ML19224A333
| ML19224A333 | |
| Person / Time | |
|---|---|
| Site: | Crane  |
| Issue date: | 11/30/1978 |
| From: | Abramson L, Moore R NRC OFFICE OF MANAGEMENT AND PROGRAM ANALYSIS (MPA) |
| To: | |
| Shared Package | |
| ML19224A328 | List: |
| References | |
| NUDOCS 7812270052 | |
| Download: ML19224A333 (16) | |
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,j' NUCLEAR REGULATORY CCMMISSION BEFORE THE ATOMIC SAFETY AND LICENSING APPEAL 30ARD In the Matter of
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METROPOLITAN EDISON COMPANY, ET AL.
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Docket No. 50-320
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(Three Mile Island Nuclea-Station,
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Unit 2)
)
TESTIMONY OF R. MCORE AND L. ABRAMSGN APPLIED STATISTICS BRANCH OFFICE OF MANAGEMENT AND PROGRAM ANALYS!S U.S. NUCLEAR REGULATORY COMMISSICN NOVEMBER 30, 1978
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AREAL CRASH DENSITY AT T?iI-2 1.
General Information The areal crash density at any point is the probability per square mile per relevant operation of a crash at the point.
A relevant operaticn is an operation which has the potential of impacting the target point.
In this chapter, a general approach for estimating the areal crash density at any point will be developed, and will then be applied using the specific data for TMI-2. A bound on the areal crash density which expresses the uncertainties will also be calculated.
The areal crash density developed here will be based on accident data for all U.S. carrier aircraft.
We use the following notation:
P = probability of an off-runway crash of a commerical aircraft engaged in a rolevant operation D(r,e)
= conditicnal crash density at any point with polar coordinates (r,9)
= probability per square mile of a crash at (r,9) given an off-runway crash of a commercial air-craft engaged in a relevant operation C(r,3)
= areal crash density at (r,9)
= probacility per square mile of a crash at ' r,-) o f a U.S. carrier aircraf t engaged in a relevant operation N ~ 02[3 By the laws of probability, C(r,9) = P D(r,9).
(1)
Since the terms in (1) are generally different for takeoffs and landings and for scheduled and unscheduled flights, we use the following system of subscripts.
P is the crash probability for operation i and type cf jj flight j, where i is either T (takeoff) or L (landing) and j is either S (scheduled) or U (nonscheduled).
Thus (1) becomes jj j(r,9).
(2)
C jj(r,e) = P D
Note that we have assumed that the crash dansity 3(r,9) depends only cr, the type of operation and not on the typ flight.
2.
Off-runway Crash Probabliities The historical off-runway crash rates for all U.S. carrier aircraft for the period 1956-1977 are tabulated in Talsle I.
(The numbers of operations and hits are taken frcm Table 9 in Read et al Testimony).
SCHEDULED li0N-SCHEDULED
' CPERATICNS' HITS! RATE (X10^5)OPERATIONS l HITS ' RATE (X10-6) i (X10 )
l (X10 )
l 0
6 I
1 l
tar:EOFFS 86.3 11 0.13 2.36 2 l 0.85 l
LANDINGS 86.3 25 0.29 2.36 13 5.5i Table I - Off-runway destruct accidents for all U.S carrier aircraft, 1956 - 1977 a
3.
Off-Runway Crash Densities The observed lccations of off-runway crashes for all U.S. carrier aircraft for the period 1956-1977 are plotted in Ficure 1 of Read et al. Testimony tabulated in Table 9.
Before using this data for estimating the crash densities, we test whether tne distance e and angle 3 are statistically independent.
To do this, it is necessary to aggregate the data in Table 9 in order to increase the frequencies in each cell.
The results are presented in Table II.
(The different intervals for e used for takeoffs and landings were chosen to represent the specific data for each.)
