Kirchhoff-Type Integrals

Among the most important types of geophysical fields are elastic wave fields or simply wave fields. These fields are studied in seismology and in seismic prospecting and they yield extremely significant information about the deep structure of the Earth which could not be gained using other geophysical methods. In view of this, the problem of extending the body of Cauchy-type integrals to wave fields is of critical value in theoretical geophysics. In this chapter we will derive the classical Kirchhoff integral and a generalized Kirchhoff integral formula for the elastic displacement vector. The Kirchhoff-type integrals introduced on the basis of the above integral formulas are a natural extension of the Cauchy-type integrals to wave fields. They can be used to develop the theory of seismic field migration which presently underlies dynamic seismology.