Weapon effectiveness and the shapes of damage functions

Sei-Hoon Moon

a 4th R&D Institute,Agency for Defense Development,Yusung,Daejeon,305-152,Republic of Korea

b Department of Mechanical and Aerospace Engineering Naval Postgraduate School,Monterey,CA,93940,USA

Keywords:Weapon effectiveness Point target Area target Cookie-cutter damage function Carleton damage function

ABSTRACT This paper provides a review of methods of assessing a fragmentation weapon’s effectiveness against a point target or an area target with keeping the focus on the necessity of using the Carleton damage function with the correct shape factor.First,cookie-cutter damage functions are redefined to preserve the shape factor of and to have the same lethal area as the corresponding Carleton damage function.Then,closed-form solutions of the effectiveness methods are obtained by using those cookie-cutter damage functions and the Carleton damage function.Finally,the closed-form solutions are applied to calculate the probability of damaging a point target and the expected fractional damage to an area target for several attack scenarios by using cookie-cutter damage functions and the Carleton damage functions with different shape factors.The comparison of the calculation results shows that using cookie-cutter damage functions or the Carleton damage function with a wrong shape factor results in quite significant differences from using the original Carleton damage function with a correct shape factor when weapon’s delivery error deviations are less than or comparable to the lengths of the lethal area and the aim point is far from a target.The effectiveness methods improved in this paper will be useful for mission planning utilizing the precision-guided munitions in circumstances where the collateral damage should be reduced.

1.Introduction

Weapon effectiveness is frequently a topic of significant interest in military combat operations.The methods of assessing a weapon’s effectiveness are mainly based on the concept of the effectiveness index(EI)to represent the effect of weapons on a wide variety of targets and to meet the computing speed of the time[1].Even though this approach seems to be oversimplified,the methods provide reasonable and consistent results.An EI is a numerical quantity that measures the effectiveness of a particular weapon against a specific target and is defined only for a given degree of damage to the target.There are various types of EIs depending on weapon-target combinations and the methodologies used.The mean area of effectiveness due to fragments(MAEF)which is often called the lethal area is used for fragment sensitive targets and is one of the most frequently used EIs[2,3].

When a MAEFis used in a methodology,it has to be represented by an adequate damage function that describes the probability of damaging a specific target when a weapon detonates at a given point around the target.There are various types of equivalent damage functions representing the same MAEFor lethal area.The most primitive form of a damage function for a fragmentation weapon is the damage matrix,which is a large set of cell-by-cell data and is generated from vulnerable areas of a target and fragmentation field data obtained from experiments[2,3].The shape of the damage matrix is asymmetric for the impact position and highly dependent on impact conditions such as the height of burst,weapon velocity,impact angle,and angle of attack.The damage matrix is not usually used for a simple methodology that should provide a closed-form solution to be built in a spreadsheet.Instead,the damage matrix is usually simplified to an equivalent function to be adequate for a method.

For fragmentation weapons,the Carleton damage function and cookie-cutter damage functions are taken for simple methodologies for getting closed-form solutions.The Carleton damage function has a form of bivariate Gaussian function and is obtained by fitting the damage matrix[2,3].Cookie-cutter damage functions are further simplifications of the Carleton damage function.They have the probability of damage of 1.0 within the lethal region and of zero outside that region.The shape of a cookie-cutter damage function is taken to be elliptical or rectangular.The Carleton damage function is primarily used in effectiveness methodologies if there exist closed-form solutions because it is closer to the damage matrix and more realistic than cookie-cutter damage functions.On the other hand,effectiveness can be calculated using a Monte Carlo simulation or an analytical approach using closed-form solutions.The latter is usually preferred from an execution time point of view but would not be available for all representations of the damage function[4,12].

Using different damage functions could result in different results,even though they have the same MAEFor lethal area.There have been several works that have investigated the characteristics of these damage functions[3,7]and compared the results of analytical studies[3,6,7,12]or Monte Carlo simulations[8,13,14,17]by employing different damage functions to estimate effectiveness on point targets or area targets.There was also a work that tried to develop skew-normal damage functions for application to some complicated combat scenarios[11].In most of those works,it was implicitly assumed that the probability of damaging a target is unity at the point of weapon detonation.That is because the effectiveness methods and the damage functions had historically been developed for weapons that are launched from air and usually have huge warheads.

However,that assumption would not be always acceptable when smaller weapons such as artillery shells are considered.Some hard targets such as armored personal carriers or main battle tanks would not always be damaged,even if an artillery shell explodes right above those targets as considered in Refs.[3].The Carleton damage function with the shape factor less than unity was considered together with a circular cookie-cutter damage function and the Carleton damage function with the shape factor of unity for use in artillery effectiveness codes.It was shown that the probability of damaging a point target resulted from the former is much less than the results calculated by using the other damage functions when the delivery error is low.However,the differences have not been seriously taken by people because the delivery error of conventional artillery shells is much larger than their lethal radii for almost all ground mobile targets and so the differences are negligible in reality.For reference,the lethal radii of conventional 155 mm artillery shells against typical ground mobile targets are at most 10 m,while they have the CEP from about 100 m at the minimum range to about 300 m at the maximum range[16].

