Martin Tygel

3-D true-amplitude finite-offset migration; discussion and reply

In a very interesting paper Schleicher et al. discuss true‐amplitude prestack depth migration of single‐fold subsets of various 3-D geometries. In the following, I would like to make some remarks, and I would like to present other candidates for inclusion in the list of seismic measurement configurations.

Fourth-Order Statistics for Parameter Estimation

Semblance is the mostly used coherence measure for parameter estimation from geophysical data. The best example is velocity analysis in which the normal-moveout velocity is extracted from commonmidpoint (CMP) gathers. In complete analogy to velocity analysis, semblance is also applied to estimate Common-Reflection-Surface (CRS) parameters by means of the hyperbolic traveltime applied to multicoverage data. In statistics, semblance is related to the so-called second moment and in optimization theory, to the least-squares solution of maximum signal energy as a characterization of reflection events. Extensions of the usual semblance can be defined by replacing second-order by higher-order quantities. Here we found encouraging results by a natural extension of semblance to fourth order. The introduced fourth-order semblance is applied to CRS parameter estimation. Numerical examples show that the search using fourth-order semblance is more reliable for high noise levels. INTRODUCTION Since the famous work of Taner and Koehler (1969), semblance has been a reliable measure of coherence in seismic processing. Many applications like stacking velocity analysis (Doherty and Claerbout, 1976; Yilmaz, 1979), migration velocity analysis (Sattlegger, 1975; Dohr and Stiller, 1975; Al-Yahya, 1989; Schleicher and Biloti, 2007), filter techniques (Reiter et al., 1993) or CRS stack (see, e.g., Höcht et al., 1999) rely on semblance to detect the shape of reflection events in seismic data. Semblance is known to depend in various degrees on operator size (aperture and window length) and noise level (Douze and Laster, 1979). Moreover, it is based on the assumption of white noise. Therefore, it sometimes shows unpredictable behaviour if the noise is actually coloured. For these reasons, many attempts have been made to find a more stable measure of coherence that depends less on the kind of noise in the data or the choice of the parameters used in the analysis. One of the most successful ones is differential semblance (Symes and Carazzone, 1991; Symes and Kern, 1994). Being a very robust and easy to calculate measure of coherence for a broad variety of situations, the second-order coherence measure semblance has survived all these attempts. Nonetheless, there exist particular situations, where other coherency measures can be advantageous. In this paper, we compare its behaviour to those of a firstand a fourth-order coherence measure. We show that while in conventional velocity analysis, there is no gain in replacing semblance by one of the other measures, for the linear search of the CRS stack (Müller, 1999), the fourth-order measure is less dependent on aperture and noise level, thus resulting in reliable estimates of the local slope more often than when using conventional semblance. INTERPRETATION OF SEMBLANCE IN STATISTICS One important step in the CMP (or CRS) stacking process is to find pre-assigned curves or surfaces (e.g., hyperbolic curves) that fit the reflection traveltimes in some best possible way. Of paramount importance is an accurate determination of the parameters that define the best-fit curves or surfaces, as these conAnnual WIT report 2007 121 vey most relevant information to be extracted from the seismic data. Of course, due to the presence of noise, these tasks can be very difficult. Therefore, it is necessary to have some measure to decide whether some curve/surface fits the traveltimes. One possibility for such a measure is the degree of alignment or coherence of the seismic traces along the trial curves/surfaces. Semblance is a quantitative measure of coherence, typically used for event characterization in noisy data sets, for example seismic data. In a certain sense, semblance represents the energy of the stacked trace divided by the energy sum of all stacking traces within a given time window. Mathematically, semblance is defined as S̃2 = w ∑ k=−w ( N ∑ i=1 ui(tk) )2 N w ∑ k=−w ( N ∑ i=1 ui(tk) ) . (1) Here, the inner summation represents the traces (index i) along which the stack is performed; the outer summation performs the stack for various time samples (index k), that fall in a given time window of width 2w + 1. The window width should be related to the length of the signal wavelet of the event. The summation enhances the signal-to-noise ratio of the resulting stack. In order to study semblance as a statistical or optimization concept, it is convenient to disregard the time-window summation. In other words, we shall, for the moment, define the local semblance as the simpler expression S2 = ( N ∑ i=1 ui )2

Amplitude-preserving MZO in laterally inhomogeneous media

A new Kirchhoff-type true-amplitude migration to zero-offset (MZO) algorithm is proposed for 2.5-D common-offset reflections in 2-D laterally inhomogeneous layered isotropic earth models. It provides a transformation of a common- offset seismic section to a simulated zero-offset section and is thus closely related to a dip-moveout correction (DMO). The simulated primary zero-offset reflections, even from curved interfaces, have the best possible signal character, i.e., the geometrical-spreading factor of an original primary common-offset reflection is replaced by that of a correct zero-offset reflection. A single weighted stacking procedure needs to be performed only similar to the familiar Kirchhoff or diffraction-stack migration. Moreover, in analogy to true-amplitude Kirchhoff migration, the weight function can be computed by dynamic ray tracing in the macro-velocity model which is supposed to be available. As the simulated zero-offset reflection amplitudes are controlled by the zero-offset geometrical-spreading factor and the (angle-dependent) offset reflection coefficients, one can thus perform a post-MZO, but pre-migration AVO analysis. If compared to correct zero-offset reflections, the simulated ones turn out to be stretched (frequency shifted) by the cosine of the reflection angle.

Seismic imaging operators derived from chained stacking integrals

Given a 3D seismic record for an arbitrary measurement configuration and assuming a laterally inhomogeneous, isotropic macro-velocity model, a unifying approach to amplitude- preserving seismic reflection imaging is provided. It consists of (a) a Kirchhoff-type weighted diffraction stack to transform (migrate) the seismic data from the (time-domain) record space into the (depth-domain) image space, and of (b) a weighted isochrone stack to transform (demigrate) the migrated seismic image from the image space back into the record space. Both the diffraction and isochrone stacks can be applied in sequence for different measurement configurations, velocity models, or elementary waves to permit a variety of amplitude- preserving image transformations. These include, e.g., (a) the amplitude-preserving transformation of a 3D constant-offset record into a 3D zero-offset record, which is known as a migration to zero offset, (b) a dip-moveout correction, or (c) the transformation (here referred to as a remigration) of a 3D depth-migrated image directly in the image space into another one for a different macro-velocity model. By analytically chaining the two stacking integrals, each image transformation can be achieved with only one single weighted stack.

Amplitude-preserving migration by weighted diffraction stacks

An amplitude-preserving migration aims at imaging compressional primary (zero- or) nonzero- offset reflections into 3D time or depth-migrated reflections so that the migrated wavefield amplitudes are a measure of angle-dependent reflection coefficients. The principal issue in this attempt is the removal of the geometrical-spreading factor of the primary reflections. Using a 3D Kirchhoff-type prestack migration approach, also often called a diffraction-stack migration, where the primary reflections of the wavefields to be imaged are a priori described by the zero-order ray approximation, the aim of removing the geometrical-spreading loss is achieved by weighting the data before stacking them. Different weight functions can be applied that are independent of the unknown reflector. The true-amplitude weight function directly removes the spreading loss during migration. It also correctly accounts for the recovery of the source pulse in the migrated image irrespective of the employed source- receiver configurations and the caustics occurring in the wavefield.