ML19199A238

Suppl Testimony of Jm Vallance & Kaplan Re ALAB-525. Clarifies Details of Spatial Distribution Model Used to Describe Airplane Hit Probabilities in Vicinity of Reactor Site.Certificate of Svc Encl

ML19199A238
Person / Time
Site:	Crane
Issue date:	03/20/1979
From:	Kaplan S, Vallance J AFFILIATION NOT ASSIGNED
To:
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NUDOCS 7904100021
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{{#Wiki_filter:. March 20, 1979 NRC PUl:!JC' In en qq pg c4 I 'd% f n 2 9 og UNITED STATES OF AMERICA p./f ff ~~ b *# NUCLEAR REGULATORY COMMISSION \\\\ p' /' 3'- BEFORE THE ATOMIC SAFETY AND LICENSING APPEAL BOARD In the Matter of ) ) METROPOLITAN EDISON COMPANY, ) Docket No. 50-320 ) et al ) ) (Three Mile Island Nuclear ) Generating Station, Unit 2) ) SUPPLEMENTAL TESTIMONY OF JOHN M. VALLANCE INTRODUCTION This testimony responds to the questions in paragraph II.A.1 of ALAB 52.. The testimony is intended to clarify certain details of the spatial distribution model used in my prior testimony in this hearing, particularly for take-off hits. DISCUSSION Basically, the spatial distribution model consists of creating approximations of hit location patterns based on fitting actual historical data. The first step is to examine the historical data to create a statement of the relative fraction of hits that occur at r = 0 (a hit on the runway), and at 0 < r < 5 miles (an off-runway hit). In this regard, note that the accident rate used in our analysis is for hits "on plus of f the runway" (Tables 7 and 8 of Vallance testimony, January 9, 1979), and the rate is stated individually for takeoffs and landings. 12'~272 1 790'11oocn>\\

Next, the radial probability density function, D (r), is created to fit the off-runway hit location data. Thus, D (r), as it was developed in my previous testimony, contains a factor to account for the fraction of the hits that are off-runway, and then a factor for the decay in the hit density with increasing radial distance from the runway (r). The off-:anway hits are further processed into hi_s occurring on the extended runway centerline, at G = 0, and hits away from the centerline, at 0<0< 180. The angular probability density function, D (0), is then created to fit the off-centerline hit location data. Thus, D(0) contains a factor to account for the fraction of the hits that are at non-zero theta values (off the extended runway centerline) and then a factor for the decay of the hit density with in-creasing G. In the angular correlation, hits occurring less than mile from the runway are excluded from the D (0) correlation base. Now let's look at the numerical values of the various hit location categories and check to be sure the integral of the probability density functions show the correct value. The analytic equations for D(r) and D(0) are given on page 22 of my January 9, 1979 testimony. Figure 3 and 5 of that testi-many (attached hereto as Exhibits 1 and 2) give a plot of the cumulative probability density of takeoff hits with r and with G. Footnotes on these figures give numerical values of the total hits in the data base and their location category. Radial 1 Consider Figure .1 of my January 9, 1979 testimony (Exhibit 1). A*?'2?3 2

The points shown an the figure represent the off-runway hits as a fraction of total hits. There are 17 points shown for off-runway hits and there are 17 points (not shown) for run-way hits (r = 0). The first off-runway hit is at a cumulative fraction of 0.5, since 50% of the hits occurred at r = 0 (on the runway). The solid line on the figure is a least mean squares fit of the off-runway hit data points and is chosen to represent the cumulative distribution of hits on a continuous basis over the interval 0 to 5 miles. This -r/1.84 miles 0.58e line is described by the equation N(r) = We take the derivative of N(r) and calculate an integration constant so that the summation of the area under D (r), in the limits of 0 to 5 miles, yields the correct value. This normalization serves to make the correlation valid over the entire range of interest. In this case, the correct value of the integral is 0.5, to reflect the fact that only of the hits in the D (r)T data base occur at values of r > 0 (see footnote of Figure 3 of January 9, 1979 testimony which states that of 34 takeoff hits in the data base, 17 of them were on the runway and 17 at 0 < r < 5 miles). Now citeck the integral of the analytic value of D (r). For this: -r/1.84 0.29e (page 22 of Vallance January 9, D (r) = 1979 testimony) -r/1.84dr D (r) dr = 0. 29 e 0 0 -5/1.84 = 1.84 x 0.29 l-e f5D (r) dr = 0. 5 So this checks out as representing the data (50% of the hits are in the region 0 < r < 5 miles). 12 ??4 3

Angular Now consider Figure 5 of my January 9, 1979 testimony (Exhibit 2). In this set of data, there are 10 points in the region 0 < 0 < 180 and there are (not shown) 4 points for values of 6 = 0 (i.e., 4 of the 14 points were on the runway centerline). Again, a least mean squares fit of the data was used to gener tn the solid line chosen to represent the continuous distribution of the non-zero hits. The equation -9/ 80 of the cumulative distribution on this basis is 0.86e An integration constant is utilized to cause the summation of the curve of its derivative to equate to the correct value. In this case, the corrcet value is x 10/14 = 0.36. The factor is due to the assumption of symmetry with respect to the number of hits on each side of the extended runway center-line and 10/14 is for the hits occurring at values of G > 0. Now check the integral of D (0). 0.44e (page 22 of Vallance D(0) = January 9, 1979 testimony) 180 -180 -0/48 0.44 e de D(0) = 0 0 -180/48 0.838 x 0.44 1-e = 180 D(0)de = 0. 36 0 So this checks out also. The correlation and the integration constant were developed based on representing the continuous function in the full range of 0 to 180 on each side of the runway centerline. It ro happens that the data base used 4 12'275

does not contain any hits for which 0 is greater than 90 However, there can be hits at 0 > 900 and, therefore, the correlation was developed on the basis of the trend line in the 0 - 900 region extending into the 90 - 1800 region with the same slope. If the value of D(0)T had been developed on the basis of assuming that no hits will occur in the 90 - 180 region and all hits will be in the 0 - 900 region, a different integration constant would result and the expression for D(0)T w uld give values about 15% larger than reported herein. It is my judgment that the integration should be done as I did it, namely, using limits of 0 to 180 Joint Probability From the above, the integral of D (r) x D(0), evaluated from 0 to 5 miles and 0 to 1800,1s 0.18. This is equivalent to saying that if a takeoff hit occurs within 5 miles of the runway, there is a 0.18 probability that it will hit some-where off the runway, off the extended runway centerline, and in one of the 0 - 1800 regions on either side of the runway. The algorithms given on page 22 of my January 9, 1979 testimony apply to off-runway, off-centerline hits only, and should have been so qualified. For hits on the runway, the joint probability density D(r) x D(0) is 0.5. For hits on the extended runway centerline, D(r) is as given on page 22 of the referenced testimony, but D(0) is 4/14. Exhibit 3, below, provides a graphic display of the estimated hit probabilities, if a hit occurs, in the various regions where the modeling details differ. The Exhibit illustrates that the estimated probabilities, calculated for each region in accordance with the model, sum to unity, as one would expect. 5 12',276

Specific Ouestions in ALAB 525 The above information answers the questions generally. Additional specific responses are indicated below: Question II.A.l.a. Why the angular normalization integral is over the range 0 to 1800 rather than 0 to 900 Answer-- This is addressed above on pages 4 and 5. I would simply repeat at this point that I cover the full 3600 range of possibilities the way I did it. If we applied this model to a plant location of, say, 1000, the results would be applicable and consistent with the data base. Questions II.A.l.b. and II.A.l.c. What is meant by the statement that crashes at 00 are allowed for. Does the spatial distribution model have validity for values at 0 = 0, in view of the treatment of 00 crashes. Answer-- The statement was intended to indicate that hits do occur at 0 = 00, and that the modeling for the non-zero hits accounts for this, as described above on pages 4 and 5. However, as noted above, the specific algorithms on page 22 of my January 9, 1979 testimony do not apply for 0=0 or for r = 0 miles. The model includes discontinuities at these regions. The adjustments incorporated in the model to permit generic application for these regions are specified above on page 5. 4.e do fc i 4 o n ~o a 6

Exhibit 1: Vallance January 9, Figure 3 1979 Testimony Cumulative Frequency Distribution with Radial Distance From Runway Takeoff Accidents j - i t',' d'l, l' l 1 1.0 - - 1 r i_-

3

= -- - = = = - =

=:

Zi -k i-r=T # 'l 1 =--E ~Mj:ti- " E h55-1 '=

= - - - - = _ =.. _ _ = _. _ = = ____ =_ _=__ _ _= :- = ~=-- ___.__;=.== t r= 1 x---+ i._. 1 0.5 __ 4 i y - -_, -w _ j-hdM - --st==L-- - +E --l{12==]f-2 0. 5 8e-f/1