On the other hand,field artillery has been increasingly employing precision-guided munitions in order to engage targets with greater accuracy[5,10].Most of them are supposed to have a CEP of less than 10 m,which is comparable to the lethal radii of 155 mm artillery shells.In this case,the probability of damage resulted from using the Carleton damage function with the shape factor less than unity is much less than the results calculated with the Carleton damage function with the shape factor of unity.Furthermore,the probability of damage from a cookie-cutter damage function shows quite a big difference from the results calculated with the Carleton damage function with the shape factor of unity as shown[3].Therefore,it is necessary to study whether it is appropriate to use other damage functions rather than the correct Carleton damage function with the shape factor less than unity for assessing the effectiveness of precision-guided munitions.

This paper reviews the methods of assessing a fragmentation weapon’s effectiveness against a point target or an area target.The review will be focused on the necessity of using the Carleton damage function with the correct shape factor rather than other damage functions.This paper will progress as follows.The next section presents the Carleton and cookie-cutter damage functions.The methods of assessing the weapon’s effectiveness against a point target or an area target are reviewed in Section 3 and Section 4,respectively.Closed-form solutions of the methods are obtained by using those cookie-cutter damage functions and the Carleton damage function.They are applied to calculate the probability of damaging a point target and the expected fractional damage to an area target for several attack scenarios by using two cookie-cutter damage functions and two Carleton damage functions with different shape factors.Then the calculation results are presented and compared.Finally,Section 5 presents the conclusions.

2.Damage functions

Various damage functions have been developed for many years,and they were mostly circular symmetric by being functions of the radial distance only,such as the circular cookie-cutter damage function,the Gaussian damage function,the exponential damage functions,and the gamma damage function[3,7,11,15].Circular symmetric damage functions are well used for blast weapons because blast from an explosion isotropically propagates along the ground surface,but they are not adequate for fragmentation weapons because fragments released from a weapon are not isotropically distributed on the ground surface in general.The shape of a damage matrix is asymmetric for the impact position and highly dependent on impact conditions.

In most of those works,it was implicitly assumed that the probability of damaging a target is unity at the point of the weapon explosion.At the time when air-launched large weapons such as GP bombs were mainly considered,there was no need to consider the shape factor less than unity because almost ground mobile targets would be killed at the weapon’s explosion site.This paper does not take that assumption because using cookie-cutter damage functions or the Carleton damage function with a wrong shape factor results in quite significant differences from using the Carleton damage function with a correct shape factor when weapon’s delivery error is less than or comparable to the lengths of the lethal area as shown in Ref.[3]and will be shown throughout this paper.

The Carleton damage function is written with the shape factor to reflect that the probability of damaging a target would not be unity at the site of the weapon explosion as in Eq.(1).Cookie-cutter damage functions are also redefined to preserve the shape factor of and to have the same lethal area as the Carleton damage function.Let the vectorsPandUbe position vectors representing a target position（p，q）and an impact point（u，v）on the ground surface,respectively.The Carleton damage function c（P，U）is then defined to represent the probability of damaging a target located atPby a weapon hitting atUon the ground surface and has a form of a bivariate Gaussian function as follows

where Rdand Rrare lethal radii in the deflection direction and the range direction,respectively.D0is the shape factor of the Carleton damage function and is a positive real number equal to or less than unity[3].It should be determined by comparison with the experimental results or the damage matrix.Note that the lethal area ALcalculated by integrating the Carleton damage function over the whole ground surface is independent of the shape factor D0and is given by

The Carleton damage function can be simplified to the elliptical cookie-cutter damage function fD，E（P，U）,which is defined by

where Rdand Rrare lethal radii used in Eq.(1).It is explicit that this damage function has the lethal area ofπRdRrbecause the probability of damage is D0within and 0.0 outside the ellipse with the geometrical radiiandHowever,it is often further simplified to the rectangular cookie-cutter damage function fD，R（P，U）with the same lethal area because it’s usually impossible to obtain a closed-form solution for the probability of damaging a point target by using that elliptical damage function.The rectangular damage function is defined by the effective lethal widthand the effective lethal lengthand is written by

This damage function also preserves the lethal area of the Carleton damage function because the probability of damage is D0within and 0.0 outside the rectangular with geometrical lengthsand,that is,=πRdRr.

3.Weapon’s effectiveness against a point target

This section reviews the methods of assessing a fragmentation weapon’s effectiveness against a point target.A point target is defined as a unitary target that its physical dimensions are small enough compared to the range for the damage mechanisms of the used weapon to reach while keeping the ability to kill the target.Closed-form solutions are obtained by applying a rectangular cookie-cutter damage function and the Carleton damage function to the methods.

An attack scenario where a weapon impacts relative to a point target is shown in Fig.1.The target is located at the pointPand the center of a lethal area represented by an elliptical damage function is at the impact pointUwhere the weapon impacts.The aim point is designated asA（≡（a，b））.Fig.2 is a view truncated along with the deflection directions and shows a target located at p,the Gaussian accuracy function relative to the aim point at a,and the damage functions centered at a weapon impact point u[2].It presents both the Carleton damage function and a cookie-cutter damage function.The probability of damaging the target by the single weapon is the value of the function at the target if the Carleton damage function is applied,while the probability is D0within the cookie-cutter boundary but 0 outside the boundary.

