On the Analysis of the Correlation Integrals of I. I. Gurvich on the Basis of Fang’s Formulas

L. A. Zashchinskii

(Presented by Professor, Doctor D. S. Mikov)

I. I. Gurvich, in his article [3], pointed out the analogy of the integrals

\[ I_1=\int_{0}^{\xi} l(\tau-\Theta,\xi)\, n(\Theta,x)\, d\Theta, \tag{1} \]

\[ I_2=\int_{0}^{\xi} l(\tau-\Theta,\xi)\, m(\Theta,x)\, d\Theta, \tag{2} \]

\[ I_3=\int_{0}^{\xi} m(\tau-\Theta,y)\, n(\Theta,x)\, d\Theta, \tag{3} \]

where \(m=m(t,y)\) is the signal;
\(n=n(t,x)\) is the temporal characteristic of the receiving region;
\(l=l(t,\Theta)\) is the temporal characteristic of the reflecting region.

In the same article the above-mentioned author emphasized the necessity of investigating the statistical properties of the characteristics \(m, n, l\), determined by their autocorrelation functions.

As an instrument for such an analysis it is natural to use the spectral representations of the corresponding random processes. This makes it possible to apply to them already developed methods for estimating the efficiency of signal separation, as was done by F. M. Goldman in the theory of interference reception [2].

In connection with the importance of the indicated problem, the choice of an apparatus for analyzing the corresponding correlation functions is of obvious interest.

At a superficial examination it may seem that this question is of little practical significance, since the Wiener–Khinchin transform

\[ W(k)=\frac{2}{\pi}\int_{0}^{\infty} R(\tau)\cos(k\tau)\,d\tau, \tag{4} \]

where \(W(k)\) is the energy spectrum of the process,
\(R(\tau)\) is the correlation function computed from a one-dimensional realization,

preserves all information about the amplitude features of the process in the sense that the inverse transform

\[ R(\tau)=\frac{2}{\pi}\int_{0}^{\infty} W(k)\cos(k\tau)\,dk \tag{5} \]

makes it possible to reconstruct the form of \(R(\tau)\), if \(W(k)\) is known [6]. This general

position in light of the problem raised by I. I. Gurvich requires an important clarification.

According to Fang’s data [7], the value \(R(\tau)\), computed from a one-dimensional realization of a multidimensional process, generally speaking, does not provide complete information about its energy spectrum, understood in the sense of Middleton [5].

Fang showed that, for a two-dimensional process, the Middleton spectrum is determined from \(R(\tau)\) by the equation

\[ W(k)=\int_{0}^{\infty} I_{0}(k\tau)\cdot R(\tau), \tag{6} \]

where \(I_{0}\) is the Bessel function of the first kind of zero order.

For a three-dimensional process the corresponding equation has the form

\[ W(k)=\int_{0}^{\infty} \tau^{2} I_{0}(k\tau) R(\tau)\,d\tau . \tag{7} \]

Equation (4) is valid for a one-dimensional process.

For isotropic stationary processes of higher dimension, correlation formulas analogous to equations (4, 6, 7) are apparently unknown.

A certain equivalent of them is given by equation (8).

\[ R(\tau)=\int_{0}^{\infty} \frac{ J_{\frac{N-2}{2}}(\lambda\tau) }{ (\lambda\tau)^{\frac{N-2}{2}} }\,dF, \tag{8} \]

\[ \begin{aligned} N&\text{ — the dimension of the process;}\\ \lambda&\text{ — an analogue of frequency;}\\ \tau&\text{ — an analogue of the shift parameter;}\\ J_{\frac{N-2}{2}}&\text{ — a Bessel function;}\\ F&\text{ — a monotone real nondecreasing bounded function.} \end{aligned} \]

The quantity

\[ f(\lambda)=\frac{dF}{d\lambda} \tag{9} \]

possesses all the formal properties of an energy spectrum and may be called, in a certain sense, an analogue of the spectrum.

However, it is obvious that \(F\) is not a spectrum in the Middleton sense, since equations (4, 5, 7) cannot be obtained from equation (8).

It is clear, however, that for computing the energy spectrum the form of the integral transformation is by no means immaterial.

