Philip S. Schultz

Velocity estimation and downward continuation by wavefront synthesis

A “wave stack” is any stack over a common shot or geophone gather in which the moveout is independent of time. It synthesizes a particular wavefront by superposition of the many spherical wavefronts of raw data. Unlike the common midpoint stack, wave stacks retain the important property of being the sampling of a wave field and, as such, permit wave‐equation treatment of formerly difficult or impossible problems. Seismic sections of field data generated by wave stacks that synthesized slanted downgoing plane waves showed a similarity in appearance to the common midpoint stacks. In signal‐to‐noise ratio they lay between the single offset section and the midpoint stack. The angle selectivity of the slanted plane‐wave stacks permitted detection of a reflector that was not visible on either the midpoint stack or the raw gathers. Simple velocity estimation in slant frame coordinates differs only in detail from standard frame coordinates. Because of the wave field character of data that have been slant plane‐wa...

Depth migration before stack

When seismic data are migrated using operators derived from the scalar wave equation, an assumption is normally made that the seismic velocity in the propagating medium is locally laterally invariant. This simplifying assumption causes reflectors to be imaged incorrectly when lateral velocity gradients exist, irrespective of the degree of accuracy to which the subsurface velocity structure is known. A finite‐difference method has been implemented for migration of unstacked data in the presence of lateral velocity gradients, where the operation of wave field extrapolation is done in increments of depth rather than time. Performing this depth migration on unstacked data results in the imaging of reflectors on the zero‐offset trace, whereupon a zero‐offset section becomes a fully imaged‐in‐depth seismic section. Such a section, in addition to being a correctly migrated depth section, shows the same order of signal amplitude enhancement as in a normal stacking process.

A method for direct estimation of interval velocities

The most commonly used method for obtaining interval velocities from seismic data requires a prior estimate of the root‐mean‐square (rms) velocity function. A reduction to interval velocity uses the Dix equation, where the interval velocity in a layer emerges as a sensitive function of the rms velocity picks above and below the layer. Approximations implicit in this method are quite appropriate for deep data, and they do not contribute significantly to errors in the interval velocity estimate. However, when the data are from a shallow depth (vertical two‐way traveltime being less than direct arrival to the farthest geophone), the assumption within the rms approximation that propagation angles are small requires that much of the reflection energy be muted, along with, of course, all the refraction energy. By means of a simple data transformation to the ray parameter domain via the slanted plane‐wave stack, three types of arrivals from any given interface (subcritical and supercritical reflections and criti...

Seismic-guided estimation of log properties; Part 1, A data-driven interpretation methodology

Seismic data are routinely and effectively used to estimate the structure of reservoir bodies but often play no role in the essential task of estimating the spatial distribution of reservoir or rock properties. Yet, for a long time, we have been using attributes or other features of seismic data to gain useful clues in the interpretation process. Since the 1960s, we have known that reflection amplitude is sensitive to the thickness of thin beds. In the 1970s, bright spots were discovered to be useful in forecasting the presence of gas sands. Then, in the 1980s, amplitude variation with offset (AVO) analysis was identified as an even more refined indicator for gas sands or other situations, giving rise to Poisson’s ratio contrasts. Other examples exist, such as predicting porosity from calibrated acoustic impedance values computed from seismic data.

Seismic‐guided estimation of log properties (Part 2: Using artificial neural networks for nonlinear attribute calibration)

We saw in Part 1 that when we have 3-D data together with a number of logged wells, we can look at possible relationships between some attributes of the seismic data and various properties measured on the logs. At multiple well locations, where we have both seismic and log data, we can look for trends in these two data types on crossplots. If we see a trend, we can quantify it with a derived or specified functional relationship. This functional relationship can be used to convert the attribute values to log properties, and when followed by a residual correction, provide a means to estimate the distribution of these properties away from the wells.

Depth migration after stack

The conventional methods for migrating a seismic section, e.g., the finite‐difference method and the Kirchhoff summation method, are inadequate in the presence of significant lateral variations in velocity. For this type of velocity distribution, the basic migration output should be in true depth, although for practical purposes it may be preferable to display it with a nonlinear depth scale. A finite‐difference method has been implemented for obtaining migrated depth sections. The concept underlying this involves all the usual assumptions of a dip line and primary reflections only, with the seismic section considered as the surface measurement of an upcoming wave field which we process with downward continuation in small increments of depth, rather than the customary increments of traveltime. The specified velocity variation laterally along a thin layer results in transmission time changes which must be corrected by a small static time shift applied to each seismic trace. This additional operation within...

Seismic-guided estimation of log properties; Part 3, A controlled study

In the first two parts of this series (published in the previous two issues of TLE), we discussed an alternative way of generating maps of rock or reservoir properties by using seismic attribute guidance. Normal procedure in generating such property maps (e.g., porosity, water or hydrocarbon saturation) takes the well data as control values, and uses 3-D seismic mainly for structural control. Using 3-D seismic attributes to guide the estimate of properties takes more effort, so there must be a reason that we choose to do it that way. In Part 1, we saw that property maps generated using attributes show greater detail. We also saw some theoretical considerations suggesting greater accuracy compared to maps from log data alone, but greater accuracy was not proven or demonstrated. Here we address the question Are the maps really more accurate? by reviewing the results of a controlled study.

Seismic data processing; current industry practice and new directions

The last ten years have seen an evolution in the state of the art for seismic data processing on a number of fronts. Data transformations investigated have made some types of analyses much more straightforward. Deconvolution has become a sophisticated process which includes statistical, model‐based, and deterministic methods. Vibroseis® processing has led to a greater understanding of the statistical limitations in recovery of the wave‐field amplitude from sign‐bit recording, and the deconvolution of Vibroseis data has improved. Multichannel filtering and analysis in transform domains have resulted in increasingly effective tools for noise reduction and signal enhancement. In statics analysis, surface consistency as a constraint remains a standard, and refraction analysis has become popular as a means of preconditioning data for residual statics estimation. Advances in stacking methodology have come mainly from addressing three effects: ray bending through lateral velocity variations, complex structure, a...

A case for larger offsets

Traditionally, when field acquisition parameters are chosen, an underlying consideration is the ability to process and image the data to be recorded. For the choice of the length of the receiver spread, a limiting factor is the root-mean-square (rms) restriction: for reflections from the depth of interest, one need not record data for which the propagation angle is so large that it cannot safely be included in an rms velocity analysis. This restriction has led to various criteria for the determination of the largest useful offset which can be recorded. For example, the farthest offset should be less than the depth of interest. Recent advances in analysis of wide-angle arrivals compel us to re-evaluate such criteria.

Simple theory for correction of marine vibroseis phase dispersion

We present an alternative correction scheme, which is simple both in concept and application, and is exact in that it is derived without approximations. This new scheme involves a data transformation to simulate a recording taken with stationary sources and receivers, as with land vibroseis. Such a simulated recording is devoid of any phase dispersion effects due to moving sources and receivers, which is the desired objective. The transformation to stationary sources and receivers reduces to a spatial filtering operation on common source and common receiver gathers, or common offset sections.