ASSESSMENT OF THE SINGLE SHOT HIT PROBABILITY AS A FUNCTION OF THE HORIZONTAL RANGE TAKING INTO ACCOUNT DIFFERENT TARGET TYPES AND POINTS OF AIM

This article presents a method for assessing the hit probability for stationary shooting targets as a function of the projectile horizontal range (PHR), the target type and the point of aim (POA) location. The proposed procedure consists of three blocks. Block I is devoted to the double integral formula taking into account changes in the point of impact (POI) location as a function of the PHR. The characteristics of bullet accuracy and precision versus the PHR are determined in block II. The basis of blocks I and II is the bi-variate uncorrelated Laplace-Gauss probability distribution. The functions of the POI location (ordinate/abscissa) versus the PHR and the functions of the characteristics of the bullet dispersion versus the PHR are represented in the form of polynomials. The description of the target silhouette contour is given in block III. Mathematically, the target contour is a piecewise function which defines the upper and lower edges of the shooting target and it also represents the limit for the double integral formula of block I. The proposed method is built on a modular basis and allows a user to change types of weapons and shooting targets. For demo calculations, the accuracy and precision characteristics of the 5.45×39 Kalashnikov assault rifle model MPi AK-74N were selected. Five types of Swiss military targets imitating an OPFOR combatant's silhouette were used as shooting targets. For illustrating the operability of the proposed method, the hit probabilities for the K, H, G, F, and E shooting targets were evaluated for the PHR from 50 to 400 m. All computations are implemented in the PTC Mathcad v.15.


Introduction
Estimation of the hit probability is an important part of the shooting effectiveness assessment (Rodney, 2012), (Svateev, 2013), (Svateev, 2014), (Peelen, 2017), (Wollschläger, 2017).In (Khaikov, 2018), we show how to estimate the single shot hit probability by shooting to a fixed target with an arbitrary contour form owing to a numerical solution of the double integral.During the calculation process, the following parameters are set: the position of the mean point of impact (POI) and the characteristics of the bullet dispersion ellipse.However, during firing, there is uncertainty about the horizontal range to the selected target; it is also necessary to have a priori information about the change in the hit probability for the interval from the minimum to the maximum distances.
Since the position of the dispersion center and the dispersion characteristics are the functions of the projectile horizontal range (PHR), the hit probability can be expressed as a function of the «target-shooter» distance 1 .
The aim of this contribution is to show how to calculate the hit probability (P hit ) as a function of the PHR with respect to different target types and the location of the POA.As a result, we get not only one (or a point) estimation of the P hit for a single (selected) value of the horizontal range, but also a change of this parameter within a certain distance interval.As in (Khaikov, 2018), all calculations will be implemented in the computer algebra system (CAS) Mathcad v.15.
In order to make the estimations original and more diverse, we will use here Swiss military targets and the external ballistics of the 5.45×39 Kalashnikov assault rifle model MPi AK-74N 2 .The information about this firearm can be found in (NVA der DDR, 1985), (TRADOC, 1975(TRADOC, , 2015)), (http://weaponland.ru,nd).

