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
bounded by a surface
, with outward pointing unit normal vector
. General position in the volume is given by
. The source position is
. The receiver group (or geophone) location is
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
The incident (or background) field
Given 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
In fact, we can recognize that the incident field is nothing more than the Green's function of the
background Helmholtz equation scaled by the Fourier transform of the time history of the source
The scattered (or perturbation) field
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
The problem that we want to solve, is to create an integral equation representation of the scattered field
for a specific field location . Here can stand for
receiver group or geophone.
To this end, we require a second Helmholtz equation written in terms of this receiver group position
Here is the Green's function of the medium.
The superscript indicates that these are formally the adjoint operator and respective
adjoint Green's function. In this simple problem, called self-adjoint .
Our formal problem is to create an integral equation for the scattered field, given the two forms of the
Helmholtz equation given by
In this case, Green's theorem may be stated as
Here, the normal derivative is given by , where is the outward
directed normal vector in this derivation.
The problem is further simplified by considering it to be unbounded, which means that the boundary
is at infinity. By the Sommerfeld radiation condition, the surface integral term
vanishes leaving the integral equation for the scattered field, when the appropriate substitutions are
The Lippmann-Schwinger equation
As is, this result is general, and represents a series of composed of terms with iterated integral operators.
This follows because the on the right hand side is, itself, representable by an integral
equation having the same form as the equation above. Physicists call this the LIppmann-Schwinger equation, though to be formally correct, the LS equation is usually written with the Schroedinger equation as the
governing equation of the problem.
The Born approximation and the Born-approximate modeling formula
We can make a simple approximation that yields a useful result called the Born approximate modeling formula.
If the strength of scattering yields a field that is smaller than the incident field, then products of
the scattered field and the perturbation , will then be lower order
than the incident and scattered field. This is called the Born approximation.
Making the Born approximation and recognizing that , and recognizing
that for the unbounded media problem, the Reciprocity theorem takes the form
Choices of Green's function
The Born-approximate modeling formula is useful only if Green's functions can be supplied.
Constant background, Born-approximate modeling formula
If the background wavespeed is a constant then we have a
closed form representation of the Green's functions. In three dimensions these are
Substituting these into the Born-approximate modeling formula yields
the constant background Born-approximate modeling formula .
The WKBJ Born-approximate Green's function
Another popular approximation is to assume that the wavelengths are short with respect to the natural lengthscales of the problem, that the background wavespeed varies with position .
Under this high frequency or large wavenumber approximation, the WKBJ or ray theoretic Green's functions may be assumed. These are
Here and are the respective travel times from the source and receiver positions to a general position in
the medium. The and are the respective ray-theoretic amplitudes.
Substituting these quantities into the Born-approximate modeling formula yields
as , which is the WKBJ Born-approximate modeling formula.
Numerous other approximations are possible given the choice of Green's function, and approximations
of the distance function. These include the Fresnel approximation and the Fraunhoffer approximation.
See Goodman (2005).
- ↑ Bleistein, N. (1984). Mathematical methods for wave phenomena. Academic Press.
- ↑ Goodman, J. W. (2005). Introduction to Fourier optics. Roberts and Company Publishers.