The applicability of dip moveout/azimuth moveout in the presence of caustics

Reflection seismic data continuation is the computation of data at source and receiver locations that differ from those in the original data, using whatever data are available. We develop a general theory of data continuation in the presence of caustics and illustrate it with three examples: dip moveout (DMO), azimuth moveout (AMO), and offset continuation. This theory does not require knowledge of the reflector positions. We construct the output data set from the input through the composition of three operators: an imaging operator, a modeling operator, and a restriction operator. This results in a single operator that maps directly from the input data to the desired output data. We use the calculus of Fourier integral operators to develop this theory in the presence of caustics. For both DMO and AMO, we compute impulse responses in a constant-velocity model and in a more complicated model in which caustics arise. This analysis reveals errors that can be introduced by assuming, for example, a model with a constant vertical velocity gradient when the true model is laterally heterogeneous. Data continuation uses as input a subset (common offset, common angle) of the available data, which may introduce artifacts in the continued data. One could suppress these artifacts by stacking over a neighborhood of input data (using a small range of offsets or angles, for example). We test data continuation on synthetic data from a model known to generate imaging artifacts. We show that stacking over input scattering angles suppresses artifacts in the continued data. Manuscript received by the Editor February 17, 2003; revised manuscript received July 9, 2004; published online January 14, 2005. 1Colorado School of Mines, Center for Wave Phenomena, Department of Geophysics, 1500 Illinois Street, Golden, Colorado 80401. E-mail: [email protected]. 2Colorado School of Mines, Center for Wave Phenomena, Department of Mathematics and Computer Science, 1500 Illinois Street, Golden, Colorado 80401. E-mail: [email protected]. 3Formerly Colorado School of Mines, Center for Wave Phenomena, Golden, Colorado: presently Université de Provence, Laboratoire d’Analyse, Topologie et Probabilités-CNRS UMR6632, Centre de Mathématiques et Informatique, 39 rue F. Joliot-Curie 13453, Marseille cedex 13, France. E-mail: [email protected]. c © 2005 Society of Exploration Geophysicists. All rights reserved. INTRODUCTION Data collected in the field are often not ideal for processing. For example, zero-offset data are important in seismic data processing, but limitations preclude collecting such data in the field. In general, we refer to methods to remedy this problem as data continuation or data mapping. Stolt (2002) gives an excellent description of why data continuation is necessary as well as a theory for performing data mapping with a constantbackground-velocity model. Patch (2002) gives an example of data continuation in medical imaging. We introduce a theoretical tool to analyze data continuation in the presence of caustics, focusing on the particular examples of dip moveout (DMO) and azimuth moveout (AMO). The mathematical formulation of this tool is given in de Hoop et al. (2003b) and de Hoop and Uhlmann (personal communication, 2004); the purpose of our paper is to discuss its interpretation, application, and computation. In comparison with previous work, we pay special attention to the case in which caustics are present in the wavefield. In this case, the operator becomes, locally, significantly more complicated. The region where the complication occurs depends on the lateral heterogeneity of the velocity model used. Unlike traditional DMO and AMO operators, our operator changes along a profile; thus, the computation becomes significantly more complex in regions with complicated velocity structure. The practical value of this tool comes in constructing the near-offset data in regions where the velocity model is complicated. We use the term data continuation to describe any act of computing data that have not been collected in the field. Early examples of data continuation using a partial differential equation can be found in Goldin (1994) and Goldin and Fomel (1995). Azimuth moveout is a special case of data