Gerard T. Schuster

Least‐squares migration of incomplete reflection data

A least-squares migration algorithm is presented that reduces the migration artifacts (i.e., recording footprint noise) arising from incomplete data. Instead of migrating data with the adjoint of the forward modeling operator, the normal equations are inverted by using a preconditioned linear conjugate gradient scheme that employs regularization. The modeling operator is constructed from an asymptotic acoustic integral equation, and its adjoint is the Kirchhoff migration operator. We tested the performance of the least-squares migration on synthetic and field data in the cases of limited recording aperture, coarse sampling, and acquisition gaps in the data. Numerical results show that the least-squares migrated sections are typically more focused than are the corresponding Kirchhoff migrated sections and their reflectivity frequency distributions are closer to those of the true model frequency distribution. Regularization helps attenuate migration artifacts and provides a sharper, better frequency distribution of estimated reflectivity. The least-squares migrated sections can be used to predict the missing data traces and interpolate and extrapolate them according to the governing modeling equations. Several field data examples are presented. A ground-penetrating radar data example demonstrates the suppression of the recording footprint noise due to a limited aperture, a large gap, and an undersampled receiver line. In addition, better fault resolution was achieved after applying least-squares migration to a poststack marine data set. And a reverse vertical seismic profiling example shows that the recording footprint noise due to a coarse receiver interval can be suppressed by least-squares migration.

Finite-difference solution of the eikonal equation along expanding wavefronts

We show that a scheme to solve the 2-D eikonal equation by a finite‐difference method can violate causality for moderate to large velocity contrasts (V2/V1>2). As an alternative, we present a finite‐difference scheme in which the solution region progresses outward from an “expanding wavefront” rather than an “expanding square,” and therefore honors causality. Our method appears to be stable and reasonably accurate for a variety of velocity models with moderate to large velocity contrasts. The penalty is a large increase in computational cost and programming effort.

Early arrival waveform tomography on near-surface refraction data

We develop a waveform-tomography method for estimating the velocity distribution that minimizes the waveform misfit between the predicted and observed early arrivals in space-time seismograms. By fitting the waveforms of early arrivals, early arrival waveform tomographyEWT naturally takes into account more general wave-propagation effects compared to the high-frequency method of traveltime tomography, meaning that EWT can estimate a wider range of slowness wavenumbers. Another benefit of EWT is more reliable convergence compared to full-waveform tomography, because an early-arrival misfit function contains fewer local minima. Synthetic test results verify that the waveform tomogram is much more accurate than the traveltime tomogram and that this algorithm has good convergence properties. For marine data from the Gulf of Mexico, the statics problem caused by shallow, gassy muds was attacked by using EWT to obtain a more accurate velocity model. Using the waveform tomogram to correct for statics, the stacked section was significantly improved compared to using the normal moveout NMO velocity, and moderately improved compared to using the traveltime tomogram. Inverting high-resolution land data from Mapleton, Utah, showed an EWT velocity tomogram that was more consistent with the ground truth trench log than the traveltime tomogram. Our results suggest that EWT can provide supplemental, shorter-wavelength information compared to the traveltime tomogram for both shallow and moderately deep seismic data.

Prestack migration deconvolution

Prestack migration in the time and depth domains is the premier tool for seismic imaging of complex structures. Unfortunately, an undersampled acquisition geometry, a limited recording aperture, and strong velocity contrasts lead to uneven illumination of the subsurface. This results in blurring the migrated image, sometimes referred to as acquisition footprint noise in the migrated image. To remedy this blurring partially, we introduce prestack migration deconvolution (MD). The MD filter is a layered-medium approximation to the inverse Hessian matrix that is applied locally to the migrated section. Both synthetic- and field-data results show noticeable improvements in reducing migration artifacts and increasing lateral spatial resolution by more than 10%. The computational expense for constructing the MD filter is related to the MD operator length. Its cost is about the same as that for migration, but opportunities exist for significantly reducing this cost. Results suggest that MD should be applied to migrated sections to optimize image quality.

Wavepath eikonal traveltime inversion: Theory

We present a general formula for the back projection of traveltime residuals in traveltime tomography. For special choices of an arbitrary weighting factor this formula reduces to the asymptotic back‐projection term in ray‐tracing tomography (RT), the Woodward‐Rocca method, wavepath eikonal traveltime inversion (WET), and wave‐equation traveltime inversion (WT). This unification provides for an understanding of the differences and similarities among these traveltime tomography methods. The special case of the WET formula leads to a computationally efficient inversion scheme in the space‐time domain that is, in principle, almost as effective as WT inversion yet is an order of magnitude faster. It also leads to an analytic formula for the fast computation of wavepaths. Unlike ray‐tracing tomography, WET partially accounts for band‐limited source and shadow effects in the data. Several numerical tests of the WET method are used to illustrate its properties.

