Explicit Exact Solution of Damage Probability for Multiple Weapons against a Unitary Target

We study the damage probability when M weapons are used against a unitary target. We use the Carleton damage function to model the distribution of damage probability caused by each weapon. The deviation of the impact point from the aimpoint is attributed to both the dependent error and independent errors. The dependent error is one random variable affecting M weapons the same way while independent errors are associated with individual weapons and are independent of each other. We consider the case where the dependent error is significant, non-negligible relative to independent errors. We first derive an explicit exact solution for the damage probability caused by M weapons for any M. Based on the exact solution, we find the optimal aimpoint distribution of M weapons to maximize the damage probability in several cases where the aimpoint distribution is constrained geometrically with a few free parameters, including uniform distributions around a circle or around an ellipse. Then, we perform unconstrained optimization to obtain the overall optimal aimpoint distribution and the overall maximum damage probability, which is carried out for different values of M, up to 20 weapons. Finally, we derive a phenomenological approximate expression for the damage probability vs. M, the number of weapons, for the parameters studied here.


Introduction
The probability of killing or damaging a target depends heavily on how close a weapon How to cite this paper: Wang, H.Y., Moten, C., Driels, M., Grundel, D. and Zhou, H. (2016) Explicit Exact Solution of Damage Probability for Multiple Weapons against a is delivered to the target.This delivery accuracy of a weapon may be affected by many components.In general, the errors are usually divided into two main groups: the dependent error and independent errors.The dependent error is related to the aiming error that results from a miscalculation of latitude, longitude, distance, wind effect, or uncertainty in locating the target position.The dependent error results in the armament impacting away from the desired target point and it affects all weapons the same way.The independent errors refer to ballistic dispersion errors, which may result from variations in bullet shape, variations in gun barrels, or variations in amount of explosive used inside each bullet [1].
Due to many uncertainties in the field of weapon effectiveness, Monte Carlo simulations have been widely employed to estimate the probability of target damage [2].Even though Monte Carlo simulations can provide reasonable estimates, exact solutions are mathematically more attractive and practically more useful.The objectives of this paper are: i) to derive explicit exact solution for the damage probability caused by multiple weapons against a single target, ii) to use the exact solution to maximize the damage probability with respect to the aimpoint distribution of weapons, with or without geometric constraint(s) on the aimpoint distribution, and iii) to study the relation of damage probability to the number of weapons when the dependent error is significant.The results obtained here can be applied to indirect fire artillery, or GPS/INS-guided weapons.
The remainder of this paper will progress as follows.Section 2 provides the detailed mathematical formulation and explicit exact solution for the kill probability.Section 3 considers the performances of various aimpoint distributions.Finally, Section 4 presents conclusions and future work.

Mathematical Formulation
We consider a single point target in the two dimensional space.We establish the coordinate system such that the target is located at the origin point . We use M weapons with dependent and independent errors to fire on the target.Due to the presence of significant dependent error, if all M weapons are aimed at the M impact points may be uniformly shifted away from the target by a significant distance, resulting in a small damage probability.To make the damage probability less susceptible to the dependent error, we aim the M weapons at M different points distributed around the target.When the dependent error shifts some impact points away from the target, it simultaneously shifts the some other impact points toward the target.
In this study all weapons are assumed to be perfectly reliable.Gross errors due to anomalies such as catastrophic weapon system failure, adverse weapon separation effects, and GPS jamming are neglected.Let • j r = the aiming point of weapon j.
• Y = miss distance from the aimpoint due to the dependent error of M weapons, affecting the impact points of all M weapons uniformly.
• j X = miss distance from the aimpoint due to the independent error of weapon j, affecting only the impact point of weapon j individually.We assume that are independent of each other and independent of random variable Y .
The impact point of weapon j is given by We model the dependent error Y as a normal random variable with zero mean: σ and 2 σ are standard deviations, respectively, in the two coordinate direc- tions, which give an indication of the spread of the dependent error in the two directions.We model each independent error j X as a normal random variable with zero mean:

X
Further, we assume that the independent errors of individual weapons are independent of each other and are independent of the dependent error Y .
We use the mathematical fact that the sum of two independent normal random variables is a normal random variable.Suppose The probability density functions of U and V are given by ( ) In terms of the probability density functions, we write Equation (1) as We rewrite the equation above in terms of expected values: ( ) ( ) Here Pr target being killed by one weapon at impact point ) This is called the Carleton damage function or the diffuse Gaussian damage function [3].The two parameters 1 b and 2 b in the Carleton damage function (3) represent the effective weapon radii in the range and deflection directions, respectively.With the impact points of the M weapons given by , 1, 2, , , the probability of the target located at the origin being killed by the M weapons is Pr target being killed by weapons at impact points , 1, 2, , , , We calculate the probability of the target being killed averaged over independent er- and averaged over the dependent error Y .For that purpose, we only need to calculate the average of each term inside the summation: , , , , have exactly the same format, we only need to derive the analytical expression for one.For conciseness, we denote ( ) ( ) simply by l j r , Y , and l j X in the calculation of ( ) Each term in the product is an average of the form on the left hand side of (2).Applying Equation (2), we write each average as ) Substituting this result into Equation ( 5), we obtain Next we average over the dependent error ( ) ~0, Y N σ .Again, the average is of the form on the left hand side of (2).Applying Equation (2), we arrive at Thus, the overall average of ( ) ( ) Similarly, the overall average of ( ) , , The probability of target being killed, averaged over independent errors and dependent error, is called kill probability, and is denoted by where ( ) are given in ( 6) and (7) above.Together, Equations ( 6)-(8), give us an explicit analytical expression for calculating the kill probability.
After the completion of the above derivation, we discovered that similar approaches had been taken separately by von Neumann [4] and by Washburn [5].

