Geophysical inversion involves the estimation of the parameters of a postulated earth model from a set of observations. Since the associated model responses can be nonlinear functions of the model parameters, nonlinear least-squares techniques prove to be useful for performing the inversion. A common type of inversion applies iterative damped linear least squares through use of the Marquardt-Levenberg method. Traditionally, this method has been implemented by solving the associated normal equations in conventional ways. However, Singular Value Decomposition (SVD) produces significant improvements in computational precision when applied to the same system of normal equations. Iterative least-squares modeling finds application in a wide variety of geophysical problems. Two examples illustrate the approach: (1) seismic wavelet deconvolution, and (2) the location of a buried wedge from surface gravity data. More generally, nonlinear least-squares inversion can be used to estimate earth models for any set of geophysical observations for which an appropriate mathematical description is available.

A review of least-squares inversion and its application to geophysical problems*

Least-squares inversion of seismic reflection waveform data can reconstruct remarkably detailed models of subsurface structure and take into account essentially any physics of seismic wave propagation that can be modelled. However, the waveform inversion objective has many spurious local minima, hence convergence of descent methods (mandatory because of problem size) to useful Earth models requires accurate initial estimates of long-scale velocity structure. Migration velocity analysis, on the other hand, is capable of correcting substantially erroneous initial estimates of velocity at long scales. Migration velocity analysis is based on prestack depth migration, which is in turn based on linearized acoustic modelling (Born or single-scattering approximation). Two major variants of prestack depth migration, using binning of surface data and Claerbout's survey-sinking concept respectively, are in widespread use. Each type of prestack migration produces an image volume depending on redundant parameters and supplies a condition on the image volume, which expresses consistency between data and velocity model and is hence a basis for velocity analysis. The survey-sinking (depth-oriented) approach to prestack migration is less subject to kinematic artefacts than is the binning-based (surface-oriented) approach. Because kinematic artefacts strongly violate the consistency or semblance conditions, this observation suggests that velocity analysis based on depth-oriented prestack migration may be more appropriate in kinematically complex areas. Appropriate choice of objective (differential semblance) turns either form of migration velocity analysis into an optimization problem, for which Newton-like methods exhibit little tendency to stagnate at nonglobal minima. The extended modelling concept links migration velocity analysis to the apparently unrelated waveform inversion approach to estimation of Earth structure: from this point of view, migration velocity analysis is a solution method for the linearized waveform inversion problem. Extended modelling also provides a basis for a nonlinear generalization of migration velocity analysis. Preliminary numerical evidence suggests a new approach to nonlinear waveform inversion, which may combine the global convergence of velocity analysis with the physical fidelity of model-based data fitting.

http://www.trip.caam.rice.edu/reports/2007/geoprosp0207.pdf

Migration velocity analysis and waveform inversion

Differential semblance optimization (DSO) is an approach to inversion of reflection seismograms which avoids the severe convergence difficulties associated with nonlinear least‐squares inversion. The method exploits both moveout and amplitude characteristics of reflections. We have implemented a version appropriate to plane‐wave (p‐tau) seismograms and layered constant‐density acoustic earth models. Theoretical and numerical analyses of this version of DSO indicate that stable and reasonably accurate estimates of both velocity trend and reflectivity can be derived. To test DSO further, we applied it to a marine data set from the Gulf of Mexico, where the method produced results which compare favorably to well‐log information. The method can be extended to incorporate laterally heterogeneous velocity models.

Velocity inversion by differential semblance optimization

A seismic attribute is a quantitative measure of a seismic characteristic of interest. Analysis of attributes has been integral to reflection seismic interpretation since the 1930s when geophysicists started to pick traveltimes to coherent reflections on seismic field records. There are now more than 50 distinct seismic attributes calculated from seismic data and applied to the interpretation of geologic structure, stratigraphy, and rock/pore fluid properties. The evolution of seismic attributes is closely linked to advances in computer technology. As examples, the advent of digital recording in the 1960s produced improved measurements of seismic amplitude and pointed out the correlation between hydrocarbon pore fluids and strong amplitudes (“bright spots”). The introduction of color printers in the early 1970s allowed color displays of reflection strength, frequency, phase, and interval velocity to be overlain routinely on black-and-white seismic records. Interpretation workstations in the 1980s provided...

Seismic attributes — A historical perspective

Seismic migration and inversion describe a class of closely related processes sharing common objectives and underlying physical principles. These processes range in complexity from the simple NMO‐stack to the complex, iterative, multidimensional, prestack, nonlinear inversion used in the elastic seismic case. By making use of amplitudes versus offset, it is, in principle, possible to determine the three elastic parameters from compressional data. NMO‐stack can be modified to solve for these parameters, as can prestack migration. Linearized, wave‐equation inversion does not inordinately increase the complexity of data processing. The principal part of a migration‐inversion algorithm is the migration. Practical difficulties are considerable, including both correctable and intrinsic limitations in data quality, limitations in current algorithms (which we hope are correctable), and correctable (or perhaps intrinsic) limitations in computer power.