From a cybernetic point of view, the choice of the “carrier function”

\[ \begin{gathered} \cos(\lambda\tau)\ \text{in equation }(4\text{—}5)\\ I_{0}(\lambda\tau)\ \text{in equation }(6)\\ \tau I_{0}(\lambda\tau)\ \text{in equation }(7) \end{gathered} \]

and, finally,

\[ \frac{ J_{\frac{N-2}{2}}(\lambda\tau) }{ (\lambda\tau)^{\frac{N-2}{2}} } \tag{10} \]

in equation (8) is equivalent to the choice of a language for transmitting information.

The latter must satisfy the properties of the information-processing channel and be selected in accordance with the dimensionality of the functions \(W(k)\) under study.

In practice, when investigating the question raised by I. I. Gurvich, it may be recommended not to confine oneself to the use of equation (4), but also to find spectra by Fang’s formulas (6–7).

The following approach is also of interest. The Fredholm equation of the first kind corresponding to transform (8) has the form

\[ R(\tau)=\int_{0}^{a} \frac{J_{\frac{N-2}{2}}(\lambda \tau)} {(\lambda \tau)^{\frac{N-2}{2}}} \, f(\lambda)\, d\lambda \tag{11} \]

The transition from integration over infinite limits in equation (5) to integration up to the upper frequency \(a\) in equation (6) is justified by the limited passband of seismic recording devices [4]. Equation [11] can be solved approximately for a finite, but sufficiently broad set of integer values of the parameter \(N=1,2,3\ldots\). Each of these values will correspond to a certain type of analogue of the spectrum

\[ f^{w}(\lambda)=f_{1}(\lambda), f_{2}(\lambda)\ldots f_{n}(\lambda). \tag{12} \]

With respect to the \(i\)-th type of analogue of the spectrum, proceeding from the properties of the information-processing channel, a criterion for the optimality of the pulse shape may be formulated (effective duration, amplitude expressiveness, etc.).

The order \(k\) of the analogue of the spectrum \(f_{k}=(\lambda)\), for which the optimality conditions are best satisfied, may be regarded as a characteristic of the stationary random process reflecting its dimensionality.

There is no basis for asserting that the quantities \(I_{1}\), \(I_{2}\), and \(I_{3}\), determined by equations (1–3), are one-dimensional. According to the definition of I. I. Gurvich, they are analogous to correlation functions computed from a one-dimensional realization of processes whose dimensionality is unknown. The quantity \(k\), determined by the method described above, may be made the basis for classifying these quantities, which, according to I. I. Gurvich, determine the conditions for applying the interpretation of the dynamic features of seismograms.

In conclusion, it should be noted that the expansion of correlation functions in Bessel harmonics is a technique that has proved itself quite well in practice. In particular, without providing a theoretical justification for it, it is widely used in analyzing the structure of microseisms [1]. This is due to the fact that Bessel functions are considerably closer in shape to real seismic pulses than trigonometric harmonics.

As a consequence, to represent the spectrum of a pulse limited in frequency and time, as a rule, a smaller number of Bessel functions is required than cosines.

The considerations set forth above can easily be generalized to other problems of seismic prospecting, as well as to problems of other geophysical methods.

REFERENCES

1. L. P. Vinnik, The Structure of Microsystems and Certain Questions of Grouping in Seismology. Moscow, “Nauka,” 1968.

1. F. M. Gol’tsman. Fundamentals of the Theory of Interference Reception. “Nauka,” 1964.

2. I. I. Gurvich. On the theoretical foundations of dynamic measurements in seismic prospecting. Geologiya i razvedka, 1970, No. 6.

3. I. I. Turvich. Seismic Prospecting. “Nedra,” 1970.

4. D. Middleton. Introduction to the Statistical Theory of Radio Communication. Moscow, “Sovetskoe radio,” 1961.

5. A. A. Kharkevich. Spectra and Analysis. Moscow, “Nauka,” 1957.

6. A. K. Fang. Remarks concerning the Wiener–Khinchin autocorrelation theorem. Proceedings of the Institute of Electrical and Radio Engineers, vol. 55, No. 4, 1967.

7. A. M. Yaglom, R. L. Dobrushin. Translators’ appendices to the book by J. Doob, Stochastic Processes. IL, 1956.