Fig.1.A scenario for a weapon to attack a point target.

Fig.2.Weapon vs.unitary target interaction geometry.

3.1.Principle of the effectiveness method against point targets

If fD（P，U）and g（U，A）represent a damage function and a delivery accuracy function respectively,the probability of damage caused to a target by a single weapon is obtained by integrating over the whole surface as follows

where R is the weapon’s reliability.If the Carleton damage function is used as a damage function,the probability of damaging the target PD1is to be the expectation value of the damage function.On the other hand,if a cookie-cutter damage function is employed,then PD1is the same as D0times the probability that the lethal geometrical region covers the point target.

The functiong（U，A）is the delivery accuracy function and can be divided into three independent parts[2]as follows

The first term of the left-hand side represents the normal distribution part.The coefficient PNMis the fraction of the number of weapons impacts according to the normal distribution and gNM（U，A）is the bivariate Gaussian distribution function written as

whereσdandσrare standard deviations of the impact point distribution in deflection and range directions,respectively.The second term represents the abnormal distribution part coming from direct hits to the aiming points.The coefficient PHITis the fraction of the direct hits that exceed the normal distribution at the aiming point.The direct hit part can be represented by the Dirac delta functionδ（U，A）.The last term is the gross error,which represents the fraction of hits outside±3σlimit[2].This error is usually neglected.It is usually supposed that PNM≤1 and PHIT=0 for unguided weapons and PNM+PHIT≤1 for guided weapons.By substituting Eq.(6),Eq.(5)becomes

where the properties of the Dirac delta function are applied in the second line.If damage functions and the Gaussian distribution function are all separable,then the integration of Eq.(5)can be separately evaluated in the deflection and range directions.Eq.(8)can be then rewritten as

On the other hand,the probability of damage caused to the target by NAmultiple attacks to multiple aim pointsAiwith i=1，…，NAis calculated according to the survival rule as follows

If all of attacks are done to an aim point,then the number of volleys NRto achieve a required probability of damage PDRis obtained as follows

3.2.Method with cookie-cutter damage functions against a point target

When an elliptical cookie-cutter damage function is available,Eq.(8)becomes the integration of the Gaussian distribution function and the delta function over the lethal elliptical region with radiiandas follows

When a rectangular cookie-cutter damage function fD，Ris employed,Eq.(9)is reduced to integration of the Gaussian distribution function and the delta function over the lethal rectangular region.Eq.(9)then becomes

The integral of the Gaussian distribution function over a finite interval does not give a closed-form solution,but it is usually expressed with the error functions or the cumulative distribution functions,which can be easily calculated by using tables or spreadsheet programs.The integral of the delta function can be expressed with the step functions.Then,Eq.(13)can be rewritten using the cumulative distribution functions and the step function as follows

where the cumulative distribution functionΦ（x，μ，σ）and the step functionθ（x）are defined,respectively,as follows

On the other hand,using the rectangular damage function equivalent to an elliptical damage function results in almost the same answers to using the latter.Let PD1NM［RDF，D0］and PD1NM［EDF，D0］represent PD1NMcalculated using a rectangular cookie-cutter damage function and an elliptical damage function with a value of D0,respectively.The left panel of Fig.3 shows plots of PD1NM［RDF，D0］,PD1NM［EDF，D0］,and the relative errors of PD1NM［RDF，D0］to PD1NM［EDF，D0］for D0=1.0 and 0.7 as functions ofσ=σd=σrfor a scenario where a weapon’s lethal radii are Rd=Rr=20 m.There are no practically significant differences between PD1NM［RDF，D0］and PD1NM［EDF，D0］regardless of the values of D0.For the example of Fig.3,the relative errors are at most 0.015 for both values of D0.Upon this fact,this paper considers only the rectangular cookie-cutter damage function in comparison with the Carleton damage function.

Fig.3.Left panel:Plots of PD1NM as functions ofσ=σd=σr for D0=1.0 and 0.7.Right panel:Plots of the relative errors of PD1NM［RDF，D0］to PD1NM［EDF，D0］as functions ofσfor D0=1.0 and 0.7.The weapon’s lethal radii are Rd=Rr=20 m.

3.3.Method with the Carleton damage function against a point target

When the Carleton damage function is employed,the integrations of Eq.(9)give explicit exact solutions.Substituting the Carleton damage function and the accuracy function into the first integral of Eq.(9)gives

The two exponents in the first integral of Eq.(16)can be combined and written as an exact square form for u as follows

After integrating on u,the first integral leaves an explicit exact expression

and the second integral in the range direction can be calculated in the same way.Inserting these results to Eq.(9)gives the probability of damage caused to the target located atPby the single weapon fired toward the aim pointAas follows

If the weapon is directly aimed toward the target,the exponents become zero and Eq.(19)gets reduced to

Note that the shape factor D0does contribute to the probability of damage,even though the Carleton damage function has the same lethal area regardless of the shape factor.

