Representing Damage Functions in Different Forms

One of the most basic, and accurate ways to show the effect of a weapon on a target is by the Lethal Area Matrix (LAM) which is an outcome of either the General Full Spray Materiel Program (GFSM) or the Joint Mean Area of Effects (JMAE) program. Generating the matrix involves keeping the weapon fixed and moving the target around the ground plane and calculating the probability of damage (PD₁) for each target location. This arrangement is shown below.

Figure 1 Weapon target geometry

If the target locations are ordered into a grid of cells, calculating the PD₁ at each cell center produces the LAM shown below.

In this table only the right hand half of the LAM is shown since is symmetrical about the range axis. If we multiply the PD₁ in each cell by the cell area and sum over the ground plane we obtain the Lethal Area, or Mean area of Effectiveness due to fragments – MAE_F.

                                                       (1)

Plotting contours of constant PD₁ produces the so-called “bug splat” form of the LAM as shown below.

Figure 2 “Bug splat” LAM contours

Also shown in the figure are smoothed contours to the bug splat which have a somewhat Gaussian form, albeit different in range and deflection. This leads to the Carleton Damage function (CDF) representation of damage, which takes the following mathematical form.

                                                                                                                    (2)

where the variables WR_r and WR_d are the weapon radii in range and deflection respectively. This function may be visualized in the following manner.

Figure 3 Carleton damage function (CDF)

For both the LAM and CDF the lethality is spread out over the ground plane however there are some representations known as “cookie cutters” which have a specific boundary inside which PD₁=1 and outside PD₁=0. The most common cookie cutter is rectangular in shape as shown below.

Figure 4 Rectangular cookie cutter damage function

A blast warhead cookie cutter damage function is similar to the one shown above, but is square. In all these representations, the aspect ratio (a) of each damage function has to be the same.

                                                                                       (3)

It turns out that this aspect ratio is a function of the weapon impact angle. For air-to-surface applications, this empirical relationship is given by the following.

                                                                    (4)

For surface-to-surface applications, the equation is different.

                                                                                                                                        (5)

Given the lethal area and impact angle, it is possible to determine specific forms of the Carleton damage function and the rectangular cookie cutter. The blast cookie cutter remains square since it is independent of impact angle.

In all forms of the damage function discussed, the lethal area has to be conserved, hence the MAE_F given by equation (1) must be the same as the integral of the CDF over the ground plane, which in turn must equal the area of the cookie cutter.

                                           (6)

Different JMEM methodologies use different representations of damage function although the basic premise is to use the highest fidelity form available consistent with mathematical tractability and speed of execution.