Shooter Games and Chance, Part 2

In my most recent blog, I talked about Precision Engagements in the blog titled "Shooter Games and Chance." The following, which is an excerpt from my book, Fundamentals of Combat Modeling, provides an example.

1.7 Direct Fire Accuracy Example

In this section, we want to determine the single-shot accuracy of a given munition against a given target with Monte-Carlo evaluation using bias and dispersion. To do this, we would need to consult AMSAA or JMEMS tables for µ_x, µ_y, σ_x, and σ_y for the given munition and situation. Next we would follow the following procedure:

1. Generate two random numbers: X ~ N(µ_x, σ_x) , Y ~ N(µ_y, σ_y)
2. Compare (X, Y) with target geometry to determine if round hit target.
3. If goal is to assess whether a given shot hit, you are done.
4. If goal is to compute P(Hit) or PH (e.g., to build a table of PH’s), repeat steps 2 and 3 to get a sufficiently large sample; PH is the fraction of rounds that hit.

Consider the infantry fighting vehicle target with frontal profile shown in Figure 1-15.

· A hit in area 1 will produce a firepower kill
· A hit in area 2 will produce a catastrophic kill
· A hit in area 3 will produce a mobility kill
· A hit in other areas will produce no permanent effect

Figure 1-15. Frontal profile of an infantry fighting vehicle target

We will assess the IFV’s vulnerability when engaged with a frontal shot whose impact point is modeled as a random variable pair (X, Y) ~ BVN(0, 0, 0.5, 0.5, 0). Using the list of pseudo random numbers (RNs) in Table 1.1 as needed, we simulate the first round to determine which type of kill, if any, occurs.

Table 1.1. Pseudo-random numbers: 0.8554 0.2287 0.6659 0.8243 0.6840 0.0430 0.8598 0.2381

We proceed by doing a Monte Carlo simulation of the impact point with origin centered on the target, then compare impact point with target profile to calculate where it hit.

Determine X coordinate of impact point (using RN 0.8554):
· Enter the Normal Table with .8554.
· Find Z ^-1 = 1.06.
· Note that Z ^-1 = (x - µ_x)/σ_x.
· Solve for x in 1.06 = (x - 0)/0.5 implying x = 0.53

2. Determine the Y coordinate of the impact point (using RN 0.2287):
· Normal Table goes from .5000 to .9999, but Normal Dist. is symmetric, so compute , and change sign of resulting Y coordinate.
· Interpolating between 0.75 and 0.74, gives Z ^-1 = 0.74.
· Solve for y in 0.74 = (y - 0)/0.05 = -0.37.

3. Assess the impact of the round. In Figure 1-16, the round hits area 4, so no kill is assessed.

Figure 1-16. Target impact assessment