus that g
g o
P is inversely proportional to t (approximately) (just as in the case ON g
where the normal distribution was assumed).
Thus, using the methodology given in the report, the longer it has been since the last offset, the smaller the probability of an offset in the next year. This is exactly the opposite of what the report desires to obtain; namely, " hazard functions which increase monotonically with time" (i.e., with t ).
g Combining the results given.above for PBS!0N ON, we see that PBSION and P l
increases proportionally, and P decreases proportionally, with t ON g
P should be relatively approximately; thus the combined term PBS!0N ON l
l insensitive to the value of t (and thus N)--a fact wnich the following g
tables bear out.
o
. 1.
Shears Bl/B3
-3 PON (x 10 '
PBS!0N
- PON(x 10 )
BSiON(x 10')
t P
g T = 210 x
r = 1.34 x 10-2000 2.93 10.00 2.934 N = T /r t 4000 5.86 5.00 2.930 U
6000 B.79 3.33 2,925 P
= 1 - (1 - C)'/N 8000 11.69 2.50 2.921 BSl0t1 10000 14.59 2.00 2.91 7 2
1,15000 21.80 1.33 2.906
-IF+t: 120000 28.96 1.00 2.896 P
=1-e ON o
o 40000 57.08
.50 2.854 (C =.9)
Table 3:
Sensitivity of PBS'ON O'l o -
U shears B-l/B-J 2.
Shear 82 3
BS!0N(x 10 )
I =10 t
P p0N(x 10')
PBSION
- P0ti(x 10 )
x g
r =.89 x 10 '
^^^^
2.92 10.00 2.923 N =. /r t i x o
5.84 5.00 2.919 2
0000 8.74 3.33 2.915
- (I- ) /N P
BSl0N B000 11.64 2.50 2.910 10000 14.53 2.00 2.905
- ( 2_+ t')
15000 21.72 1.33 2.896 1
t P
= 1 -e o
o 20000 28.85 1.00 2.885 ON 40000 56.87
.50 2.944 (C =.9)
Table a: Sensitivity of P
- shear B-2 BS !0tl ON o
l 2
The accuracy of the approximation PON 7 is shown by these tables.
The o
approximation PBSl0tl ~ r to in (1 - C )is not quite as good since, for example, I
x for t = 8000, U
1 f i - C) in so that the higher order
>> 8000
=
T /r t
'o g
1 terns of (I" (1 - C)) cannot be neglected as easily as the higher order terms T /r tg
-la-of(h).
Ncnetheless, the difference between Pgg;g Pg g
for t between o
2,000 and 40,000 years is less than 3% for botn snears, and less than 1", for t between 4,000 and 15,000 years (the limits on t which the report uses).
o g
Thus the results of Approach I are almost completely insensitive to what value of t is chosen.
(See Table 6 for a summary of these results.)
o
-- Accroach $2 Section 2.5 Analysis of PgPBSION In Apprcach 2, the only differences from Approach #1 are tnat a value for N is assumed (thus yielding an imediate value for PBS!Ct;). and that Pg is obtained by integrating over all possible values of i, rather than by using the MLE for t.
Using A:proach =2, P
is given by equation (4-9) in the g
I Oft *I) re:crt:
e
'P N t 't N)dt on 1
0(tg)
_r
=
o s
~
where Q(t) = e (i.e., tne Weibull distribution is assumed for tne time between offsets),
p(t, t jC' N' E) PIiIO' N) g s
p(, tit, t, o, N) =
""d g
3 p(t, t3;c, N) o x
P(t, t lc, N) =
p(t, t lc, N, i) p(iic, N) di o
3 g
3 P(t, t ic, N, t).is assumed to be given by g. s I
((t t ) - ;N-1)t)'
t K
't K
3 o o
o 1
l P(t, t !c, N, t) = e (F) f (w) = e ( T) 2(N-1) c
~
o s
'2 -
I l
l(
a Since p(i :. N) is assumed to have a diffuse districution, we see that P(1,,
t., :, N) = C:
P(t, t lc, N, i) 0 2
0 s
P ft'c,N) wnere C1=x
.' P(t,ts:,N,t) P (t.:,N)dt a
0
~
= (.' P ( to,t !:,N,t)dt) ;
3 O
where in this excression is just a du :ny variable of integration.
