Kirchhoff Imaging Research Articles

Optimal waveform design for array imaging

We introduce and analyze several algorithms for optimal illumination in array imaging. We consider time reversal and Kirchhoff migration imaging in homogeneous media, in regimes where the signal-to-noise ratio is high (infinite). Extensions to coherent interferometric imaging in clutter are described briefly. We show with numerical simulations that the optimal illumination algorithms image selectively closely spaced point scatterers and extended scatterers with considerably better resolution than without the optimization. We analyze the imaging algorithms in the Fraunhofer diffraction regime for small and extended scatterers. Using the prolate spheroidal wavefunctions we also derive analytic expressions of optimal illuminations for imaging strips.

Crosswell imaging by two-dimensional oriented wave path migration

SUMMARY We apply a new imaging method, called oriented wave path migration, to 2-D prestack crosswell imaging. Compared to conventional crosswell imaging techniques, for example, Kirchhoff migration, which requires separation of the data into upgoing and downgoing waves prior to migration, oriented wave path migration automatically separates the wavefields during the migration process. Results with synthetic data show that oriented wave path migration can effectively locate the steeply dipping fault boundaries and noticeably improve the image resolution compared to the Kirchhoff migration method. Both oriented wave path and Kirchhoff migration images are suitable for geological interpretation. Results with field data show that oriented wave path migration generates quality images similar to the Kirchhoff migration method, all of which roughly correlate with the P-wave sonic logs.

Wave Equation versus Kirchhoff pre-stack depth migration algorithms ? An Australian case study

Oil companies demand for depth imaging technologies to be applied as part of their data processing sequence is quickly growing. Amongst depth imaging technologies, also rapidly evolving, are the ways seismic data are pre-stack depth migrated once a reliable depth model is available.Until a few years ago Kirchhoff type algorithms have been widely used to perform pre stack depth migration (PSDM). Although making approximations on the imaged data, they are economical, reliable, and easy to implement and understand. Advances in computing however, have now made the long known Wave equation type algorithms an economically feasible alternative for pre stack depth migration as they have shown advantages over the Kirchhoff type particularly under the salt structures of the Gulf of Mexico. Rare are the opportunities, however, to encounter salt structures such as the ones seen in the Gulf of Mexico on Australian datasets. Still Wave equation algorithms have theoretical advantages over Kirchhoff that we review in the first part of the article.In order to determine if those theoretical advantages improve the imaging of the structures seen around Australia, it is on the Australian dataset of the Skua and Swift fields in the Bonaparte basin, where both Kirchhoff and Wave equation pre stack depth imaging were performed, that we analyse and compare the two methods. We show that superior continuity, imaging and resolution of the seismic reflectors are obtained using the Wave equation algorithm.

Crosscorrelogram migration of inverse vertical seismic profile data

We present the theory of crosscorrelogram migration of ghost reflections, also known as interferometric imaging, to delineate reflector geometries from inverse vertical seismic profile data. The theory includes the equations for forward modeling, migration, asymptotic inversion, and model resolution of crosscorrelgrams. Rather than using primary reflections, crosscorrelogram migration can use ghost reflections to reconstruct the reflector geometry. Other multiples can be used as well, including pegleg multiples and higher-order multiples. Its main advantages over conventional Kirchhoff migration are (1) both source location (e.g., drill-bit position) and source wavelet need not be known (such as when using a drill bit as a source in a deviated well), (2) it is insensitive to source-related static errors, and (3) it has wider subsurface illumination than conventional Kirchhoff migration of primary reflections. Crosscorrelation migration can effectively widen the source-receiver aperture by more than 50% compared to standard inverse vertical seismic profile (IVSP) migration. The primary disadvantages are (1) it uses ghost reflections for imaging, which can be weaker (or sometimes more distorted) than primary reflections; (2) crosscorrelation creates virtual multiples that can sometimes appear as coherent noise in the final image; and (3) it has poorer horizontal resolution than standard IVSP migration. Results from imaging simulated IVSP traces show that crosscorrelation migration produces reflectivity-like images that correlate well with the actual reflector geometry of a layered fault model. These images are almost completely immune to static errors at the source location and have wider subsurface illumination than conventional IVSP migration images. We also apply crosscorrelation migration to IVSP data recorded at a Friendswood, Texas, test site. Results show that the crosscorrelation image correlates better with the well log and wider subsurface illumination than a conventional Kirchhoff migration image.

