Timothy H. Keho

Paraxial ray Kirchhoff migration

A rapid nonrecursive prestack Kirchhoff migration is implemented (for 2-D or 2.5-D media) by computing the Green’s functions (both traveltimes and amplitudes) in variable velocity media with the paraxial ray method. Since the paraxial ray method allows the Green’s functions to be determined at points which do not lie on the ray, two‐point ray tracing is not required. The Green’s functions between a source or receiver location and a dense grid of thousands of image points can be estimated to a desired accuracy by shooting a sufficiently dense fan of rays. For a given grid of image points, the paraxial ray method reduces computation time by one order of magnitude compared with interpolation schemes. The method is illustrated using synthetic data generated by acoustic ray tracing. Application to VSP data collected in a borehole adjacent to a reef in Michigan produces an image that clearly shows the location of the reef.

Imaging with deep-water multiples

Acquisition of on‐bottom hydrophone data recording of a near‐surface source provides an opportunity to treat water column multiples as useful signal. A ray‐equation based Kirchhoff depth migration is used to image primary reflections and deep‐water multiples recorded on an Ocean Bottom Hydrophone (OBH). The image of the subbottom sediments is shown to be improved by inclusion of the deep‐water multiple in the imaging process. Field data, jointly acquired by Woods Hole Oceanographic Institute and University of Texas Institute for Geophysics at Austin and consisting of an OBH (2300 m depth) recording a 10 800 cubic inch air gun array, are used to illustrate the feasibility of the technique. Images are obtained from both the primary reflections and from energy that has undergone an additional passage through the water column. Comparison of these images reveals an excellent correlation of reflectors with the predicted polarity reversal observed in the multiple’s image. Synthetic data are used to examine the d...

The paraxial ray method

The paraxial ray method is an economical way of computing approximate Green’s functions in heterogeneous media. The method uses information from the standard dynamic ray‐tracing method to extrapolate the seismic wave field at receivers in the neighborhood of a ray so that two‐point ray tracing is not required. Applicability conditions are explicit: they define where asymptotic (high‐frequency) methods are valid, and how far away from the ray the extrapolation remains accurate. Increasing the density of the ray fan improves accuracy but increases computation time. However, since reasonable accuracy is obtained with relatively few rays, the method yields results similar to the two‐point ray‐tracing method, but at a fraction of the cost. Examples of wave‐field extrapolation from a ray to neighboring receivers show that traveltime extrapolation is more accurate than amplitude extrapolation. Accuracy, robustness, and efficiency tests, comparing paraxial ray synthetic seismograms with acoustic finite‐difference...

The impact of migration on AVO

Amplitude variation with offset (AVO) analysis is often limited to areas where multidimensional propagation effects such as reflector dip and diffractions from faults can be ignored. Migration-inversion provides a framework for extending the use of seismic amplitudes to areas where structural or stratigraphic effects are important. In this procedure, sources and receivers are downward continued into the earth using uncollapsed prestack migration. Instead of stacking the data as in normal migration, the prestack migrated data are used in AVO analysis or other inversion techniques to infer local earth properties. The prestack migration can take many forms. In particular, prestack time migration of common-angle sections provides a convenient tool for improving the lateral resolution and spatial positioning of AVO anomalies. In this approach, a plane-wave decomposition is first applied in the offset direction, separating the wavefield into different propagating angles. The data are then gathered into common-angle sections and migrated one angle at a time. The common-angle migrations have a simple form and are shown to adequately preserve amplitude as a function of angle. Normal AVO analysis is then applied to the prestack migrated data. Examples using seismic lines from the Gulf of Mexico show how migration improves AVO analysis. In the first set of examples, migration is shown to improve imaging of subtle spatial variations in bright spots. Subsequent AVO analysis reveals dim spots associated with dry-hole locations that were not resolvable using traditional processing techniques, including both conventional AVO and poststack migration. A second set of examples shows improvements in AVO response after migration is used to reduce interference from coherent noise and diffractions. A final example shows the impact of migration on the spatial location of dipping AVO anomalies. In all cases, migration improves both the signal-to-noise ratio and spatial resolution of AVO anomalies.

Elastic Kirchhoff Migration For Vertical Seismic Profiles

Elastic Kirchhoff migration is implemented for the VSP recording geometry. The resulting migration formula requires measurement of the stress as well as the displacement. Since stress is not measured in a VSP, and in many cases the horizontal component of displacement is not measured, approximate migration formulas are given for these cases. The elastic migration formula for the case where only the vertical components are available, is the same as the acoustic migration formula, where the pressure data are replaced by the magnitudes of the elastic data as reconstructed from the vertical components, and the acoustic Green’s functions are replaced with either the P or S wave elastic Green’s functions. Two expressions for migration of two component displacement data are presented. In the first, the terms involving traction data are simply ignored. In the second, an improved backpropagation operator for the displacement field is obtained by replacing the traction data in the Kirchhoff integral by displacement data using Hooke’s law. The migration expressions for the cases where two component data are available produce images which are less contaminated by artifacts than the migration images of one component data.

Using multiparameter Born theory to obtain certain exact multiparameter inversion goals

Because no exact method exists for multiparameter, multidimensional seismic inversion, approximate methods based on variations of the Born approximation are significant. Artifacts which result from its linear nature limit the usefulness of multiparameter Born inversion. These artifacts can be identified by inferring from the differential equation that portion of the Fourier transformed data that is most consistent with the approximate linear Born relationship. When this portion of the data is inverted, the true parameter changes may be distinguished from linear artifacts.

Marine source wavelet and radiation pattern estimation

We demonstrate the capability of the method derived by Weglein et.al. (1990) to estimate a marine source signature (wavelet and radiation pattern) for the case of a cosg amplitude variation. This method ls based on a Kirchhofflike integral for wavefield separation and extrapolation of hydrophone dual streamer data (one streamer above the other). We also propose a reciprocal application of thii method ss an economical way to accurately determine the source signature in shallow water immediately preceding acquisition. The reciprocal experiment consists of deploying a disposable vertical array of three or four hydrophones on site and towing the source array over it to measure the source signature.