# Seismic Applications of Acoustic Reciprocity

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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 system is dealt with in a similar fashion; however the spatial interaction needs a different approach. The reciprocity theorem handles this interaction by identifying one state with the spatial impulse function, also known as the Green's function, while the other state is connected with the actual source distribution. In general, the resulting integral representation is not a spatial convolution. Moreover, the systematic use of the reciprocity theorem leads to a hierarchical description of the seismic experiment in terms of increasing complexity. Also from an educational point of view this approach provides a hierarchy and the student learns to break down the seismic problem into constituent partial solutions.This book should contribute to the understanding that the reciprocity theorem is a powerful tool in the analysis of the seismic experiment.

## Table of Contents

- 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 power reciprocity. Redatuming of seismic data.**11. Deghosting.**Decomposition based on field reciprocity. Decomposition based on power reciprocity. The surface-related vertical particle velocity. Receiver deghosting. Source deghosting. Deghosting in the strip configuration.**12. Removal of Surface Related Wave Phenomena.**Reciprocity between the actual and desired state. Auxiliary reciprocity relations. Alignment of vertical positions. Actual multiple-removal procedure. Multiple removal in the strip configuration.**13. Boundary Imaging.**The boundary-integral representation. Fourier transform with respect to source coordinates. Fourier transform with respect to receiver coordinates. Angular-frequency-domain analysis. Imaging. Analytic solution for a planar reflector. Imaging of a disc. Two-dimensional case. Imaging of the strip configuration.**14. Domain Imaging.**The domain-integral representation. Fourier transform of source and receiver coordinates. Angular-frequency-domain analysis. Imaging. Two-dimensional case. Two-dimensional scattering by a circular cylinder. Imaging of the circular cylinder.**15. Seismic Inversion.**The domain-integral representation. Simultaneous minimization. Inversion algorithm. Bibliography. Index.

## Product details

- Language: English
- Copyright: © Elsevier Science 1993
- Published: January 27, 1993
- Imprint: Elsevier Science
- eBook ISBN: 9781483291208

## About the Authors

### J.T. Fokkema

#### Affiliations and Expertise

Department of Petroleum Engineering and Technical Geophysics, Faculty of Mining and Petroleum Engineering, Delft University of Technology, Delft, The Netherlands

### P.M. van den Berg

#### Affiliations and Expertise

Laboratory for Electromagnetic Research, Faculty of Electrical Engineering, Delft University of Technology, Delft, The Netherlands

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