Geophysics
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A case for upward continuation as a standard separation filter for potential‐field maps Separation filtering is incomplete even under the ideal synthetic condition of known power spectra of the regional and residual fields. I have designed some Wiener filters, which minimize the inevitable separation error, from previous statistical source models of Naidu, and Spector and Grant. This formulation includes the classic separation filters of Strakhov and of Elkins as Wiener filters. A proposed generalization of Wiener filters, denoted as uniformly suboptimum filters, quantitatively supports the statement that a wide span of separation problems may be solved adequately using some convenient, small standard filter family. A uniform random‐source model without assumed vertical correlations invokes upward continuation filters. In addition to this role as a Wiener filter, the upward continuation operator is given by elementary functions in both space and wavenumber domains, is numerically stable, and is also physically comprehensible when applied to real, nonrandom anomalies. In view of these distinguishing features, I propose to use the upward continuation operator to build a convenient, standard family of separation filters.
Geophysics - Tập 52 Số 8 - Trang 1138-1148 - 1987
A generalization of the Fourier pseudospectral method The Fourier pseudospectral (PS) method is generalized to the case of derivatives of nonnatural order (fractional derivatives) and irrational powers of the differential operators. The generalization is straightforward because the calculation of the spatial derivatives with the fast Fourier transform is performed in the wavenumber domain, where the operator is an irrational power of the wavenumber. Modeling constant-[Formula: see text] propagation with this approach is highly efficient because it does not require memory variables or additional spatial derivatives. The classical acoustic wave equation is modified by including those with a space fractional Laplacian, which describes wave propagation with attenuation and velocity dispersion. In particular, the example considers three versions of the uniform-density wave equation, based on fractional powers of the Laplacian and fractional spatial derivatives.
Geophysics - Tập 75 Số 6 - Trang A53-A56 - 2010
A stable approach for <i>Q</i>-compensated viscoelastic reverse time migration using excitation amplitude imaging condition The earth [Formula: see text] filtering causes poor illumination of the subsurface. Compensating for the attenuated amplitude and distorted phase is crucial during elastic reverse time migration (ERTM) to improve the imaging quality. Conventional [Formula: see text]-compensated ERTM ([Formula: see text]-ERTM) methods tend to boost the attenuated energy to inverse the [Formula: see text] effects. These methods usually suffer from severe numerical instability because of the unlimited amplification of the high-frequency noise. Low-pass filtering is generally used to stabilize the process, however, at the expense of precision. We have developed a stable compensation approach in this paper. Based on the decoupled fractional Laplacians viscoelastic wave equation, two compensation operators are obtained by extrapolating wavefield in the dispersion-only and viscoelastic media. Because no explicit amplification is included, these two operators are absolutely stable for implementation. To improve the division morbidity for calculating the compensation operators, we use the excitation amplitude criterion and embed the operators into a vector-based [Formula: see text]-compensated excitation amplitude imaging condition. With the derived imaging condition, we could compensate for the absorption accurately without needing to concern the stability issue. The [Formula: see text]-ERTM results for noise-free data are carried out over a simple layered model and a more realistic Marmousi model with an attenuating area to verify the feasibility of the proposed approach. The migration results for noisy data from the Marmousi model further prove that the proposed method performs better fidelity, adaptability, and antinoise performance compared with conventional compensation method with low-pass filtering.
Geophysics - Tập 83 Số 5 - Trang S459-S476 - 2018
ATTENUATION OF SHEAR AND COMPRESSIONAL WAVES IN PIERRE SHALE Attenuation measurements were made near Limon, Colorado, where the Pierre shale is unusually uniform from depths of less than 100 ft to approximately 4,000 ft. Particle velocity wave forms were measured at distances up to 750 ft from explosive and mechanical sources. Explosives gave a well‐defined compressional pulse which was observed along vertical and horizontal travel paths. A weight dropped on the bottom of a borehole gave a horizontally‐traveling shear wave with vertical particle motion. In each case, signals from three‐component clusters of geophones rigidly clamped in boreholes were amplified by a calibrated, wide‐band system and recorded oscillographically. The frequency content of each wave form was obtained by Fourier analysis, and attenuation as a function of frequency was computed from these spectra. For vertically‐traveling compressional waves, an average of 6 determinations over the frequency range of 50–450 cps gives α=0.12 f. For horizontally‐traveling shear waves with vertical motion in the frequency range 20–125 cps, the results are expressed by α=1.0 f. In each case attenuation is expressed in decibels per 1,000 ft of travel and f is frequency in cps. These measurements indicate, therefore, that the Pierre shale does not behave as a visco‐elastic material.
Geophysics - Tập 23 Số 3 - Trang 421-439 - 1958
An analytic signal-based accurate time-domain viscoacoustic wave equation from the constant-<i>Q</i> theory We have introduced a concise time-domain wave equation to accurately simulate wave propagation in viscoacoustic media. The central idea behind this work is to dismiss the negative frequency components from a time-domain signal by converting the signal to its analytic format. The negative frequency components of any analytic signal are always zero because we can construct the viscoacoustic wave equation to honor the relaxation property of the medium for positive frequencies only. The newly proposed complex-valued wave equation (CWE) represents the wavefield with its analytic signal, whose real part is the desired physical wavefield, whereas the imaginary part is the Hilbert transform of the real component. Specifically, this CWE is accurate for weakly and strongly attenuating media in terms of the dissipation and dispersion, and the attenuation is precisely linear with respect to the frequencies. The CWE easily and flexibly models dispersion-only, dissipation-only, or dispersion-plus-dissipation seismic waves. We have verified these CWEs by comparing the results with analytical solutions, and we achieved nearly perfect matches. When extending to heterogeneous media, the results are consistent with those computed from the nonstationary operator-based Fourier integral method and from the standard linear solid equations.