Creating a mathematical model
Let us assume that the attacking opposing force (OPFOR) is represented by running or lying combatants whose silhouettes can be replaced by plane (or 2D) shooting targets with a certain equivalent area 3 1 In this article, the expressions "the target-shooter distance" and "the projectile horizontal range" are synonymous. 2MPi AK-74N -(in German) Mashinenpistole-Automat Kalaschnikow AK-74 für Nachtsichtgerat.Assault rifles MPi AK-74N were manufactured on "VEB Fahrzeug und Waffenfabrik Ernst Thälmann", Suhl (Thuringia). 3As a rule, shooting targets are a set of geometric primitives: rectangles, trapezoids, triangles, and squares.
target types and points of aim, pp.(Fig. 1a).Human silhouette targets of different types are used for military and police firearms training.More schematized targets are applied for firearm sports competitions.
Let it be necessary to estimate the hit probability for several targets at the distances x I , x II , and x III (Fig. 1b).All shooting targets at different distances are the same and in Fig. 1b  When firing at a shooting target, depending on the PHR, the mean bullet (ballistic) trajectory4 will pass through the target or above (below) it (Fig. 1).The difference in the PHR between the shooter and the shooting target (x I , x II , x III ) determines the difference in the POI location and the constantly increasing parameters of the dispersion ellipse for each target.The location of the POA in Fig. 1  Let us define that the shooting mode (single shots) and the aiming parameters are not changed by shooting.The ABCD curve (Fig. 1b) is the mean ballistic trajectory (MBT) with respect to the specified angle of departure (or number/clicks of sight) and the previously specified zero range.Uncertainty in the PHR magnitude is transformed into a vertical projectile dispersion at a chosen shooting target.Naturally, the hit probability value (P hit I , P hit II , P hit III ) will change with the variations of the POI location and with the change in the dispersion ellipse characteristics 5 .
The mathematical model of the hit-probability vs the PHR will consist of three blocks.Block I will be devoted to the double integral formula taking into account the changes in the POI location and bullet dispersion.Block II will include the bullet dispersion characteristics vs the horizontal range.Finally, block III will describe the functions of a target silhouette contour.
Block I.This block of the mathematical model is based on the results of (Khaikov, 2018).If the T region (letter "T" means a target) is a plane figure, and the hit probability of random variables Z and Y is determined by the 2D uncorrelated Laplace-Gauss distribution   y z f , , then the target hit probability is defined by formula (1) (Venttsel', 2006, p.196) where: Tthe region of integration in the zy-plane; z, ythe Cartesian coordinates;   y z f , -the integrand and the uncorrelated bi-variate Laplace-Gauss distribution (2): 5 The first important factor is the type of weapon and the type of cartridge used.In the second place, the POI location depends on the target type.For the pre-selected weapon, the cartridge and the shooting target, the POI location and the characteristics of the dispersion ellipse are the functions of the PHR.In the case under consideration, skills of the shooter and combat stress as psychophysical conditions are not taken into account.
target types and points of aim, pp.13-35 (2) In (Khaikov, 2018, pp.739-756), it is shown that if the target silhouette is described by two functions (the function of the bottom contour line of the target φ 1 (z) and the function of the upper line of the target's contour) φ 2 (z) and two constants a and b (a<b), then the hit probability can be expressed as ), we get a formula that realizes the transition from for the assault rifle MPi AK-74N will be described in block II.
The trajectories of the steel core bullet of the 5.45×39 Kalashnikov cartridge (bullet weight -3.4 g; muzzle velocity -960 m/s) under the line of departure and over/under the line of sight for mechanical sight from 1 to 10 are described in (NVA der DDR, 1985, pp.90-93) as tabulated data.Let us visualize this table and obtain the approximation formula (Fig. 2).
The data corresponding to sight number 3 (NVA der DDR, 1985, pp.90-93) can be approximated using a polynomial.As a result, we obtain the function  where x is the PHR («shooter-target» distance) in meters (50 ≤ x ≤ 400 m) and m y is the ballistic trajectory profile in meters for the sight setting 3 (position 3 for standard mechanical sight of the Kalashnikov assault rifle).The red curve in Fig. 2 shows the change of the POI height.At 50 m, shots will be about 6 centimeters=0.06 m high.The vertex of the flight path is 27.66 cm = 0.267 m, observed for the PHR 168.3 m (Fig. 2a).The zero distance is 300 m.Beyond 300 m (PHR>300 m), the bullet trajectory will drop rapidly to about 43 centimeters ≈ 0.427 m low at 400 m.The right-hand bullet drift or the bias of the POI to the right side due to the bullet gyroscopic stabilization can be expressed by the polynomial model where x is the PHR in meters (50 ≤ x ≤ 400 m) and z m is the right-hand bullet drift in meters (the blue dotted curve in Fig. 2a).
The bullets dispersion in the vertical ( In order to use formula (4), we need to determine the functions φ 1 (z), φ 2 (z) and the constants a and b.
Depending on an OPFOR combatant silhouette seen by a shooter, there are «head», «head-shoulders», «head-burst», «head-hip», and «man sized» military targets (Fig. 3)6 .The target contour can be designated in two ways (Khaikov, 2018, pp.739-756): in option I, the upper (lower) contour should be identified as the y(z) functions or, in option II, the right (left) contour should be identified as the z(y) curves.For all five targets in Fig. 3, the upper contour lines (UCL) are red, and the bottom counter lines (BCL) are yellow.If we consider the UCL of the target K, then we use the notation UCL K .This procedure for the BCL of the target E is similar: BCL E .
The geometry analysis and the contour lines identification for the G target are shown in Fig. 4. It is a flat shooting target which can be described as the ABCDEFGHIJ polygon.The G target has an axis of symmetry that passes through the EJ (Fig. 4a).The left side of the G target is the ABCDEJ polygon, and the right-hand side is the EFGHIJ polygon.The areas of these polygons are the same.The presence of the right-hand bullet drift causes the POI to be located on the right side (the red ellipse in Fig. 4b ).a) b)