TAKEOFFS NI 0 U 0 -5
!20c-500 l90'-100 l
! 0-1 ! 2 f2.0, I 2 (2.23 I? /i
' l-2 l1 (1.0) 2 li.2) ! O f0.3) l2 i 2-2 l 0 (0.~' (0.:' l
/ 0. 5' l2i 3a i J (0. 3) i 1 (0.21 l 0 f0.3' l'j i J-i 2 R.0) } O (i.2) I 1 (0.E' I 3 I
!5
!c l2 l15 LANDING 5
! N I C -10c 0
0 I gso-i n o l l10-30 0
0-1 I 72 (11.5T I a (3.33 !1 (1.7' I'-
I l 6 (6.8) ! 2 (2.3) l 2 (1.0) I 10
(:..4.
Ie r,..o s !
- r. n.. '.
} 7_, ;
i (0.9) {0(0.;)
l 3-a 3
l2.7) i
!2 i
i (0.73 i 0 (0.2' 0 ( 0. i
i Tacle II.
Freauencies of off-runway hits for all U.S. carrier aircraft, 1956-1977
('iumcers in parentheses are expected numbers of hits under independence)
. --nom af O."- 4
The numbers in parentheses are the expected numbers of hits if the dis-tance r of a hit is statistically independent of its angle 9 By inspection, the expected frequencies agree very well with the observed frequencies (a chi-square test would be highly non-significant).
Accord-ingly, we shall assume that r and e for off-runway hits are distributed independently.
By definition, the crash density D(r,5), given an off-runway nit.
Con-sider the shaded differential region with sides dr and rds and with area rdrds, where e is measured in radians (see Figure 2).
If dQ is the probability of a hit in that region, then d
D(r,9) = rcra 9 Now let g(r) and h(9) be the crash densitiec of r and e, rescect ely, conditional on an off-runway crash.
By the assumed independence of r and 9, dQ = [g(r)dr] - [h(9)di].
It follows that D(r,9) = g(r) h(9),
(3) r
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FIGURE 2 DIFFERENTIAL HIT REGI0tl Scurce: Testimony of John M. Vallance Figure 9
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- 3. '.S There are two approaches to estimating g(r) and h(9).
The first, which has been taken by the applicant, is to fit the observations to a model which is consonant with the observed data.
The second, wnich we will adopt, is to assume a uniform density in the vicinity of the point of interest and then estimate its value directly from the observed number of hits.
This approach has the advantages of not relying on any assumed model or curve fitting and of yielding confidence limits for the densities.
For any point (r,0), the procedure for estimating g(r) and h(9) is as follows.
To estimate g(r) for landings consider the interval in Table 9B of Re2.d et al. Testimony which contains r and the intervals on either side.
If x is the total number of hits in these three intervals, then we estimate gL(r) by [0,where40isthetotalnumberofoff-runway landing accidents.
The estimate of gT(r) for takeoffs is calculated in a similar manner using the total number of hits in the three intervals surrounding r in Table 9 of Read et al. Testimony.
To estimate h(3) for landings, use the same procedure as above witn the data from Table 9B of Read et al. Testimony, but divide the answer by 2.
This division by 2 stems from the assuttption that, since planes tend to make a straight-in approach on landing, they have the same probability of crashing to the left as to the right.
Hence the angular density obtained from Table 98 of Read et al. Testimony where all the angles are taDulated in one quadrant is double the density h (9).
To estimate h(9) for takeoffs, use the same procedure as above with the data from Table 9A but do not divide the answer by 2.
The resultant estimate of 3 339 of h ( ) is conservative.
It would be unbiased if planes always took T
off to the left or the right and then always crashed in the quadrant over which they took off.
Although this is r.ot the case, no credit is t ken in the estinate of h (S)*
T We now estimate g(r) and h(s) at TMI-2, i.e., for r = r = 2.7 and 9 =
g U
34.
From Table 9A of Read et al. Testimony for takeoffs we assume e =
g that g(r) is uniform for 2 1r1 3.5 with 3 hits in this interval and that h(9) is uniform for 25' 191 40 with 2 hits in this interval.
The estimate values are then 3
gT(r ) = TI G =.13 per mile,
g 2
180 n (3 )
- IT ' 15 =.51 per eidian.
t 0 For landings, from Table 9B of Read et al. Tastimony, we assume that g(r) is uniform for 2.0 1r1 3.5 with 9 hits in th's interval and h(9) is uniform for 25' < s < 40* with 3 hits in this interval.