• O 45 1~-

=- y g .== -__,d= ---_- ~ a /4 s y c g m_,- I t j t - 7 '4 c 1,, , i ' i i,*- i i ,i l i i t t 1 i e i it e s 3 i4i e i e i e i i e M + f f 4 I e t l l t i ._I l l 1 1 j t i I i} M I 5 it I } [ r I i I t i i l it i i l i i i iiii t i N d. i i i i i t l t i iit j i i i i i i i iii i i ii aiii i i i i i i i i i i i i i, i i i i N I I I ' II' I ,'==.'=;- - -

= _:

==r:-- =E M 0.1 O

=

x

=.- - = =

=.

u -- ni -V-_-}p dM-un3d g_ ~

1. f -

g , gr x - 7 _- - -- g : x- - 0 l ___r_=.. = =_ g U ~ [ ~-~- g _...__ _t--- N 0.05 o x .a ..__y=---- a =_.-. c H c E c U -r i 2, I 1 I l l l 1 l 4 i 1 3 I I I 5 I l 4 i i ( ) i i i iii r i i l I l t i i i i, t ; i e i iii! i i e i i < i 0.01 0 1 2 3 4 5

Radius, r, miles The construction of this figure incorporates

'4 data points, of which 17 occurred at 0 < r< 5 miles.

Exhibit 2: Ficure 5 Val. lance January 9, 1979 Testimony Cumulative Frequency Distribution with Angle from Runway Centerline Takeoff Accidents I i 1.0 _p

==_ = -. - _ = - - - EEiMEE

=hEH=5I

r;rM M NFE E _=-SEm h=E

-

-_v-- . --J___-. -g-= = _At-.m,, _ c 1 i= -9/48 - ~ ~ u.__;- - w + r-i 0.06e _ -+. 0.5 -n

r e

y _ _ _ _ - 1=-#jt#4= ---- =1 zUG M =- ~-7' =MEE-T -~ TdfE5 5E ^'

--r-------=

g g p nqgg o = -{Z 1

---

CM_\\.

] = = x n i x e m y i m .x o 3 N -e4 i i 1 ,ll' -t i, ; li ' -N ;l, , i, i .i i i i, ii3 i i,,, i i ii i i i i i i i i i iiii i i i i i ii i;,i i i is i i i i i Tii itiT i O i tN il c 0.1 iii; i i ii iiii iii it i i i i i i i i i lii i i ' i iliPyi - : = : j _ ____ __ __ s O Fittk 441

-i= 44+

=-r:- =4=i+

-= -13E== .---_.---_ G_ _ -- e ei-i = =

= = =

u .u. o > 0.05 e4a C a ,E_._._ = _ - -. - - _ = - - - =- = a Ec i o ~ i_ t i ~m y, l',. i i 1 ' i 1 j i, 9 +,i j , e i ,e i a e t i __ { t i e e e t i , l t t i i l I i i e I i i 1, l l l ) t 6 } l l I 4 i l 1 i l l l l l I O ii 3 ' I E ' l il I I ' ' I 1 l i I I l { l l l l l i l l l l l i l 8 i 4 ! i i i' 0.01 i i i i ' i ' i i'i i ' i i , ' i i. 0 30 60 90 1.T R?B

angle, G, degrees The construction of this figure includes 14 data points, of which 10 occurred at 0 <G

< 90

Exhibit 3 Integrated Probability of a Hit in Various Regions Within Five Miles of the Runway, If a Takeoff Hit Occurs Within Five Miles of the Runway N s N N Region 4 -,'x s. 'sN' 5 mi. N \\' '\\ 'N N. 'N x x s NN N x\\ Region 1 \\ sxx' x N l (runway) N m s Region 2 '//, 77 ~ 7- ~ ~ ~' ',/,/7, ,/ /, '/ ./ / Region 3 f' ~ / /, /',// /* ,/, / s ,,// ,/ ,/,l / For 1.0 Takeoff Hits at r - 5 miles: 0.50 hits in Region 1 (runway hit) 0.14 hits in Region 2 (centerline hit) 0.18 hits in Region 3 0.18 hits in Region 4 1.00 1.'? ~ 260

FRCTI"UTlt :T :ENTltOOM 20 March 1979 s (4 4 l ' A.<> c y N le s @Q4 2 L ~ 4 # UNITED STATES OF AMERI t. NUCLEAR REGULATORY COMMISSION BEFORE THE ATOMIC SAFETY AND LICENSING APPEAL BOARD In the Matter of ) ) METROPOLITAN EDISON COMPANY, ) Docket, No. 50-320 et al ) (Three Mile Island Nuclear Generating ) Station, Unit 2) ) SUPPLEMENTAL TESTIMONY OF DR. STANLEY KAPLAN 1T RDt

This supplement to our prior tcstimony responds to issues raised in paragraph II.A.2 of ALAB-525. The Board there discussed the method employed by Applicants in specifying the precision of the hit probability values and noted that "the results seem to imply that the variables whose probability was represented by the individual frequency distributions were assumed to be either independent in the statistical sense or at least not correlated in an insalubrious manner." ALAB-525, slip op. at p. 11. In particular, the Board requested clarification of the rationale for regarding the histograms for crash rate, radial crash density and angular density as being "indepenaent" distributions. The purpose of the present testimony is to explain the basis for Applicants' multiplication of the probability distributions rs independent. At the outset, we acknowledge as correct the Board's observation that the method of histogram multiplication employed by Applicants to combine individual probability distributions for the crash rate and radial and angular crash densities in order to determine the areal crash density was based on a determinatica that these individual probability distributions were independent. A full explanation of the basis for this determination of independence and its place in our methodology unavoidably requires rather lengthy and complex articulation, much of it mathematical in nature. This full explanation is provided in the attached Annex to this testimony. For the convenience of the Board, we begin by summarily stating the main points of the analysis. l ',, ~ '<? n?a ~

2-It can be demonstrated mathematically (see Annex, pp. 10-14) that an assumption of separable functional forms for the variables r and 9 necessarily implies that the probability hirtograms for the radial and angular derivatives are independent. Thus, the key inquiry is whether the assumption of separable functional forms is itself justified; i.e., whether r and 8 are separable. The NRC Staff statisticians have demonstrated to their satisfaction, using traditional statistical techniques, thac r and 0 are independant, or separable in present terms. Analysis from a Bayesian perspective (see Annex, pp. 9-10, 14-17) results in the same conclusion. There is no a priori basis for viewing r and 6 as nonseparable. The historical crash data conform more closely to separable functional forms than to a reasonably specified set of nonseparable forms. Thus, the actual data support the selection of separable functional forms for the anal / sis. The use of such separable functional forms provides assurance of the required independence of the histograms, justifying application of the independent histogram multiplication methodology to determine what this Board has termed "the precision of the hit probability values." ALAB-525, slip op. at ;. 10. Thus, the results presented on the basis of this methodology are believed to be accurate. 1r 203 As a double-check on the sensitivity of the results to the determination of independence, further calculations have been performed which demonstrate that a-alternative and more general specification of the model permitting a substantial degree of nonseparability of r and 9, i.e., not dependent on the determination of independence, has a fairly small effect on the final probability curves. (See Annex, pp. 15-27 and Figures 7.1 and 7.2.) This reinforces our confidence in the validity of our results. 1.3 204

ANNEX 1. INTRODUCTION AND FJRPOSE In its Memorandum and Order of February 1, 1979[1) (see page 10 and 11, Item 2) the Board questioned the manner in which the probability histograms are multiplied together in Vallance's testimony (as revised December 8, 1978, page 24). In particular, the Board questioned the rationale for regarding the histograms for crash rate, radial crash density and angular density as being " independent" distributions in this multiplication. The purpose of the present testimony is to answer this que s tion. 2. OUTLINE OF APPROACH We remark at the outset that this question is subtle and probes to the heart of fundamental matters. It will therefore be necessary to prepare the backdrop with considerable care before proceeding to answer the question itself. We beg the reader's indulgence, therefore, in this preparation and we shall endeavor to be as concise and as clear as pos sible. Next we note also, upon reflection, that this question really contains three subquestions, subtly interwoven. The first of these is basically a conceptual question, having to do with the fundamental meaning of proba-bility and with whether separability of the spatial variables necessarily implies independence of the probabilities. It can be answered defini-tively either with a qualitative, semi-intuitive approach or an explicit analytical approach. We shall do both in the following tions. Th second subquestion has to do with the crash itself and asks I whether this data supports the belief that the spatial crash distribution is separable in the radial and angular variables. This question has been looked at by the NRC statisticians (see testimony of Moore & Abramson, November 30, 1978, pages 3 and 4) who apparently satisfy themselves that the variables are independent. This question also can be addressed very satisfactorily and very thoroughly using a Bayesian approach and a class of nonseparable fitting functions. The amount of computational work involved here could become very large depending on the class chosen. We therefore adopt a 'first approximation' treatment which we feel catches the majority of the nonseparable effect and yet keeps computer time within reason. The third subquestion concerns the separability of space and time. That is, is the spatial distribution of crashes a function of time ? Intui-tively one might expect some dependence here since type of aircraft, in s trumentation, etc., varie s with time. A look at this question is shown in Figures 8.1 through 8. 4 which give the scatter diagrams for crashes before and after January 1, 1965. Needless to say the data, 12 28S 1

