On the application of phase-frequency characteristics in seismogram processing
Abstract Generated abstract
This paper examines the use of phase-frequency characteristics in processing seismic records, particularly for detecting reflected waves when amplitude-based frequency filtering is insufficient. The authors compute amplitude and phase spectra for finite seismogram segments using Fourier-based formulas, comparing segments containing reflected waves with segments known not to contain them in the 40 to 100 Hz range. Their analysis of data from the Tomsk Geophysical Trust suggests that amplitude spectra may sometimes coincide for useful and non-useful segments, while the phase-frequency characteristic of reflected waves shows a more stable, recognizable form. The study proposes phase spectra as an additional criterion for identifying reflected waves and estimating their time parameters, while noting the need for further testing on larger data sets.
Full Text
On the Application of Phase-Frequency Characteristics in Seismogram Processing
V. P. Ivanchenkov, V. A. Shlotgauer
(Presented by the scientific seminar of the Research Institute of Nuclear Physics, Electronics, and Automation)
Spectral representations began to be used in seismic prospecting from the time of the application of the reflected-wave method, when the need arose for frequency filtering in order to distinguish reflections against the background of other oscillations. But in interpreting seismic-observation data, for a long time mainly the kinematic characteristics of waves were used: travel times and the propagation velocities determined from them. Dynamic characteristics, understood as the totality of dependences determining the character of oscillations of particles of the medium in time and space during wave propagation, were almost not used in the interpretation process. However, the development of seismic exploration methods under conditions of more complex types of media has made it necessary, in interpretation, to involve dynamic characteristics along with kinematic ones [1]. The amplitude and phase spectra of a seismic wave are its most important dynamic characteristic. The spectra of a seismic wave and the form of its recording are related by the well-known Fourier relations, according to which any function satisfying the Dirichlet conditions can be represented by means of Fourier integrals in complex form:
\[ F(t)=\frac{1}{2\pi}\int_{-\infty}^{\infty} S(\omega)e^{j\omega t}\,d\omega, \tag{1} \]
\[ S(\omega)=\int_{-\infty}^{\infty} F(t)e^{-j\omega t}\,dt. \]
The function \(S(\omega)\) is a complex function of the argument \(\omega\) and is called the complex spectrum of the oscillation \(F(t)\).
\[ S(\omega)=\Phi(\omega)e^{j\varphi(\omega)}. \tag{2} \]
The modulus of the complex spectrum \(\Phi(\omega)\) is called the amplitude spectrum, and the argument \(\varphi(\omega)\) the phase spectrum of the oscillation \(F(t)\).
\[ \Phi(\omega)=\sqrt{A^{2}(\omega)+B^{2}(\omega)}, \tag{3} \]
\[ \varphi(\omega)=\operatorname{arctg}\frac{B(\omega)}{A(\omega)}, \tag{4} \]
\[ A(\omega)=\int_{-\infty}^{\infty} F(t)\cos \omega t\,dt, \tag{5} \]
\[ B(\omega)=\int_{-\infty}^{\infty} F(t)\sin \omega t\,dt. \tag{6} \]
In the majority of cases encountered in seismic prospecting practice, the oscillation curve is recorded in the form of a graph, and an analytical solution of the Fourier integrals is difficult; therefore, methods for the approximate calculation of the integral relations (1) become very important. The calculation of integrals (5) and (6) can be considerably simplified if it is assumed that the oscillation \(F(t)\) is limited in time by the interval \((0,\tau)\), and that the main part of its complex spectrum is contained in the region \((0,\omega_{в})\). Seismic pulses satisfy this assumption well. Taking these assumptions into account, calculation of the spectrum of a seismic oscillation reduces to calculation of integrals over finite limits. In addition, according to Kotelnikov’s theorem, a function with a limited spectrum and limited duration can be fully characterized by \(n+1\) values of the complex spectrum for the discrete frequency values \(0,\Delta\omega,2\Delta\omega,\ldots,n\Delta\omega\) [2].
