Average Damage Caused by Multiple Weapons against an Area Target of Normally Distributed Elements

This paper investigates the effect of launching multiple weapons against an area target of normally distributed elements. We provide an analytical form of the average damage fraction and then apply it to obtain optimal aimpoints. To facilitate the computational efforts in practice, we also consider optimizations over given constrained patterns of aimpoints. Finally, we derive scaling laws for optimal aimpoints and optimal damage fraction with respect to the radius of the area target.


Introduction
The theory of firing, which mainly concerns aiming, kill probability and allocation of munitions, was inspired by World War II and has been progressed significantly in the past decades [1]. A brief history of firing theory can be found in Washburn and Kress's book [2] where the authors also presented a detailed discussion on shooting without feedback or with feedback. Another good reference on weaponeering is given by Driels [3].
In this paper we are interested in studying the effect of precision-guided munitions such as Excaliburs. These coordinate-seeking munitions are usually guided by radio, radar, or laser and launched by a cannon. They are intended to hit a target accurately and cause minimal collateral damage to civilians, friendly result from inaccuracies associated with identifying a target's location. In contrast, the ballistic dispersion errors are caused by random weapons effects, which may vary from one weapon to another and are assumed to be independent from shot to shot. When a single weapon is fired, it is natural to aim it at the expected center of the target. However, when multiple weapons are launched against a unitary target, the probability of damaging the target can be improved significantly by spreading the aimpoints around the target and the optimal distribution of aimpoints has been investigated in [4] and [5]. Our goal here is to extend our previous studies to estimate the probability of destroying an area target of normally distributed elements with multiple weapons. We will seek optimal aimpoints for various number of weapons.
The plan of this paper is to first review our previous analytical results for the case of multiple weapons against a single target in Section 2. Section 3 introduces the mathematical problem of multiple weapons being released against an area target consisting of normally distributed elements. Exact solution for the average damage fraction is then derived. Section 4 calculates the optimal aiming points and examines the relation among the radius of area target, the number of weapons and the optimal (maximum) damage fraction. In addition to the unconstrained overall optimization of the damage fraction, we also study empirical, fast and robust constrained optimization over several prescribed patterns. The goal is to reduce the computational complexity of optimization and to compute a set of nearly optimal aimpoints efficiently. Section 5 provides scaling laws for optimal aimpoints and optimal damage fraction with respect to the radius of area target. Section 6 highlights conclusions.

Review of Our Previous Analytical Results for the Case of Multiple Weapons against a Single Target
Even though the world is three-dimensional, most targets are known to be on the surface of the Earth and therefore the targets are assumed to be in a two-dimensional ground space. Conventionally, we use two coordinates to define this ground plane: the range direction and the deflection direction. The range direction is defined by the direction of the weapon's velocity vector, whereas the deflection direction is perpendicular to the range direction.
Previously [5] we have studied the case of multiple weapons with both dependent and independent errors against a single target positioned at target = x ( ) 0, 0 . For reader's convenience, we review briefly the mathematical formulation of the problem. Let • j r = the aiming point of weapon j for • Y = the dependent error of M weapons, affecting the impact points of all M weapons uniformly. For example, Y is the part of error associated with identifying the target location incorrectly/inaccurately. We assume Y is a normal random variable. • j X = independent error of weapon j, affecting only the impact point of weapon j individually. For example, j X is the part of error associated with aiming and firing weapon j. We assume that { } , 1, 2, , j j M = X  are normal random variables, independent of each other and independent of normal random variable Y . We model the dependent error Y as a normal random variable with zero mean: σ and 2 σ are standard deviations, respectively, in the range and the deflection directions, which give an indication of the spread of the dependent error in these two directions. We model each independent error j X as a normal random variable with zero mean: The impact point of weapon j is given by We use the Carleton damage function described below to model the probability of killing by an individual weapon. Let Pr target being killed by one weapon at impact point This is the well-known Carleton damage function or the diffuse Gaussian damage function [2]. The two parameters 1 b and 2 b in the Carleton damage function (1) represent the effective weapon radii in the range and deflection directions, respectively.
The probability of a target being killed by the M weapons, averaged over all random errors (i.e., dependent and independent errors), is called the kill probability and is mathematically denoted by ( ) kill target, weapons p M . Note that in the notation for the kill probability, the target identity is explicitly included. This will be very convenient later in the discussion of an area target with multiple target elements, in which we can study the kill probability for each individual element.
With the impact points of the M weapons given by the random variable where function G is defined as , , , , , , , 1, 2, , , , , , Mathematically, , of all factors involving only components in the range direction (i.e., 1 , of all factors involving only components in the deflection direction (i.e., 2 ). Together, Equations This analytical solution will be used in next section to calculate the damage fraction of an area target consisting of normally distributed target elements.