Resolving the K-2 salt structure in the Gulf of Mexico

While potential field techniques have long been applied in regional studies and for mineral exploration, the recently-commercialized full tensor gradiometry (FTG) system possesses the resolution and sensitivity required in detailed mapping for oil and gas exploration, even for deep objectives. Used in combination with prestack depth imaging, this provides a potent mapping capability especially in areas of structural complexity. We illustrate this with a subsalt case study from the Gulf of Mexico where an integrated approach using wave-equation depth imaging and FTG in-version succeeds in resolving the base of salt in an area where Kirchhoff depth imaging alone fails.

Synergies in geophysical, medical, and space imaging: Overview and introductory session

The workshop's first session reviewed seismic, medical, and space imaging technology with two presentations from each field—one to introduce basic concepts and demystify the jargon, and one to describe the state of the art. Bob Stolt's introductory seismic overview presented the basic concepts of 2- and 3D, pre- and poststack, time and depth migration, and gave simplified tutorials on different integral- and difference-equation migration techniques. “Seismic depth imaging of complex geological structures” by Larry Lines emphasized the complexity of the earth as a propagating medium, and the need for sophisticated processing to produce accurate structural images. Complex geologic structures (which often involve steeply dipping beds, folded structures, and anisotropic velocities) usually invalidate many conventional processing methods. However, according to Lines, Kirchhoff, f-x , and reverse-time prestack depth imaging perform adequately (at a minimum) when a reasonable velocity model and data with coherent signals are available. Figure 1 shows the final interpretation for a prestack depth migration of a triangle zone feature in the Alberta Foothills. A complete description of the data and imaging can be found in “Seismic imaging and velocity analysis for an Alberta Foothills seismic survey” by Yan and Lines (Geophysics, 2001). Figure 1. The final interpretation for a prestack depth migration of a triangle zone …

Fast Kirchhoff migration in the wavelet domain

We present results of the application of the wavelet transform method to seismic imaging. The objective of this research is to develop 3D seismic Kirchhoff imaging in the wavelet-transform domain, making use of the time-frequency property of wavelets. We propose to migrate the wavelets as units rather than single samples as in conventional Kirchhoff migration implementation. In practice, the wavelet transform of each trace from a seismic section is carried out, then the significant wavelet coefficients are migrated according to their time location in a manner similar to the conventional migration of samples. These operations are then followed by a proper reconstruction. Since the number of significant wavelet coefficients is much fewer than the number of samples in the time domain, and the migration procedure in fact does not change, computation time is significantly reduced. We conducted a series of 2D and 3D experiments with artificial and real data to check whether the migration of significant wavelet coefficients leads to a correct result. Results from synthetic data as well as field data have shown that the migration in the wavelet domain significantly reduces computation time while maintaining good image quality.

Kirchhoff image propagation

Seismic imaging processes are, in general, formulated under the assumption of a correct macrovelocity model. However, seismic subsurface images are very sensitive to the accuracy of the macrovelocity model. This paper investigates how the output of Kirchhoff inversion/migration changes for perturbations of a given 3‐D laterally inhomogeneous macrovelocity model. The displacement of a reflector image point from a perturbation of the given velocity model is determined in a first‐order approximation by the corresponding traveltime and slowness perturbations as well as the matrix. of the Beylkin determinant. The required traveltime derivatives can be calculated with ray perturbation theory.Using this result, a new, computationally efficient Kirchhoff inversion/migration technique is developed to predict in parallel a series of subsurface images for perturbations of a given macrovelocity model during a single inversion/migration process applied to the unmigrated seismic data. These images are constructed by superposition of the seismic data at predicted image point locations which lie on surfaces that expand from the initial image point as a function of the velocity perturbation. Because of the analogy to Huygens wavefronts in wave propagation, the technique is called Kirchhoff image propagation.A 2‐D implementation of Kirchhoff image propagation requires about 1.2 times the computation time of a single migration to calculate a set of propagated images. The propagated images provide good approximations to remigrated images and are applied to migration velocity analysis.

