Upgrading effectiveness of fire estimation.

One of the most important tasks of military science is to develop methods of estimating effectiveness of employment of weapons. Of considerable importance are methods of estimating the effectiveness of employment of conventional weapons and, most importantly, methods of estimating the effectiveness of fire on multiple area targets.

Existing analytical methods of estimating the effectiveness of conventional fire on area targets are based on the assumption of even distribution of individual targets inside multiple targets. This is understandable because these methods were developed in the first part of the 20th century. This assumption was valid at that time because multiple targets were suppressed or destroyed during the Great Patriotic War of 1941-1945 and in the postwar period as a rule by suppressing and destroying troops and weapons that with equal probability could be stationed anywhere inside the area taken up by multiple targets. Finding the real location of troops was not possible as a rule, for which reason fire had to be delivered on the entire area taken up by such targets and that substantially increased consumption of ammunition and time needed for fire missions.

The nature of multiple targets and of individual targets that constitute them has greatly changed since. There are now long-range and armored weapons located in a variety of ways and variously determining operating efficiency of multiple area targets, mobile command posts, receiving and transmitting centers, radars, etc. They are frequently more vulnerable than the personnel who is protected by armor and military equipment. This makes it possible to suppress or destroy multiple area targets not only by destroying their personnel but by knocking out pieces of military equipment whose number is small but their contribution to operating efficiency of multiple area targets is big on the whole. (1)

The growing capabilities of reconnaissance assets and systems make it possible to reliably determine the location of such individual targets, which comprise multiple area targets, as weapons and military equipment. It is of course desirable to destroy such targets using precision weapons but it is possible to also use conventional weapons. It is simply wasteful to fire on entire areas instead of on individual targets whose location is known. There are, or course, recommendations to fire on separate sections of multiple area targets where groups of individual targets could be found. Using existing methods, it is possible to determine the kill effectiveness and calculate the consumption of ammunition, but the effect of ammunition on other individual targets (groups of targets) can be determined only approximately through very complicated calculations. In addition, existing methods are unsuitable for estimation purposes in firing on such multiple targets with precision weapons.

It follows that there is a contradiction between existing mathematical techniques for estimation of effectiveness of conventional weapons used against multiple area targets and modern methods of destruction of such targets the effectiveness of which cannot be estimated with the use of existing mathematical techniques.

There is also a second contradiction--between existing mathematical techniques for estimation of effectiveness of conventional weapons used against multiple area targets and capabilities of modern reconnaissance and weapon assets to precisely locate individual targets, which comprise multiple area targets, and to score their assured destruction.

During the decades following the development of existing methods for estimation of effectiveness of conventional weapons, our researchers have proposed a number of ways to update existing methods and widen the permissible areas of their application, but that led to additional errors. In the meantime the capability of the computing assets of modern automated fire control systems makes it possible to cope with more complicated calculations which makes unnecessary numerous simplifications that are usually made in using existing methods for estimation of effectiveness of conventional weapons. The latter methods besides overestimate the target kill rates (or underestimated ammunition consumption rate). This is because these calculations yield not realistic but what is known as "ideal" fire density whose form is far removed from the form of practically achievable fire density. Hence a third contradiction consisting in the imbalance between the tools used for estimation of effectiveness of conventional weapons that yield only approximate solutions and the capabilities of the computing assets of modern automated fire control systems.

Naturally, many of the above contradictions could be eliminated with the use of statistical methods of estimation, but their "blindness" and cumbersome nature make them little suited for research and development of practical recommendations in the employment of conventional weapons. There are many reasons militating against the use of statistical methods in special software of automated fire control systems.

Thus, the problem of updating mathematical techniques for estimation of effectiveness of conventional weapons consists in developing methods that could take full account of the nature of targets, the capabilities of reconnaissance and weapon assets and the potential of modern computing assets.

One possible solution is to use optimal search methods. (2) Their use in solving "direct" and "reverse" problems of estimating the effectiveness of conventional weapons made it possible to obtain a system of integral equations which takes into account parameters that can be taken into account using existing methods and makes it also possible to take into account the nature of distribution of individual targets inside multiple targets (the target density function).

The system of integral equations imposes no constraints on either the function of optimal density of fire or the function of density of distribution of targets (inside multiple targets) whereas existing methods require continuity and differentiability of the latter density. This approach fully takes into account the capabilities of modern reconnaissance assets and, based on a single methodological approach, solves "direct" and "reverse" problems of estimating effectiveness of destruction of individual targets and multiple targets by conventional and precision weapons. Regardless of the type of problem, solutions under given conditions will always be close to the optimal, i.e., the target kill effectiveness with the specified ammunition consumption will be as high as possible and the required ammunition consumption will be minimal when the specified target kill effectiveness is achieved.