Figure 4 -Analysis of the geometry of the G shooting target (a). A location of the dispersion ellipse (b) Рис. 4 -Анализ геометрии стрелковой мишени G (a). Положение эллипса рассеивания (b) Слика 4 -Анализа геометрије мете за гађање G (а). Локација елипсе дисперзије (б)
target types and points of aim, pp.13-35 Not taking into account the symmetry of the G target, its UCL can be described as the BCDEFGH line.When symmetry is taken into account, the geometric formula of the left-hand side is BCDE.Similar reasoning applies to the description of the BCL: AJI (without symmetry) and AJ (the left side of the figure with symmetry taken into account).
It is necessary to determine the coordinates of the vertices of the ABCDEFGHIJ polygon and to obtain an analytical expression of the sides of the G target.The coordinates of the points for the G target contour and its analytic functions are collected in Table 1.The contour line (CL) CL G (z) of the G shooting target (the UCL (BCDEFGH) UCL G (z) and the BCL (AJI) BCL G (z) ) can be described by a piecewise function: Since the geometry of the G target has an axis of symmetry (EJ axis) and if the POI lies on the EJ axis, then the hit probabilities for the right and left parts are the same.If the POI location is not on the EJ axis, then the hit probabilities for the right and left parts of the G target are different.The side of the target where the POI is located has a higher hit probability.In view of the fact that the bullet gyroscopic stabilization causes a shift of the POI to the right, the right-hand side of the G target has a larger hit probability than the left one.
The coordinates of the points for the H target contour line are collected in Table 2.The analysis of Table 2 allows us to describe the contour needed for integrating the double integral (4) and determining the hit probability.The contour of the H shooting target (the UCL (BCDEFGH) and the BCL (AJI)) can be described by a piecewise function: The coordinates of the points for the F target contour line are presented in Table 3.The contour of the F shooting target (the UCL (BCDEFGH) and the BCL line (AJI)) can be described by a piecewise function: The coordinates of the points for the E target contour line are listed in Table 4.The contour of the E shooting target (the UCL (EFGHIJK) and the BCL (DCBAPONML) (Fig. 5)) can be described by a piecewise function: The hit probability of this target can be found as an algebraic subtraction of the hit probabilities of its three domains P hitK = P hit (ABEFGHILM) -P hit (BEDC) -P hit (KJIL) .
The coordinates of the vertices of the K target as a geometric figure and the analytical functions of its sides are collected in Table 5.The sequence stepsthe development of the calculation program in the CAS Mathcad will be described in the next section.

Calculation in the CAS Mathcad
The projectile horizontal range (in meters) is defined in the Mathcad as a range of variable x with the initial value of 50, the end value of 400 and the increment of 51 (the step size is 1 meter) x:= 50, 51 .. 400 .
target types and points of aim, pp.13-35 Using the constant a, the my vector can be displaced vertically in the up-down direction.If a>0, then the zero range increases (>300 m).Usually, a is equal to 0 (zero range = 300 m).Thus, the parameter a makes it possible to simulate the position of the POA along the height of the shooting target.
For each value of x, set the x th element of the vector variables σy and σz, and my and mz with a corresponding value of x.The tables of numbers for the vector variables σy, σz and my, mz are of the same size.Each of them consists of 351 numbers.Using the variables sYi and sZi, we determine the center of the shooting target.For the K target, its center will be defined as (sYK, sZK).For the H target, as (sYH, sZH), and so on.The data on the position of the centers of the K, H, G, F, and E targets are shown in Table 6 (in meters).For the K target, the following expression can be proposed: P hitK = P hit (ABEFGHILM) -P hit (BEDC) -P hit (KJIL) = P hit A -P hit L -P hit R, here P hit (ABEFGHILM), P hit Aprobability of hitting the ABEFGHILM domain as a function of the PHR; P hit (BEDC), P hit ALprobability of hitting the BEDC domain as a function of the PHR; P hit (KJIL), P hit Rprobability of hitting the KJIL domain as a function of the PHR.
Due to the conditional operator if, the UCL and the BCL of the constituent elements for the K shooting target are defined as: The assessment of the hit probabilities of the elements of the K target is realized by the following formulas The hit probability assessment for the H target is The formulas for the G and F targets are given in the appendix of this paper.The hit probability assessment for the E target is