The estimated values are then 9
1 gL(#o) * ~40 ' l.5 =.15 per mile, f=.14perradian.
h (e ) =
g
~'s^331 From (3), the estimates of the conditional crash densities for takeoffs and landings at TVI-2 are then D (#o'i)=(
)
) =.075 per square mile, T
o D (r, s ) = (.15 )l4)=.0078persquaremile.
L g g
From (2), the estimated areal crash densities are calculated by multiplying the estimated conditional crash densities by the crash rates frcm Table I.
The results are presented in Table III.
SCHEDULED NONSCHEDULED TAKE0FFS 3.3 x 10"9 2.1 x 10-8
-9 LANDINGS 2.3 x 10 4.4 x 10-9 Table III.
Estimated areal crash densities at TMI-2 for U.S. carrier aircraft engaged in a relevant operation (probability cer square mile)
The uncertainties in the estimated areal crash densities in Table III can be characterized by confidence limits.
The procedure for generating these confidence limits is discussed in the Append:x and results are presented in Table IV for three cases.
The particJlar confidence levels of 70%, 855 and 97'.' result from the choice of 905, 955 and 995 corFidence levels, respectively, for each of the factors entering into the estimates of the areal crash densities.
Note that the confidence levels given in Table IV are conservative.
- f C02 SCHEDULED l
Confidence Limits (X10-9) f 975 f
ESTIMATED 70%
85%
-9 3.3 x 10
<29 l <42
<88 l
f TAKEOFFS LANDINGS 2.3 x 10
<11
<l5
<25 i
-9 i
l l
NONSCHEDULED l
l l
Confidence Limits (X10-8) l85%
975 ESTIMATED 7 0
TAKEOFFS l
2.1 x 10-0
<34 l<55 l
<l26
-0 LANDINGS l
4.4 x 10
<22 l<31 l
<57 i_
Table IV.
Areai crash densities at TMI-2 for U.S. carrier aircraft engaged in a relevant operation (probability per sauare mile)
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Lt
- y
APPENDIX ON CONFIDENCE LIMITS The treatment of data arising from cour. ting failures is often based upon the following binomial assumptions:
( a )' There are T " identical" trials.
(b)
Each trial results in one of two outcomes:
" failure" or " success" (c) The probability of failure is the same for each trial.
(d)
The trials are independent.
Upon completion of the T trials, the number of failures i'. Counted.
Let x denote the number of observed failures.
Then x takes on one of the values 0, 1, 2, --, T, Suppose p denotes the probability of failure in this binomial model.
Then the data -- namely x and T -- provide an unbiased estimate of the binomial parameter p by setting p=f.
(A1)
Moreover, an upper one-sided confidence limit with confidence coefficient (1 - a) is obtained from
_ (x + 1)f)_,(2(x + 1), 2(T - x))
(A2)
~
p T-x+(x+1)F),j2(x*1),2(T-x))
'l~3.?4 where F),,(2(x + 1), 2(T - x)) is the 100 (1 - a)th percentile of the F distribution with 2(x + 1) and 2(T - x) degrees of freedom.
An inter-pretation of p is as follows.
If the true value of p r eeds p, then the probability of observing x or fewer failures in T trials is less than a.
In other words, if 2.is'small and if p exceeds p, then an improbable event has occurred.
Hence an upper confidence limit for p is often interpreted as an effective upper bound for p.
When the number of trials T is large compared with the number of failures x, the formula (A2) can be considerably simplified by using the Poisson approximation to the binomial distribution.
The confidence limit can then be approximated by 2x, (2(x + 1))
_pO (A3) 2T where the symbcle means "approximately equal to" and x (2(x + 1))
is the 100(1 - 2)th percentile of the chi-scuared distribution with 2(x + 1) degrees of freedom.
[With some change of notation to deal with the airplane crash data at hand, all of this is based on a development given by K. A. Brownlee, Statis tical Theory and Methodolocy in Science and Encineerina, 2nd edition, Wiley, 1965; see especially Section 3.5 and 3.14. ]
Of course, the formula (A3) can also be used wnen the subject, rather than being binomial, is tha; of dealing with " numb!rs of occurrences
% ' ' O.93
. in T units of experience" and the Poisson distribution is directly applicable.
