R. Gerhard Pratt

Seismic waveform inversion in the frequency domain; Part 1, Theory and verification in a physical scale model

Seismic waveforms contain much information that is ignored under standard processing schemes; seismic waveform inversion seeks to use the full information content of the recorded wavefield. In this paper I present, apply, and evaluate a frequency‐space domain approach to waveform inversion. The method is a local descent algorithm that proceeds from a starting model to refine the model in order to reduce the waveform misfit between observed and model data. The model data are computed using a full‐wave equation, viscoacoustic, frequency‐domain, finite‐difference method. Ray asymptotics are avoided, and higher‐order effects such as diffractions and multiple scattering are accounted for automatically. The theory of frequency‐domain waveform/wavefield inversion can be expressed compactly using a matrix formalism that uses finite‐difference/finite‐element frequency‐domain modeling equations. Expressions for fast, local descent inversion using back‐propagation techniques then follow naturally. Implementation of ...

Efficient waveform inversion and imaging: A strategy for selecting temporal frequencies

Prestack migration and/or inversion may be implemented in either the time or the frequency domain. In the frequency domain, it is possible to discretize the frequencies with a much larger sampling interval than that dictated by the sampling theorem and still obtain an imaging result that does not suffer from aliasing (wrap around) in the depth domain. The selection of input frequencies can be reduced when a range of offsets is available; this creates a redundancy of information in the wavenumber coverage of the target. In order to optimize the use of this information, we define a new discretization strategy that depends on the maximum effective offset present in the surface seismic survey: the larger the range of offsets, the fewer frequencies are required. The strategy, exact in a homogeneous 1D earth, selects frequencies by making use of the well-known effect of image stretch in normal-moveout (NMO) correction and in migration (usually considered detrimental for the imaging). The strategy is also useful in more general earth models: we apply it to the 2D Marmousi model and recover a continuous range of wavenumbers using only three input frequencies. The Marmousi inversion result accurately predicts all other data frequencies, demonstrating the redundancy of the data.

Seismic waveform inversion in the frequency domain; Part 2; Fault delineation in sediments using crosshole data

A crosshole experiment was carried out in a layered sedimentary environment in which a normal fault is known to cut through the section. Initial traveltime inversions produced stable but low-resolution images from which the fault could be only vaguely inferred. To image the fault, wavefield inversion was used to produce a velocity model consistent with the detailed phase and amplitude of the data at a number of frequencies. Our wavefield inversion scheme uses a classical, descent-type algorithm for decreasing the data misfit by iteratively computing the gradient of this misfit by repeated forward and backward propagations. Our propagator is a full-wave equation, frequency-domain, acoustic, finite-difference method. The use of the frequency-space domain yields computational advantages for multisource data and allows an easy incorporation of viscous effects. By running wavefield inversion on the field data, a quantitative velocity image was produced that yielded a significantly improved image of the fault (when compared with the original traveltime inversions). Because the original field data were noisy and contained a high degree of multiple scattering (from the layering of the sediments), the transmitted arrivals were selectively windowed to enhance the image. The sediments at the site were strongly attenuating; we therefore used a viscoacoustic model during the modeling and the inversion that correctly simulated the observed decrease in amplitude with increasing frequency and source-receiver offset. Furthermore, since the traveltime inversion indicated a high degree of anisotropy at the site, a fixed, homogeneous level of anisotropy was used during the inversion. Tests at varying levels of anisotropy confirmed the improvement in image quality and in data fit when anisotropy was incorporated. The final image was verified by examining the distribution of the residuals in the frequency domain, by comparing time-domain modeled wavefields with the observed data, and by direct comparison with borehole logs.

Frequency‐domain elastic wave modeling by finite differences: A tool for crosshole seismic imaging

The migration, imaging, or inversion of wide-aperture cross-hole data depends on the ability to model wave propagation in complex media for multiple source positions. Computational costs can be considerably reduced in frequency-domain imaging by modeling the frequency-domain steady-state equations, rather than the time-domain equations of motion. I develop a frequency-domain approach in this note that is competitive with time-domain modeling when solutions for multiple sources are required or when only a limited number of frequency components of the solution are required.

