It is often useful to construct integral equations as modeling formulas. One method of creating such
an integral equation representation is the application of Green's theorem to a wave equation. A general integral equation formalism may be obtained using the notion of Scattering theory. That is, we assume
that the medium may be decomposed into a known background wavespeed profile plus a perturbation called the scatterer. The wavefield, similarly may be decomposed into an background wavefield, also called the reference or the incident field, plus a perturbation field also called the scatterer.
The scatterer may be thought of as a volume scatter, or a surface scatterer. In this formalism we will
consider the perturbation in the wavespeed profile to be a volume scatterer. We will also consider wave
propagation to be governed by the scalar wave equation.
The scalar wave equation and the scalar Helmholtz equation
The scalar wave equation is given by
Here, is general position in the medium, is the source position, is general time , is the time history
of the source, , is the wavespeed of the medium, and is the wavefield due to a source
located at initiated at time .
Applying the forward Fourier transform in time to the scalar wave equation yields the scalar Helmholtz equation
We assume that the medium consists of a volume enclosed in a surface . For
an unbounded medium we will allow this boundary surface to be at infinity. We further consider that the
medium suggests of a background or incident model, represented by the velocity function , plus a scatterer, represented by a perturbation which is a deviation from the background velocity model.
One way of representing this that preserves the form of the Helmholtz equation is
Correspondingly, we consider that the wavefield is similarly decomposable into an incident wavefield
, which is the field in the absence of the scatterer, plus the scattered field
, such that
Substituting these items into the Helmholtz equation, we obtain
We can make sense of this by rearranging the terms
Give the way that we have stated the problem, the incident field is described by the Helmholtz equation written in terms of the background wavespeed profile,
with a source term
The scattered field is represented by the
same background Helmholtz operator with background wavespeed, but with a source function composed of
the interaction of the incident and scattered fields and the scatterer represented by the perturbation
We will consider the Helmholtz equation to be the governing equation of our problem. Because we are interested
in recording at a specific position, there is a second Helmholtz equation that
is of interest
Here is the Green's function of the medium.
The superscript indicates that these are formally the adjoint operator and respective
Green's function. In this case we have a self-adjoint problem.