I* 1
't +1
.t
_. t -t )-(N-l)
(s t
0 1 -
o o
4 o; 1
ON 2h f'e'
- O""..' ='
C ' 1 " e
Iqus P d
=
v 3. v.,-
ic "o',
'f t -t )-(N-1):
-,o."
+
s e m.
=
C., e;u u
+
.e
.u t
4 o..-r.-1).:
o wnere C: -
C.-
Ig2-5 =.5 again implies tnat K = 2, and tnat : =f 1
u f(1 7') implies u =
ct, whare c = 1.127 t
f(l*I)
Inus we have tnat 2t,3,)) a-L( o)
,t 2 (It -;o) - (N-l)E) o s
+
] d *"
,e
= C ',. ( 1 - e (c t) ct (N-1) t-ON n
Suestituting in for C: and simplifying yields:
2
.' R(i)
S(t) di (1)
P
=
ON
=. S(t) dt e--
.' 2 -
. 2t +1 g
where R(t) = 1 - e' (cg):
(t /c )2 + 2(t -t ):/(N-1) 4(t -t )]
s o o
5 o S(i) = e- [
E2 t
and To find out which values of R(i) are most heavily weighted by S(t), we wi'l find the I which maxici:es 5(i).
(cal' this value l')
Maximizing 5(i) is clearly equivalant to minimizing the bracketed quantity in the exponent of S(i).
Thus, we want to nin g(i) = a__, a t
t 2(t -t )
wnere a = (to/C); -
5 3 n-1 and b = 4(t -t )
3 o g (i) = - 2_a, 1,_
33 2
(t /C);
(t -t )
g'(t)=0*t'=h=2(t-t) ii h (2)
+
s g The second derivative test verifies that this value of t does indeed yield a ninimum. The following table shows t', PON, i /E ("Best Case"), and P /P ON ON
("Best Case") for various values of N, t, and t.
g 3
s l
-17 i
V
- cn
.e t
t
{*
P ai0 t'(Best Case)
)g3t5est Case) o s
CN i
"Sest Case" 12e 8,000 1,600,000 12,551 3.17 1.00 1.00 1
256 6.259 12.8 499 4.04 64 25,296
.78 2.01
.246 16,000 12.535 6.35
.999 2.00 4,000 12.571 1.5a 1.00
.498 3.200,000 25,142
.79 2.00
.249 600,000 6,268 12.7 499 4.01 Table 5 4
Sensitivity of P to Approach 22 parameters ON Thus we see that, essentially, doubling N cuadruples POT 1, doubling t doubles g
I PON, and doucling t reduces P by a factor of 4.
Closer analysis of equations s
ON
]
(1) and (2) verifies that these are approximately the relations which i
should exist cetween P ano N, t and t.
For example, doublino N aporoxinately ON g
g nalves i' (since tne (tg /c) /2(t - t ) term in equation (2) is very small 3
g l
compared to the (t - t )/(N - 1)' term) and thus values of R close to s
g l
R(I'/2) are the most heavily weighted in the integral in equation (1), rather tnan values close to R(t') (which was the case before N was doubled).
I 2t 'l o
g 4((cW )
p(7'/2 )
1 - e- (ct'/2);
)
,g 2t +1 2t +1 R( t ')
g g
1 - e, (ct')2 1 - e_ ((ci');)
(using Taylor's expansion) 2t +1' 2t +1 0
4(
) - higher order terms in (
)
(t')2
-(ct'F 2t +1 2t +1
(
'O (ct ')
0
)
(ci');) - higher order terms in. (
i i-i h
j
-.-,,m.
y.
e
~
r
-1B-is small (it equals Bx10~0 for ne "Best Case" 0
=4 since (ci')
values of t and I.')
o to approximately quadruple, so we would excect that doubling N would cause PON since the values of R which are most heavily weighted are quadrupled.
1 (as before), we see that P P
is app-oximately Since PBSiON*
ti ON BSl0N g.,2,approximately).
t pro:ortional to N, (since by Tacle 5, Pg p'. =
,2
's See Table 7 for a summary of tnese results.
