Forward and Adjoint Modeling Using Green’s Functions: Forward modeling of the wave equation is defined as computing the propagating wavefields for a given source-receiver distribution, the source wavelet, and a velocity-density model. This chapter presents the forward-modeling approach with the Lippmann-Schwinger (LS) equation, a summation of weighted Green’s functions over the model coordinates, in which the weights are the reflection-like coeficients in the model. Another name for this is diffraction-stack modeling under the weak scattering approximation, otherwise known as the Born approximation.
Reverse Time Migration: In this chapter, we derive the equations for the reverse time migration (RTM) algorithm. This algorithm is used heavily by the oil industry for subsalt imaging of reflectors. It also is used to migrate residuals iteratively for both LSM and FWI. Compared to other migration methods such as diffraction-stack migration discussed in Chapter 10, RTM accounts for all the arrivals in the wavefield, including both primaries and multiples if the velocity model is suficiently accurate. This can lead to much better resolution in the image, but it is at the cost of being more sensitive to errors in the migration velocity model. For frequencies less than 20 Hz, RTM is the migration method of choice for imaging below complex geology such as salt bodies.
Figures & Tables
This book describes the theory and practice of inverting seismic data for the subsurface rock properties of the earth. The primary application is for inverting reflection and/or transmission data from engineering or exploration surveys, but the methods described also can be used for earthquake studies. I have written this book with the hope that it will be largely comprehensible to scientists and advanced students in engineering, earth sciences, and physics. It is desirable that the reader has some familiarity with certain aspects of numerical computation, such as finite-difference solutions to partial differential equations, numerical linear algebra, and the basic physics of wave propagation (e.g., Snell’s law and ray tracing). For those not familiar with the terminology and methods of seismic exploration, a brief introduction is provided in the Appendix of Chapter 1. Computational labs are provided for most of the chapters, and some field data labs are given as well. Matlab and Fortran labs at the end of some chapters are used to deepen the reader’s understanding of the concepts and their implementation. Such exercises are introduced early and geophysical applications are presented in every chapter. For the non-geophysicist, geophysical concepts are introduced with intuitive arguments, and their description by rigorous theory is deferred to later chapters.