that is the number of crashes, is far too sparse to allow a time pattern to be definitively seen. The question of time dependence of the spatial pattern can be pursued in a manner similar to that used to address nonseparability at r,0 (i. e., the second subquestion). This pursuit, however, becomes computationally even more complicated, especially when combined with spatial nonseparability. Moreover, we expect that the result of a Bayesian application here will yield essentially the same result, i.e., the same crash frequency at the TMI location, as our previous calcula-tion. Therefore, we do not pursue this matter any further at this time, but of course stand ready to do so should the Board consider it worth-while. To understand the discussion which we shall give of the subques-tions it is essential to understand the sense in which we use the word " probability," and the distinction we make between the idea of probability and that of " frequency. " It is essential also to be clear on the notion of " probability of frequency" which in our December 8 testimony, was the conceptual context we chose within which to addres s the Board's original request f or a recalculation of crash likelihood including estimate of unce rtainty. To make sure, therefore, that we are fully understood we shall begin in the next section by defining these basic words and concepts, and then review the overall pattern of the argument in the December 8 te stimony. This will then allow us to. proceed to the subquestions which are the subject of the present note. l 3. PROBABILITY. FREQUENCY, AND PROBABILITY OF FREQUENCY We adopt the following definition of probability, given by E. T. Jaynas: I " Probability theory is an extension of logic, which describes the inductive reasoning of an idealized being who represents degrees of plausibility by real numbers. The numerical value of any probability (A/B) will in general depend not only on A and B, but also on the entire background of other proposi-tions that this being is taking into account. A probability assignment is ' subjective' in the sense that it describes a state of knowledge rather than any property of the 'real' world; but it is completely ' objective' in the sense that it is independenc of the personality of the user; two beings faced with the same total background of knowledge must assign the same probabilities. " 12 206 2

Thus, as we shall use it, probability is a numerical measure of our state of knowledge or state of certainty. It is thus by definition a subjective or ' soft ' number. The word ' frequency, ' on the othe r hand, we shall use to refer to the result of an experiment involving repeated trials. It is thus a 'hard' or ' objective' number and is so, at least in concept, even if the experiment is only a thought experiment er an experiment to be done in the future. Now with the above definition of probability it would make no sense to ask for the uncertainty of a probability because the probability is already the numeric.a1 expression of uncertainty and we would thus be asking for the uncertainty of the uncertainty. Neve rthele s s, there is a valid thought behind this question. To provide a linguistic framework for handling this thought we introduce the notion of ' probability of frequency' as follows. Suppose, with espect to the Three Mile Island site, we envision a thought experiment m which millions and milli.ns of planes fly into and out of the Harrisburg Airport. At the end o..ais time we look over the records and ask what fraction of these flights crashed into the TMI plant. In other words, what was tt.e frequency, F, of collisions with the plant? At least in concept this quantity F, collisions per flight, is a tangible number which would be known at the end of this experiment. Our problem to. lay is to predict what this number would be. Since we do not know this number, we naturally express our prediction in the form of a probability distribution against F. 4.43D PF(F) o ;; a a A C bC a F FIGURE 3.1 - PROBABILITY OF FREQUENCY CURVE 1.,3 207 3

Thus we come to the notion of the ' probability of frequency' curve as a way of expressing our state of knowledge. Denoting this curve by pF(F), the expected value of the frequency is: (3.1) F pF(F) FdF = 0 This number is the " expected frequency. " It is also the probability we assign to the prospect of a collision on the next flight. From the curve we can also get percentile values. e.g., F9 0 "" F 10'

• F 90

.90 = pF(F)df 0 F 10 .10 = pF(F)dF 0 We could then describe ourselves as being 80 percent confident that the frequency F, if we were to actually measure it, would lie between F10 and F90 These confidence bounds give a measure or indication of our certainty in the prediction of F.* The full story of our certainty is expressed by the entire curve pF(F) its elf. t This notion of probability of frequency thus constitutes an appro-priate framework for answering the Board's original request. We therefore adopted it in our December 8 te

mony, and it therefore is also part of the backdrop for the present te.stimony.

4. RECAP OF DECEMBER 8 ANALYSIS In the December 8 analysis, the number of crashes per year is expressed in the form: 1/2 E N f S (r,0 ) A c = K KKK o o K Note that we use the term "corfidence" or " confidence bounds" in a Bayesian sense [2) which is slightly different from the way the orthodox statistician uses the term (see for example [3], p. 387, and [4], p. 29). 12 208 4

Here K denotes a category of flight operation, e. g., scheduled takeoff, N is the number of operations of that category per year at the relevant g end of the Harrisburg runway, f is the crash rate (i. e., crashes per g operation) for that category, and A is 6e eHeche target area for Gat g category. S (r,0) is the spatial density, crashes per square mile, of g crashes for that category, and S bo,00) is the spatial density at the g coordinates ro,0 of the TMI plant. Finally the value 1/2 is in effect 0, a reduction of the N to account for two-si' mmeuy abom Ge g runway center line. The Board's current question relates to these numbers S (ro,00). g We next review the December 8 derivation of these numbers in such a way as to form a basis for the analysis in the present testimony. For this purpose introduce: fraction of crashes occurring beyond 6(r,0) a radius r and angle 0. (4.1 ) More explicitly 6(r,0) represents, of all crashes occurring within five miles of the end of the runway and on the right side of the runway, that fraction of crashes which occurs in the shaded area of'the following diagram: f 6 I 5 miles 4 / o Runway 12 289 5

Note that this fraction could be different for each category, so that & should have a subscript K. For simplicity however, we shall omit the subscripts from here on and regard them as understood. In terms of 4, the crash density is: 1 8 8 6(r,0) (4. 2) S(r,0) - r ar 80 (where e is measured in radians). In the December 8 testimony the fraction & is assumed separable in r and 0, that is: R(r)O(0) (4. 3) 6(r,0) = so that: 1 de (0) R(r) S(r,0) O = r dr 1-D(r) D(0), (4. 4) s r and 1 -D(r ) D(0 ) (s.5) S(r,0 ) = o o r o o o The quantities D(rg), D(O ) were obtained from the data in the g form of probability histograms by the following procedure: 4.1 The Quantity D(r ) g Since D(rg) is just the derivative of R(r) at r, we regard our o problem as having to infer what the function R(r) is from the data on crash radii. Since the data is insufficient to define R(r) precisely, we need to obtain in some form a probabilistic statement expressing our state of knowledge of what R(r) is. We do this by conceiving of R(r) as being embedded in, being one point in, a set or " space" of functions M: R(r) EM (4. 6) We then seek, based on the crash data, to erect a probability distribu-tion over the function spaced to express our knowledge of where R is. 12~200 6

Since M is an infinite dimensional space we approximate this procedure by establishing in d a grid as follows: -A.r 3 R..(r) = a.e R..(o)

1. 0 (4. 7)

= lj 1 IJ where a., A. are selected from a discrete set of possible values 1 j {a } = {a a ' "3' ***' "I ( 2 p,} p1, x ^ Ad (4 8b) = 2 3 Our experimental data consists of the set of crash radii, which set we label B, {r B s ff rg, r3'

• In light of the information B, we next ask ourselves: W1.at is our confidence that the "true" function R(r) has the "value" R..(r)?

