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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.

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 [

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 forces and infrastructure, especially hospitals, schools, churches, and residential homes. The precision-guided weapons are in general subject to target-location errors and ballistic dispersion errors. The target-location errors, or aiming errors, 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 [

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.

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 [

・ r j = the aiming point of weapon j for j = 1 , 2 , ⋯ , M .

・ 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.

・ X j = independent error of weapon j, affecting only the impact point of weapon j individually. For example, X j is the part of error associated with aiming and firing weapon j. We assume that { X j , j = 1 , 2 , ⋯ , M } 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:

Y ~ N ( ( 0 0 ) , ( σ 1 2 0 0 σ 2 2 ) )

where σ 1 and σ 2 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 X j as a normal random variable with zero mean:

X j ~ N ( ( 0 0 ) , ( d 1 2 0 0 d 2 2 ) )

The impact point of weapon j is given by

w j = r j + Y + X j

We use the Carleton damage function described below to model the probability of killing by an individual weapon. Let w = ( w ( 1 ) , w ( 2 ) ) be the impact point of a weapon where w ( 1 ) and w ( 2 ) denote respectively the range component and the deflection component of the impact point from the target. In Carleton damage function, the probability of the target being killed by a weapon at impact point w is mathematically modeled as

Pr ( target being killed by one weapon at impact point w ) = e x p ( − w ( 1 ) 2 2 b 1 2 ) e x p ( − w ( 2 ) 2 2 b 2 2 ) (1)

This is the well-known Carleton damage function or the diffuse Gaussian damage function [

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 p kill ( target , M weapons ) . 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 { w j = r j + Y + X j , j = 1 , 2 , ⋯ , M } , we derived an analytical expression for the kill probability (averaged over the distribution of impact points) as a function of quantities

( σ 1 , σ 2 , d 1 , d 2 , b 1 , b 2 , { r j , j = 1 , 2 , ⋯ , M } ) .

p kill ( target , M weapons ) = G ( σ 1 , σ 2 , d 1 , d 2 , b 1 , b 2 , { r j , j = 1 , 2 , ⋯ , M } ) (2)

where function G is defined as

G ( σ 1 , σ 2 , d 1 , d 2 , b 1 , b 2 , { r j , j = 1 , 2 , ⋯ , M } ) = − ∑ k = 1 M ( − 1 ) k ∑ ( j 1 , ⋯ , j k ) E [ F 1 ( j 1 , ⋯ , j k ) ] E [ F 2 ( j 1 , ⋯ , j k ) ] (3)

E [ F 1 ( j 1 , ⋯ , j k ) ] = ( b 1 2 b 1 2 + d 1 2 ) k 2 ( ( b 1 2 + d 1 2 ) / k ( b 1 2 + d 1 2 ) / k + σ 1 2 ) 1 2 × exp ( ( ∑ l = 1 k r j l ( 1 ) / k ) 2 − ∑ l = 1 k r j l ( 1 ) 2 / k 2 ( b 1 2 + d 1 2 ) / k − ( ∑ l = 1 k r j l ( 1 ) / k ) 2 2 ( ( b 1 2 + d 1 2 ) / k + σ 1 2 ) ) (4)

E [ F 2 ( j 1 , ⋯ , j k ) ] = ( b 2 2 b 2 2 + d 2 2 ) k 2 ( ( b 2 2 + d 2 2 ) / k ( b 2 2 + d 2 2 ) / k + σ 2 2 ) 1 2 × exp ( ( ∑ l = 1 k r j l ( 2 ) / k ) 2 − ∑ l = 1 k r j l ( 2 ) 2 / k 2 ( b 2 2 + d 2 2 ) / k − ( ∑ l = 1 k r j l ( 2 ) / k ) 2 2 ( ( b 2 2 + d 2 2 ) / k + σ 2 2 ) ) (5)

Mathematically, F 1 ( j 1 , ⋯ , j k ) is the product, over k weapons { j 1 , ⋯ , j k } , of all factors involving only components in the range direction (i.e., σ 1 , d 1 , b 1 , r ( 1 ) ) while F 2 ( j 1 , ⋯ , j k ) is the product, over k weapons { j 1 , ⋯ , j k } , of all factors involving only components in the deflection direction (i.e., σ 2 , d 2 , b 2 , r ( 2 ) ). Together, Equations (2), (3), (4) and (5) form an explicit analytical expression for the kill probability of a point target, p kill ( target , M weapons ) . This analytical solution will be used in next section to calculate the damage fraction of an area target consisting of normally distributed target elements.

Now let us examine the situation where M weapons are used against an area target centered at x target = ( 0,0 ) , consisting of N discrete elements, normally distributed around the center. Let

・ Z k = the location of element k of the area target, for k = 1 , 2 , ⋯ , N .

