# Seismic Applications of Acoustic Reciprocity

## 1st Edition

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The seismic applications of the reciprocity theorem developed in this book are partly based on lecture notes and publications from Professor de Hoop. Every student Professor de Hoop has taught knows the egg-shaped figure (affectionately known as "de Hoop's egg") that plays such an important role in his theoretical description of acoustic, electromagnetic and elastodynamic wave phenomena.

On the one hand this figure represents the domain for the application of a reciprocity theorem in the analysis of a wavefield and on the other hand it symbolizes the power of a consistent wavefield description of this theorem.

The roots of the reciprocity theorem lie in Green's theorem for Laplace's equation and Helmholtz's extension to the wave equation. In 1894, J.W. Strutt, who later became Lord Rayleigh, introduced in his book *The Theory of Sound* this extension under the name of Helmholtz's theorem. Nowadays it is known as Rayleigh's reciprocity theorem.

Progress in seismic data processing requires the knowledge of all the theoretical aspects of the acoustic wave theory. The reciprocity theorem was chosen as the central theme of this book as it constitutes the fundaments of the seismic wave theory. In essence, two states are distinguished in this theorem. These can be completely different, although sharing the same time-invariant domain of application, and they are related via an interaction quantity. The particular choice of the two states determines the acoustic application, in turn making it possible to formulate the seismic experiment in terms of a geological system response to a known source function.

In linear system theory, it is well known that the response to a known input function can be written as an integral representation where the impulse response acts as a kernel and operates on the input function. Due to the temporal invariance of the system, this integral representation is of the convolution type. In seismics, the temporal behaviour of the syste

Preface. Introduction. **1. Integral Transformations.** Cartesian vectors. Integral-transformation methods. Discrete Fourier-transformation methods. **2. Iterative Solution of Integral Equations.** The integral equation. Direct minimization of the error. Recursive minimization of the error. Selfadjoint operator LT. The Neumann expansion. Special choices of the operator T. Operators of convolution type. **3. Basic Equations in Acoustics.** The acoustic wave equations. The acoustic equations in the Laplace-transform domain. **4. Radiation in an Unbounded, Homogeneous Medium.** Source representations in the spectral domain. Source representations in the *s*-domain. Far-field radiation characteristics in the *s*-domain. Source representations in the time domain. Far-field characteristics in the time domain. The Cagniard-de Hoop method. The acoustic wavefield of point sources. **5. Reciprocity Theorems.** The *s*-domain field reciprocity theorem. The time-domain reciprocity theorem of convolution type. The *s*-domain power reciprocity theorem. The time-domain reciprocity theorem of correlation type. **6. Field Reciprocity between Transmitter and Receiver.** Point-transducer description. Volume-transducer description. Surface-transducer description. **7. Radiation in an Unbounded, Inhomogeneous Medium.** The volume-source problem. The surface-source problem. **8. Scattering by a Bounded Contrasting Domain.** The domain-integral equation formulation. The boundary-integral equation formulation. **9. Scattering by Disk.** Scattering by a planar object of vanishing thickness. Disk in a homogeneous embedding. Analytic solution for a pressure-free plane. Disk in a homogeneous halfspace. Two-dimensional scattering by a strip. **10. Wavefield Decomposition.** Decomposition based on field reciprocity. Decomposition based on p

- Language:
- English

- Copyright:
- © 1993

- Published:
- 27th January 1993

- Imprint:
- Elsevier Science

- Electronic ISBN:
- 9781483291208

- Print ISBN:
- 9780444890443

@qu:The authors have done a good job of describing increasingly complex wave problems, starting from simple cases and systematically applying the principle of dynamic reciprocity. The book is recommended for libraries and research workers in the field.
@source:Pure and Applied Geophysics