3.4.Calculation results and analyses for example scenarios

This section presents some calculation results for a few scenarios of a weapon’s effectiveness against a point target and compares them to see how significant differences there are.It shows that using cookie-cutter damage functions and the Carleton damage function not equipped with the correct shape factor will result in totally wrong answers when a weapon’s delivery errors are less than or comparable to the lengths of the lethal area.

One of the simplest scenarios is the case that PNM=0.0 and PHIT=1.0,and that the weapon is aimed at the target.Then,PD1HIT=RPHITD0for both methods of using the Carleton damage function and a rectangular cookie-cutter damage function,as can be seen from Eq.(13)and Eq.(19).If D0is not preserved and set to be 1.0,as usually done,then using the Carleton damage function and a rectangular cookie-cutter damage function results in PD1HIT=RPHIT,which is 1/D0times larger than that from using damage functions with the shape factor D0.

For the next analyses,let PD1NM［CDF，D0］represent PD1NMcalculated by using the Carleton damage function with a specific value of the shape factor D0.Two values of the shape factor D0are supposed to be 1.0 and 0.7.Furthermore,the Carleton damage function with D0=0.7 is supposed to be the most realistic one,that is,the closest damage function to the damage matrix of a weapon to a point target.It is also assumed that R=1.0,PNM=1.0 and PHIT=0.0.The target is supposed to lie at the origin of a coordinate system:p=q=0.0.Calculation results are presented in Figs.4-6 as plots of PD1NM［RDF，1.0］,PD1NM［RDF，0.7］,PD1NM［CDF，1.0］,and PD1NM［CDF，0.7］and plots of their relative errors to PD1NM［CDF，0.7］,that is,（PD1NM［XDF，D0］-PD1NM［CDF，0.7］）/PD1NM［CDF，0.7］,where XDF means RDF or CDF.

Fig.4 displays the results from the scenario that the weapon is aimed at the target.For simplicity and clarity of analysis,impact points are assumed to be circular-symmetrically distributed and so σd=σr=σ.The weapon’s lethal radii are chosen to have the same lethal radii in the deflection and range directions,i.e.,Rd=Rr=20 m and so WET=LET=m.Explicitly,both PD1NM［RDF，1.0］and PD1NM［CDF，1.0］show substantial differences from PD1NM［CDF，0.7］when the weapon’s delivery accuracy is less than or comparable to a weapon’s lethal radii while PD1NM［RDF，0.7］shows a relatively small difference.The relative error of PD1NM［RDF，1.0］to PD1NM［CDF，0.7］represented by the green line in the right panel of Fig.4 reaches up to 0.660 atσ≅8.622 and is larger than 0.1 in the range ofσin Fig.4.The relative error of PD1NM［CDF，1.0］to PD1NM［CDF，0.7］is 3/7 whenσ=0 and decreases monotonically.That can be observed in Eq.(20)because the relative error of the Carleton damage function with D’0to the Carleton damage function with D0is given by（D’0/D0-1）/（1+2D’0σ2/）.The relative error of PD1NM［RDF，0.7］to PD1NM［CDF，0.7］reaches up to 0.281 atσ≅0.523.This result tells that using damage functions not preserving D0will result in wrong answers regardless of the kind of damage functions,when the delivery accuracyσis less than or comparable to the lethal radii and shows that preserving the shape factor even in the cases that a cookiecutter damage function should be used would result in better answers.

Fig.4.Left panel:Plots of PD1NM as functions ofσ=σd=σr.Right panel:Plots of the relative error of PD1NM［XDF，D0］to PD1NM［CDF，0.7］as functions ofσ.The weapon’s lethal radii are Rd=Rr=20m.The aim point lies on the top of the target at(0.0,0.0).

Fig.5 shows that the above observations are valid for the case when the lethal radii and the delivery errors are different in the deflection and range directions.It presents plots of PD1NMand its relative error as functions of deviationsσdandσr.Here,the weapon’s lethal radii are chosen to be Rd=30 m and Rr=20 m.

Next,the case that the aim point is off-targeted is considered.The aim point is supposed to lie along the line a=b.Impact point distribution is assumed to be circular-symmetric and the weapon’s lethal radii are chosen to have the same lethal radii Rd=Rr=20 m as done for cases in Fig.4.In Fig.6,the damage probabilities and their relative errors are plotted as functions ofσand a.

In the extreme case thatσgoes to zero,the damage probabilities resulted from using cookie-cutter damage functions behave exotically and show very large differences from PD1NM［CDF，0.7］.When the aim point is closer to the target than the distance of WET/,PD1NM［RDF，1.0］and PD1NM［RDF，0.7］keep constant values of 1.0 and 0.7,respectively,then suddenly drop to 0.0 at the distance.That is because a cookie-cutter damage function has a sharp boundary contrast to the Carleton damage function.When the aim point is farther than the distance WET/,the lethal geometrical region always misses the target,while the Carleton damage function always covers the target even though it has small values at the target position.As the deviation parameterσis not zero,PD1NM［RDF，1.0］and PD1NM［RDF，0.7］are not zero,even if the aim point is farther than the distance WET/.However,their values are less than 0.00052 for D0=1.0 and 0.00037 for D0=0.7 as a is farther than WET/+2σ.The relative errors of PD1NM［RDF，1.0］and PD1NM［RDF，0.7］reach up to 3.290 and 3.811,respectively,at a=WET/andσ=0.The relative errors approach to-1 when the aim point a is farther than WET/+2σfrom the target.