1J This question is answered simply, using Bayas' theorem as described in Appendix B of the December 8 testimony. The result is a probability value, pij, assigned to each i,j combination. The set of doublets: {(p..,R..(r))} (4.10) lj lj thus may be thought of as constituting a probability histogram erected over the function space M. From this it is very simple to derive a probability histogram for the quantity D(rg): [(P ;, Dij(r))} (4.11) i o where: D..(r )= -R..(r) (4.12 )

1) o dr ij o

Thus in (4.11) we have ar. t PD, a finite probability distribution or probability histogram, expressing our state of knowledge of the quantity D(r ) based on the information B, the t t of crash rad] - oqi o

r. ~

7

Dispensing now with the double index, we can rewrite (4.11) in the form: {(p,D(r ))} {p

1. 0 (4.11a)

= n n n n

4. 2 The Quantity D(0)

A similar process was applied for the angular variable, re s ulting in an FPD over a mesh of angular functions: {(q.. O..(0))} (4.13) lj lj and similarly an FPD for the desired quantity D(O )o {(q.., D..(0 ))} (4.14 ) 1j 13 o where O..(0) (4.15 ) D..(e ) - d6 lj

1) O

-0=0 o This FPD expresses our state of knowledge of the quantity D(e ) b: sed o on the set of angles {0, 02,..., O } at which crashes occurred. The 1 p details are in Appendix B of the December 8 testimony. Again dispensing with the double subscript we rewrite (4.14) in the form: {(q Dm(0 ))}

1. 0 (4.14a)

= q m o m m

4. 3 Multiplication of the Distributions To determine the spatial density S(rg, O ) we need to multiply the g

quantities D(r ), D(O ) as in (4. 5). These quantities are expressed as g g probability histograms in (4. lla) and (4.14a). If these histograms are now regarded as independent they may be multiplied to yield a histo-gram for S(r,0 ) according to the simple convolution rule: o o {(pq ,S )} (4.15a) S(r,O ) = g where S D (r )Dm(6) (4.15b) = nm n o o 1,3 7 0 2 8

e This is what was done in the December 8 testimony, page 24, and this is what is being questioned by the Board. We now therefore turn to this question directly. 5. SUBQUESTION 1. QUALITATIVE DISCUSSION OF INDEPENDENCE We now ask Subquestion 1: Are the probability histograms (4. lla) and (4.14a) independer.t ? What is the precise meaning of this question? To nail it down, construct a table as follows: TABLE 5.1 q q 2 m D (6 ) D (6 ) D (6 ) P 1(# 11 12 1 o P D (#o) M ~ 2 2 (5.1) P D (r ) a n n o nm 1 Now let {anm} represent a joint probability distribution on D(r ), D(e ). That is, let a express our state of confidence in the o g nm pair (D (r ), D (e )) (5. 2) Also, of course:

1. 0 (5. 3)

= a nm r, m 1.T 203 9

9 Now the question is: In Table 5.1, does a = P9 nm nm for all pairs n, m? If it does, this is what we would mean by saying the hictograms (4. Ila) and (4.14a) are independent. Otherwise, they are " dependent" or " correlated. "" So, given that the pn egresses our confidence in the D (ro) and n the c ur confidence in the D (Oo), do we have any reason or evidence 1m n to a ssign to the combination (D (ro), Dm(O )) any probability value other n o than the product pn9m? If we do not, and do not for all n, m combina-tions, then the histograms are independent. Intuitively, this is the ca se. No such evidence or reason comes to mind. Therefore, we assign the joint probabilities as in (5. 4) and thus obtain an FPD for S(r,0 ) as: o o {(pq Snm(r,0 ))} (5.Sa) nm o o with s 1 Snm(r, 0 ) = - D (ro) Dm(0) (5. 5b) o o r n o o which is the same as (4.15). 6. SUBQUESTION 1, ANALYTICAL APPROACH L In the previous section we qualitatively justified the independent multiplication in (4.15) by simply noting that we had no reason for assigning a joie probability a any different than the product Pn9m-nm We have thus argued that the absence of knowledge of dependency is in eHect a statement of the independence of our two FPDs for D(ro) and D(0 ). 0 In the present section we address the same question analytically. The essence of the question is whether the assumptio,. of a separable form (4. 3) necessarily iinplies that our state of knowledge distributions for D(ro) and D(Oo) are independent. It is not obvious that it does, salubriously or otherwise. 12~294 10

although it seems so intuitively. It turns out that this implication is indeed valid under one further, rathe r mild condition. We shall demonstrate this by repeating the Bayesian argument of Appendix B but keeping the radial and angular variables combined. Thus although we assume separability in the function 6, as in Equation (4. 3), we do not assume independence of our probability distributions. We will see, rathe r, that this independence follows as a consequence of separabi-lity in 6 if also our prior state of knowledge with respect to 6 is separable. Exactly what this means will become clear in the following. Demonstration What was done in Appendix B can be viewed as approximating the 'true' functivn 6(r,0) with a four parameter family of functions: = R (r)Og(0)

• (6.1) 6 (r,0, a, A, a,4 )

g wher 1.O, r=0 l l R..(r) q - A.r (6. Za) =

• 3 J, r>0 a.e

,2

1. O,

=0 0 ?. (6. 2b) Og(0) = ] p + b, 0< 0 < r/2 aek l The functions R, O were chosen deliberately discontinuous at r=0, 0=0, in order to represent the data properly at those points. The value b was assigned so that the curve would have the right asymptote at 0=900 To each member of this four parameter family there corresponds a spatial crash density according to (4. 6) as follows: A S(r, 0, i, j, k, f ) D..(r ) Dkf(0 ) (6. 3) = o o r ij o o o where D (r ) and Dg(0 ) are as in (4.12) and (4.15).

• More precisely the true 6(r,0) is thought of as embedded in a space of functions, within which the family (6.1) is a finite subset. Hopefully, this subset comes close to the true 6(r,0).

b.$ ?N 11

We now seek to establish over this four parameter family a probability distribution which expresses our state of knowledge in light of the actual crash location experience. The information we have on crash locations is summarized in the set B, as follows: H M N 5; {r }h=1 ;{(r, 0) }m= 1 ; {(r,0)}n=1 B= m n n V {(r, U/2)} (6. 4) where ( Number of crashes occurring at r=0; = Number of crashes occurring at r<0. 5 mile; H = M= Number of c rashes occurring a t 0 =0, ra0. 5 mile; Number of crashes occurring at U/2 > 0 >0, ra0. 5 mile; N = Number of crashes occurring at 0 = r/2, ra0. 5 mile; V = (r,0 ) = Coordinates of ith crash point. j We now wish to establish a probability distribution over the family s -' ' unctions (6.1) in light of the evidence (6. 4). For this purpose we write Bayes' theorem in the form: p(Bli, j, k. () p(1, y, k, f lB) p (1, j, k, () = p (i, j, k, f ) p(Bli, j, k, f ) g i, j, k,f (6. 5) Here po(i, j, k,() is the prior probability that we assign to the combination ai, Aj, ak, Ilf and p(Bli, j, k, f) is the likelihooc that we would have experienced evidence B given this combination. 12 IS~2"b

In light of (6.1), (6. 2), (6. 3), (6. 4), and also (4. 2), we may write this likelihood as proportional to: (M N H V p(Bli, j, k, () x (1 -a.)( (a A.) M+N+HtV j m n h v/ e i t,) N ~# (1 b)M (a 4 )N f On 1 (6, 6 ) e 7 The first bracketed term on the right is a function of i, j only and the seccand if of k, f only. Thus the likelihood is itself separable in the fo rm-p(Bji, j, k, f) x p (Bji, j) p 0 Putting this in (6. 5) we have: p (i, j, k, f) p (Bli, j) p0( } g p(i, j, k, f lB) = P (i, j, k, f) p (Bli, j) p ( o 0 i, j, k, f (6. 8) Now if the prior is separable P (i, j, k, f ) p (i, j) p (' = g oO then from (6. 8) or(i,j) P (BIi,j) p )P( P r o0 0 p(i, j, k, flB) = or(i' ) P (Bli, j) [pos, ) P (Blk,f) P r 0 _1, J k, f (6.10) The bracketed quantities on the right are exactly the posterior probabilities obtained in Appendix B, i. e., p(i, jlB) p(k, flB) (6.11) p(i, j, k, f l B) = 12 297 13

or, in terms of the notation of Sections 4.1 and 4. 2 p(i, j, k, f IB) p q g (6. lla) = Thus if the prior is separable, (6. 9), then so also is the posterior (6.11). In this case our histogram for the spatial density {(p(i, j, k, f l B), S(r, e, i, j, k, f )) } (6.12) o o become s: {(p(i, j jB) p(k,( l B), 1 D (r )D g(e))} (6.13) o But according to our convolution rule (4.15) for multiplying FPDs this histog ram, (6.13 ), is the product of the histograms: (P(i, j jB), f D ))f - (p(k, f lB), f(0 ))f (6.14) (r D 9 o or in the notation of Section 4 ij - ij(r hq f ' kf(O ' f(P o k o o which reproduces the result of Section 4. 3. Thus we have shown that our probability distributions for the radial and angular derivatives are independent if our prior state of knowledge over the family (6.1) is separable. The prior chosen in Appendix B of course was separable. We have no reason now, even from the broader perspective of this section, to do otherwise. Thus we can conclude for all essential purposes that the assumption of the separable form (4. 3) already implies that the probability histograms for the radial and angular derivatives are independent. 7. SUBQUESTION 2, SPATIAL SEPARABILITY In the previous two sections we have shown that if the spatial variables are assumed separable then the uncertainties in the r and 0 derivatives are independent. We now turn to the question of whether the experimental data actually supports the assumption of separability. 12~2DS 14