・ r j , X j , 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, Z k , the location of element k of the area target, is modeled as a normal random variable with zero mean:

Z k ~ N ( ( 0 0 ) , ( s 1 2 0 0 s 2 2 ) ) (6)

We assume that { Z k , k = 1,2, ⋯ , N } are independent of each other, and are independent of X j and Y .

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

w j ( k ) = r j + Y + X j − Z k ≡ r j + Y ( k , eff ) + X j (7)

where the effective dependent error of the M weapons relative to element k is defined as

Y ( k , eff ) ≡ Y − Z k

Note that Y ( k , eff ) is a normal random variable with zero mean

Y ( k , eff ) ~ N ( ( 0 0 ) , ( σ 1 2 + s 1 2 0 0 σ 2 2 + s 2 2 ) )

The kill probability of element k caused by the M weapons (averaged over random independent errors { X j , j = 1,2, ⋯ , M } , over the random dependent error Y , and over the random element location Z k ) is given by

p kill ( element k , M weapons ) = G ( σ 1 2 + s 1 2 , σ 2 2 + s 2 2 , d 1 , d 2 , b 1 , b 2 , { r j , j = 1,2, ⋯ , M } ) (8)

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 σ 1 2 be replaced by ( σ 1 2 + s 1 2 ) and σ 2 2 be replaced by ( σ 2 2 + s 2 2 ) .

Let χ k 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.

q damage ( area target , M weapons ) ≡ 1 N ∑ k = 1 N χ k

The average damage fraction has the expression

E [ q damage ( area target , M weapons ) ] = E [ 1 N ∑ k = 1 N χ k ] = 1 N ∑ k = 1 N p kill ( element k , M weapons ) = G ( σ 1 2 + s 1 2 , σ 2 2 + s 2 2 , d 1 , d 2 , b 1 , b 2 , { r j , j = 1 , 2 , ⋯ , M } ) (9)

Expression (9) gives the exact solution for the case of an area target of normally distributed elements.

When both the independent error ( X j , j = 1 , 2 , ⋯ , M ) and the dependent error ( Y ) of the M weapons are absent, random variables { χ k , k = 1 , 2 , ⋯ , N } are independent of each other. In this situation, we can calculate analytically the standard deviation of damage fraction (random variable). The variance of damage fraction has the expression

var [ q damage ( area target , M weapons ) ] = var [ 1 N ∑ k = 1 N χ k ] = 1 N 2 ∑ k = 1 N var [ χ k ] = 1 N G ( 1 − G ) (10)

where shorthand notation G is defined as

G ≡ G ( σ 1 2 + s 1 2 , σ 2 2 + s 2 2 , d 1 , d 2 , b 1 , b 2 , { r j , j = 1 , 2 , ⋯ , M } ) (11)

The standard deviation of damage fraction is

std [ q damage ( area target , M weapons ) ] = 1 N G ( 1 − G ) (12)

While the average damage fraction tells us the average number of target elements killed out of total N target elements, the standard deviation of damage fraction describes the fluctuations/uncertainty in the actual number of target elements killed in individual realizations. Large standard deviation means large swing in the actual number of target elements killed from one realization to another.

Note that this expression for the standard deviation is valid only in the absence of dependent and independent errors, for which we have ( σ 1 , σ 2 , d 1 , d 2 ) = ( 0 , 0 , 0 , 0 ) . 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 ( σ 1 , σ 2 , d 1 , d 2 ) . We demonstrate this behavior numerically in ^{6} runs for each set of parameter values and in each of the two situations below (i.e. without or with firing error), yielding accurate numerical results to compare with theoretical predictions.

In

In

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” [

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 times expensive.

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

・ Pattern A1: M points on an ellipse, uniform in parameter angle.

Specifically, the M aiming points are mathematically described by

θ j = θ 1 + ( j − 1 ) M 2 π , j = 1 , 2 , ⋯ , M (13)

( x j , y j ) = R ( η cos θ j , 1 η sin θ j ) (14)

This constrained pattern has three parameters: θ 1 , R and η , over which we are going to optimize the average damage fraction. Here θ 1 is the parameter angle for weapon 1. R is the effective radius of the ellipse, satisfying

R = area of ellipse π

whereas η is the aspect ratio of the ellipse, satisfying

η = major axis minor axis

・ Pattern A2: 1 point at center and ( M − 1 ) points on an ellipse, uniform in parameter angle.

One aiming point is placed at the center. The rest ( M − 1 ) aiming points are distributed along an ellipse, uniformly in parameter angle θ , as described by Equations (13) and (14) where M is replaced by ( M − 1 ) . This constrained pattern has three parameters: θ 1 , R and η .