Fig.5.Left panel:Plots of PD1NM as functions ofσd andσr.Right panels:Plots of the relative errors of PD1NM to PD1NM［CDF，0.7］as functions ofσd andσr.The weapon’s lethal radii are Rd=30 m and Rr=20m.The aim point lies at the top of the target at(0.0,0.0).

Fig.6.Left panel:Plots of PD1NM as functions ofσ=σd=σr and a=b.Right panel:Plots of the relative errors of PD1NM to PD1NM［CDF，0.7］as functions ofσand a.The weapon’s lethal radii are Rd=Rr=20 m.

On the other hand,in the region of the parameter space thatσ＞a≫WET/,PD1NM［RDF，1.0］and PD1NM［RDF，0.7］are larger than PD1NM［CDF，0.7］,but the differences are not so big.Roughly,whenσ

The above examples and observations tell that it is necessary to employ the Carleton damage function with the correct shape factor rather than cookie-cutter damage functions and the Carleton function with a wrong shape factor when a weapon’s delivery errors are less than or comparable to the lethal radii,and when the aim point is closer to the target than the lethal radii.They also suggest that preserving the shape factor would result in better answers even in the cases that a cookie-cutter damage function should be used.

4.Weapon’s effectiveness against an area target

In this section,we deal with area target methods and generalize it into the case that the aiming point does not coincide with the center of an area of targets.An area target is defined as a set of identical targets uniformly distributed in a defined area.An area target could contain anything from ground troops to tanks[2].

4.1.Principle of the effectiveness method against area targets

An example of an area target is shown in Fig.7.There,an area of targets is centered at the point P and assigned length and width dimensions of LAand WA,respectively.In that figure,the area target consists of 48 target elements,which are uniformly distributed inside the target area dimensions.The amount of damage done to an area target is measured with a quantity called the fractional damage,which is concerned with what fraction of these target elements is damaged for a given weapon impact[8].For example,6 of 48 targets in Fig.7 are covered by the weapon’s lethal area,and so the fractional damage is 6/48.The fractional damage is proportional to the fractional coverage,which is the ratio of the area covered by the lethal area to the whole area of targets because it is assumed that the target elements are uniformly distributed within an area target.In this case,the problem of calculating fractional damage is reduced to a problem of calculating the fractional coverage.

Fig.7.Weapon vs.area target interaction geometry.

In the area target method,a weapon’s effectiveness is measured by the expected fractional damage that is the expectation value of the fractional damage.The expected fractional damage by a single attack can be defined as

wherePis the position vector of the center of the area target.The term FD（P，U）represents the fractional damage,which is the ratio of the number of damaged targets to the number of total targets inside the area of targets.When the lethality of a weapon is represented by a cookie-cutter damage function,the fractional damage is proportional to the coverage FC（P，U）,the ratio of the area covered by the damage function to the area of targets.On the other hand,it is equal to the average of the Carleton damage function over the area of targets when it is used to represent the lethality of weapon as discussed in Ref.[2].

On the other hand,the expected fractional damage by NAmultiple attacks is calculated according to the survival rule as done for the case of a point target:

whereAiwith i=1，…，NAare aim points.When the aim points are coincident and all weapons are fired independently,the required number of sorties or volleys NRto achieve a required expected fractional damage EFDRis then given by

4.2.Method with a rectangular cookie-cutter damage function against an area target

In the area target method employed in the joint munitions effectiveness manual(JMEM),the rectangular cookie-cutter damage function is used instead of the Carleton damage function[2].When a cookie-cutter damage function represents the lethal area of a weapon,the fractional damage is proportional to the area covered by the damage function,as discussed in the previous paragraph.The fractional damage function is then able to be written by the product of the conditional probability and the fractional coverage FC（P，U）as follows

The calculation of the expected fractional damage is then reduced to calculating expectation values of the fractional coverage

Inserting the delivery accuracy function(6)into Eq.(25)gives

The second term in the parentheses is the fractional coverage by the weapon hit the aiming point and E［FC］NM（P，A）is the expected fractional coverage by the weapon impacting according to the normal distribution,that is,

If the fractional coverage in each direction can be separated as follows

then Eq.(26)can be rewritten as

In Eq.(28),FCd（p，u）and FCr（q，v）are fractional coverages in the deflection and the range directions respectively.

If a side of the lethal area is smaller than the corresponding side of the area target,then the lethal area is expanded to cover the area target fully for the convenience of calculating FD（P，U）.In this case,the probability of damage within this area is not D0as given by the rectangular cookie-cutter damage function Eq.(4),but the conditional probability of damage given by the ratio of lethal area to the expanded damage area

Fig.8.Cookie-cutter damage function against an area target in the deflection direction.

For a situation where the center of an area target is fixed at the point p and the weapon hit around the target as shown in Fig.8,there will be a partial coverage of the target area by the lethal area.Then,the fractional coverage function in the deflection direction looks as shown in Fig.9,and can be expressed in the same way described in Ref.[2].

where s≡（WEP+WA）/2 and t≡（WEP-WA）/2.