T r
P P
o 0ti BS ON ON BS'ON t
T r
A B
AB g
x l
Ki A/K K'O AB n
A B/K AB/K y...
Kr A
KB K AB to the par: meters of Table 6: Sensitivity of P P
ON BSl0N Approach
- l.
(K is any positive constant-within certain bounds which contain 1).
l t-t N
P P
PON PBSl0N o
3 ON BSiON t
t N
A B
AB o
s K{
KA B
K. AB c
2 A/K B
AB/K K[3 2
KN KA B/K K
AB Table 7 :
Sensitivity of P P
to the parameters 0tt BSl0M of. Approach 82. (K is any positive constant-within certain bounds which contain 1).
a Section 2.6 Analysis of.PRBiBS A unifom distributior was assumed for the location of a future offset between the sheces, given that such an offset occurs.
This produces a value of Ppg.BS equal to [ [ l with 2, b, and b as defined on page 2.
It is stated that various symmetric distributions were investigated, each of them yielding than was obtained by assuming a uniform distribution.
a smaller value for PRBlBS Thus, the report states, "it was assumed that P should be based on a gg gg uniform distribution to produce maximum conservatism".
It shoula be noted that the uniform distribution does not in fact yield "mayimum conservatism",
even from among the class of symmetric distributions.
The only symmetric distribution which can claim this is one which gives a.5 probability of the offset occuring under the reactor, and under the spot the same size as the reactor an equal distance away from the existing shear farthest away from tne reactor (to make it symmetric). Of course, such a distribution would be unreasonably conservative but it is mentioned just to make sure that the casual reader of the report does not believe that he is getting something which in fact he is not.
In fact, we believe that assuming a uniform distribution in order to obtain P
is probably reasonable.
RBjBS Section 2.7 Sumary The results of sections 2.3 through 2.6 are summarized below:
The procedure given by Approach #1 in the report is essentially equivalent to the following procedure.
=-
2 2
P P
+P P
l In P = Pg7 Pggig7 PRBlBS*E0fi BS Oft RBlBS 0tt BS Ot1 gg;gg P
take:
I I
l C
Pg7 Pgg jg77 = In
- wnicn is an upper bound (by a very small amount)enh(1-(1-C)
") -- an ex-pression which is extremely insensitive to n P 'i 1.6 ' 0" 2
. depending on whether the normal or Weibull C"' * *'o t
distribution is assumed for the time between o
offsets on existing shear i.
l 1
i 1
o in (1 C) l BS O., = in ( A )
r t P
=
a
,3 x
- +b RB:BS " L - b
- where ; = 72 ft, b = 4 ft, L = 1,320 ft.
1 P
30 P *
- C-
+C-in
)_c RB/Bt (3) x x) i (where C = 1.6 assuming the normal distribution, and C = 2 for the Weibull 2
2 distribution).
tJote.that P is independent of t and ti.
As a check on this o
expression, its value was calculated using the values given in the report for Approach 41.
The results are given below:
4 l
. Eqn.
Report (3) s y
'x rx10*(ft/yr)
Resul t result Case Distribution t*
B-i/8-3 5-2 B-1/3-3 E-2 C x 10' x 10' Best Case Weibull 160,000 210 140 1,34
.89
.9 1.2 1.2 1
1.1 1.1 tiornal 4
1.45
.97 1.2 1.2 5
1.28
.86 1.1 1.16 6
128,000 1.4 1.4 7
195,000 1.0 1.0 8
52 88 2.18 1.45
?.3 2."
9 420 280
.99
.66 1.0 1.0 10
.95 1.5 1.5 11
.10
.05
.05 Table 8 Comsarison of the results of Approach =1 (in the report) an.
Equation (3)
Thus, tne report's results are almost exactly duplicated by equation (3).
i (The fact that the two deviations from the report are positive is expected since by equation (3), PB5i0ii (the term whose approximation is least accurate) is actually less than (h) rt (the expression which 3
g in (q) in
=
fi
'x was used in obtaining equation (3).) rionetheless, the results are very close.
The result given by Approach 42 is the same as for Approach al except that j
t fl' J
o P
is given by a term which is approximately proportional to 0t1 t-s
)
= 1 - (1 - C)
).