・ Pattern A3: 2 points on the x-axis and ( M − 2 ) points on an ellipse, uniform in parameter angle.

Two aiming points are placed, respectively, at ( x a ,0 ) and ( − x a ,0 ) . The rest ( M − 2 ) aiming points are distributed along an ellipse, uniformly in parameter angle θ , as described by Equations (13) and (14) where M is replaced by ( M − 2 ) . This constrained pattern has 4 parameters: θ 1 , R , η , and x a .

・ Pattern B1: M points on an ellipse, uniform in polar angle.

The M aiming points are mathematically described as follows. Their polar angles ϕ j are given by

ϕ j = ϕ 1 + ( j − 1 ) M 2 π , j = 1 , 2 , ⋯ , M (15)

Their parameter angles θ j are determined by the condition that the two vectors

#Math_129# (16)

Consequently, the aiming points on the ellipse can be described as

( x j , y j ) = R ( η cos θ j , 1 η sin θ j ) (17)

This constrained pattern contains three parameters: ϕ 1 , R and η .

・ Pattern B2: 1 point at center and ( M − 1 ) points on an ellipse, uniform in polar angle.

One aiming point is placed at the center. The rest ( M − 1 ) aiming points are distributed along an ellipse, uniformly in polar angle ϕ , as described by Equations (15), (16) and (17) where M is replaced by ( M − 1 ) . This constrained pattern has three parameters: ϕ 1 , R and η .

・ Pattern B3: 2 points on the x-axis and ( M − 2 ) points on an ellipse, uniform in polar angle.

Two aiming points are placed, respectively, at ( x a ,0 ) and ( − x a ,0 ) . The rest ( M − 2 ) aiming points are distributed along an ellipse, uniformly in polar angle ϕ , as described by Equations (15), (16) and (17) where M is replaced by ( M − 2 ) . This constrained pattern includes four parameters: ϕ 1 , R , η , and x a .

An example of Pattern B3 (2 points on the x-axis and rest of points on an ellipse, uniform in polar angle) is shown in the left panel of

In numerical simulations below, we choose the parameter values as follows:

M = 1 to 12 , number of weapons;

( b 1 , b 2 ) = ( 60,100 ) , parameters in Carleton damage function;

( σ 1 , σ 2 ) = ( 5 , 5 ) , standard deviation (s.d.) of dependent error in firing errors;

( d 1 , d 2 ) = ( 5,5 ) , s.d. of independent errors in firing errors;

s = s 1 = s 2 = 15 to 300 , radius of area target (s.d. of elements distribution).

We compare the results of optimization over constrained patterns with those of the overall optimization.

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 M = 12 weapons (

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

discrepancy between these two sets of optimal aiming points, the corresponding damage fractions are still very close to each other: the optimal damage fraction for Pattern B3 is p { opt , PatternB 3 } = 0.813372 while the overall optimal damage fraction is p { opt , unconstrained } = 0.814997 . 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 M = 12 weapons has 24 variables, which converges much slower than the constrained optimization.

The optimal aiming points constrained to Pattern B1 for M = 6 , the optimal aiming points constrained to Pattern B2 for M = 7 and the corresponding unconstrained optimal aiming points are displayed in

the number of weapons, M; for a fixed value of M, the optimal damage fraction decreases as the radius (s) of area target is increased (i.e., damage fraction is lower for a larger area target). Both of these results are reasonable and consistent with our intuition.

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

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

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, we define the size of a set of M aiming points { r j , j = 1 , 2 , ⋯ , M } mathematically as

L AP ≡ 1 M ∑ j = 1 M | r j | 2 (18)

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

r j ( scaled ) ( s ) ≡ 1 s r j ( s ) (19)

The right panel of

r j ( scaled ) ( s ) is invariant with respect to s (20)

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

an area target of typical/representative radius: { r j ( s 0 ) , j = 1,2, ⋯ , M } . We use s 0 = 150 in our study. For an area target of radius s, we simply calculate/predict a set of nearly optimal aiming points from { r j ( s 0 ) } using the scaling law.

r j ( s ) = s s 0 r j ( s 0 ) (21)

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 ) .

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.

The authors would like to thank TRAC-Monterey for supporting this work and Center for Army Analysis (CAA) for bringing this problem to our attention. The views expressed in this document are those of the authors and do not reflect the official policy or position of the Department of Defense or the U.S. Government.

Wang, H.Y., Labaria, G., Moten, C. and Zhou, H. (2017) Average Damage Caused by Multiple Weapons against an Area Target of Normally Distributed Elements. American Journal of Operations Research, 7, 289-306. https://doi.org/10.4236/ajor.2017.75022