Reflection waveform inversion using local descent methods: Estimating attenuation and velocity over a gas-sand deposit

Prestack seismic reflection data contain amplitudes, traveltimes, and moveout information; waveform inversion of such data has the potential to estimate attenuation levels, reflector depths and geometry, and background velocities. However, when inverting reflection data, strong nonlinearities can cause reflectors to be incorrectly imaged and can prevent background velocities from being updated. To successfully recover background velocities, previous authors have resorted to nonlinear, global search inversion techniques. We propose a two-step inversion procedure using local descent methods in which we perform alternate inversions for the reflectors and the background velocities. For our reflector inversion we exploit the efficiency of the back-propagation method when inverting for a large parameter set. For our background velocity inversion we use Newton inverse methods. During the background velocity inversions it is crucial to adaptively depth-stretch the model to preserve the vertical traveltimes. This reduces nonlinearity by largely decoupling the effects of the background velocities and reflectors on the data. Nonlinearity is further reduced by choosing to invert for slownesses and by inverting for a sparse parameter set which is partially defined using geological reflector picks. Applying our approach to shallow seismic data from the North Sea collected over a gas-sand deposit, we demonstrate that the proposed method is able to estimate both the geometry and internal velocity of a significant velocity structure not present in the initial model. Over successive iterations, the use of adaptive depth stretching corrects the pull-down of the base of the gas sand. Vertical background velocity gradients are also resolved. For an insignificant extra cost the acoustic attenuation parameter Q is included in the inversion scheme. The final attenuation tomogram contains realistic values of Q for the expected lithologies and for the effect of partial fluid saturation associated with a shallow bright spot. The attenuation image may also indicate the presence of fracturing.

The application of diffraction tomography to cross‐hole seismic data

Previously published equations for diffraction tomography do not solve the “two and one‐half dimensional problem” (point source illumination of two‐dimensional geology) if sources and receivers are confined to linear arrays. In spite of this lack of a formal solution, useful images can be formed by the application of two‐dimensional formulas to such problems. The estimation of difference fields, of crucial importance in diffraction tomography, reduces to the problem of estimating the source function. Using assumptions about the consistency of the source behavior, we extract the source function in a statistical fashion from cross‐hole data. Using this technique, the difference fields are computed directly from the recorded wave fields for two experiments and diffraction tomographic images are obtained. In the first experiment, the data are generated using a two‐dimensional finite‐difference modeling algorithm. In the second, a physical scale model of a crosshole experiment is performed in an ultrasonic mod...

Waveform tomography at a groundwater contamination site: Surface reflection data

Wehaveappliedacoustic-waveformtomographyto452D seismic profiles to image the 3D geometry of a buried paleochannel at a groundwater-contamination site at Hill Air Force Base in Utah. The paleochannel, which is incised into an alluvium-covered clay aquitard, acts as a trap for dense nonaqueous-phase liquids DNAPLs that contaminate the shallowest groundwater system in the study area. The 2D profiles were extracted from a 3D surface reflection data set. First-arrival traveltime tomography provided initial velocity models for the waveform tomography. We inverted for six frequency components in the band 30‐90 Hz of the direct and refracted waves to produce 45 2D velocity models. The flanks and bottom of a channel with a maximum depth of about 15 m were well modeled in most of the 45 parallel 2D slices,whichallowedustoconstructa3Dimageofthechannel by combining and interpolating between the 45 image slices. The 3D model of the channel will be useful for siting extraction wells within the site remediation program.The alluvium that fills the channel showed marked vertical and lateral velocity heterogeneity. Traveltime tomography and waveform tomography can be complementary approaches. Used together, they can provide high-resolution images of complicatedshallowstructures.