As mentioned earlier, Moore and Abramson have apparently given an affirmative answer to this question using statisticians' methods. A person of Bayesian persuasion would approach this matter by not treating the question as if it had only two possible answers, yes or no, separable or not. Rather he would embed the question in a context within which there was a continuum of separability ranging from totally separable to highly nonseparable. He would then allow the experimental evidence to dictate where we fall within this continuum--or more accurately to erect a probability distribution over this continuum. This is the approach we shall follow in the present section. To carry out this program we return to the four parameter family of fitting functions (6.1) and now embed this family in a larger, five parameter family as follows: cD(r,0, a, A, a, p, Y ) R (r)Og,(0, r) (7.1) = p where now: 1.O,r=0 i R..(r) q -A.r ( (7. Za) =

• 3 3, r>0 a.e 1

1. 0, 0 = 0

= ae +b 1+7 (r-2. 5) 0<0 sr/2 (7. 2b) Og(0, r) I 0, 6 > r/2

• Observe also that in (7. 2b) we have treated the situation at 0 =r/2 slightly differently. In the December 8 and January 9 testimony, in order to handle the fact that the experimental crash density undergoes a precipitous drop to zero at 0 =r/2, we regarded curselves as simply fitting the data in the range Os0<r/2, and chose the value b to force the right asymptotic value to the fit as 0 -+ r/2. In the present testimony since we are doing the work over again anyway, we take the opportunity to make a slight improvement over this procedure. We now regard the fitting function itself as having a step at 0 =r/2.

The discontinuity at this step results in a delta f metion for the crash density at 0 =r/2. This delta function is then included in our likelihood calculation in exactly the same way as the deltas at r=0 and 0 =0. This procedure makes little difference numerically over what was done in the December 8 and January 9 testimonies but it is more satisfying conceptually. ~3~299 15

Observe that the fifth parameter, 7,, that we have added is in effect a 'nonseparability index. ' For Yw=0 (7.1) reduces to the separable case (6.1), so that the family of functions (6.1) is a subset of the family (7.1), i. e., 6 C@ (7. 3) Observe also of course that we have not considered the entire space of nonseparable functions. This would be much too big a space to work with. We have taken one first step into the nonseparable wilderness, from the safety of (6.1), by allowing the constant term b, in (6. 2b), to now be in effect a function of r. We do this in (7. 2b) by multiplying b by the r dependent term [1+7 (r-2. 5)]. The specific form w of this term is chosen so that the average asymptote, b, .vould obtain at the average r value 2. 5 miles. Thus O now becomes a function of r also through the nonseparable, or " coupling" te rm, and the size of the parameter 7, determines the degree of nonseparability or coupling of the r,0 variables. We next use the data itself to assign a probability distribution against 7 as well as w against the other four parameters. We do this of course using Bayes' theorem and the fact that the spatial crash density is, from (7.1): 1 - -- 4(r,0) (7. 4) S(r,0) = r ar 80 l 1 1 R(r) O(0,r) + R(r) O(0, r) = _r B r ,80 _r _8 r 8 0 (7. 5) } With this crash density, and the forms (7. Za, b) it follows that the probability of experiencing the set of crash locations B, (6. 4), is proportional as follows: )(a, A ) a(a,gf)[(A, a,7 p(Bla, A, a, ,y ) oc ) (f( A, a, j,7,) (7. 6) where ? and (7 are the same separated radial and angular terms as in (6. 6) LT 300 16

M N H V -A.[r +[r +[ rh+[1 r 3 1 i n 1 ? (a, A ) - (1-a ) ga A ) e i j (7. 7) N -4f 0

1. (a,pf) a (1-a -b)M(a p )N 1 n e

(7. 8) g is the correction for nonseparability, M M 'l-A(r -2.5) [( A., a, y ) =h1+ (7. 9) J k w (1 -a-b) m= 1 and 20 is a term accounting for the step function at 0 =r/2. V -42 by j Li(A,a. 4, y ) = ae +b 1-A (r -2.5) k (7.10) The derivation of (7. 6) is given in the appendix. It remains only to do the numerical work. For this purpose we choose the same a, A,a,4 grids as in our January 9 suppismental testimony and get of ourse the same 1, a tableaus. We therefore do not reproduce them here. The I' tableaus are shown in Tables 7.1 through 7.10. It is interesting to note that for both landings and takeoffs the A' matrix is the largest for 7 =0, which seems to say, at least as far as A*is concerned, that the data prefers the separable fit to any other. The probability curves for spatial crash density at the TMI location are shown in Figures 7.1 and 7. 2, and compared there with the curves that are obtained using the separable fitting functions of the December 8 and January 9 testimonies. It is seen that the inclusion of nonseparabi-lity, by way of the b terms at least, has a fairly small effect on the final probabihty curves. 1,T '.3 C1. 17

TAB LE 7.1. ITABLEAU FOR LANDING CRASIIES y = 0. 0 w a A -2000E+00 .3000E+00 .6000E+00 .5000L+00 .6000L+00 .7000E+00 .13 3 3L + 01 .1000L+01 .1000E+01 .1000E+01 .1000L+01 .1000L+01 .1000E+01 .100ut+01 .1000L+0i .10 0 0 L + 01 .1000L+01 .1000L+01 .1000L+01 .1000L+01 .dOOOL+00 .1000L+01 .1000E+c1 .10000+01 .1000L+01 .1000t+01 .10001+01 .obo/L+00 .1000L+01 .10 0 0 6 +01 .1000E+01 .10000+01 .1000L+01 .1000E+01 . > l t 4L + 00 .1000E+01 .1000E+ul .1000E+01 .1000L+01 .1000L*01 .1000L+01 .5000L+00 .1000E+01 .1000E+01 .1000E+01 .1000L+01 .1000L+01 .1000E+01 .4464L+00 .1000b+01 .1000E+01 .1000L+01 .1000L+01 .1000L+01 .10000+01 .4000t+00 .10006+01 .10 0 0 E +u l .10uoE+01 .1000L+01 .1000L+01 .1000L+01 g .Jo30t+00 .1000t+01 .10 0 0 E +01 .1000E+01 .1000L+01 .1000L+01 .10001+01 .J333E+00 .1000L+01 .1000E+01 .1000E+01 .1000L+01 .1000L+01 .1000L+01 .307/L+00 .1000E+01 .1000E+01 .1000E+01 .1000L+01 .1000L+01 .1000E+01 M 10 e CN

1 a 00=00000000 o 3000o00o000 + + + + + + + + + + + + w U d & S & N ad 4 Y No 40 con N OoN # 2 o N Omd mO-Om#N o DC4? ToCN?t" N CCD4 44mmNNN e e e e e e e e e e o e o o0000000000 o o0000o00000 + + + + + + + + + + + + 4 'a d ad aa4 a4 J J d o 407740 C?*#N o 4 c??" " "a n "0 ? " 4 a B C% ?2%?% O N N J4ac## #mm 4 e . e e e e e. e. e e O O COOO OOO OOOO O O C0000000000 Z + + + + + + + + + + + + ~ a aawaaaaaawa h O ? JM SNmO9 W # O Z O M NmeoO#m# 00 O O cNme-Or~?~ _3 P D W N DD D b. D P 4 4 m e e e e e e e e e e O O O OOO OO)oC) h o O C0000000000 i + + + + + + + + + + + + W WWWWWWW WW WW O O?N =N n~c0 Cu g >p O C N 7 0A A N ?mN e 1 g a 900 MON #a?Of ~~ neNN~ cccrer A e .. e e..... e e

+ + + + + + + + + W J WWWWWWWWWW o OD">M DD7T4% s N o NaNmC?mcfM? O 09aN fM ? OfN? g m O O V N N .N c444A g .... e e e. e e. A Q o o000o000o00 o cooococcooO p + + + + + + + + + + + + W WWWWaaWWWWW e Do?DT?NM NTN o Om4N cN7NDmm o N ONoN aNot 4# N C OCCNN NN 4 C. 4 e e e e o e e e e. e M M OOOOOO OOO C0000000o00 D + + + + + + + + + + + W aaaaaaWaan M OON f0 7 0 09N moo Od oWomM N g moo ON OTO OMO ~4 A DB B 4 Tm1 9 e e e e e e e o e e e {k JL., t f L( 19 9 O O OOC COC00OOO + +, + + + + + l l l l w -esw-w waaaa O ?NNOcNONPON O N#7#N ON md 4N O N4% N D4 " m4DC ?. % N 4mNN~m m P 4 O C0000000000 C 00000200000 + + + + + + + + + + + + E 4 4 4 4 4 4 4 4 4 S d O mM OO DQOm ?E N O PNC "NONO"h a O POM 44 % mCN TN U D