Taking all the above into account, the calculation of the amplitude-frequency and phase-frequency characteristics of portions of a real seismogram was carried out according to the following formulas:
\[ A_k=\frac{2}{t''-t'}\int_{t'}^{t''} F(t)\cos \frac{2k\pi t}{t''-t'}\,dt, \tag{7} \]
\[ B_k=\frac{2}{t''-t'}\int_{t'}^{t''} F(t)\sin \frac{2k\pi t}{t''-t'}\,dt, \]
\[ \Phi_k=\sqrt{A_k^2+B_k^2}, \]
\[ \varphi_k=\operatorname{arctg}\frac{B_k}{A_k}, \]
where \(t''-t'\) is the duration of the analyzed portion of the seismogram.
The purpose of the calculations was to obtain the spectral characteristics of a section of a seismic trace containing a reflected wave, and the subsequent analysis of these characteristics in order to determine the possibility of using them both for detecting the reflected wave and for determining its time parameters. In the analysis of the spectral characteristics, special attention was given to the phase-frequency spectrum. This is explained by the fact that, when seismic oscillations are recorded, one most often has to deal with superposed or interfering waves, since the waves arriving at the seismic receiver propagate along different paths, with different velocities, and overlap one another. This phenomenon causes the greatest difficulties in the separation of reflected waves. At present, frequency filtering of seismic oscillations, based on knowledge of the amplitude spectra of waves, is used to separate the useful wave. But if the spectra of the superposed waves are close to one another, then their resolution by means of frequency filtering becomes very difficult.
Under these conditions, the informational significance of the phase-frequency spectrum increases, and the task of considering phase spectra consists in finding additional features by means of which it would be possible, from the phase spectrum of a function, to judge its form in the time representation. To compare the results of spectral analysis on a seismic trace, two types of segments were taken: a segment recording a reflected wave, and a segment that was known in advance not to contain a reflected wave. Taking into account the pass band of the input filters of the seismic receivers, the decomposition was carried out in the range from 40 to 100 Hz. The time interval was taken equal to 40 msec, which is approximately equal to the duration of the reflected wave. All calculations were performed by formulas (7) using the BESM-4, based on data from the Tomsk Geophysical Trust. Spectral characteristics were obtained for 10 segments containing a reflected wave and 10 segments not containing a reflected wave. Analysis of the results obtained showed that the spectral characteristics of segments containing a useful wave are distinguished by constancy in the form of the graphs, that is, the ratio of the amplitudes and phases of the harmonics making up the reflected wave is unchanged. In Fig. 1, a segment of the seismic trace containing a reflected wave is shown; in Fig. 1, b its generalized amplitude-frequency characteristic is shown, and in Fig. 1, c the phase-frequency characteristic is shown.
Fig. 1. Segment of a seismic trace with a useful wave and its spectral characteristics.
Conversely, the spectral characteristics of segments not containing a useful wave do not exhibit any recurrence of form. In Fig. 2, a and b show two segments not containing a reflected wave and their spectral characteristics.
Fig. 2. Segments of seismic traces not containing a useful wave, and their spectral characteristics.
Comparison of the amplitude-frequency characteristics of different segments showed that in some cases the amplitude spectrum of a segment not containing a useful wave coincides in shape with the amplitude spectrum of the reflected wave. However, no such coincidence was observed for the phase spectra.
It should also be noted that the phase-frequency characteristic of the reflected wave has the characteristic form shown in Fig. 1c. The experiments carried out on detecting the useful wave in a real seismogram by tracking the form of the phase-frequency curve, as well as determining its time parameters, yielded good results.
From the foregoing the following conclusions may be drawn:
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The amplitude-frequency characteristic cannot unambiguously indicate the presence of a useful wave.
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The phase-frequency characteristic of the reflected wave has a characteristic form, making it possible to use it for detecting the useful wave and determining its time parameters.
In conclusion, it should be said that the results presented are an attempt to use the phase-frequency characteristic in processing seismic information. Further studies on more extensive material will make it possible in the future to refine the limits of applicability of phase characteristics in the interpretation of seismograms.
REFERENCES
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I. S. Berzon, A. M. Epinateva, G. N. Pariyskaya, S. P. Starodubrovskaya. Dynamic characteristics of seismic waves in real media. Moscow, Publishing House of the Academy of Sciences of the USSR, 1962.
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A. A. Kharkevich. Spectra and Analysis. Moscow, Fizmatgiz, 1962.