Mathematical Formulation: Multiple Weapons against an Area Target of Normally Distributed Elements
, and Y are the same as defined in Section 2. They are respectively, the aiming point, the independent error, and the dependent error of weapon j.
In this situation, k Z , the location of element k of the area target, is modeled as a normal random variable with zero mean:  are independent of each other, and are independent of j X and Y . Figure 1 shows a sample distribution for an area target of 20 elements normally distributed with 1 2 4 s s = = .
To study the damage fraction caused by the M weapons on the area target, we examine the kill probability of element k caused by the M weapons. The impact point of weapon j relative to element k of the area target is given by where the effective dependent error of the M weapons relative to element k is defined as The kill probability of element k caused by the M weapons (averaged over random independent errors { } , 1, 2, , j j M = X  , over the random dependent error Y , and over the random element location k Z ) is given by   2  2  2  2  1  1  2  2  1  2 1 2   element , weapons   , , , , , , , 1, 2, , where function G is defined in Equations (3), (4) and (5). Notice that in the case of an area target of normally distributed elements, the kill probability of element k has exactly the same form as in the case of a single target at ( ) 0, 0 with the exception that all instances of 2 1 σ be replaced by ( ) Let k χ be the Bernoulli random variable indicating whether or not element k is killed ("1" corresponding to "killed"). The damage fraction (random variable) of the area target is the number of elements killed normalized by the total number of elements.
Expression (9) where shorthand notation G is defined as The standard deviation of damage fraction is Note that this expression for the standard deviation is valid only in the absence of dependent and independent errors, for which we have . When either the independent errors or dependent error or both are present, the standard deviation of damage fraction is larger than the value predicted by applying Equations (11) and (12) with non-zero ( ) . We demonstrate this behavior numerically in Figure 2 and In Figure 3, the radius of area target is fixed at 1 2 10 s s = = ; the damage fraction is affected by the firing error (the total effect of dependent and independent errors; for a single shot, there is no need to distinguish dependent and independent errors). In this case, only the predicted mean of damage fraction is valid. The predicted standard deviation calculated by applying Equations (11) and (12) with non-zero ( ) is invalid since Equations (11) and (12) are derived based on the assumption of zero firing error. The right panel of

Optimal Aiming Points for an Area Target
Next we investigate the optimal aiming points for the case of multiple weapons against an area target of normally distributed elements. We apply MATLAB built-in function "fminsearch" [6] on Formula (9) to find aiming points which produce the largest damage fraction. Technically MATLAB "fminsearch" yields only a local optimum. To find the global optimum, in each optimization we start MATLAB "fminsearch" with 5 random initial vectors. In all cases of our simulations, all 5 random initial vectors lead to the same optimum, indicating that the optimum found is very likely the global optimum.
In this study, in addition to finding the unconstrained overall optimal aiming positions, we also consider optimizations over a set of given constrained patterns of aiming points. The goal is to find simple and efficient "empirical" methods for calculating nearly optimal aiming positions. This approach greatly simplifies the numerical complexity of finding the optimal aiming points at the price of obtaining an approximate optimum. Based on our observations in simulations, computationally the overall optimization is at least two orders of magnitude (100 times) more expensive than the "empirical" optimization over a constrained pattern even when the significant additional cost of starting with multiple random vectors is excluded. In a sequential computing environment, starting the overall optimization with 5 random initial vectors makes the computation 5 We consider 6 constrained patterns of aiming positions as listed below. These constrained patterns are motivated by the results of overall unconstrained optimization, some of which are shown in Figure 6, Figure 7 and Figure 10.
• Pattern A1: M points on an ellipse, uniform in parameter angle. Specifically, the M aiming points are mathematically described by ( ) This constrained pattern has three parameters: 1 θ , R and η , over which we are going to optimize the average damage fraction. Here 1 θ is the parameter angle for weapon 1. R is the effective radius of the ellipse, satisfying area of ellipse π R = whereas η is the aspect ratio of the ellipse, satisfying Consequently, the aiming points on the ellipse can be described as This constrained pattern contains three parameters: 1 φ , R and η . We compare the results of optimization over constrained patterns with those of the overall optimization. Figure 4 plots the difference in optimal (maximum) damage fraction ( opt p ) between constrained and unconstrained optimizations as a function of area target radius (s) for various numbers of weapons (M). For small s, the difference in damage fraction is small among various constraints. This is intuitive and reasonable since for small s, the fatal area of each weapon is capable of covering the whole area target. For moderate to large s, the difference in damage fraction compares the performance of optimization constrained over each given pattern. For 6 M ≤ (Figure 4(a)), the best approximate optimal damage fraction is achieved by distribute aiming points over an ellipse, uniformly in polar angle (Pattern B1 described above). At 7 M = (Figure 4(b)), the best approximate optimal damage fraction is achieved by placing an aiming point at center and placing the rest six aiming points over an ellipse, uniformly in polar angle (Pattern B2). This is also true for 8 M = and 9 M = . As the number of weapons increases, at 10 M = (Figure 4(c)), the best approximate optimal damage fraction is achieved by placing two aiming points on the x-axis  and the rest eight aiming points over an ellipse, uniformly in polar angle (Pattern B3). The constrained optimum over Pattern B3 remains very accurate at 12 M = weapons (Figure 4(d)).
In summary, as M increases, the best pattern of aiming points for obtaining approximately the highest damage fraction goes from Patterns B1 to B2 to B3. This transition is clearly demonstrated in Figure 5 where the difference in optimal damage fraction ( opt p ) between constrained and unconstrained optimizations is shown as a function of M at 150 s = (radius of area target). Figure 6 compares the unconstrained optimal aiming points and the optimal aiming points constrained to Pattern B3, respectively for 10 M = and 12 M = at 150 s = . At 10 M = , the optimal aiming points of Pattern B3 match almost exactly the unconstrained aiming points. At 12 M = , the optimal aiming points of Pattern B3 deviate from the unconstrained aiming points. Despite the apparent The difference between these two damage fraction values is less than 0.2%. It is important to point out the difference in computational complexity between these two optimizations. While the constrained optimization over Pattern B3 has 4 variables, the unconstrained optimization for 12 M = weapons has 24 variables, which converges much slower than the constrained optimization.
The optimal aiming points constrained to Pattern B1 for 6 M = , the optimal aiming points constrained to Pattern B2 for 7 M = and the corresponding unconstrained optimal aiming points are displayed in Figure 7.  A practical question regarding resource allocation is the following: Given the radius of area target (s), what is the minimum number of weapons needed to achieve a given threshold of damage fraction? This question is answered in Figure 9. Figure 9 shows that for any given threshold of damage fraction, the minimum number of weapons needed is an increasing function of the area target radius (i.e., larger area target requires larger number of weapons), which again is reasonable and consistent with our intuition.