1. 4 4m? N W

% N ". M. ". W o 0000 3 200 000 O C0000000000 + + + + + + + + + + + + u 4 4 4 d 4 4 4 4 4 44 O ~re-?rN e~nc O ? 4BNmO"P4P T Z O msm em N m? omo O r. B. D P 4 4. m m N N. N N. Z<a O 20000000000 o ao200000000 g + + + + + + + + + + + + ca w wuswwwo vaww O O DDOkN #74DT4 N O O % ?4a"?oB NND I O ? nnm: 4"% 4m O D 44eaee emMMN y W 3 4 h O C0000000000 C a 000000000 00 + + + + + + + + + + + + g w owwwwwaxwww O co-~<?OcosO O ?N ?MM M ACOVO O 9Dm??4 % 4CN P M N CDDDB44 4mm g O C0000000000 W G C0000000000 g + + + + + + + + + + + + a awwaaawaaaa c O f@ M4mNm##NO O M fCO?NN#MfC b O N NOn? JN?4MO N ~ ~ cononeere s 4M OO 20OOOOO C0000000000 q + + + + + + + + + + + u a w mu w a a _u.u w ; m0O% 404O DR N g 9CO4*O#OmmN 9 00 CN O#OOMO w*t 0a3# #9 99 12 304 20 O OOO~H ma m .4 aN O OOOOOOOCOOO + + + + 1 i l i l i l I w wwwww waww

• O TTDD?

NO4NT O 4N c?N f? Ommt O DDmP% % NNPO 2 NM " C 4m%"~ C e o e e e e e e e e e e O O300COOM """ O OCOOOOOOOOO + + + + + + + + l t l l w awaa aaawawa O 4,am~ND"NNmP O Nm?N #OmOf #m ,a r C ON40 0M O*m? O g D <mNN"A "D 24m m e e e e e e e e o e e e W<e O C000000003-C C00000 00000 g + + + + + + + + + + + 3 a waamawawaaa Q ~ O <O r# # cemm-r D C m% "D% 2"N % 2N O O N 4ODM DDNO D Q p f9 mN NM ww w Z t. e e e o e e e o e o e <A O 30000000000 W O OO 30O330O 3O + + + * * * * * * * + + O w WwwwwwwwwwW g O O B OT"? DP"N O O 8 y O CDM J O9pNNN # ll O N Of Cm?DN? Of 4 PD4mmNNNm "e W ) e. e e.. e e o e. A C O 000 33000000 O 300 30 00 3000 g + + + + + + + + + + + + w wwwwwwwwwww O M NDDRV4f74N O " BDM RTOPmPQ O mDODOPN QDNO y m DDD44mmN NNN e e e e e e e e, e e e N N O C0000000000 O OOOC330000O + + + + + + + + + + + + g a wwwwwwwaaow O DNDDD"M DO4 D O O MNDDD CN"??N O NM DOPM N #OND N DDDD4%mmmNN e e o e o e e e o e e e M M OOOVOOOO O C0000000000 D + + + + + + + + + + + annawaawawa MOON #O#O 4mN 4 mOOOd CeommN M OOON O# O Om3 M aD CD BZ Tmm# e e e e e e e o e e e 1.Y 305 21 O O""~~M NNNNm C 00000000000 + + 6 l l 1 l l l l 1 I w wwwus o uwwww O memNDN tmomO O M OM CmNON N3" O P O O ?= N"N m#mr N M Q mN& N 4%M C s O OCOOM M M M e M M C 0C000000000 + + + + l l I s l l 3 daaa4 a.JJaa m can: ce4 rNmD y eM omeco-M NM m O ? M P 4 2N 4 43 D D N%M QB4 ~. ' ~ d U a 3,30 ,3-m mm-C O CJ000000000 ... +. + i s l e s g a aAwaJ aawaaa m O 3?M PDN POm34 Q O N&N ON-ONOO g O 7M #7DNODCBf m N. M ?. N D 4. m. A M M A M v O C00000000-M o O 00000000000 4 g + + + +.. + + + + i e JWwA WW W WWWWW p a O 2NcM cNm>ONm O NBT4% CDN PT4 N N O N ?NNNCANOOO A > -M M mN e m m N. N M g <b C C00e0000000 Ng O. 3 0 0 3 30 00 000 + + + + w wawwwwwwwww O NNTCCDM DDMM m O DM DOM 7mM 4 -0 M r O mD?4744 QDnM a#MMNNN.MM M M 5 m g Am C 00000000000 O 00000000000 4 t + + + + + + + + + + W WW MWWWWWWW W C eOM occeC N c C ConNC04NN4m O DMD7% 3DRCND N Db#m1mNNN& M e 4 M C00000000 00000000000 + + + + + + + + + + ^ g w 4.u w a a.a a.a m a R OON#Of009N 4 MOO 4M o#OmmN mCC4N O<C Cmc 4 M D DBbrimmn ....... e...

• cy -

^ t1 : 22 O ------~~~~~ O O000OOOOOOO + + + + + + + + + + + + a .a a .J a A aaaAaw 0 0 0 0 0 0 0 0 0 ') C0 0 00000000000 0 00000 ' 00000 0-~~~~~~~a-- gW I O -~~~~~~~~~~ W C 00000000000 + + + + + + + + + + + + g a w a a a.a awanaa C 00000 000000 g 0 00000000000 L 0 00000000000 p a-~~-a.- m--- N OWg C ~~~~~~~~~~ C 00000000003 + + + + + + + + + + + + b w o wwwwwwww w a 3 00 0000 0000 d O C 00 "JO0000 000 O 0 0 000000000 g O < --~~~~~~~~~ ,. e........ O -~~~~~~~~~~ N b 0 00000000 303 .1 + + + + + + + + + + + + c w a w w w w w W.a w w a 0 00000000000 g O O000000O000 b 0 000000 00000 m -~~~~~~~~~~ .x O -~~~~~~~~~4 C 00000000000 \\ 6 + + + + + + + + + + + + y w 2awaJDwwawA 0 00000000000 0 00000000000 Q 0 00000000000 y -~~~~~~~~~~ g ~~O00000000 d 30000000000 + + + + + + + ++++ 'a w 'i w 4 w W d w 4 2 9CONfo#C omN X mOOO-OeOm9N 9 CO ONOfCO9 0 ~~C ODaff999

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23 a O OOCOOOOOOCO C 000000007 0 0 + + + + + + + + + + + 4 4 44 4 4 44 4 4 44 0 40-N?-09 NN O C m4Cft cO OmOD C m O M P N O D D m

• 4 2

"scrNNN :- O W M 2 0030000 0 0 00 = 0 00000000000 g + + + + + + + + + + + + a d a444 3aJ a D a g o rnnm ucom g C N e-eN omcGeu U C PmNOr% DmNOD P ?? L ?. '. 3 2 2 C. D. D % 4 Og a 00000000000 0 000 0 000 3 000 Z + +. + + +, + + + + + a vwwwwaww_ww g c omOuN e-emom 3

• BNDB N3% 4 NO d

C OA fNM O?N CA # g O ". ?=>@ ===nx=* g a I D u 0 00000000000 0 33 W 3 3,00000000 + + + + + + + + + + + a N w awwwwwwwwwo O mc&mOe-u% cN O ffmMNNNM wd W 0 N OA4mNM OFDN S m 22.&e77&7 a o.t W. 0 00000000000 C 00000000000 + + + + + + + + ++++ b uJ W w 4 uJ W u L;J w W g. uJ C CDNT~4kN tan N C TCM.%4D4%O M N M C NN OOfmNw CO? Q N. TT?.??????? n 4 H ^ ~ ~C00000000 OOOOOOOQOOO g + + + + + + + ++++ 4W4444wAW44 mCON #OfCCmN

f. MOO 4m040mmN mOCCNOfcamo m4 U D D D r ~* m * -

e... - a 13 308 24 I 9 e 2 200000000o0 0 00000 00020e + + + + + + + + + + + + 4 4 i.d 4 44 4 4 44 u O 4? DP 4 PPPR ?O C -ONONOCNO-? O D-O-N m? D-D4 2 D DN N O2bbb< < .......... e W o ao20 0 ocacoo r] C 00000000000 + + + + + + ++++++ 4 .-.a a a a w w a a a.a.a O D? #? 9 0 ?O o? D U O N~e-O?e-me-2 -N

