Seismic waves travelling through inelastic media are attenuated by the conversion of elastic energy into heat. Upon being attenuated, the travelling wave changes: amplitude is reduced, travelling waveform is modified due to high-frequency content absorption, and phase is delayed. Attenuation is usually quantified through the quality factor Q: the ratio between the energy stored and lost in each cycle due to inelasticity. The energy attenuation and phase distortion caused by the absorbing medium can be removed by inverse Q filtering. In this paper we introduce a method in time frequency domain to compensate the attenuation based on non-stationary linear filters proposed by Margrave (1998).
Constant-Q attenuation model: The theory of constant-Q model (Kjartansson, 1979) predicts an amplitude loss given by:
where Q is the attenuation parameter, is the angular frequency, is the velocity, is the initial amplitude, and is the amplitude at the travelled distance x. A dispersion relation, for the velocity with respect to the frequency, is an essential element of the Q-constant theory. For the examples we used in this paper, the following dispersion relation (Aki and Richards, 2001) has been used:
which gives the phase velocity at any frequency, , in terms of the velocity at a reference frequency . A linear filter is entirely characterized by its impulse response. In the theory of constant-Q model the earth is considered a linear filter, the attenuating earth impulse response is a fundamental result. Kjartansson (1979) shows that the Fourier transform of the attenuating medium impulse response is:
A nonstationary convolutional model for an attenuated seismic trace can be established by combining equations (2) and (3), then by nonstationary convolving the attenuated impulse response with a reflectivity function and, finally, by convolving the result with an arbitrary wavelet (Margrave and Lamourex ,2002).
where the ‘hat’ symbol indicates Fourier transform, is the reflectivity function, is the wavelet and is the time-frequency exponential attenuation function,
in which the real and imaginary components in the exponent and connected through the Hilbert transform H, result that is consistent with the minimum phase characteristic of the attenuated pulse.
Inverse-Q filtering: Nonstationary convolution can be expressed in domain as follows:
which is transfer function
Regarding to transfer function characteristic in nonstationary convolution equation in time-frequency domain, if is input in frequency domain, then:
Where is the output in time domain.
In two equations (6) and (8), transfer function in time-frequency domain, has nonstationary filter characteristic.
The filter operators defined in these two equations are called pseudodifferential operators (Saint-Raymond, 1991). Here, denotes pseudodifferential operator. Such operators are more efficient for nonstationary filters and essentially for inverse Q filtering. We tested the performance of the method on both real and synthetic seismic data.