Results of the calculation
The curves characterizing the hit probabilities functions of five different types of targets (K, H, G, F, and E) that were determined using the MBT (5) are shown in Fig. 7.The space under the curve in the interval 50-400 m is highlighted in yellow.For all five targets, the POA (red square) was the same locationthe middle of the BCL.The shooting conditions were considered to be ideal.The hit probability change vs the PHR for all five targets is a classical decreasing curve (Shereshevskiy et al, 1979).The dashed lines in Fig. 7a show the ranges that correspond to the hit probabilities of 0.9, 0.8, and 0.7 for the K target.
Changing the shape for the hit probability vs the PHR with respect to the F target and the location of the POA (red square) is shown in Fig. 8. Fig. 8 shows that if the range to the shooting target is 200 meters and if the POA is positioned on the BCL (see the red square as the graphical symbol), then the hit probability is 0.92 (Fig 8a).If the POA is 0.5 m higher, then the hit probability for the PHR=200 m is increased to be 0.98 (Fig. 8b).If the POA is increased to 1.0 meters, then the hit probability decreases to 0.08.Thus, if the distance to the shooting (real) target is known as well as the target geometric dimensions and the ballistic characteristics of the bullet flight path (MBT and bullet dispersion), the POA determining algorithm for a remote weapon station of a ground combat vehicle can be proposed.

Conclusions and Implications
This paper represents a method for assessing the single shot hit probability for a stationary (fixed) shooting target as a function of the PHR, the selected target type and the location of the POA.The proposed procedure has three blocks.The proposed method is built on a modular basis and allows a user to change types of weapons and types of shooting targets.
The single shot hit probability depends on five variables: height of the POI; lateral displacement of the POI relative to the target; bullet dispersion in the height (vertical direction); lateral bullet dispersion (horizontal direction); and the forms of the shooting target contour.To order these variables, three auxiliary quantities can be introduced.These are the projectile horizontal range (PHR), the altitude of the point of aim and the lateral displacement of the point of aim.The last two variables can be combined into the «POA location».
For demo calculations, the external ballistics and the bullet dispersion characteristics of the 5.45×39 Kalashnikov assault rifle model MPi AK-74N were selected.Five types of Swiss military targets imitating OPFOR combatant silhouettes were used as shooting targets.For illustrating the operability of the proposed method, the hit probability for the K, H, G, F, and E shooting targets was evaluated as well as for the PHR from 50 to 400 m.The calculations have shown that the PHR and the form of the target cause a complex dependence between the hit P and the shooter-target distance.
The developed mathematical model can be used in tactical computer interactive simulators that emulate combat situations of a tactical strike.Likewise, the proposed mathematical models of shooting targets can be used for improving digital target range simulators or in firecontrol systems of remote weapon stations.

Formulas for the hit probability assessment of some shooting targets
The hit probability hit P vs the PHR for the G and F targets can be expressed as

Figure 3 -
Figure 3 -Military shooting targets (cibles de campagne): K, H, G, F, and E Рис. 3 -Армейские мишени K, H, G, F, E для стрельб Слика 3 -Војне мете за гађање K, H, G, F и E Among the five shooting targets, the K and H targets have the minimum height of 33 centimeters (0.33 m); conversely, the E target has the maximum height of 1.65 m.The target area is increased in the Notes: The coordinates of the points are given according to Fig.3.The symmetry of points is shown with the orange and blue background.The locations of the vertices for the G target can be used in the analysis of the H target and the F target.
Notes: The coordinates of the points are given according to Fig.3.The symmetry of the points is shown with the orange and blue background.
Coordinates of the points for the F target contour line Таблица 3 -Координаты точек контура мишени типа F Табела 3 -Координате тачака контуре мете типа F coordinates of the points are given according to Fig.3.The symmetry of the points is shown with the orange and blue background.
coordinates of the points are given according to Fig.3.The symmetry of the points is shown with the orange and blue background.

Table 1 -
Coordinates of the points for the G target contour line

Table 2 -
Coordinates of the points for the H target contour line