Geophysics - Tập 86 Số 3 - Trang T117-T126 - 2021
Absorbing boundary conditions for elastic waves Absorbing boundary conditions are needed for computing numerical models of wave motions in unbounded spatial domains. The boundary conditions developed here for elastic waves are generalizations of ones developed earlier for acoustic waves. These conditions are based on compositions of simple first‐order differential operators. The formulas can be applied without modification to problems in both two and three dimensions. The boundary conditions are stable for all values of the ratio of P‐wave velocity to S‐wave velocity, and they are effective near a free surface and in a horizontally stratified medium. The boundary conditions are approximated with simple finite‐difference equations that use values of the solution only along grid lines perpendicular to the boundary. This property facilitates implementation, especially near a free surface and at other corners of the computational domain.
Geophysics - Tập 56 Số 2 - Trang 231-241 - 1991
Novel first-order <i>k</i>-space formulations for wave propagation by asymmetrical factorization of space-wavenumber domain wave propagators Wave-equation simulation based on the k-space method produces nearly dispersion-free wavefields and enhances simulation stability. However, for simulation in heterogeneous media, the conventional first-order k-space method requires many mixed-domain operators, which are the most expensive part of the wave-extrapolation process. We have analyzed and summarized the problem of the conventional k-space method as symmetrical factorization of the wave propagators. Based on this analysis, we develop a novel asymmetrical factorization-based k-space method that can significantly reduce the number of mixed-domain operators without compromising modeling accuracy. By using this method, the number of mixed-domain operators is reduced by half, and thus, the computational cost decreases significantly. Furthermore, we have compared our method to the conventional pseudospectral method. The comparison finds that, at comparable accuracy, our method is more efficient due to its ability to use a larger time step. Acoustic and elastic examples demonstrate the correctness and effectiveness of our method.
Geophysics - Tập 87 Số 6 - Trang T417-T433 - 2022
Theory and modeling of constant-<i>Q</i> P- and S-waves using fractional time derivatives I have developed and solved the constant-[Formula: see text] model for the attenuation of P- and S-waves in the time domain using a new modeling algorithm based on fractional derivatives. The model requires time derivatives of order [Formula: see text] applied to the strain components, where [Formula: see text] and [Formula: see text], with [Formula: see text] the P-wave or S-wave quality factor. The derivatives are computed with the Grünwald-Letnikov and central-difference fractional approximations, which are extensions of the standard finite-difference operators for derivatives of integer order. The modeling uses the Fourier method to compute the spatial derivatives, and therefore can handle complex geometries and general material-property variability. I verified the results by comparison with the 2D analytical solution obtained for wave propagation in homogeneous Pierre Shale. Moreover, the modeling algorithm was used to compute synthetic seismograms in heterogeneous media corresponding to a crosswell seismic experiment.
Geophysics - Tập 74 Số 1 - Trang T1-T11 - 2009
Incorporation of attenuation into time‐domain computations of seismic wave fields The only numerically tractable way yet found to incorporate attenuation into numerical time‐domain computations of seismic wave fields is to approximate the viscoelastic modulus by a low‐order rational function of frequency. The coefficients of this function can be determined by the Padé approximation. Our test computations show, however, that this approximation generally is of poor quality. Therefore, we suggest a new approach which is based on the rheological model of the generalized Maxwell body, which has a modulus of the desired rational form. We choose the relaxation frequencies logarithmically equidistant in the frequency band of interest, and determine the weight factors by simple numerical curve fitting to an arbitrary Q law. This approach is superior to the method above both in accuracy and in computational efficiency. For most practical applications, approximations of orders 2 or 3 are sufficient. The computing time and memory requirements for a finite‐difference calculation are then approximately twice those of a purely elastic calculation. As a first application of the method, we compute SH channel waves in discontinuous coal seams with Q = 50 within the coal. The results show that the high‐ frequency Airy phase is strongly attenuated. This indicates that care has to be taken in comparing the results of purely elastic model calculations of the propagation of seam waves with experimental data.
Geophysics - Tập 52 Số 9 - Trang 1252-1264 - 1987
Pseudo-analytical finite-difference elastic-wave extrapolation based on the k-space method Cost-effective elastic-wave modeling is the key to practical elastic reverse time migration and full-waveform inversion implementations. We have developed an efficient elastic pseudo-analytical finite-difference (PAFD) scheme for elastic-wave extrapolation. The elastic PAFD scheme is based on a modified pseudo-spectral method, k-space method, in which a pseudo-analytical operator is used to ensure the high accuracy of elastic-wave extrapolation. However, the k-space method is motivated for a pure wave mode, and thus its application in coupled first-order elastic-wave equations may cause the elastic pseudo-analytical operators to suffer from crosstalk between the P- and S-wavefields. The approaches presented attempt to overcome these shortcomings by introducing two improvements to achieve the goal. This is done, first, by performing a predictor-corrector strategy in first-order elastic-wave equations to eliminate those errors during wave extrapolation. Considering the massive computational cost in the spectral domain, we have developed an efficient elastic PAFD implementation, in which an innovative model-adaptive finite-difference coefficient-predicted scheme is provided to reduce the computational cost of elastic pseudo-analytical operator differencing. Dispersion analysis demonstrates the flexibility with varying velocity and superior performance of our PAFD scheme for spatial and temporal dispersion suppression than the existing Taylor-expansion-based scheme. Under the same simulation parameters, several numerical examples prove that the elastic PAFD scheme can provide more accurate simulation results, whereas the conventional scheme suffers from spatial or temporal dispersion errors, even in complex heterogeneous media.
Geophysics - Tập 83 Số 1 - Trang T1-T14 - 2018
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