The expected fractional coverage by the weapon hit the aim point is able to be directly calculated by inserting the aim point coordinates into Eq.(31).On the other hand,the expected coverage ratio due to the normal distribution in the deflection direction can be represented in terms of the cumulative distribution functions and the normal distribution functions after integrating by parts

Fig.9.Fractional coverage function in the deflection direction.

The fractional coverage in the range direction FCr（q，v）is obtained by replacing s and t with new parameters s’≡（LEP+LA）/2 and t’≡（LEP-LA）/2 in Eq.(32).The corresponding expected fractional coverage can be also obtained by changing s and t with s’and t’and by using b andσrinstead a andσdin Eq.(32).Then,the expected fractional damage is obtained by inserting these results into Eq.(29)as follows

In the sense that the cumulative distribution functions and the normal distribution functions in Eq.(33)can be quickly calculated by using a normal distribution table or a spreadsheet program,Eq.(33)can be termed as a closed-form solution even though it includes numerical integrals on Gaussian functions.

4.3.Method with the Carleton damage function against an area target

Although the Carleton damage function is more accurate and preferred for fragmentation weapons,a rectangular cookie-cutter damage function rather than the Carleton damage function is employed in the JMEM area target method.This is because the method with a rectangular cookie-cutter damage function has known to have a closed-form solution and so can quickly output the expected fractional damage using a spreadsheet,as shown in the previous section.On the contrary,Monte Carlo simulations only had been employed in the area target method with the Carleton damage function[2]because a closed-form solution had not been known until the author pointed the way to get a closed-form solution with employing the Carleton damage function in Ref.[9].Even though Monte Carlo simulations can provide reasonable estimates,closed-form solutions are mathematically more attractive and practically more useful.In this section,the area target method using the Carleton damage function introduced in Ref.[2,9]is reviewed and is generalized to consider a case that the aim point and the center of an area target do not coincide.

Fig.10.Damage functions against an area target.

Since the Carleton damage function continuously varies on an area target as shown in Fig.10,the fractional damage of the area target is not able to be represented as a geometric coverage,unlike the method where a cookie-cutter damage function is used.As discussed in Ref.[2],the fractional damage is the same with a mean value of the Carleton damage function over the area of targets（P，U）and the fractional damage can be obtained by integratingc（X，U）over the area target and dividing with the target area

where p±≡p±WA/2 and q±≡q±LA/2.The expected fractional damage by a weapon is then given by

Let EFD1NM（P，A）and EFD1HIT（P，A）represent the first term and the second term of the second equation in Eq.(35).Since both the Carleton damage function and the delivery accuracy function are separable in the deflection and range directions,they can be calculated separately for each direction.The contribution from a weapon impacting the aim point and excessing the normal distribution represented with EFD1HIT（P，A）is given by

where the definition WET≡is used.（q，b）is expressed in the same way.Therefore,the expected fractional damage by weapons hit the aim point can be calculated in the spreadsheet program as follows

On the other hand,the first term EFD1NM（P，A）corresponding to the contribution of a weapon impacting around the aim point along with the normal distribution can be rewritten as

Each integral of Eq.(39)contains a double integral as shown in the following integral

The double integral does not seem to be separable and integrable because the two variables x and u in the Carleton damage function are intertwined.However,it was shown that Eq.(41)has a closed-form solution in that the expected fractional damage is represented as a combination of single integrals of Gaussian functions as done in the method of using a rectangular cookie-cutter damage function in Ref.[9].Eq.(41)can be rewritten by changing the integration order,gathering the exponents of Gaussian functions,and making in an exact square form to u as follows

where

Since the first integral of Eq.(48)is the integration of a Gaussian distribution function over the whole u space,it is given byEq.(42)can then be written as a single integral of a Gaussian function

or,in terms of the cumulative distribution functions

Thus,the overall expected fractional damage can be represented as a combination of single integrals of Gaussian functions as done in the case of using the rectangular cookie-cutter damage function,that is,

This solution has a much simpler form than that of the method using a rectangular cookie-cutter damage function and can be easily calculated with a weaponeering spreadsheet.

4.4.Calculation results and analyses for example scenarios

This section presents some calculation results of the expected fractional damage for a few scenarios and compares them as done for the methods against a point target in the previous section.It also discusses the necessity of employing the Carleton damage function rather than cookie-cutter damage functions in the area target method.It also shows that the shape factor D0should be kept for the method of using cookie-cutter damage functions as well as for the method using the Carleton damage function.

The simplest analysis again comes from the case that PNM=0.0 and PHIT=1.0,and that the weapon is aimed at the target,i.e.,A=P.Then,the fractional coverage becomes 1.0 in the method of using an rectangular cookie-cutter damage function,and the expected fractional damage is given by

For the method of using the Carton damage function,the expected fractional damage then becomes,from Eq.(36)and Eq.(40),

Unlike the case for a point target,the expected fractional damages EFD1HITresulted from different methods for the same D0are not the same.That is because it is not determined by the value of the Carleton function at the center of the target,but averaged over the area of targets.Notice that the above two formulas are reduced to that of the case of a point target when the target size is much smaller than the lethal radii:EFD1HIT≈R PHITD0.

