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Abstract

The one-dimensional Fourier transform
$$f\left( {t,x,y,z} \right) = \frac{1} {{2\pi }}\begin{array}{*{20}c} \infty \\ \smallint \\ { - \infty } \\ \end{array} S\left( {\omega ,x,y,z} \right)e^{iwt} d\omega$$
(1)
$$S\left( {\omega ,x,y,z} \right) = \begin{array}{*{20}c} \infty \\ \smallint \\ { - \infty } \\ \end{array} f\left( {t,x,y,z} \right)e^{ - i\omega t} dt. $$
(1’)
is used in the analysis of a number of seismic problems where the frequency spectrum is of interest [1, 3], In addition to the one-dimensional Fourier transform, it is sometimes advantageous to make use of a multidimensional Fourier transform, particularly for describing the properties of seismic waves which are functions of several variables. The following function may be defined formally as the k-th degree Fourier transform of a function of n variables:
$$\begin{array}{*{20}c} {f\left( {x_1 , \ldots ,x_n } \right) = \frac{1} {{\left( {2\pi } \right)^k }}\begin{array}{*{20}c} { + \infty } \\ \smallint \\ { - \infty } \\ \end{array} \cdot \cdot \cdot \begin{array}{*{20}c} { + \infty } \\ \smallint \\ { - \infty } \\ \end{array} B\left( {\omega _{x_1 } , \ldots ,\omega _{xl} ,x_{k + 1} , \ldots ,x_n } \right)e^i \mathop \sum \limits_{l = 1}^l \omega _{xl} x_{l_{d\omega _{x1} , \ldots ,d\omega _{xl} ,} } } \\ {B\left( {\omega _{x_1 } , \ldots ,\omega _{xk} ,x_{k + 1} , \ldots ,x_n } \right) = \begin{array}{*{20}c} { + \infty } \\ \smallint \\ { - \infty } \\ \end{array} \cdot \cdot \cdot \begin{array}{*{20}c} { + \infty } \\ \smallint \\ { - \infty } \\ \end{array} f\left( {x_1 , \cdots ,x_n } \right)e^{ - i} \mathop \sum \limits_{l = 1}^l \omega _{xl} x_{l_{dx_1 , \ldots ,dx_l .} } } \\ \end{array}$$
(2)
.

Keywords

Phase Velocity Seismic Wave Interference Pattern Inverse Fourier Transform Dispersive Wave 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Literature Cited

  1. 1.
    I. S. Berzon, A. M. Epinat’eva, G. N. Pariiskaya, and S. P. Starodubrovskaya, Dynamic characteristics of seismic waves in real media, Izd. Akad. Nauk SSSR, 1962.Google Scholar
  2. 2.
    V. L. Dutkin and A. P. Prudnikov, Operational Computations in Two Variables and Their Application, Fizmatgiz, 1964.Google Scholar
  3. 3.
    L. L. Khudzinskii, Frequency analysis of interference patterns of seismic waves, Tr. Inst. Fiz. Zemli, No. 6:173 (1959).Google Scholar

Copyright information

© Consultants Bureau, New York 1969

Authors and Affiliations

  • S. A. Kats

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