Stability and efficiency are two issues of general concern in inverse Q filtering. This paper presents a stable, efficient approach to inverse Q filtering, based on the theory of wavefield downward continuation. It is implemented in a layered manner, assuming a depth-dependent, layered-earth Q model. For each individual constant Q layer, the seismic wavefield recorded at the surface is first extrapolated down to the top of the current layer and a constant Q inverse filter is then applied to the current layer. When extrapolating within the overburden, instead of applying wavefield downward continuation directly, a reversed, upward continuation system is solved to obtain a stabilized solution. Within the current constant Q layer, the amplitude compensation operator, which is a 2-D function of traveltime and frequency, is approximated optimally as the product of two 1-D functions depending, respectively, on time and frequency. The constant Q inverse filter that compensates simultaneously for phase and amplitude effects is then implemented efficiently in the Fourier domain.

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A stable and efficient approach of inverse Q filtering

A principal limitation on seismic resolution is the earth attenuation, or Q -effect, including the energy dissipation of high-frequency wave components and the velocity dispersion that distorts seismic wavelets. An inverse Q -filtering procedure attempts to remove the Q -effect to produce high-resolution seismic data, but some existing methods either reduce the S/N ratio, which limits spatial resolution, or generate an illusory high-resolution wavelet that contains no more subsurface information than the original low-resolution data. In this paper, seismic inverse Q -filtering is implemented in a stabilized manner to produce high-quality data in terms of resolution and S/N ratio. Stabilization is applied to only the amplitude compensation operator of a full inverse Q -filter because its phase operator is unconditionally stable, but the scheme neither amplifies nor suppresses high frequencies at late times where the data contain mostly ambient noise. The latter property makes the process invertible, differ...

/pdf/inverse-q-filter-for-seismic-resolution-enhancement-4vitt2j9mf.pdf

Inverse Q-filter for seismic resolution enhancement

Estimates of the quality, Q, factor are commonly obtained from vertical seismic data or stacked surface seismic data. This paper describes a method that allows Q-factor to be estimated directly from common midpoint (CMP) gathers. Absorption of the wavefield is dependent on three parameters: frequency, traveltime in the medium, and medium Q-factor. Assuming that the amplitude spectrum of the seismic source signature may be modeled by that of a Ricker wavelet, we derive an analytical relation between Q-factor and seismic data peak frequency variation both along offset and vertical time direction. The Q-factor is estimated from CMP gathers using a layer-stripping approach.

Estimation of quality factors from CMP records

Reduced amplitude and distorted dispersion of seismic waves caused by attenuation, especially strong attenuation, always degrades the resolution of migrated images. To improve image resolution, we evaluated a methodology of compensating for attenuation (∼1∕Q) effects in reverse-time migration (Q-RTM). The Q-RTM approach worked by mitigating the amplitude attenuation and phase dispersion effects in source and receiver wavefields. Source and receiver wavefields were extrapolated using a previously published time-domain viscoacoustic wave equation that offered separated amplitude attenuation and phase dispersion operators. In our Q-RTM implementation, therefore, attenuation- and dispersion-compensated operators were constructed by reversing the sign of attenuation operator and leaving the sign of dispersion operator unchanged, respectively. Further, we designed a low-pass filter for attenuation and dispersion operators to stabilize the compensating procedure. Finally, we tested the Q-RTM approach on a simple layer model and the more realistic BP gas chimney model. Numerical results demonstrated that the Q-RTM approach produced higher resolution images with improved amplitude and phase compared to the noncompensated RTM, particularly beneath high-attenuation zones.

Q-compensated reverse-time migration

We have extended the method of stationary spiking deconvolution of seismic data to the context of nonstationary signals in which the nonstationarity is due to attenuation processes. As in the stationary case, we have assumed a statistically white reflectivity and a minimum-phase source and attenuation process. This extension is based on a nonstationary convolutional model, which we have developed and related to the stationary convolutional model. To facilitate our method, we have devised a simple numerical approach to calculate the discrete Gabor transform, or complex-valued time-frequency decomposition, of any signal. Although the Fourier transform renders stationary convolution into exact, multiplicative factors, the Gabor transform, or windowed Fourier transform, induces only an approximate factorization of the nonstationary convolutional model. This factorization serves as a guide to develop a smoothing process that, when applied to the Gabor transform of the nonstationary seismic trace, estimates the magnitude of the time-frequency attenuation function and the source wavelet. By assuming that both are minimum-phase processes, their phases can be determined. Gabor deconvolution is accomplished by spectral division in the time-frequency domain. The complex-valued Gabor transform of the seismic trace is divided by the complex-valued estimates of attenuation and source wavelet to estimate the Gabor transform of the reflectivity. An inverse Gabor transform recovers the time-domain reflectivity. The technique has applications to synthetic data and real data.

Gabor deconvolution: Estimating reflectivity by nonstationary deconvolution of seismic data

Although some attention has been paid to the idea that seismic migration is equivalent to a type of deconvolution (of the spatial wavelet), less thought has been given to the opposite perspective: that deconvolution (of the earth Q filter) might itself be equivalent to a form of migration. The key point raised in this paper is that a dispersive 1-D backward propagation can form the basis of a number of different algorithms for inverse Q filtering, each of which is akin to a particular migration algorithm. An especially efficient algorithm can be derived by means of a coordinate transformation equivalent to that in the Stolt frequency-wavenumber migration.This fast algorithm, valid for Q constant with depth, can be extended to accommodate depth-variable Q by cascading a series of constant Q compensations, as in cascaded migration. By combining a cascaded phase compensation with a windowed approach to amplitude compensation, we obtain an algorithm that is sufficiently efficient to be used routinely for prestack data processing. Data examples compare the results of conventional processing with the more stable phase treatment that can be obtained by including prestack inverse Q filtering in the processing.

Neil D. Hargreaves

Papers

Inverse Q filtering by Fourier transform