Theory of nonstationary linear filtering in the Fourier domain with application to time variant filtering

A general linear theory is presented which describes the extension of the convolutional method to nonstationary processes. Two alternate extensions are explored. The first, called nonstationary convolution, corresponds to the linear superposition of scaled impulse responses of a nonstationary filter. The second, called nonstationary combination, does not correspond to such a superposition but is shown to be a linear process capable of achieving arbitrarily abrupt temporal variations in the output frequency spectrum. Both extensions have stationary convolution as a limiting form. The theory is then recast into the Fourier domain where it is shown that stationary filters correspond to a multiplication of the input signal spectrum by a diagonal filter matrix while nonstationary filters generate off-diagonal terms in the filter matrix. The width of significant off-diagonal power is directly proportional to the degree of nonstationarity. Both nonstationary convolution or combination may be applied in the Fourier domain, and for quasi-stationary filters, efficiency is improved by using sparse matrix methods. Unlike stationary theory, a third domain which combines time and frequency is also possible. Here, nonstationary convolution expresses as a generalized forward Fourier integral of the product of the nonstationary filter and the time domain input signal. The result is the spectrum of the filtered signal. Nonstationary combination reformulates as a generalized inverse Fourier integral of the product of the spectrum of the input trace and the nonstationary filter which results in the time domain output signal. The mixed domain is an ideal domain for filter design which proceeds by specifying the filter as an arbitrary complex function on a time-frequency grid. Explicit formulae are given to move nonstationary filters expressed in any one of the three domains into any other. INTRODUCTION A common occurrence in geophysical research and data processing is the need to apply convolutional operators which somehow depend on both variables of a Fourier transform pair. Time variant filtering is a typical example. Filters are convolutional operators which shape the spectrum of a time series therefore a time variant filter must both shape the spectrum and change with time. Another example is the vertical extrapolation of a wavefield through a laterally variable velocity structure. The kinematics of wavefield extrapolation can be handled by a phase shift which depends on horizontal wavenumber and velocity, or equivalently by a spatial convolution over the lateral coordinate. Thus, when velocity varies laterally, a convolution is desired which depends on both the lateral coordinate and the horizontal wavenumber. Ordinary convolutional filters are incapable of directly handling these and other similar situations since they assume a "stationary" impulse response. By stationary it is meant that the filter's properties do not change with time or space. Since the convolution theorem (see any good text on signal processing, for example Karl, 1989, p 88, or Brigham ,1974, p 58) states that stationary convolution is a multiplication of Fourier spectra, it is commonly assumed that Fourier methods are also incapable of