Jörg Schleicher

Pulse distortion in depth migration

When migrating seismic primary reflections obtained from arbitrary source‐receiver configurations (e.g., common shot or constant offset) into depth, a pulse distortion occurs along the reflector. This distortion exists even if the migration was performed using the correct velocity model. Regardless of the migration algorithm, this distortion is a consequence of varying reflection angle, reflector dip, and/or velocity variation. The relationship between the original time pulse and the depth pulse after migration can be explained and quantified in terms of a prestack, Kirchhoff‐type, diffraction‐stack migration theory.

A comparison of imaging conditions for wave-equation shot-profile migration

The application of a deconvolution imaging condition in wave-equation shot-profile migration is important to provide illumination compensation and amplitude recovery. Particularly if the aim is to successfully recover a measure of the medium reflectivity, an imaging condition that destroys amplitudes is unacceptable. We study a set of imaging conditions with illumination compensation. The imaging conditions are evaluated by the quality of the output amplitudes and artifacts produced. In numerical experiments using a vertically inhomogeneous velocity model, the best of all imaging conditions we tested is the one that divides the crosscorrelation of upgoing and downgoing wavefields by the autocorrelation of the downgoing wavefield, also known as the illumination map. In an application to Marmousi data, unconditional division by autocorrelation turned out to be unstable. Effective stabilization was achieved by smoothing the illumination map.

Data stacking beyond CMP

Stacking has been used in seismic data processing for a long time. In fact, stacked sections (or volumes in 3D) are standard deliverables in the industry, and the concepts of common-midpoint (CMP) gathers and normal moveout (NMO) correction are mentioned in almost every textbook on seismic processing. Although the general trend is toward prestack imaging (either in time or in depth), the construction of stacked sections remains an important step within the seismic processing flow, since they are almost always the first available interpretable images of the subsurface.

Determination of Fresnel zones from traveltime measurements

For a horizontally stratified (isotropic) earth, the rms‐velocity of a primary reflection is a key parameter for common‐midpoint (CMP) stacking, interval‐velocity computation (by the Dix formula) and true‐amplitude processing (geometrical‐spreading compensation). As shown here, it is also a very desirable parameter to determine the Fresnel zone on the reflector from which the primary zero‐offset reflection results. Hence, the rms‐velocity can contribute to evaluating the resolution of the primary reflection. The situation that applies to a horizontally stratified earth model can be generalized to three‐dimensional (3-D) layered laterally inhomogeneous media. The theory by which Fresnel zones for zero‐offset primary reflections can then be determined purely from a traveltime analysis—without knowing the overburden above the considered reflector—is presented. The concept of a projected Fresnel zone is introduced and a simple method of its construction for zero‐offset primary reflections is described. The pr...

Obliquity-correction imaging condition for reverse time migration

The quality of seismic images obtained by reverse time migration (RTM) strongly depends on the imaging condition. We propose a new imaging condition that is motivated by stationary phase analysis of the classical crosscorrelation imaging condition. Its implementation requires the Poynting vector of the source and receiver wavefields at the imaging point. An obliquity correction is added to compensate for the reflector dip effect on amplitudes of RTM. Numerical experiments show that using an imaging condition with obliquity compensation improves reverse time migration by reducing backscattering artifacts and improving illumination compensation.

Eigenwave Based Multiparameter Traveltime Expansions

Three different 2-D traveltime approximations for rays in the vicinity of a fixed zero-offset ray are presented and analyzed. All traveltimes are given as three-parameter expansions involving the emergence angle of the zero-offset ray with respect to the surface normal, as well as two wavefront curvatures associated with the zero-offset ray, namely the normal wave and normal-incidence-point wave. A comparison of all three multiparameter traveltime expansions is carried out. INTRODUCTION Traveltimes of rays in the (paraxial) vicinity of a fixed (central) ray can be described by certain parameters which refer to the central ray only. The traveltime approximations directly obtained from paraxial ray theory are the parabolic and the hyperbolic expansions ((Schleicher et al., 1993)). An appealing alternative traveltime description, has been recently proposed by (Gelchinsky et al., 1997). In this new representation, the paraxial rays can be specified so as to focus at a certain point of the zero-offset ray or at an extension to this ray. For this reason, Gelchinskys expression has been referred to as the multi-focus traveltime. In the second-order approximation the multi-focus traveltime agrees with its parabolic and hyperbolic counterparts. In this paper we provide simple derivations of all above mentioned formulas and examine their behaviour for a synthetic model. THE TRAVELTIME EXPANSION FORMULAS In the following, we refer to Figure 1. 1email: Thilo.Mueller@physik.uni-karlsruhe.de 61