Performances of Various Aimpoint Distributions of Multiple Weapons against a Single Target
Now we apply the exact solution to examine the kill probability corresponding to various distributions of the aimpoints of M weapons.
Let L A denote the weapon lethal area or the fragmentation mean area of the effec- tiveness.It describes the effect of a warhead against a target and includes the effects of direct hit, blast, and fragmentation.We can calculate L A from the Carleton damage function (3) as The aspect ratio of the weapon radii of the Carleton damage function ( ) where θ is the impact angle.
Once the lethal area L A and the aspect ratio a are given, one can calculate the wea- pon radii for the Carleton damage function (3) as follows: For all the cases considered in this paper, we choose We first consider the case of M weapons with aimpoints uniformly distributed on a circle as formulated below ( ) ( ) where r is the radius and θ the phase off-set angle of the distribution.These are pa- rameters that we can tune to maximize the kill probability.
For each value of M, we maximize the kill probability with respect to ( ) , r θ .This unconstrained nonlinear optimization can be achieved by using MATLAB built-in function fminsearch which is based on a direct search method of Lagarias et al. [6].The results are listed in Table 1.
Note that the Carleton damage function we use is not isotropic.It has different effective radii in the range and deflection directions.To accommodate this anisotropic property of the Carleton damage function, we consider the case of M weapons with aimpoints distributed on an ellipse as formulated below ( ) ( ) where η is the aspect ratio of the ellipse.In the formulation above, we elongate one axis by η and simultaneously shrink the other axis by the same factor.In this way, the area of the ellipse is maintained at 2 πq , independent of the aspect ratio η .Para- meter q has the meaning ( ) ( ) area of ellipse major axis minor axis π q = = × From q and η , we can determine the major axis and the minor axis as The asterisks reflect that when r = 0, θ is not meaningful, meaning that θ is arbitrary and irrelevant.
major axis q η = minor axis q η = We should point out that parameter φ is not the polar angle of the aimpoint of weapon 1. φ is the angular value used in the parametric equation of the ellipse to cal- culate the aimpoint of weapon 1. φ is the phase angle before the major axis is elon- gated and before the minor axis is shrunk.
For each value of M, we maximize the kill probability with respect to ( ) , , q φ η .We obtain the results in Table 2.
In the above, we calculated the performance of placing the aimpoints of M weapons along a circle or an ellipse.We now examine the case of aiming one weapon at the center and aiming the rest ( ) weapons at positions distributed on an ellipse.The aimpoints of M weapons are distributed as formulated below.
( ) ( ) For each value of M, we maximize the kill probability with respect to ( ) , , q φ η .The optimal results are reported in Table 3.
Next, we fully optimize the distribution of M aimpoints without constraining them The asterisks reflect that when r = 0, θ is not meaningful, meaning that θ is arbitrary and irrelevant.
Table 3.The optimal distribution for M aimpoints when one of them is aimed at the origin while the rest of aimpoints are uniformly distributed around an ellipse, and the corresponding probability of kill.Here φ is the off-set value in the parametric equation of the ellipse.For the cases of 5 M ≤ , the kill probability is not improved by moving one of the M aimpoints to the center.on a circle or an ellipse.We represent the M aimpoints in polar coordinates.

( )
, , , The optimal solutions for are listed in Table 4.
Figure 1 shows the optimal distributions of aimpoints for  5.As M (the number of weapons) increases, the optimal distribution of aimpoints has more layers, covering a larger area with a more uniform distribution over the area.In    In the presence of dependent error, however, the situation is completely different.
The same dependent error affects all M weapons.The outcomes of individual weapons are no longer independent of each other.As a matter of fact, when the M weapons are all aimed at the same position, the outcomes of individual weapons are highly correlated with each other.As an example, we examine the case of aiming all M weapons at the origin.The averages of ( ) The kill probability is     In the presence of dependent error, to simplify the analysis, we assume that the independent errors are zero ( ) and assume that vertical components of { } j X and Y .Since the horizontal components and vertical components are independent of each other, we have

1 M
= (yellow circles) and 2 M = (blue squares) while the optimal distributions for the cases of

Figure 3
Figure 3 illustrates the optimal distributions of aimpoints for 5 M = (yellow cir-

Figure 5
Figure 5 displays the optimal distributions of aimpoints for 9 M = (yellow circles)

Figure 7 ,
Figure 7, we plot the optimal distributions of aimpoints for 15 M = (yellow circles)

−
all M weapons are aimed at the same positon, geometrically with M. With the optimal distribution of aimpoints for M weapons, we may expect that in the case of aiming all M weapons at the same position.Indeed, as demonstrated in the left panel of Figure8, when the M weapons are aimed according to the optimal distribution of aimpoints, the case of aiming all M weapons at ( ) 0,0 .The right panel of Figure8shows the enhancement in the decay of survival probability distribution of aimpoints.Specifically, in the right panel of Figure8, we plot the quantity below as a function of M Even with the optimal distribution of aimpoints, however, the log survival probabili-

Table 1 .
The optimal distribution for M aimpoints when they are uniformly distributed around a circle and the corresponding probability of kill.Here r is the radius and θ is the phase off-set an-

Table 2 .
The optimal distribution for M aimpoints when they are uniformly distributed around an ellipse and the corresponding probability of kill.Here φ is the off-set value in the parametric equation of the ellipse.The cases of