dependencies or correlations is well understood scientifically and mathematically, ^{2} but many DOD models do not do so adequately. In what follows we give examples of such effects from other domains, primarily to illustrate their generic character and to thereby increase acceptance of the need to address them.
In the mid-1970s, an important military issue was whether emerging Soviet ICBMs would be able to destroy U.S. Minuteman ICBMs in their silos. The answer depended heavily on the effects of targeting a given silo with two (or more) reentry vehicles, since neither accuracies nor reliabilities were high enough to assure high probabilities of kill with a single RV.
The naive calculation was to assess the probability of a silo's destruction D by n RVs as follows:
D = 1 - (1 - RP_{k})^{n},
where R is the reliability of a single RV and P_{k} is the single-shot kill probability for a reliable RV attacking a given silo (a function of the RV's accuracy and yield, and the silo's characteristics). The equation treats the RVs as independent. The second term is the probability that n independent RVs fail to destroy the silo.
The first problem with the naive calculation is that the reliability of a given RV is correlated with the reliability of its sister RVs on a given ICBM: the principal failure mode was not in fact the RV, but the missile. Thus, if a missile failed, all of its RVs would fail. As a result, it was often assumed that a nuclear attack plan would “cross-target” weapons so that a given silo would be attacked by RVs coming from different missiles. In that case, for two RVs the equation would be
D = 1 - (1 - RP_{k})(1 - RP_{k}).
In reality, the problem is much more complicated because the effects of the successive RVs are not independent: the first RV, if it arrives and detonates, may create shock waves and send dirt and other debris into the air through which the second RV must penetrate. On the other hand, partial damage from the first RV may reduce the strength of the silo to a second, and so on. On a larger scale,
^{2 } |
In statistical mechanics, the term “correlations” is often used to mean what we refer to here as “probabilistic dependencies.” In some fields, however, “correlations” refer to only a subset of the many possible dependencies. Thus, we have avoided the term here. |