The Kirchhoff–Helmholtz integral for anisotropic elastic media

Abstract The Kirchhoff–Helmholtz integral is a powerful tool to model the scattered wavefield from a smooth interface in acoustic or isotropic elastic media due to a given incident wavefield and observation points sufficiently far away from the interface. This integral makes use of the Kirchhoff approximation of the unknown scattered wavefield and its normal derivative at the interface in terms of the corresponding quantities of the known incident field. An attractive property of the Kirchhoff–Helmholtz integral is that its asymptotic evaluation recovers the zero-order ray theory approximation of the reflected wavefield at all observation points where that theory is valid. Here, we extend the Kirchhoff–Helmholtz modeling integral to general anisotropic elastic media. It uses the natural extension of the Kirchhoff approximation of the scattered wavefield and its normal derivative for those media. The anisotropic Kirchhoff–Helmholtz integral also asymptotically provides the zero-order ray theory approximation of the reflected response from the interface. In connection with the asymptotic evaluation of the Kirchhoff–Helmholtz integral, we also derive an extension to anisotropic media of a useful decomposition formula of the geometrical spreading of a primary reflection ray.

Wide-angle FD and FFD migration using complex Padé approximations

Seismic migration by downward continuation using the one-way wave-equation approximations has two shortcomings: imaging steep-dip reflectors and handling evanescent waves. Complex Pade approximations allow a better treatment of evanescent modes, stabilizing finite-difference migration without requiring special treatment for the migration domain boundaries. Imaging of steep-dip reflectors can be improved using several terms in the Pade expansion. We discuss the implementation and evaluation of wide-angle complex Pade approximations for finite-difference and Fourier finite-difference migration methods. The dispersion relation and the impulsive response of the migration operator provide criteria to select the number of terms and coefficients in the Pade expansion. This ensures stability for a prescribed maximum propagation direction. The implementations are validated on the Marmousi model data set and SEG/EAGE salt model data.

Time-migration velocity analysis by image-wave propagation of common-image gathers

Image-wave propagation or velocity continuation describes the variation of the migrated position of a seismic event as a function of migration velocity. Image-wave propagation in the common-image gather (CIG) domain can be combined with residual-moveout analysis for iterative migration velocity analysis (MVA). Velocity continuation of CIGs leads to a detection of those velocities in which events flatten. Although image-wave continuation is based on the assumption of a constant migration velocity, the procedure can be applied in inhomogeneous media. For this purpose, CIGs obtained by migration with an inhomogeneous macrovelocity model are continued starting from a constant reference velocity. The interpretation of continued CIGs, as if they were obtained from residual migrations, leads to a correction formula that translates residual flattening velocities into absolute time-migration velocities. In this way, the migration velocity model can be improved iteratively until a satisfactory result is reached. Wi...

Seismic modeling by demigration

Kirchhoff-type, isochron-stack demigration is the natural asymptotic inverse to classical Kirchhoff or diffraction-stack migration. Both stacking operations can be performed in true amplitude by an appropriate selection of weight functions. Isochron-stack demigration is closely related to seismic modeling with the Kirchhoff integral. The principal objective of this paper is to show how demigration can be used to compute synthetic seismograms. The idea is to attach to each reflector in the model an appropriately stretched (i.e., frequency-shifted) spatial wavelet. Its amplitude is proportional to the reflection coefficient, transforming the original reflector model into an artificially constructed true-amplitude, depth-migrated section. The seismic modeling is then realized by a true-amplitude demigration operation applied to this artificially constructed migrated section. A simple but typical synthetic data example indicates that modeling by demigration yields results superior to conventional zero-order ray theory or classical Kirchhoff modeling.