Thedifficulties here lie in the fact that the surfaces of population distribution anddis-
tribution of bomb damage have not been made to mesh properly. For example, in the case
of one bomb drop the ideal fit for the population of the city would be best at the aiming
point and so weighted as to give diminishing importance to points farther out, in accordance
with Result 2. To this extent, then, the three components of the problem discussed at the
outset are not independent. They will of course be treated as if they were, and the city
will be fitted only once with a binormal surface; thus circumspection must be exercised in
the use of the formulas if they are not to be pushed beyond their proper range of validity.
Since the expected level of casualties springs from integrating Result 2 over the entire
city, inaccuracies of fit should be smoothed out in the process.
The numerical details in-
volved in getting the fit to the city are postponed until the next section; assume now that
it has been obtained and that the map has been rotated so that the principal axes of the
city are aligned r-w and n-S,i.e., the city is the frame of reference. If its standard devia-
tions are o, and 1, its distribution function is
(1/2ae.r.) exp [— (x?/20?) ~ (y?/272)].
(A9)
Assume for simplicity that a bomb is aimed at its center, with standard deviations os
and 7s, and tilted through an angle w with respect to the urban population (see Fig. Al).
Determining the effectiveness of this attack, although a straightforward application of
Eq. A8, involves some rather lengthy algebra; only the answeris giverhere.
Bomb pattern
City pattern
Fig. Al — Relation of Bomb Pattern to City Pattern when Bomb
Is Aimed at Center of City
Result 3. If one bomb characterized by Eq. A2 is delivered according to the setup
pictured in Fig. Al the expected level of casualties is
e+ {{1+ A(oR + 2)101 + (7B + 72] + M203 — 72) (02 — 72). sin? w}
It is perhaps worth noting that when oz > Ts and g, > Tf,this result is greater than
e+ {{1+ A(oR + 72)J01 + A(73 + 6)J}
(A10)
c+ {[1 + M(oR+ 02)]C1 + A(72 + 72)]}
(All)
and less than
ORO—R-—17 (App B)
,
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