Types of signal models

When analyzing physical data, two main approaches are used to create mathematical signal models.

The first approach operates with deterministic signals, the values ​​of which at any moment of time or at an arbitrary point in space (as well as depending on any other arguments) are a priori known or can be sufficiently accurately determined (calculated). This approach is convenient in direct problems of geophysics (field calculations for given models of media), in problems of active impacts on the environment with previously known parameters and shape of the impact signal (vibration seismic exploration, electromagnetic logging methods, etc.), as well as when using well-known and reliable geological and geophysical data. To describe non-random signals, quasi-deterministic models in which the values ​​of one or several parameters are a priori unknown, and are considered random variables with a small random component, the influence of which can be neglected.

The second approach assumes a random nature of signals, the law of change of which in time (or in space) is of a random nature, and which take on specific values ​​with a certain probability. The model of such a signal is a description of the statistical characteristics of a random process by specifying the laws of probability distribution, correlation function, spectral energy density, etc.

Randomness can be due to both the intrinsic physical nature of the signals, which is typical, for example, for methods of nuclear geophysics, and the probabilistic nature of the recorded signals, both in time or place of their appearance, and in content. From this point of view, a random signal can be considered as a reflection of a process that is random in nature or the physical properties of an object (process), which are determined by random parameters or a complex structure of the geological environment, the measurement results in which are difficult to predict.

There is no sharp boundary between these two types of signals. Strictly speaking, deterministic processes and deterministic signals corresponding to them do not exist in nature. Even signals that are well known at the entrance to the environment (with external influence on it), at the place of their registration, are always complicated by random interference, the influence of destabilizing factors and a priori unknown parameters and structure of the environment itself. On the other hand, the random field model is often approximated by the method of superposition (addition) of signals of a known shape. Deterministic models can also be used to study purely random processes if the level of the useful signal in this process is significantly higher than the level of statistical fluctuations, which occurs, for example, when registering ionizing radiation from rocks.

The choice of a mathematical model of the field in a particular method of geophysics is also influenced to a large extent by the complexity of the mathematical apparatus for signal processing and the established traditions of geological interpretation of observation results. It is not excluded that the model changes, as a rule, with a transfer from probabilistic to deterministic, in the process of accumulating information about the phenomenon or object under study.