Acoustic wave-equation traveltime and waveform inversion of crosshole seismic data

A hybrid wave‐equation traveltime and waveform inversion method is presented that reconstructs the interwell velocity distribution from crosshole seismic data. This inversion method, designated as WTW, retains the advantages of both full wave inversion and traveltime inversion; i.e., it is characterized by reasonably fast convergence which is somewhat independent of the initial model, and it can resolve detailed features of the velocity model. In principle, no traveltime picking is required and the computational cost of the WTW method is about the same as that for full wave inversion. We apply the WTW method to synthetic data and field crosshole data collected by Exxon at their Friendswood, Texas, test site. Results show that the WTW tomograms are much richer in structural information relative to the traveltime tomograms. Subtle structural features in the WTW Friendswood tomogram are resolved to a spatial resolution of about 1.5 m, yet are smeared or completely absent in the traveltime tomogram. This sugg...

A theoretical overview of model-based and correlation-based redatuming methods

We review the equations for correlation-based redatuming methods. A correlation-based redatuming method uses natural-phase information in the data to time shift the weighted traces so they appear to be generated by sources (or recorded by geophones) shifted to a new location. This compares to model-based redatuming, which effectively time shifts the traces using traveltimes computed from a prior velocity model. For wavefield redatuming, the daylight imaging, interferometric imaging, reverse-time acoustics (RTA), and virtual-source methods all require weighted correlation of the traces with one another, followed by summation over all sources (and sometimes receivers). These methods differ from one another by their choice of weights. The least-squares interferometry and virtual-source imaging methods are potentially the most powerful because they account for the limited source and receiver aperture of the recording geometry. Interferometry, on the other hand, has the flexibility to select imaging conditions that target almost any type of event. Stationary-phase principles lead to a Fermat-based redatuming method known as redatuming by a seminatural Greens function. No crosscorrelation is needed, so it is less expensive than the other methods. Finally, Fermats principle can be used to redatum traveltimes.

Resolution limits of migrated images

Far-field formulas are derived for the point scatterer responses, the horizontal resolution limits, and the dynamic ranges of 2-D and 3-D migrated images. Numerical tests verify the validity of these formulas for practical CDP geometries. Our results show that in the far field, where the depth is much greater than the width of the recording aperture, the horizontal resolution limit for migrated images is proportional to the depth of the scatterer and inversely proportional to the wavenumber and aperture size. Given the same recording aperture size, the 2-D prestack migration image has the same horizontal resolution limit as the corresponding 2-D poststack image. For the same aperture size, the 3-D poststack migration image has twice the resolution as the 3-D prestack migration with sources and receivers distributed in fixed areas. Also, prestack migration images have a better dynamic range than poststack migration images, and dynamic range improves with an increase in width of the recording aperture. The resolution limits are based on a Rayleigh-like zero crossing criterion.

Elastic wave equation traveltime and waveform inversion of crosswell data

A method is presented for reconstructing P‐ and S‐velocity distributions from elastic traveltimes and waveforms. The input data consist of crosswell hydrophone records generated by a piezoelectric borehole source. Borehole effects are partially accounted for by using a low‐frequency Greens function to simulate the pressure generated in the fluid‐filled receiver well. The tube waves in the borehole are ignored, on the assumption that they can be removed from the field data by median filtering. In addition, the source‐radiation pattern is partially taken into account by inverting for the equivalent stress components acting on the earth at the source location. The elastic wave equation traveltime and waveform inversion (WTW) method is applied to both synthetic crosswell data and the McElroy field crosswell data. As predicted by theory, results show that elastic WTW tomograms provide a sharper interface image than delineated in the traveltime tomograms. The spatial resolution of the McElroy traveltime tomogr...

An efficient multiscale method for time-domain waveform tomography

This efficient multiscale method for time-domain waveform tomography incorporates filters that are more efficient than Hamming-window filters. A strategy for choosing optimal frequency bands is proposed to achieve computational efficiency in the time domain. A staggered-grid, explicit finite-difference method with fourth-order accuracy in space and second-order accuracy in time is used for forward modeling and the adjoint calculation. The adjoint method is utilized in inverting for an efficient computation of the gradient directions. In the multiscale approach, multifrequency data and multiple grid sizes are used to overcome somewhat the severe local minima problem of waveform tomography. The method is applied successfully to 1D and 2D heterogeneous models; it can accurately recover low- and high-wavenumber components of the velocity models. The inversion result for the 2D model demonstrates that the multiscale method is computationally efficient and converges faster than a conventional, single-scale method.