The drawback of the system of integral equations, however, consists in that the optimal pattern of density of fire resulting from solving this system of equations repeats in a certain way the distribution pattern of individual targets around the area of multiple targets while at the same time local sections of the area engaged by fire should be continuous. The optimal density of fire is mainly dictated by the number of pieces of weapons (mortars, launcher vehicles) available to a unit performing the fire mission. In practical terms therefore the number of aiming points (in artillery, the number of angles of elevation and deflection settings) is always equal to or at least multiple of the number of weapons performing the fire mission. This makes it necessary to estimate the effectiveness of target engagement considering the fact that the method of firing on it, which is being optimized, is of a discrete fashion. In the ideal case, therefore, the optimal number of aiming points should be equal to the actual number of the weapons if they are employed individually. This can be taken into account if we transform the system of integral equations to a system of balance equations of the following form:

[k.summation over (i=1)][P.sub.i] - k[P.sub.k+1][greater than or equal to]P>[k.summation over (i=1)][P.sub.i] - k[P.sub.k]; [1/[alpha]][([k.summation over (i=1)][P.sub.i] - k[P.sub.k+1])][greater than or equal to]N>[1/[alpha]][([k.summation over (i=1)][P.sub.i] - k[P.sub.k+1][P.sub.k])]

which can be solved provided

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

where P--average kill probability of one individual target (expectation of the relative number of destroyed individual targets in multiple targets);

N--the number of pieces of ammunition needed to destroy an individual target;

[P.sub.i]--probability of finding an average trajectory inside the ith individual zone of the [DELTA][S.sub.i] area of fire equal in size to the computed kill area (zone) per one piece of ammunition (computed area of an individual target);

f(x, z)--the distribution density of recurrent firing errors based on errors in locating individual targets;

k--the number of individual zones where target kill is optimal;

[alpha]--integral parameter that takes account of the power of one piece of ammunition.

The system of equations is a "balance" system because the true values of P or N in solving the "direct" and "reverse" problem of estimating the effectiveness of fire lie in the interval between solutions of two inequalities with variables [P.sub.k] and [P.sub.k+1].

The solving of this system of equations consists in finding the optimal area of firing expressed by the number of zones k where individual targets are to be hit, and the number of pieces of ammunitions to be expended on each target. In solving the "direct" problem, it also consists in determining the probability of target kill (expectation of the relative number of destroyed individual targets). The main distinction of the system of balance equations from the system of integral equations is that, in the former, the optimal zone of fire on a target is broken up into k individual optimal zones [DELTA][S.sub.i], each equal in size to computed sizes of the individual targets.

The system of balance equations makes it possible: to take into account the discrete nature of individual targets locations and accuracy in determining their coordinates and the discrete nature of density of fire on multiple targets when fire is delivered by a limited number of weapon assets with a limited number of aiming points; not to fire on the entire area of multiple targets (to fire on every individual target separately) and not to fire on multiple targets as if they were separate targets and to take stricter account of the dispersion of hits (it can be taken into account approximately using existing methods). Taking appropriate account of the dispersion of hits affords the possibility of getting solutions that are fully in line with real conditions when the fire on one of the individual targets can hit some other elements of multiple targets. Moving over from firing on entire areas to selective firing and stricter account of dispersion of hits helps reduce the consumption of ammunition in firing on multiple targets (Table).

As is evident from the Table, the existing method underestimates ammunition consumption in firing on individual targets and overestimates it in firing on multiple targets. Furthermore, it exhibits practically no sensitivity to the number of individual targets. Yet the existing and the proposed methods have practically the same accuracy in estimation of the effectiveness of firing on a separate target. In the examined case, the difference in ammunition consumption manifests itself as soon as hits are scored on assemblies of small-size targets and grows considerably when hits are scored on multiple targets located on a considerable area when the effects of selective target kill and dispersion of pieces of ammunition are in full evidence.

The solution based on the system of balance equations is at first sight the same as the solution obtained on the basis of existing methods. These solutions, however, belong to qualitatively different modes of delivery of fire on the target. In existing modes, fire is delivered on the entire area of a multiple target with a considerable proportion of ammunition being wasted and owing to this the number of projectiles calculated with the use of existing methods is not sufficient to score the required result of firing. Using a different mode fire is delivered on concrete individual targets and the specified number of ammunition proves sufficient for the suppression of the multiple target.

Thus, the proposed approach is based on a uniformly optimal target-kill strategy and makes it possible to obtain solutions to estimation problems that accord as far as possible with the real firing conditions. Being selective with regard to engaging multiple targets, this approach is a key to developing methods to estimate effectiveness of fire on dissimilar targets and to estimate effectiveness of strikes with precision weapons.

The proposed approach also points out ways to improving conventional weapons the control of whose fire should comply with the programs generated by on-board computers changing range and the line of fire and insuring optimal distribution of bursts inside the areas of multiple targets in accordance with the individual target location data. Overall, the approach makes it possible to take full account of the growing capabilities of reconnaissance and weapon assets and thus to increase the effectiveness of conventional fire.

     Target          Target     Target   Number
                      type       size      of
                               (length,  targets
                                width)
                               meters

Battery of          Multiple    700 by      8
self-propelled      targets      500
armored guns
on position                                 6

Platoon of          Multiple    300 by      4
self-propelled      targets      200
armored guns
on position                                 3

Self-propelled     Individual     -         1
antiaircraft
mount on APS
in position

                               Formulas
     Target        "Ideal"   System of   System of
                     fire    integral     balance
                   density   equations  eqautions

Battery of          1310       1270         1110
self-propelled
armored guns
on position         1310       1255         1005

Platoon of           410        410         350
self-propelled
armored guns
on position          410        410         350

Self-propelled       190        205         205
antiaircraft
mount on APS
in position

Note. Range--10 km; type of projectile--OF-2; full initial firing data
processing.

NOTES:

(1.) Voennaia mysl', No. 4, 2002, pp. 56-62.

(2.) See: B. Koopman, "The Optimum Distribution of Effort," The Theory of Search, No. 111. Operat. Res., No. 5, 1957, pp. 613-626.

Col. N.I. ZOLOTOV (Res.)

Candidate of Technical Sciences