We analyze the kinematic properties of offset-domain Common Image Gathers (CIGs) and Angle-Domain CIGs (ADCIGs) computed by wavefield-continuation migration. Our results are valid regardless of whether the CIGs were obtained by using the correct migration velocity. They thus can be used as a theoretical basis for developing Migration Velocity Analysis (MVA) methods that exploit the velocity information contained in ADCIGs.
We demonstrate that in an ADCIG cube the image point lies on the normal to the apparent reflector dip, passing through the point where the source ray intersects the receiver ray. Starting from this geometric result, we derive an analytical expression for the expected movements of the image points in ADCIGs as functions of the traveltime perturbation caused by velocity errors. By applying this analytical result and assuming stationary raypaths, we then derive two expressions for the Residual Moveout (RMO) function in ADCIGs. We verify our theoretical results and test the accuracy of the proposed RMO functions by analyzing the migration results of a synthetic data set with a wide range of reflector dips.
Our kinematic analysis leads also to the development of a new method for computing ADCIGs when significant geological dips cause strong artifacts in the ADCIGs computed by conventional methods. The proposed method is based on the computation of offset-domain CIGs along the vertical-offset axis (VOCIGs) and on the ``optimal'' combination of these new CIGs with conventional CIGs. We demonstrate the need for and the advantages of the proposed method on a real data set acquired in the North Sea.
With wavefield-continuation migration methods being used routinely for imaging project in complex areas, the ability to perform Migration Velocity Analysis (MVA) starting from the results of wavefield-continuation migration is becoming essential to advanced seismic imaging. As for Kirchhoff imaging, MVA for wavefield-continuation imaging is mostly based on the information provided by the analysis of Common Image Gather (CIGs). Most of the current MVA methods start from Angle-Domain CIGs (ADCIGs) Biondi and Sava (1999); Clapp and Biondi (2000); Liu et al. (2001); Mosher et al. (2001), though the use of more conventional surface-offset-domain CIGs is also being evaluated Stork et al. (2002).
Both kinematic and amplitude properties Sava et al. (2001); Wapenaar et al. (1999); de Bruin et al. (1990); de Hoop et al. (2002) have been analyzed in the literature for ADCIGs obtained when the migration velocity is accurate. On the contrary, the properties of the ADCIGs obtained when the migration velocity is inaccurate have been only qualitatively discussed in the literature. This lack of quantitative understanding may lead to errors when performing MVA from ADCIGs. In this paper, we analyze the kinematic properties of ADCIGs under general conditions (accurate or inaccurate velocity). If the migration velocity is inaccurate, our analysis requires only a smooth migration velocity function in the neighborhood of the imaging point. We discuss this condition more extensively in the first section. The application of the insights provided by our analysis may substantially improve the results of the following three procedures: a) measurement of velocity errors from ADCIGs by residual moveout (RMO) analysis, b) inversion of RMO measurements into velocity updates, and c) computation of ADCIGs in the presence of complex geologic structure.
Our analysis demonstrates that in an ADCIG cube the image point lies on the normal to the apparent reflector dip passing through the point where the source ray intersects the receiver ray. We exploit this result to define an analytical expression for the expected movements of the image points in ADCIGs as a function of the traveltime perturbation caused by velocity errors. This leads us to the definition of two alternative residual moveout functions that can be applied when measuring velocity errors from migrated images. We test the accuracy of these alternatives and discuss their relative advantages and disadvantages. Furthermore, the availability of a quantitative expression for the expected movements of the image points is crucial when inverting those movements into velocity corrections by either simple vertical updating or sophisticated tomographic methods. Therefore, our results ought to be incorporated in velocity updating methods.
Our theoretical result also implies that ADCIGs are immune, at least at first order, from the distortions caused by image-point dispersal. Image-point dispersal occurs when migration velocity errors cause events from the same segment of a dipping reflector to be imaged at different locations Etgen (1990). This inconsistency creates substantial problems when using dipping reflections for velocity updating; its absence makes ADCIGs even more attractive for MVA.
The computation of ADCIGs is based on a decomposition (usually performed by slant-stacks) of the wavefield either before imaging Mosher et al. (1997); Prucha et al. (1999); Xie and Wu (2002), or after imaging Biondi and Shan (2002); Rickett and Sava (2002); Sava and Fomel (2002). In either case, the slant stack transformation is usually applied along the horizontal subsurface-offset axis. However, when the geologic dips are steep, this ``conventional'' way of computing CIGs does not produce useful gathers, even if it is kinematically valid for geologic dips milder than 90 degrees. As the geologic dips increase, the horizontal-offset CIGs (HOCIGs) degenerate, and their focusing around zero offset blurs. This limitation of HOCIGs can be sidestepped by computing offset-domain CIGs along the vertical subsurface-offset axis (VOCIGs) Biondi and Shan (2002). Although neither set of offset-domain gathers (HOCIG or VOCIG) provides useful information for the whole range of geologic dips, an appropriate combination of the two sets does. Our analysis of the kinematic properties of ADCIGs suggests a simple and effective method for combining a HOCIG cube with a VOCIG cube to create an ADCIG cube that is immune to artifacts in the presence of arbitrary geologic dips.
The plan of attack for covering the broad, but interrelated, set of issues that are relevant to the use of ADCIGs for MVA is the following. We start by briefly reviewing the methodology for computing offset-domain and angle-domain CIGs by wavefield-continuation migration. The second section analyzes the kinematic properties of CIGs and ADCIGs, and contains the main theoretical development of the paper. The third section exploits the theoretical results to define a robust algorithm to compute ADCIGs in the presence of geological structure and illustrates its advantages with a real-data example. The fourth section verifies the theoretical analysis by using it to predict reflector movements in the migrated images of a synthetic data set. Finally, the fifth section derives two expressions for the RMO function to be applied for measuring velocity errors from migrated images.