Figure I.5 Setting up the coordinate system on the picture plane.
Source: From Wentzel [2].
As a law of dispersion, all types of firing and bombing are generally subject to normal law. This is due to the fact that the firing error on each of the axes can be represented as the sum of a large number of elementary errors resulting from various factors. If one of the coordinate axes (usually 0X) is at least close to the direction of firing, such axes are called the principal axes. In this case, the dispersion law will take the simplest form:
(I.10)
where
In the practice, it is not the values of σx, σy that are usually used, but the so‐called probable (median) deviations of 0X, 0Y axes, which are denoted by Ex and Ey , respectively:
Median deviations are convenient because they correspond to the principal half‐axis of the dispersion ellipse, within which exactly half of all hits lie. The law of dispersion, in this case, takes the following form:
It is usually assumed that errors along the firing line (Ex) do not depend on errors across the firing line (Ey). Therefore, these values can be considered independently of each other.
I.3.4 Scheme of Two Groups of Errors
Random shot error consists of several components of random errors: target coordinate error, correction for meteorological and ballistic factors, technical dispersion caused by differences in weight and shape of projectiles, etc. In a single shot, each component of the total shot error is repeated only once, in which case it is said that there is one group of errors. When several shots are fired at the same target, some components of the total shot error caused by common sources are repeated, while other components caused by different sources are not repeated. For example, when shooting at an unobserved target from the same gun with the same scope setting, all the shots will repeat the target coordinate error and will not repeat the error caused by variations in the shape and weight of the projectiles. Please note that in this case, the target coordinate error, although repeated from shot to shot, is an accidental rather than systematic error.
Shot errors that have at least one common source and contain at least one common component are called dependent. Shots fired under these conditions are referred to as dependent shots. In real‐world conditions, shots are always dependent, so we have to consider a complex error system consisting of at least two error groups.
If we return to the case of firing from one gun at an unobserved target (uncontrolled firing), we can consider two groups of random errors: data preparation errors (Ex0, Ey0) – group error, and technical dispersion (Bd, Bs) – individual error. Then the total median error of the shot along the firing line
(I.12)
across the firing line
(I.13)
The group error makes the salvo as a whole deviate from the target (Figure I.6). Individual errors create dispersion inside the salvo.
An estimate of the values of aggregated median errors when an artillery division is firing after complete preparation of the initial data can be obtained from Table A.1 in the Appendix.
The degree of dependence between the shots can be characterized by the correlation ratio of the hit points coordinates. In the scheme of two error groups, the correlation ratio for each pair of shots is equal to the ratio of the square of the group error probable deviation to the square of the total error probable deviation:
(I.14)
(I.15)
Figure I.6 Effect of the group and individual shooting errors.
Source: From Wentzel [2].
In practice, most often these two ratios differ slightly and are averaged by the formula