MCMECHAN, G.A. 1983, Migration by Extrapolation of Time-Dependent Boundary Values, Geophysical Prospecting 31,413-420. Migration of an observed zero-offset wavefield can be performed as the solution of a boundary value problem in which the data are extrapolated backward in time. This concept is implemented through a finite-difference solution of the two-dimensional acoustic wave equation. All depths are imaged simultaneously at time 0 (the imaging condition), and all dips (right up to vertical) are correctly migrated. Numerical examples illustrate this technique in both constant and variable velocity media.

The dispersive waves in a common‐shot wave field can be transformed into images of the dispersion curves of each mode in the data. The procedure consists of two linear transformations: a slant stack of the data produces a wave field in the phase slowness‐time intercept (p — τ) plane in which phase velocities are separated. The spectral peak of the one‐dimensional (1-D) Fourier transform of the p — τ wave field then gives the frequency associated with each phase velocity. Thus, the data wave field is linearly transformed from the time‐distance domain into the slowness‐frequency (p — ω) domain, where dispersion curves are imaged. All the data are present throughout the transformations. Dispersion curves for the mode overtones as well as the fundamental are directly observed in the transformed wave field. In the p — ω domain, each mode is separated from the others even when its presence is not visually detectable in the untransformed data. The resolution achieved in the result is indicated in the p — ω wave field by the width and coherence of the image. The method is applied to both synthetic and real data sets.

The U.S. Geological Survey conducted an extensive seismic refraction survey in the Imperial Valley region of California in 1979. The Imperial Valley is located in the Salton Trough, an active rift between the Pacific and North American plates. Forty shots fired at seven shot points were recorded by 100 portable seismic instruments at typical spacing of 0.5–1 km. More than 1300 recording locations were occupied, and more than 3000 usable seismograms were obtained. We analyzed five profiles using a standard ray‐tracing program, constructed a contour map of reduced travel times from our most widely recorded shot point, and modeled an existing gravity profile across the Salton Trough. Results are itemized: (1) All models have in common a sedimentary layer (Vp = 1.8–5.0 km/s), a “transition zone” (Vp = 5.0–5.65 km/s), a basement (Vp = 5.65 km/s in the Imperial Valley, 5.9 km/s on the bordering mesas), and subbasement (Vp = 7.2 km/s). (2) The sedimentary layer ranges in thickness along the axis of the Salton Trough from 3.7 km (Salton Sea) to 4.8 km (U.S.‐Mexican border). On the bordering mesas it is quite variable in thickness. (3) The “transition” zone is about 1 km thick in most places. In the Imperial Valley there are no marked velocity discontinuities in this zone between the sedimentary layer and basement. On the bordering mesas, however, there is a discontinuity at the top of this zone. (4) There are apparently two types of basement. On the bordering mesas, basement is crystalline igneous and metamorphic rocks. In the Imperial Valley, basement is mostly lower‐greenshist‐facies sedimentary rocks, based primarily on the smooth transition in character from sediment to basement arrivals, the low value of basement velocity, and the fact that deep (4 km) wells in the valley penetrate only the upper part of the known Cenozoic stratigraphic column for the Salton Trough. (5) The subbasement, or intermediate crustal layer, ranges in depth along the axis of the Salton Trough from 16 km (Salton Sea) to 10 km (U.S.‐Mexican border). Gravity modeling requires that this layer deepen and/or pinch out beneath the bordering mesas and mountain ranges. Based on its high velocity and the presence of intrusive basaltic rocks in the sedimentary section in the Imperial Valley, the subbasement is thought to be a mafic intrusive complex similar to oceanic middle crust. (6) Several structures are seen that affect basement, transition zone, and deeper parts of the sedimentary layer. They include a scarp along the Imperial fault, as much as 1 km down to the northeast, and a scarp passing roughly along the topographic boundary between the Imperial Valley and the bordering mesa to the west, as much as 3½ km down to the east. We interpret the latter scarp to be the suture, or rift boundary, between the older crystalline basement on the mesa and the younger metasedimentary basement in the Imperial Valley. (7) On a contour map of reduced travel time from our most widely recorded shot point, subtle patches of early arrivals among otherwise late ar...

A 40-channel wide‐aperture ground penetrating radar (GPR) data set was recorded in a complicated fluvial/aeolian environment in eastern Canada. The data were collected in the multichannel format usually associated with seismic reflection surveys and were input directly into a standard seismic processing sequence (filtering, static corrections, common‐midpoint gathering, velocity analysis, normal‐ and dip‐moveout corrections, stacking and depth migration). The results show significant improvements, over single‐channel recordings, in noise reduction and depth of penetration (by stacking), and in spatial positioning and reduction of diffraction artifacts (by migration). These characteristics increase the potential for reliable interpretation of structural and stratigraphic details. Thus, without having to develop any new software, GPR data processing technology is brought to the same level of capability, flexibility, and accessibility that is current in seismic exploration.

To apply reverse‐time migration to prestack, finite‐offset data from variable‐velocity media, the standard (time zero) imaging condition must be generalized because each point in the image space has a different image time (or times). This generalization is the excitation‐time imaging condition, in which each point is imaged at the one‐way traveltime from the source to that point. Reverse‐time migration with the excitation‐time imaging condition consists of three elements: (1) computation of the imaging condition; (2) extrapolation of the recorder wave field; and (3) application of the imaging condition. Computation of the imaging condition for each point in the image is done by ray tracing from the source point; this is equivalent to extrapolation of the source wave field through the medium. Extrapolation of the recorded wave field is done by an acoustic finite‐difference algorithm. Imaging is performed at each step of the finite‐difference extrapolation by extracting, from the propagating wave field, the amplitude at each mesh point that is imaged at that time and adding these into the image space at the same spatial locations. The locus of all points imaged at one time step is a wavefront [a constant time (or phase) trajectory]. This prestack migration algorithm is very general. The excitation‐time imaging condition is applicable to all source‐receiver geometries and variable‐velocity media and reduces exactly to the usual time‐zero imaging condition when used with zero‐offset surface data. The algorithm is illustrated by application to both synthetic and real VSP data. The most interesting and potentially useful result in the processing of the synthetic data is imaging of the horizontal fluid interfaces within a reservoir even when the surrounding reservoir boundaries are not well imaged.