Kirchhoff imaging beyond aliasing

Crucial image resolution may be lost when spatially aliased data are imaged with Kirchhoff algorithms that employ standard antialiasing methods. To maximize resolution, I introduce a method that enables the proper imaging of some aliased components in the data, while avoiding aliasing artifacts. The proposed method is based on a detailed analysis of the different types of aliasing that affect Kirchhoff imaging. In particular, it is based on the observation that operator aliasing depends on the dip spectrum of the data. A priori knowledge on the characteristics of the dip spectrum of the data, in particular on its asymmetry, can thus be exploited to enable “imaging beyond aliasing.” The method is not of general applicability, but it successfully improves the image resolution when a priori assumptions on the data dips are realistic. The imaging of salt‐dome flanks in the Gulf of Mexico has been enhanced by the application of the proposed method.

A beam approach to Kirchhoff depth imaging

Although Kirchhoff migration is currently the tool of choice for 3-D seismic prestack depth imaging, full volume imaging using Kirchhoff methods is costly for large surveys or interactive processing. This article describes a beam methodology for Kirchhoff imaging that leads to a theoretical speedup of up to two orders of magnitude over traditional implementations of Kirchhoff migration. Beam methodology has been used routinely in Amerada Hess for generating 3-D full-volume prestack depth images for the last several years.

Kirchhoff imaging as a tool for AVO/AVA analysis

Kirchhoff‐type weighted stacking methods are used in an ever more sophisticated way with the aim of aggregating amplitude information into imaged seismic sections. This is, for instance, the case of true‐amplitude prestack depth migration (PreSDM), in which amplitudes of migrated primary reflections essentially represent a measure of offset‐dependent reflection coefficients. Application of true‐amplitude PreSDM to several individual common‐offset sections generates an ensemble of migrated sections directly amenable to an amplitude‐variation‐with‐offset (AVO) analysis.

Imaging complex geologic structure with single‐arrival Kirchhoff prestack depth migration

We compare various forms of single‐arrival Kirchhoff prestack depth migration to a full‐waveform, finite‐difference migration image, using synthetic seismic data generated from the structurally complex 2-D Marmousi velocity model. First‐arrival‐traveltime Kirchhoff migration produces severe artifacts and image contamination in regions of the depth model where significant reflection energy propagates as late or multiple arrivals in the total reflection wavefield. Kirchhoff migrations using maximum‐energy‐arrival traveltime trajectories significantly improve the image in the complex zone of the Marmousi model, but are not as coherent as the finite‐difference migration image. By carefully incorporating continuous phase estimates with the associated maximum‐energy arrival traveltimes, we obtain single‐arrival Kirchhoff images that are similar in quality to the finite‐difference migration image. Furthermore, maximum‐energy Green's function traveltime and phase values calculated within the seismic frequency band give a Kirchhoff image that is (1) far superior to a first‐arrival—based image, (2) much better than the analogous high‐frequency paraxial‐ray Green's function image, and (3) closely matched in quality to the full‐waveform finite‐difference migration image.

Maximum energy traveltimes calculated in the seismic frequency band

Prestack Kirchhoff migration using first‐arrival traveltimes has been shown to produce poor images in areas of complex structure. To avoid this problem, I propose a new method for calculating traveltimes that estimates the traveltime of the maximum energy arrival, rather than the first arrival. This method estimates a traveltime that is valid in the seismic frequency band, not the usual high‐frequency approximation. Instead of solving the eikonal equation for the traveltime, I solve the Helmholtz equation to estimate the wavefield for a few frequencies. I then perform a parametric fit to the wavefield to estimate a traveltime, amplitude, and phase. The images created by using these parameters in a Kirchhoff imaging algorithm are comparable in quality to those created using full‐wavefield, finite‐difference, shot‐profile migration.