In this case, however, the Poisson parameter is not con-strained to lie between 0 and 1 and is to be considered a " rate of occurrence per unit of experience" instead of a probability as in the binomial case from the Poisson case so that clear lines of argument can be preser:ted.
Table Al is a table of upper 90%, 95% and 99% confidence limits for pT for x = 0(1)30(5)50, based on the Poisson approximation.
An upper confidence limit for p can be calculated by dividing the tabular entries by 1.
For example, supcose that r = 11 crashes were observed in T =
6 86.3 x 10 o erations.
Then the unbiased estimate of p is 0.13 x 10
'=
=
86.3 x 10 and the 95% upper confidence limit for p is 0.21 x 10
-=
=
86.
x 10
[This example is based on the data for scheduled takeoffs (cf. Table I), which will be used for illustration throughout this Appendix.]
2 Ci%
UPPER CONFIDENCE LIMITS FOR pT x
S0t 95%
995 0
2.30 3.00 4.61 1
3.89 4.74 6.64 2
5.32 6.30 8.41 3
6 98 7.75 10.05 4
7.9' 9.15 11.60 5
9.27 1 0.51 13.11 6
10.5 11.8 14.6 7
11.8 13.1 16.0 8
13.0 14.4 17.4 l
9 14.2 15.7 18.8 10 15.4 17.0 20.1 l
11 16.6 18.2 21.5 12 17.8 19.4 22.8 13 19.0 20.7 24.1 14 20.1 21.9 25.4 15 21.3 23.1 26.7 16 22.4 24.3 28.0 17 23.6 25.5 29.3 18 24.8 26.7 30.6 19 25.9 27.9 31.9 20 27.0 29.1 33.1 21 28.2 30.2 34.4 22 29.3 31.4 35.6 23 30.4 32.6 36.8 24 31.6 33.8 38.1 25 32.7 34.9 39.3 26 33.3 36.1 40.5 27 35.0 37.2 41.8 26 36.1 38.4 43.0 29 37.2 39.5 44.2 i
30 38.3 40.7 45.4 35 43.9 46.4 51.4 40 49.4 52.1 57.4 45 54.9 57.7 63.2 50 60.4 63.3 69.1 Source:
Formula (A3) x = nunber of failures p = pecbability of a failure T = number of trials TABLE Al.
UPPER CONFIDENCE LIMITS FOR THE EXPECTATION CF A POISSON 5
VARIABLE 3
r.i h
I The Poisson approximation to the binomial distribution is an excellent one for the numbers of operations and hits in this analysis.
Hence Table Al yields accurate confidence bounds for off-runway accident rates.
For values of T as small as those used to estimate the conditional crash density D(r,9), the Poisson approximation yields confidence limits which are larger than the exact binomial values.
Hence, Table Al can be used to calculate conservative upper.onfidence limits for the densities g(r) and h(s).
As an example, T = 5 for takeoffs ind x = 3 for estimating gT(Fo) and x = 2 for estimating h ( 0).
From Table Al, con-servative 95'; upper T
confidence limits for gT(r ) and h (r ) are g
T g gT("o) #
55
' l.5 =.34 per mile, Ii (00) < jf
= 1.60 per radian.
T
[The exact 95". upper confidence limit for gT (r ) from A2 is.29.]
g Since the areal crash density C(r, e) is a product of three factors, an upper confidence limit for C(r, e) can be calculated by multiplying the upper confidence limits for each of the three factors.
If the confidence level for each of the factors is 100(1 - a)% then it can be shown that the product is a conservative IC0(1-3a)% conficence limit for C(r, 9)
[cf. Multivariate Statistical Methods by Donald F. Morrison, 2nd edition, McGraw-Hi.,1976, page 33.] For example, if 95% upper confidence limits are used for each of the factors, the product is a conservative 85", uoper v
f'
~
h e,og conficence limit.
Thus, a conservative 85". upper confidence limit for the area density at TMI-2 for..
aled takeoffs is C
-b) (.34)(1.6)
TL (r s ) = (.21 x 10 g,
g 2.7
-8
= 4.2 x 10
[The factor of 2.7 came from the equation (3) for D(r,3).]
..y29
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