William S. Harlan

Tomographic estimation of shear velocities from shallow cross‐well seismic data

A cross-well, shear-wave data set posed two problems: 1) wells are separated by only 12.5 wavelengths, so thin raypaths are a poor approximation; 2) data are recorded through a high-velocity limestone surrounded by low-velocity shales. Waves are bent considerably, so either refracted or direct arrivals can arrive first. Only first-arrivals can be picked with any reliability. Ray tracing is avoided entirely by extrapolating traveltimes explicitly from sources or receivers to every point in a region of interest. Rather than search for a single fastest raypath, this method finds Fresnel regions containing all paths that add constructively to first-arrivals. This method also robustly finds minimum traveltimes, whether direct or refracted. The resolution of the estimated velocity model is controlled explicitly with basis functions of adjustablewidth. Wavepaths and velocities are estimated alternately: as the accuracy of paths improves, velocities are allowed to introduce sharper detail. No more two-dimensional detail is introduced than necessary.

Simultaneous velocity filtering of hyperbolic reflections and balancing of offset-dependent wavelets

Hyperbolic reflections and convolutional wavelets are fundamental models for seismic data processing. Each sample of a “stacked” zero‐offset section can parameterize an impulsive hyperbolic reflection in a midpoint gather. Convolutional wavelets can model source waveforms and near‐surface filtering at the shot and geophone positions. An optimized inversion of the combined modeling equations for hyperbolic traveltimes and convolutional wavelets makes explicit any interdependence and nonuniqueness in these two sets of parameters. I first estimate stacked traces that best model the recorded data and then find nonimpulsive wavelets to improve the fit with the data. These wavelets are used for a new estimate of the stacked traces, and so on. Estimated stacked traces model short average wavelets with a superposition of approximately parallel hyperbolas; estimated wavelets adjust the phases and amplitudes of inconsistent traces, including static shifts. Deconvolution of land data with estimated wavelets makes wa...

Tomographic correction of transmission distortions in reflected seismic amplitudes

A tomographic inversion adjusts seismic reflection amplitudes to remove distortions caused by spatial variations in the transmission properties of the overlying earth. These variations create distinctive patterns on displays of reflection amplitude versus source-receiver offsets and midpoints. These patterns are inverted for amplitude corrections that remove the transmission distortions. The methodology is demonstrated on a strong “bright spot” reflection under a large filled channel in the Gulf of Mexico. Transmission anomalies are defined at each point in a depth model as a fractional increase or decrease in wave amplitude. These changes in amplitude scale multiplicatively along raypaths. An inversion of transmission anomalies (i) minimizes errors between modeled and picked amplitudes, (ii) uses a quadratic objective function for easy optimization, and (iii) distinguishes reflectivity changes from transmission anomalies. Reflection are estimated by reflection tomography for interval velocities. The effects of channel irregularities greatly obscured the observed amplitude versus offset in the Gulf of Mexico The transmission anomaly model reconstructed recorded amplitudes accurately and removed the corresponding interference patterns. Most transmission anomalies were imaged near the top of the channel. Local focusing and defocusing of waves by velocity variations can explain these perturbations of amplitudes. An anomaly does have different effects on reflections at different depths.

Introduction to the supplement on velocity estimation for depth imaging

In the last decade, much of 3D seismic signal processing has been replaced by the single imaging process called prestack depth migration. Rapid advances in resolution and illumination have changed the way we explore, delineate, and monitor reservoirs. We now see more attention turning to the velocity models that make such depth imaging possible. For this supplement to the September-October 2008 issue of GEOPHYSICS, we invited summaries of existing practices, descriptions of new approaches, and case studies that illustrate the practical aspects of building and updating velocity models. Velocities are implicit in every imaging or moveout-sensitive processing step. Velocities still consume the most hours of human labor and interpretation during seismic processing. We cannot simply delegate the geologic implications of a velocity model to a later interpretive step. We need velocities for a seismic image, an image for interpretation, and interpretation velocities.

Common‐offset depth migrations and traveltime tomography

Many methods of depth migration velocity analysis emphasize Well-focused images. Others linearize and invert the effect of perturbed velocities on migrated images. We prefer to use developed methods of reflection traveltime tomography by converting picked migrated reflections into equivalent multi-offset traveltimes. Migration benefits prestack picking by simplifying reflections and diminishing noise. Depth migration does not add information to reflections, however. In fact, the bias of a poor velocity model should be removed. Conventional dynamic ray methods, or extrapolated traveltime tables suffice for the estimation of prestack traveltimes (geometric modeling). We need only find the midpoint hat reflects from a migrated point at the correct angle and offset. Constant-offset sections of a North Sea line were independently migrated in depth and viewed on a 3D interpretive workstation. One reflection at the base of chalk imaged at inconsistent depths over offset. This and ather reflections were picked over a range of offsets. Equivalent prestack traveltimes were modeled through the migration velocity model. The chosen method of traveltime tomography implicitly encouraged consistency in commonreflection points for raypaths at various offsets. The final estimated velocity model showed an increase in velocities near the base of the chalk, then a decrease in velocities below. Remigration of the data with the revised velocities greatly increased the visibility of the reflection at the base of the chalk.

Separation of signal and noise applied to vertical seismic profiles

Inversion of the band‐limited one‐dimensional VSP response is nonunique because impedance functions with very different statistics produce equivalent responses. Least‐squares methods of inversion linearly transform noise and tend to produce impedance functions with a Gaussian distribution of amplitudes. I modify a least‐squares inversion procedure to exclude nonzero impedance derivatives that are significantly influenced by noise. The resulting earth model shows homogeneous intervals unless the data have reliable information to the contrary. The data are modeled with a one‐dimensional wave equation and three invertible functions: acoustic impedance, a source wavelet, and the traces’ amplification. First, a linearized least‐squares inverse perturbs the source function to model the downgoing wave. A relinearized inverse finds perturbations of all three modeling functions to account for first‐order reflections. Further iterations explain higher order reflections. To estimate the reliability of impedance pert...

Traveltime tomography with multioffset common‐reflection points

Reflection traveltime tomography has evolved away from layered models toward independent parameters for velocities and reflectors. We introduce a simple method of optimizing interval velocities and common-reflection points simultaneously. Interval velocities are parametrized as a smooth function of spatial coordinates, independently of common-reflection points. Dynamic ray methods and explicit traveltime extrapolations identify common-reflection points that best model prestack traveltimes. The error between a modeled and measured traveltime is scaled by the cosine of a raypath’s angle of reflection. This scaled traveltime error is equivalent to the error of a reflection at normalincidence, or zero-offset. Velocities are revised to minimize the variance of these equivalent errors for all offsets of a common-reflection point. A North Sea seismic line was particularly unsuitable for a layered velocity model. Salt interrupted reflections, and chalk velocities increased rapidly with depth. The tomographically estimated velocities showed strong lateral changes. Prestack depth migration confumed that the velocity model accurately explained traveltimes.