Chapter 9: Seismic Imaging
In this chapter, we describe quantitatively the actual asymptotic inverse process for diffraction-stack migration – namely, isochron-stack demigration. We show that the diffraction-stack integral (DSI) and the isochron-stack integral (ISI) constitute an asymptotic transform pair that is interlinked well by the duality theorems that were derived in Chapter 5. This transform pair can be used to solve a multitude of true-amplitude target-oriented seismic-imaging (or image-transformation) problems, including, e.g., the dynamic counterparts of the kinematic map-transformation examples that Chapter 2 discussed qualitatively. All image-transformation problems can be addressed by applying the two stacking integrals in sequence, whereby the macrovelocity model, the measurement configuration, or the ray code of the considered elementary reflections may change from step to step. Alternatively, the two stacking procedures can be combined mathematically into a single process, which leads to weighted (Kirchhoff-type) summations along certain stacking surfaces (or inplanats). This chapter provides the general formulas for the stacking surfaces and the true-amplitude weights of these new Kirchhoff processes. To demonstrate the value of the proposed imaging theory, which is based on analytically chaining the two stacking integrals, we solve the true-amplitude configuration-transform and remigration problems for the case of a 3D isotropic, laterally inhomogeneous medium.
In this important chapter of our book, we present the theoretical background for the geometrically motivated mapping and imaging concepts that were discussed in Chapter 2. The diffraction-stack theory, as presented in Schleicher et al. (1993a) and discussed in detail in Chapter 7 of this book, serves as our point of departure for presenting the theory of true-amplitude isochron-stack demigration along very similar lines. These Kirchhoff-type migration and demigration operations provide the basis for the unified approach to seismic-reflection imaging (Hubral et al., 1996a; Tygel et al., 1996) that is the central subject of this book. In the last section of this chapter, we show how to chain the DSI and the ISI to solve various seismic-image-transformation problems. As in the papers just cited, we present this unified theory here in the time domain. Bleistein and Jaramillo (2000) and Bleistein et al. (2001) discuss an equivalent frequency-domain theory.