Migration and inversion of seismic data

A point-source seismic recording can be decomposed into a set of plane-wave seismograms for arbitrary angles of incidence. Such plane-wave seismograms possess an inherently simple structure that make them amenable to existing inversion methods such as predictive deconvolution. Implementation of plane-wave decomposition (PWD) takes place in the frequency-wavenumber domain under the assumption of radial symmetry. This version of PWD is equivalent to slant stacking if allowance is made for the customary use of linear recording arrays on the surface of a three-dimensional medium. An imaging principle embodying both kinematic as well as dynamic characteristics allows us to perform time migration of the plane-wave seismograms. The imaging procedure is implementable as a two-dimensional filter whose independent variables are traveltime and angle of incidence.

Plane‐wave decomposition of seismograms

Most of the seismic data processing procedures are divided into 2-D, 2.5-D, crooked lines or 3-D versions dictated by the differences in the shot and receiver configurations. In this paper, we introduce a tomographic approach that overcomes these geometrical difficulties and provides stable statics solutions from picked first-break times. We also show that the first-break picks contain both the short and the long wavelength surface statics. The solutions are obtained by solving a set of generalized surface-consistent delay-time equations using the method of weighted least squares and conjugate gradient. While iterating, each first-break pick is evaluated to ensure its consistency with the least-squares solution. Based on consistency, we weight the traveltime picks and use them in the next iteration. These weights also serve as an indicator of anomalous picks to the user. We show that long wavelength solutions leave large residual errors in the least-squares solutions. We also use the expected length of the Fresnel zone to differentiate between short and long wavelength static solutions. After removing the influence of long wavelength statics, we apply short wavelength statics to reduce the residual errors further. We demonstrate the validity of our unified method by applying it to actual data examples. The removal of both long and short wavelength statics improves the initial data set that produces a more consistent set of velocities and leaves only the short wavelength residual reflection statics, which are generally less than quarter wavelet period delays. This removes the most probable cause of the leg jump contamination and poor velocity estimates from the residual statics computations, especially from the 3-D data.

A unified method for 2-D and 3-D refraction statics

A Neural Network was trained to pick first breaks in seismic reflection data. The network was trained by selecting a few records of seismic data sorted in common source format, manually picking the first break times, computing the four seismic trace attributes (Taner, 1988) of peak amplitude, mean power level, power ratio between a forward and reverse sliding window, and envelope slope for the picked peek and its two adjacent peaks. This analysis was done separately for each trace with no attempt to capture the spatial trace-to-trace or record-to-record pattern of seismic first break picks. The resulting network consisted of 12 input units (4 attributes X 3 peaks), a single hidden layer with 5 units, and a single output unit. Using a backpropagation algorithm (Rumelhart et al., 1986; Parker, 1985; Werbos, 19741, we adjusted the weights between the nodes of the network until the single output unit recorded a high value (near unity) when evaluating the attributes of the ‘ correct’ peak. We call adjusting the weights ‘ training the network’ . Once trained, the network examines the remaining traces and picks first breaks for peaks whenever the output network value exceeds 0.8. To date, we have achieved 90% or greater accuracy using the backpropagation neural network to pick first break arrivals from surface sources.

First Break Picking Using Neural Networks

The autocepstrum of a reflection seismogram is defined by the cepstrum of its autocorrelation function. Using the autocepstrum extends the basic deconvolution method for removing a minimum‐phase source wavelet to unmask subsurface reflectivity. When we record only the seismic trace and assume a minimumphase source wavelet, deconvolution reduces to estimating the wavelet autocorrelation. In practice, a portion of the seismic trace autocorrelation is used as an estimate of the wavelet autocorrelation. This can be justified by assuming a random reflectivity series with a white power spectrum. However, in cases where the reflectivity spectrum is not white, a preferred wavelet autocorrelation may be obtained by low‐pass windowing the trace autocepstrum. This approach liberates the selection of various deconvolution parameters such as filter length and design window length that are typically chosen to reinforce the assumption of a white reflectivity spectrum. For problems that require short, deconvolution‐filte...

D. E. Wagner

Papers

Plane‐wave decomposition of seismograms

A unified method for 2-D and 3-D refraction statics

First Break Picking Using Neural Networks

Deconvolution by autocepstral windowing

Nuclear Yields from Rayleigh Waves