For the next analyses,let EFD1NM［RDF，D0］and EFD1NM［CDF，D0］represent the expected fractional damages calculated using a rectangular cookie-cutter damage function and the Carleton damage function together with a specific value of the shape factor D0,respectively.Two values of the shape factor D0are supposed to be 1.0 and 0.7.The Carleton damage function with D0=0.7 is supposed to be the most realistic one.It is also assumed that R=1.0,PNM=1.0 and PHIT=0.0.The target is supposed to lie at the origin of a coordinate system:p=q=0 as done in the previous section.Calculation results for some scenarios are presented in Fig.11,Fig.12,Fig.13 as plots of EFD1NM［RDF，1.0］,EFD1NM［RDF，0.7］,EFD1NM［CDF，1.0］,and EFD1NM［CDF，0.7］and their relative errors to EFD1NM［CDF，0.7］,i.e.,（EFD1NM［XDF，D0］-EFD1NM［CDF，0.7］）/EFD1NM［CDF，0.7］,where XDF means RDF or CDF.

Fig.11 displays the results of scenarios that the weapon is aimed at the center of a target.For simplicity and clarity of analysis,impact points are assumed to be circular-symmetrically distributed and so σd=σr=σ.The weapon’s lethal radii are chosen to have the same lethal radii in both directions,i.e.,Rd=Rr=20 m and so WET=m.Three area targets with different sizes are considered and their sizes are assumed to be WA=LA=（2/3）WET,（3/3）WET，and（4/3）WET.

Fig.11(a)shows the results for the case that WA=LA=（2/3）WET.There,the expected fractional damages EFD1NMas functions ofσbehave in similar ways to the damage probabilities PD1NMs for a point target.A difference is that the damage functions with the same D0don’t result in the same EFD1NMwhenσ=0 unlike in the case of the point target.That’s because it is not determined by the value of the Carleton function at the center of the target,but it is averaged over the area of targets.Explicitly,EFD1NM［RDF，1.0］,EFD1NM［RDF，0.7］,and EFD1NM［CDF，1.0］show considerable differences from EFD1NM［CDF，0.7］whenσis less than or comparable to a weapon’s lethal radii.The relative errors of EFD1NM［RDF，1.0］and EFD1NM［RDF，0.7］to EFD1NM［CDF，0.7］reach up to 0.716 atσ≅3.967 and 0.288 atσ≅9.567,respectively,and are larger than 0.1 in large ranges ofσ.The relative error of EFD1NM［CDF，1.0］to EFD1NM［CDF，0.7］goes up to 0.339 atσ=0 and decreases monotonically.

Fig.11.Left panels:Plots of EFD1NM as functions ofσ=σd=σr.Right panels:Plots of the relative errors of EFD1NM［XDF，D0］to EFD1NM［CDF，0.7］as functions ofσ.The weapon’s lethal radii are Rd=Rr=20m.The aim point lies at the center of the area target,at(0.0,0.0).

Fig.11(b)shows the case that the target lengths are the same with the lethal area lengths,i.e.,WA=WETand LA=LET.In this case,the expected fractional damages EFD1NMbehave in a similar manner to the case the target size is smaller than the lethal area lengths besides that they decay more rapidly.The relative errors of EFD1NM［RDF，1.0］and EFD1NM［RDF，0.7］to EFD1NM［CDF，0.7］reach up to 1.008 atσ=0 and 0.414 atσ≅1.943,respectively,and are larger than 0.1 in relatively large ranges ofσ.The relative error of EFD1NM［CDF，1.0］to EFD1NM［CDF，0.7］is relatively small but still goes up to 0.253 atσ=0.

Fig.12.Left panels:Plots of EFD1NM as functions ofσd andσr.Right panels:Plots of the relative errors of EFD1NM［XDF，D0］to EFD1NM［CDF，0.7］as functions ofσd andσr.Rd=30m and Rr=20m.The aim point lies at the center of the area target at(0.0,0.0).

Fig.11(c)presents plots for the case that WA=LA=（4/3）WET.In this case,EFD1NM［RDF，1.0］and EFD1NM［RDF，0.7］coincide with each other.That’s because,in the area target method of using rectangular damage functions,the shape factor is reflected only in the expanded lethal lengths in the forms ofandLA）.The dependency on D0disappears from EFD1NMwhenand.The relative errors of EFD1NM［RDF，1.0］and EFD1NM［RDF，0.7］to EFD1NM［CDF，0.7］reach up to 0.424 atσ=0,and are larger than 0.1 in large ranges ofσ.The relative error of EFD1NM［CDF，1.0］to EFD1NM［CDF，0.7］goes up to 0.167 atσ=0 and decreases monotonically.The expected fractional damages EFD1NMstill show quite big differences with each other when the delivery accuracy is less than or comparable to the lethal radii,even though the relative errors are smaller than those from the previous cases.

Fig.12 presents plots of EFD1NMand its relative error as functions of deviation parametersσdandσr.Here,the weapon’s lethal radii are chosen to be Rd=30 m and Rr=20 m.It shows that the above observations are valid for the case where the lethal radii and the delivery errors are different in the deflection and range directions.