(and Pgg j g;;
74 P
Tables 6 and 7 (page 18) sumarize the approximate sensitivities of P0r4 BS { 0ti to the-parameters of the two approaches, as develooed in sections 2.3 through 2.6.
L
' i j
In this memorandum, an analysis of the properties and actual behavior of the models proposed in the report has been attempted.
This analysis shows l
that, for the parameter ranges considered in the report, the models in the i
report function in a relatively simple way in spite of their apparent i
complexity. The fairly simple relations between the parameters of the I
model and the overall result need to be examined for plausibility and for consistency with geological and seismic theory.
For example, Approach #1 yields a hazard function (PON) which monotonically decreases with time since the last offset (t ) and an overall result which is insensitive to t,
o o
t whereas Approach #2 yields a hazard function which monotonically increases with t and an overall result which is sensitive to t as well as N (in o
o l
the manner described above).
The correctness or incorrectness of these i
properties, as well as the other properties which are discu: sed in this memorandum, need to be judged.
1 I
e b
i APPENDIX A In the report, it is stated, in calculating Pg3lgp, that:
The parameter, P, of the binomial distribution was selected at a confidence level, C, based on observing zero offset events in t*
years, that is:
P = 1 - (1 - C) /t*
The reasoning behind the use of this expression is investigated below:
j Solving for C in the equation:
1 i
p = 1 - (1 - C) /t*
we obtain:
3 l
C = 1 - (1 - p)t*
1 - P(t* failures out of t* Binomial trials with parameter p) u P(at least one success in t* Binomial trials with parameter p)
=
Thus C is not strictly a " confidence level", i.e., C does not equal the probability that the true value of p is less than or equal to the given value 1
of p.
i C is, however, related to a confidence level, as is shown below, If we take a uniform prior distribution for p, then, since the likelihood i
function for.p, given t* failures and 0 successes, is:
4 l
(gt*) p(1 - p)t* = (1 - p)t*,
i we have that the posterior distribution for p is given by:
t*
k*
(t* + 1)-(1 - p)t*
~
=
/11 - (1 - po)' dpo o
t L
Appendix A where p3 is a dumy variable of integration. Thas, the confidence level C3 must satisfy (t* + 1) (1 - p3)t* dpa = P(true p is < p) = Co o
Thus Co = [-(1 - po)*+* + 1]P o
= 1 - (1 - p)t' + I (lA)
Thus C = 1 - (1 - p)t* is less than Co, and so C, as defined on the report, is a lower bound on the actual confidence level, i.e., if p = 1 - (1 - C) /t*,
the probability that the true p is less than p is actually greater than C.
Note that by equation (lA),
2 p = 1 - ( 1 - C o ) / ( *'* + 1 )
and so if the C used in the report is in fact interpreted to be the con-fidence level, then the expressions such as 1 - (1 - C) /t* and 1 - O - C) I" which are used in the report should be 1 - (1 - C) /(t* + 1) and 1 - (1 - C) /(N + 1)
These expressions, and 1 - (1 - C) /(n + 1), are used in parts of this memo.
For large values of t*, N, and n, the difference between these expressions is very small, and the report's expressions in fact yield slightly larger than would be obtained by using the correct -
values for PBSlGi and PBSION expressions. Thus, given the model the report uses, the report's expressions are reasonable.
3 r
APPENDIX 3 The value of S, (1 - (1 - C) II"'I)) and the ratio of this value to I
1 - (1 - C)l/(t"I)'is given below for various values of n (assuming that C =.9 and t* = 128,000).
i I
l S. (1 - (1 - C)III"+I))x105 E.(1-(l-C)III"*I) n I - (1 r)I/(t*^l) 0 0
0 1
1
.534
.297 2
.837
.465 a
1.15
.641 10 1.48
.820 50 1.72
.959 200 1.78
.989 1000 1.795
.998 1
f 10000 1.799
.9998 50000 1.799
.99997 100,000 1.799
.999995 128,000~
1.799 1.0 l
Thus we see that (, (1 - (1 - C) II"+I)) is an increasing func. ion of n and so is less than 1 - (1 - C)II"I) for n < t*.
x
~