Waveform tomography at a groundwater contamination site: VSP-surface data set

Application of 2D frequency-domain waveform tomography to a data set from a high-resolution vertical seismic profiling (VSP) experiment at a groundwater contamination site in Hill Air Force Base (HAFB), Utah, reveals a surprisingly complicated shallow substructure with a resolution of approximately 1.5 m. Variance in the waveform misfit function is reduced 69.4% by using an initial velocity model from first-arrival traveltime tomography. The waveform tomography model suggests (1) a low-velocity layer at 1 to 4 m depth, (2) a high-vertical-velocity gradient of 80 m/s/m on average, and (3) severe lateral variations — velocity contrasts as large as about 200 m/s occur in a distance as short as 1.5 m. The model is well correlated with lithologic logs and is interpreted geologically. A Q-value of 20 is estimated for the target area. The extreme lateral and vertical variations of the subsurface compromise many standard seismic processing methods.

Sound-speed and attenuation imaging of breast tissue using waveform tomography of transmission ultrasound data

Waveform tomography results are presented from 800 kHz ultrasound transmission scans of a breast phantom, and from an in vivo ultrasound breast scan: significant improvements are demonstrated in resolution over time-of-flight reconstructions. Quantitative reconstructions of both sound-speed and inelastic attenuation are recovered. The data were acquired in the Computed Ultrasound Risk Evaluation (CURE) system, comprising a 20 cm diameter solid-state ultrasound ring array with 256 active, non-beamforming transducers. Waveform tomography is capable of resolving variations in acoustic properties at sub-wavelength scales. This was verified through comparison of the breast phantom reconstructions with x-ray CT results: the final images resolve variations in sound speed with a spatial resolution close to 2 mm. Waveform tomography overcomes the resolution limit of time-of-flight methods caused by finite frequency (diffraction) effects. The method is a combination of time-of-flight tomography, and 2-D acoustic waveform inversion of the transmission arrivals in ultrasonic data. For selected frequency components of the waveforms, a finite-difference simulation of the visco-acoustic wave equation is used to compute synthetic data in the current model, and the data residuals are formed by subtraction. The residuals are used in an iterative, gradient-based scheme to update the sound-speed and attenuation model to produce a reduced misfit to the data. Computational efficiency is achieved through the use of time-reversal of the data residuals to construct the model updates. Lower frequencies are used first, to establish the long wavelength components of the image, and higher frequencies are introduced later to provide increased resolution.

Weighted-Averaging Finite-Element Method for 2D Elastic Wave Equations in the Frequency Domain

We present a weighted-averaging frequency-domain finite-element method for an accurate and efficient 2D elastic wave modeling technique. Our method introduces three kinds of supplementary element sets in addition to a basic element set that is used in the standard finite-element method. By constructing global stiffness and mass matrices for four kinds of element sets and then averaging them with weighting coefficients, we obtain a new global stiffness and mass matrix. With optimal weighting coefficients determined by a Marquardt–Levenberg method to minimize grid dispersion and grid anisotropy, we can reduce the number of nodal points per shear wavelength from 33.3 (using the standard finite-element method) and 20 (using the eclectic method) to 5, with the errors of group velocities no larger than 1%. By reducing the number of grid points per wavelength, we achieve a 97% and 75% reduction of computer memory required to store the complex impedance matrix for a band-type matrix solver and a nested dissection method, respectively, compared with those of the eclectic method. Our method gives approximate solutions compatible with exact solutions for an infinite homogeneous, a semi-infinite homogeneous (Lamb9s problem), and a horizontal two-layer model with fewer grid points than the standard and the eclectic method. A major advantage of the weighted-averaging finite-element method for the elastic wave equation is that it provides solutions very close to correct solutions for Lamb9s problem economically, unlike most of the displacement approaches. In addition, our scheme makes the complex impedance matrix symmetric, which satisfies reciprocity. Seismic forward modeling techniques that satisfy reciprocity are of critical importance in seismic imaging and inversion because we can economically calculate a Jacobian matrix using the reciprocity. Successful simulation of a large-size model shows that our method can be used for the simulation of wave propagation in the geological model needed in the reverse-time migration or seismic inversion.