Next,the case that the aim point is off-targeted is considered for three targets with different sizes.The aim point is supposed to lie along the line a=b.Impact point distribution is assumed to be circular-symmetric,and the weapon’s lethal radii are chosen to have the same lethal radii Rd=Rr=20 m，as done for results in Fig.11.In Fig.13,the expected fractional damages and their relative errors are plotted as functions ofσand a.Fig.13(a)shows the results for the cases that WA=LA=（2/3）WET.The expected fractional damages EFD1NMbehave very similarly to PD1NMin Fig.6 as functions of bothσand a.In the extreme case thatσgoes to zero,EFD1NMresulted from using cookie-cutter damage functions behaves exotically and shows large differences from EFD1NM［CDF，0.7］.When the aim point is closer to the target than the distanceWET,EFD1NM［RDF，1.0］and EFD1NM［RDF，0.7］keep constant values of 1.0 and 0.7,respectively,then decay along withWET-a］2,and finally,become zero atWET.These behaviors are the same as those of R PCDPNMFCd（a，0）FCr（a，0）because the normal Gaussian function becomes the Dirac delta function in the extreme case that σgoes to zero.The relative errors of EFD1NM［RDF，1.0］and EFD1NM［RDF，0.7］to EFD1NM［CDF，0.7］reach up to 0.854 at a=5.916 and 0.517 at a=9.412.They become negative and then finally-1 as the aim point goes away.On the other hand,the relative error of EFD1NM［CDF，1.0］to EFD1NM［CDF，0.7］goes up to 0.339 at a=0.

In the region of the parameter a＞σ＞+2/3）WET,EFD1NM［RDF，1.0］and EFD1NM［RDF，0.7］are not zero,even if the aim point is farther than the distance+2/3）WETfrom the center of the target.However,they get negligible as a is farther than （WET+2σ,for example,they become less than 0.00005 for D0=1.0 and 0.00004 for D0=0.7 whenσ=20 m.The relative errors of EFD1NM［RDF，1.0］and EFD1NM［RDF，0.7］reach up to 0.854 and 0.517,respectively,whenσ=0.The relative errors close to-1 when the aim point a is farther thanWETfrom the target.On the other hand,in the region of the parameter space that2/3）WET,EFD1NM［RDF，1.0］and EFD1NM［RDF，0.7］are a little bit larger than EFD1NM［CDF，0.7］.

Roughly,in the region thatσ

Fig.13(b)shows the results for the cases that WA=LA=WET.The expected fractional damages EFD1NMbehave very similarly to those in Fig.13(a)besides thatWETbecomes 0 for D0=1.0.Fig.13(c)presents plots for the case that WA=LA=（4/3）WET.In this case,EFD1NM［RDF，1.0］and EFD1NM［RDF，0.7］coincide with each other for the same reason as in Fig.11(c),but the overall behavior of EFD1NMis similar to the results in Fig.13(a).

The above examples and observations tell that it is necessary to employ the Carleton damage function with the correct shape factor rather than cookie-cutter damage functions and the Carleton function with a wrong shape factor when a，σ

5.Conclusions

This paper reviewed methods of assessing a fragmentation weapon’s effectiveness against a point target and an area target by keeping the focus on the necessity of using the Carleton damage function with the correct shape factor.In those methods,the effect of a fragmentation weapon to a target is represented by the Carleton damage function or cookie-cutter damage functions.In this paper,the damage functions were rewritten to incorporate the shape factor D0of the Carleton damage function and to preserve it in a cookie-cutter damage function.

Then,the effectiveness methods against a point target were extended to incorporate those damage functions and to be applicable to the scenario where the aim point is off-targeted.The method was applied to calculate the probabilities of damaging a point target for several attack scenarios by using four damage functions,the Carleton and a rectangular cookie-cutter damage functions with D0=1.0 and the ones with D0=0.7.

This paper also reviewed the conventional effectiveness method against an area target and extended the method to preserve D0in a rectangular cookie-cutter damage function and to be applicable to the scenario where the aim point is off-centered from an area target.It also provided an alternative area target method to use the Carleton damage function rather than a cookie-cutter damage function and presented a closed-form solution that can be applicable to the scenario where the aim point is off-centered.The two area target methods were employed to calculate the expected fractional damages of an area target for several attack scenarios by using four damage functions as done in the case of the method against a point target.

The calculation results were presented as functions of the deviations of delivery errors and the distance between the aim point and a target.The comparisons of the calculation results for both a point target and an area target suggest that the Carleton damage function with a correct shape factor D0should be used when the deviations of delivery errors are less than or comparable to the lethal radii,and when the aim point is much farther than the deviation of delivery accuracy from targets.Observations of the comparisons additionally tell that it would be better to keep the shape factor even if a cookie-cutter damage function has to be used.

Finally,the improved effectiveness methods to use the Carleton damage function with a correct shape factor will be greatly useful for mission planning utilizing the precision-guided munitions in circumstances where the collateral damage should be reduced.That is because those methods can provide more accurate results than the conventional ones for precision-guided weapons and allow them to select an adequate aim point for reducing collateral damage.

Declaration of competing interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

Acknowledgment

A part of this work was supported by the Engineer and Scientist Exchange Program between the Republic of Korea and the United States.