Scaling Laws for Optimal Aiming Points and Optimal Damage Fraction with Respect to Area Target Radius
Finally, we study how the optimal aiming points change with s, the radius of area target, and explore if there is a scaling law relating sets of optimal aiming points at different values of s. We start by examining the optimal aiming points for 4 different values of area target radius. The 4 panels in Figure 10 show the optimal aiming points for M = 10 weapons, respectively, for  of area target is increased, the set of aiming points needs to cover a larger region.
On the other hand, to maximize the damage fraction, the killing areas associated with individual weapons also need to maintain a certain degree of overlapping with each other. These two needs contradict each other and cannot be both accommodated simultaneously with a fixed number of weapons (M) as the area target radius is increased. Thus, it is expected that as the radius of area target is increased, the spread size of optimal aiming points will increase less than linearly.
Here we avoid using the term "radius of optimal aiming points" because the distribution of aiming points is not circularly symmetric.
For the purpose of investigating the spread size of aiming points quantitatively, To explore how the size of optimal aiming points scales with the area target radius, we plot these two quantities against each other in a log-log plot in the left panel of Figure 11, which also includes a fitting function of the form AP L s ∝ .
The log-log plot along the fitting function indicates that the size of optimal aiming points ( AP L ) approximately is proportional to the square root of area target radius ( s ). These simulation results lead us to the empirical conclusion that the size of optimal aiming point distribution scales as the square root of area target radius. Based on this key observation, we introduce the scaled aiming points as The right panel of Figure 11  This scaling property gives us an even more efficient way of calculating optimal aiming points. We only need to calculate the optimal aiming points for We evaluate the performance of this efficient method by examining the damage fraction values achieved by these sets of nearly optimal aiming points.
Specifically, for each area target, we calculate the damage fraction values corresponding respectively to three sets of aiming points: • aiming points calculated in the unconstrained optimization; • aiming points calculated using scaling law (21); • all aiming points = ( ) 0, 0 . corresponding to all weapons aiming at ( ) 0, 0 is much lower. Therefore, we conclude that scaling law (21) is an efficient and accurate method for calculating a set of nearly optimal aiming points.

Concluding Remarks
We have studied the average damage fraction of an area target caused by multiple weapons. The area target was assumed to consist of normally distributed elements. Using the analytical expression of the average damage fraction, we compared various distribution patterns of the aimpoints and gave optimal patterns for different number of weapons. Scaling laws for optimal aimpoints and optimal damage fraction with respect to the radius of the area target were derived. One prospective future research is to extend our current work to an area target of uniformly distributed elements. Another avenue for future research is to consider an area target where the elements are assigned different values and seek optimal aimpoints in order to minimize the total average surviving value.