CQONELDENFT14L
Result 1. To determine the parameters c and , which characterize the bomb, compute the zeroeth and second moments of the empirical distribution and set
c= V(2/m) - (ub/eB), X= bo/be.
Before considering the joint implications of Eqs. Al and A2it is instructive to note a
possible improvement of the latter in the case of a ground burst. Owing to the so-called
‘crater effect’’ the observed curve of bomb effectiveness will be very flat at the origin.
Nowexp [— (A/2)r?] = 1+ Ofr?), but
[1/(A — x)][A exp (— xr?) — « exp (— Ar*)]
(A4)
= 1+ O(r*) is flatter at the origin and should in this case give a better fit.
If the curveis
fitted by the first and third moments, a particularly simple expression results:
hh = f [1/(A — «)JDA exp (— xr?) — « exp (— Ar’) Jr dr = A E(1/A) + (1/4),
ps = [CAVA «VILA exp (— ert) —« exp (— Art)]* dr = HS [(1/02) + (1/dK) + (1/0
so that
2pa — Bui = 4 ((1/¥*) — (2/Ax) + (1 /e*)],
and from the first and last of these equations,
AK = 1/ (uy V2— 32).
(A5)
By making the coefficients in Eq. A4 independent parameters, instead of having them depend on the exponents, a better fit yet may be realized, at the expense of a more complicated analysis. Equation A2 rather than Eq. A4 will be used throughout the sequel, since
it results in simpler expressions. Any results based on Eq. A2 mayeasily be extended to
the analogous expression for Eq. A4.
Suppose now that a bomb of the form Eq. A2 is delivered according to the distribution Eq. Al. If the bomb lands at (z, y) the proportion of casualties at another point
(€, v) will be
c-exp {— (A/2)[(x — £)? + (y— v)*]};
however, the probability of this occurring is given by Eq. Al, and hence the expected level
of casualties at (£, v) is
c/(2moars)|"{*” exp (— (a*/20b) — (y*/278) — (A/2)L(a— 8+ (y-v)*]} dedy. (AB)
This is typical of the sort of expression that must be evaluated repeatedly in the
course of the problem: the definite integral of the negative exponential of an inhomogene-
ous quadratic form in several variables. Fortunately a single calculation can be done.
Write the integral as
[° exp (— XAX’ — 2LX') dX,
+e
where X = (21, %,---, 2n), - dX means /
positive definite matrix, and L is a (row) vector.
+o
+0
(A7)
.
:
dxi-:-dz,, A is a symmetric
It is well known that for such A there eX-
ists a real nonsingular matrix P such that A = PP’ and hence A! = P’—'P-! and |P| = v/|A|.
ORO—R-17 (App B)
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