1. -41-O OM O N

N ?. D. D. N D. D. a D ? D OW 0 00000 : 00o00 o cocoocoonco + + + + + + + + + +++ g w wwwwwww_-- a, c-o#9 oreno m M o o Do47 4 o% # No """*""***~' O 4 O.

x.. D x x D s

-N~ z p u o oOOO Dococco 3 N 3 o 30000000000 + + + + + + + + + +++ g w wwwwawawwaw O N NON -DDNDD 'n o ?V DN'. Vo?-N# b o # N o " .'.f N o ? N A m - x1

o. n D.~

~ ~ sq 6 0 0000n000000 C ocoom o0000o + + + $ + +++++++ ad d4wdw&oD4w N O'DCCDk-NDD?< a O ewmeemoo<em O P 2 nod %D4NOT y

n. :.172 c o. n n. o n N 8

--ooooooooo 00000000000 d + + + + + + + + + + + .a.u a w.a a a x u a w 900N #o#c4MN X MOO 4-o#OMMN moo 4N o#O4mo wwD4AA#19 m9 ........ e. e 12 3C9 25 0 00000000000 O 000 OOC 00O00 + + + + + + + + + + + + a aaada Jaadua O nw JN w dM #NmJ O ##DN " DOD # a~ O CM f4mN mV #ON O N N con ##mmmN Es.1 a coccooccooc w 2 00 000000000 + + + + + + + + + + ++ H* s 4 4 4 s.J J 4 4 J Lad a 4 o n amO nDm anvP U O P P2m?t?"DH T g O C" CN% M 72NTP W D. n. u. k. D. D. D P N D# # OW v o o0000000000 o ccoCocooD oo + + + + + + + + + +++ b o usswwawawa-m o m 4 mDAc?nNN N d 6 o N #N MN fwoos 9 O O O ceNeoN? cmON W J & mxNNN 4JO4A = C D< 0 00o00000000 W 3 2 o00o000o00o A 7, + + + + + + + + + + + + g w w a w w a tu a w w a w k o NecMCOON M NT C MNN9 4 N O9Vm4 H O sro.moNeN n# m ecuecNNN o y, .ce b 0 0000000000o 0 00000000000 5 + + + + + + + +++++ g h aa4ww.uwADwD O e-NONOmOm-= A O CM PM Dm@h#mM Q O m"%DmM BC#No 4 N 7. ? c J CVNNN NN g ~ ~o0000o000 00000000OOo D + + + + + + +++++ awwmaawawa4 MOON #c#c4mN X MOO 4M O#OmmN M oO4NO#C OMO MM cJOA ## mm9 14T'e ~ 0 $ ,Q t t L.v 26 2 o0o0c000000 0 000000o0oC 0 . + + + + + +.... a d a d a4 4 44 44 4 o N # 9 4-wNoo?N 0 9N&DD TN NN 44 2 0A n4047?M M 4 4 % CD4 4 m N N N ". ". W= c ococococo30 m 2 o000000 0000 .... +.... +. g a aa-aaa-aaaa o% m D.' O 0% D% cN N o DNmN PDok % cb g o -DOM o9 7 #CN 9 4 D. D. D. P. 44 4. m m ? D% O Ng c ocococcocco 2 o. 0 0 o. 2 0 0 a 000 +..... E awwwwawowww _o Omoy, <,oD o< d O o ON H O9 N 9 00 C# ~ O 4 2D% % D 2 BDDT4 o c-~N orocm? O k o, .... e...... C 0 00o0000o000 N N o 00

a. 3 3 0 0 0

o. 0 c.

A > + + + + m v owwwawwwwww 0 ?CMOONN#NNm O D4NQ47DnNNm O o ?D4 D #O% #H MD m DDD% NN 4O CDD X O' 0 o0000000000 C o0000000000 + + + + +.. + + b w wwwwwwwwWwW 0 %TM NNNTDM 4D y C eomeDDPNOmN A O "O4M D DNTN 4" (C

1. oOoNN N O44O N.

4 B ~ ~ ococoeoco o0c00 Q00000 D + + + + + + +. +. wwamwwawwww 9 CON IOfoO9N X MOO 4d C#OmmN moccNoecemo 4 w D CDr## 9m 9 1,, 27 Nonseparable Case v coA .b Ni \\ Separable Case u M~ 1 1 I i i i i i i e i i i I .5

1. 0

1. 5 x 10-S(r,0 ) Crashes per Square Mile o

o I FIGURE 7.1. CRASH DENSITY AT TMI LOCATION LANDINGS 12' ~ 312 28 Nonseparable Case I b E Q b 3 c5 Separable Case u4 i i i i i i e i e i i i i I -4 .5

1. 0

1. 5 x 10 S(r,0 ) Crashes per Square Mile c

o FIGURE 7. 2. CRASH DENSITY AT TMI LOC ATION TAKEOFFS 12 3*3 29 8. SUBQUESTION 3. TIME DEPENDENCE This question relates to the possibility that the spatial distribution of crashes is itself time dependent, so that we would have nonseparability of space and time. In principle this question can be pursued in the same way as the spatial nonsepar ability by now allowing R and O to be functions of time also. This could be done computationally, for example, by adding terms linear in time to the decay constants A,4, and then setting the coefficient of these linear terms 'oy Bayes' theorem using the crash data. The computational work however would now become quite exten-sive. More important is the fact that there have been simp 1v too few crashes to allow any meaningful inference to be made on the time depen-spatial shape. An attempt to do so would therefore not yield dence m useful results in our opinion, and so we dc not pursue this approach any furthe r at this time. 9. CONCLUSION In summary our answer to the Board's question is as follows: a. If fitting functions are chosen which are separable in r and 0 it follows that our probability curves for the r and 0 derivatives can be combined as independent distributions. b. The actual data on crash location is reasonably in accord with the postulate of separability. This was the conclusion of the NRC statisticians, and is also supported by the Bayesian [ analysis of the present testimony, as reflected in the final curves of Figures 7. I and 7. 2. t c. Scatter plots of the crash points do not reveal to the eye any obvious time dependence of the spatial crash distribution. 10. REFERENCES [1] U. S. Nuclear Regulatory Commission, Atomic Safety & Licensing Appeal Board Decision, July 19, 1978, Docket No. 50-320, page 24, Item (3). [2] Li dley, D. V., " Introduction to Probability & Statistics From n a Bayesian Viewpoint," Cambridge University Press, 1970, Part 2, page 15. & ~ W.6 30 ~ N tN 2 180 ~ / \\. \\ I N / i x \\ 0 g i40 j [ . 220 i40o x \\ X \\ / i \\ go \\ / / / 130 \\ \\ -f / 230 130 \\~ \\ s s\\ / \\ N' \\ //esr x g

,K

\\il N x / \\ 4 110 2w' / 250* \\ 110' \\' \\ N U/,/ 5 5 MK i x. s x \\ 'r = 1 MI L'E o, 0 70* 290 [ h 2 MILE / 3:0 30, a 50 3 MILE 310 \\ / / / 31 POINTS TOTAL 4 MILE \\ 16 POINTS r < 0.5M go 320 10 POINTS r = 0 320 / 5 MILE 3# 3:0*

.50 0

10 20 30 30 20" 10 350* 340 330" FIGURE 8.1. SCATTER PATTERN FOR LANDING ACCIDENTS (1956-1964) 1.'t'3*6 31 0 180 O g 140* \\

f'NN Ol X

\\ \\ *j / s s 2., 7 ,m. s 8 'x / \\ ,\\ 2Wo / 110 110 \\ 2% 0 \\ / 5 'y \\1

• b

^ 2 = i o = r = 1 MILE \\ f 70* 290* h ( 2 MILE \\ 3 MILE .? 33 POINTS TOTAL ,17 POINTS r < 0.5M ' g 13 POINTS r = 0 5 MILE 5 FIGURE 8. 2. SCATTER PATTERN FOR LANDING ACCIDENTS (1965-1977) 13 ~ 33 32 k 2N3 180 \\ < l / / s \\ y \\ i i ,/ f 140* 0 \\ 220 x \\ s 1 / / 'a /* \\ \\ j \\ / / N 'N \\ x' \\ \\ \\ 12, ; x 1 \\ / [ '\\ SI E 120 240 240 0 .y j 120 s \\ \\ \\ { ,/,. \\ / 15 / N /

