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      The integration is performed over the whole area, within which the coordinate law does not turn zero.

      For nuclear warheads, the specified damage zone is a circle with an area S_sp = πR ² , where R is the radius of the specified damage zone.

      For conventional ammunition, the specified damage zone is often represented as a rectangle, and therefore, its area is S_sp = 4l_x l_y , where the 2l_x and 2l_y are the depth and front of the specified target sizes.

      The probability of damaging an elementary target with several shots when using the concept of a specified damage zone can be calculated as the probability of at least one ammunition hitting this zone. In this case, the probability is calculated using the same formulas as when assessing the effectiveness of contact ammunition.

I.3 Evaluation of the Effectiveness of Shooting

      I.3.1 The Concepts of Combat Effectiveness of Weapons

      A detailed study of the theory and methods of assessing the combat effectiveness of various types of weapons and ammunition is not included in this textbook. However, consideration of the basic concepts will certainly be useful for a better understanding of what follows. Using simple models of combat operations, we will illustrate how generalized characteristics of the damaging effect are used in assessing the approximate combat effectiveness.

      As a rule, the combat effectiveness of a weapon is assessed by the amount of damage inflicted on enemy objects (targets). The effectiveness of firing depends on a large number of different factors: type, size, mobility, and vulnerability of the target; characteristics of the dispersion and destruction effect of ammunition; the amount of ammunition (the number of guns, missiles, batteries, divisions, aircrafts, etc.) involved in the destruction of targets; methods of firing; reliability of weapons; time of staying in position; counteraction of the enemy, etc. In particular, according to the army data, 6 seconds after firing begins, the first soldier already reaches a shelter, and 11 seconds afterward, the last soldier leaves the exposed area.

      I.3.2 Classification of Targets, Typical Efficiency Indicators

      To assess the effectiveness of the shooting, the whole variety of targets can be represented by three typical kinds [2].

      A single target is an elementary, usually small target that provides certain functions (aircraft, ship, tank). The task of firing is its damage. As an indicator of efficiency, the probability of damage W = P(A) is often used, where A – the damage of the target.

A group target is a target consisting of several single targets, such as the position of an anti‐aircraft missile system, a tank convoy, or a group of aircraft or ships united by a common combat mission. When firing, it is necessary to damage the group as a whole and thus prevent it from fulfilling its combat mission. According to the experience of World War II, the enemy's refusal to continue the attack is directly related to the number of losses. Loss of 50–60% tanks means refusing to continue the attack with the probability of 0.90–0.95. The average number of damaged targets in the group M_d = M[K_d], where the random value of K_d is the number of damaged targets, is usually taken as an indicator of the shooting efficiency. In some cases, a more specific task is set before a group target is fired, for example, to disable the enemy's antiaircraft missile system. Damaging the position of the antiaircraft missile system may mean either damaging all missile launchers or damaging the target illumination radar. Then the indicator of the effectiveness of firing on a group target will be the probability of completing the task A: W = P(A).

      An area target is a target consisting of a set of objects distributed within a certain area in an obscure way, for example an accumulation of personnel and battle equipment. Typical for an area target is that it is not the individual objects that are targeted, but the entire area as a whole. As an efficiency indicator, the expected value of the portion of the damaging area M = M[U] is used, where U = S_d/S_t is the ratio of the damaging area to the full target area.

      I.3.3 Dispersion During Shooting

      There is always dispersion in any type of firing, both with unguided and guided missiles. The main reasons for the dispersion of projectiles during firing are an inaccurate determination of target coordinates; aiming error; the influence of meteorological factors (wind, change of atmospheric pressure, humidity); fluctuations of the launcher; production tolerances during ammunition manufacturing.

      All errors affecting the deviation of the projectile from the target can be divided into systematic and random errors. Systematic errors from shot to shot do not change, they can be measured and taken into account later (e.g. the deviation of the aim point from the center of the projectile dispersion) and the effect of low atmospheric pressure when firing at altitudes other than sea level. Random errors cannot be measured, as they vary from shot to shot. As a result of the combined effect of all firing errors, the actual trajectory of the projectile never coincides with the calculated trajectory, and the point of hit (or blast) of the projectile inevitably deviates from the calculated point to which the projectile was directed. This phenomenon is called “dispersion.”

The law of distribution of random values characterizing the point of hit (or blast) of ammunition is called the law of dispersion. For contact ammunition or remote one with flat dispersion, this law is presented as the law of distribution of the two coordinates (x, y) of points of hitting. Usually, the distribution is given as the value of the probability density φ(x, y). The value φ(x, y)·dx·dy is the probability of hitting an area with dimensions dx·dy adjacent to the point with coordinates (x, y) (Figure I.4).

      Similarly, for projectiles with volumetric dispersion, the dispersion law is the distribution of the three coordinates of the blast point (x, y, z) and is characterized by the probability density φ(x, y, z), with the value of φ(x, y, z)·dx·dy·dz being the probability of the projectile blast in the elementary volume dx·dy·dz adjacent to the point (x, y, z).

Let's consider the case of flat dispersion as a simpler one. Imagine shooting a contact projectile or a remote one with flat dispersion. First of all, we must choose a flat surface on which we will study the dispersion of hit points. This surface is commonly referred to as a picture plane or a dispersion plane. When shooting at land or sea targets with remote ammunition, this is usually the surface of the ground or sea. When a land or sea target is shot contact projectiles, it is usually considered to be on a vertical dispersion plane. In the case of air targets, the picture plane is most often drawn through the point of hit perpendicular to the vector of the relative velocity at which the projectile meets the target.

Figure I.4 Area with dimensions of dx·dy, adjacent to a point at the coordinates (x, y).

      Source: From Wentzel [2].

When the picture plane Q is fixed, the rectangular X0Y coordinate system is selected. Figure I.5 shows the picture plane and the coordinate system for the case of shooting an air target.

      Usually, when a remote projectile is fired at sea or ground targets, the X‐axis is at least

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