g g

s t 25 >g . lh = = 2g {, j g v r'= 1 MIL 7# 290* / \\ h \\ 2 MILE \\ l l / / 3if 500 5# 3 MILE 310* 320 -/ 17 POINTS TOTAL 4 MILE 6 POINTS r < 0.5M 4ca 'O 5 POINTS r = 0 320 _ 5 MILE 3 0 3 3J0 340 3W 0 10 20 30 0 0 3 30 20 10 350 340 330 FIG URE 8. 3. SCATTER PATTERN FOR TAKEOFF ACCIDENTS (1956-1964) 33 e e e 210 200 190 170 160' 150* 150 160 170* 180 190 200* 210 0 0 / / x s i 140 's N -/ / / \\ \\g 0 ' 220: ,.0 ,,0o I / 230 \\ k 130 0 130 \\ / / 230 1 / \\ \\ 2:::/ N \\\\ hee 8 \\ I ' \\ \\ \\ }l / N \\ x x ~ 0 0 100 260 0 jgo i 260 < '/ 0 0 E 90 270 270 80* Xil / } 280* I 80* 0 a0 280 \\ r = 1 MIL $ 70* 2900 h f 2 MILE o ,n \\ 0 310 50 50' 3 MILE 3'o \\ 16 POINTS TOTAL N 4 MILE 13 POINTS r < 0.5M \\ o 320 4a \\ 12 POINTS r = 0 320o 40 / s u ltE 330 340* 350 0 10 20" 30" 0 0 30 20 10" 350 340 330 0 0 0 FIGURE 8. 4. SCATTER PATTERN FOR TAKEOFF ACCIDENTS (1965-19 7) 12~338 34 [3] Benjamin, J. R. and C. A. Cornell, " Probability, Statistics and Decision for Civil Engineers," McGraw-Hill, 1970. [4] Snedecor, G. W. and W. G. Cochran, " Statistical Methods, " Sixth Edition, Iowa State University Press, 1967. 1'?~3t.9 35 APPENDIX From (7. 2) and (7. 5) and dropping subscripts for simplicity we evaluate the likelihood as follows: A) For r 20. 5 mile and 0 > 0, we have f(a A) e (A.1) S(r,0) (a ) e = i nis is the crash density per square mile and therefore the likelihcod of experiencing the set of crash points N (r,0 " " n=1 is proportional to the product: N -Ar -p0 ] [ (aA) e " (a ) e n=1 N N -y[e" -A r" 1 (aA)N (ap)N (A. 2) 1 1 =- e e t @] r \\ l n ( j B) For r 2 0. 5 and 0 =0, since a + b[1+7(r-2. 5)] (B.1) lim O(0, r) = 6- 0+ and O(0, r)

1. 0 (B. 2)

= we have O(0, r) l = -{1 - a - b[1+7(r-2. 5)]}6(0) (B. 3) 0=0 12 320 A-1 where 6 (0) is the Dirac delta. Therefore 1 -Ar -(aA) e 1 -a-b[1+y(r-2. 5)] 6(0) S(r,0) = r +1 ~ # ae bY6(0) r Ar by 1-(a A) e (1 -a-b) + 7[1 -A(r-2. 5)] 6(0) (B.4) = Therefore the likelihood of experiencing the set of crash points ,0) M (B. 5) r* m=1 is proportional to the product: M -Ar* ] (a A) e (1 -a-b) + [1-A(r

2. 5)]

m= 1 M 1 l A [1-A(r -2. 5)])I -A

1. m M

I 1 (aA)M (1 -a-b)M ] g,g m e 1+ g M j TI r m= 1 1J kl m) (B. 6) i C) At r=0 we similarly have a delta function S(r,0) = (1-a)6(r) C.1) and therefore 'he likelihood of experiencing ( crashes at r=0 is (1-a)5 (C. 2) D) For crasbes in the range 0< r< 0. 5 mile we regard the angle as not being measured meaningfully. Thus we simply ;onsider that we have no e values for these points. The probability of a strike at a value r in this range is therefore simply 1 9.' . T 4-A-2 A (r) (a A) e ' # (D.1) R ~ = dr H and therefore the likelihood of experiencing the set of crash radii r is proportional to: h=1 H -A r" (aA)H (D. 2) 1 e E) For r 2 0. 5 and 0 =r/2, since = ae + b[1+a(r-2. 5)] (E.1) lim O(0, r) 0-f-and lim O(e, r)

0. 0 (E. 2)

= e-p + we have 8 a + b[1+7(r-2. 5)] 5(0 7) (E. 3) g O(0, r) = - ae 0=r/2 and n ~N 1 -Ar 2 7 + b[1+7(r-2. 5)] 5(0 7) S(r, r/2) = - (aA) e ae 1 ae" by 5(0-7/2) r = ^ e'# ae +b [1-A(r-2. 5)] 5(0-f) ~ (E.4) 12 ~ 322 A-3 Therefore the likelihood of experiencing the set of crash points: f(r, r/2 ) (E.5) v=1 is proportional to: V - A b#v V r (LA)V 1[ ae +b -[1 - A(r -2.5)] 1 FI 2 by e V v=1 (E.6) F) Thus, putting ( A. 2), (11. 6 ), (C. 2), (D 2), and (E. 6) together we see that the likelihood of experiencing the full set of crash data is proportional to N M H V )_ -{r + {r + {r -A ~r (1 - a)((a A)N+M+H+V 1 1 1 1 e x t N -4 On (1 -a-b)M (ag)N 1 e M ba V 7 - [1- (r -2. 5)] h -g-b b b#[1-A(r -2. 5)] 1x x ae +b x (1- -b) A y J{ rn=1 v=1 (F.1) which are the four terms given in (7. 6) through (7.10). ig 323 A -4 March 20, 1979 UNITED STATES OF AMERICA NUCLEAR REGULATORY COMMISSION BEFORE THE ATOMIC SAFETY AND LICENSING APPEAL BOARD In the Matter of ) ) METROPOLITAN EDISON COMPANY, ) Docket No. 50-320 et al ) ) (Three Mile Island Nuclear ) Generatina Station, Unit 2) ) CERTIFICATE OF SERVICE I hereby certify that copies of " Supplemental Testimony of John M. Vallance," dated March 20, 1979, and " Supplemental Testimony of Dr. Stanley Kaplan," dated March 20, 19 9, were served upon those persons listed on the attached Service List this 20th day of March, 19'i9, Those persons whose names are marked with an asterisk were served personally, and those unmarked were served by deposit in the United States mail, postage prepaid. 8 AT.an J. eisbald Dated: March 20, 1979 1lC3M -. s = C,NIe S.ve=,m_S C.p 3.* m C*4 -w s.'sCr-~ v u w e.: u--, r e..v Cevu a~a rswd o Eefere che Accric Safe-7 and Licensine Acceal Ecard In the Matter of ) ) ME"'.RCPCL' TAN EOI5CN CC:GA2r?, ec a'. ) Occhec Nc. 50-320 ) ("'hree Mi' a 'C ' d Nuclear Statica, ) Unit 2) ) _SEWICE LIST

• Alan S. Ecsenthal, Esq.,

C'- ' i - '

• "'.awrence J. Chandler, Esq.

Ac--ic Safe 7 and Licensing Appeal Office of the Executive Legal Direc:0 Ecard U.S. Nuclea-Regniaccry C: " s s '.c n U.S. Biuclear Regulater1 C. dssien Washing:cn, D.C. 20555 Washing ca, D.C. 20535 Dr. Chauncey R. Kepford

• Dr. W.

Reed Jchnsen, Merher 433 Criando Ave ne Atcmic Safety and Licensing Appeal State Colleca, Pennsylvania 16801 Ecard U.S. Nuclear Regulatory Ccmdssion Karis W. Ca.rter, Esq., Assistant 'ashingten, D.C. 20555 Attorney Genera]. Office of Enforcement

• -Jercme E.

Shar*'-, Esq., Mecher Depa t-.ent of Enviren= ental Reac,: ces At=mic Safe:f and Licensing Appeal 709 Health and Welfare Building Ecard Harrisburg, Pennsylvania 17120 U.S. Nuclear Regulatory Cc:=nissicn Washing:On, D.C. 20555 Atomic Safety and Licensing Appeal 3 card Panel Edward Luton, Esq., chm-'N U.S. Ncclear Regulatcry :cmnissica Atcmic Safety and Licensing Ecard Washing:On, D.C. 20555 U.S. Nuclea-Regulaccry C d ssicn Washing =n, D.C. 2555 Atemdc Safety and Licensing Ecard Panel Mr. Gustave A. Linenberger U.S. Nuclear Regulatcry Cecmissien Atcmic Safet7 and Licensine.,.3 card Washincten, D.C. 20555 U.S. Nuclear Regulatcry Cc==issicn Washington D.C. 20555 Dccheting and Service Section r Office of the Secre:2-J 7 es: C. Salc U.S. Nuclea-Regulaccry Cc i ssien Professer, Fisheries Research Washing cn, D.C. 20535 Institute, WE-10 University of Wasid gren Seattle, Washing cn 98195 I' d} ~ Q 4] d t..t, F}}