# Difference between revisions of "Born-approximate modeling formula"

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

${\displaystyle \left[\nabla ^{2}-{\frac {1}{v^{2}({\boldsymbol {x}})}}{\frac {\partial ^{2}}{\partial t^{2}}}\right]U({\boldsymbol {x}},{\boldsymbol {x}}_{s};t)=-f(t)\delta ({\boldsymbol {x}}-{\boldsymbol {x}}_{s})}$.

Here, ${\displaystyle {\boldsymbol {x}}\equiv (x_{1},x_{2},x_{3})}$ is general position in the medium, ${\displaystyle {\boldsymbol {x}}_{s}\equiv (x_{s1},x_{s2},x_{s3})}$ is the source position, ${\displaystyle t}$ is general time , ${\displaystyle f(t)}$ is the time history of the source, ${\displaystyle \nabla \equiv (\partial /\partial x_{1},\partial /\partial x_{2},\partial /\partial x_{3})}$, ${\displaystyle v({\boldsymbol {x}})}$ is the wavespeed of the medium, and ${\displaystyle U({\boldsymbol {x}},{\boldsymbol {x}}_{s};t,t_{0})}$ is the wavefield due to a source located at ${\displaystyle {\boldsymbol {x}}={\boldsymbol {x}}_{s}}$ initiated at time ${\displaystyle t=0}$.

Applying the forward Fourier transform in time to the scalar wave equation yields the scalar Helmholtz equation

${\displaystyle \left[\nabla ^{2}+{\frac {\omega ^{2}}{v^{2}({\boldsymbol {x}})}}\right]u({\boldsymbol {x}},{\boldsymbol {x}}_{s};\omega )=-F(\omega )\delta ({\boldsymbol {x}}-{\boldsymbol {x}}_{s})}$.

### Perturbation theory

We assume that the medium consists of a volume ${\displaystyle V}$ enclosed in a surface ${\displaystyle S}$. 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 ${\displaystyle c({\boldsymbol {x}})}$, plus a scatterer, represented by a perturbation ${\displaystyle \alpha ({\boldsymbol {x}})}$ which is a deviation from the background velocity model.

One way of representing this that preserves the form of the Helmholtz equation is

${\displaystyle {\frac {1}{v^{2}({\boldsymbol {x}})}}\equiv {\frac {1}{c^{2}({\boldsymbol {x}})}}\left[1+\alpha ({\boldsymbol {x}})\right]}$.

Correspondingly, we consider that the wavefield is similarly decomposable into an incident wavefield ${\displaystyle u_{I}}$, which is the field in the absence of the scatterer, plus the scattered field ${\displaystyle u_{S}}$, such that

${\displaystyle u({\boldsymbol {x}},{\boldsymbol {x}}_{s},\omega )\equiv u_{I}({\boldsymbol {x}},{\boldsymbol {x}}_{s},\omega )+u_{S}({\boldsymbol {x}},{\boldsymbol {x}}_{s},\omega )}$.

Substituting these items into the Helmholtz equation, we obtain

${\displaystyle \left[\nabla ^{2}+{\frac {\omega ^{2}}{c^{2}({\boldsymbol {x}})}}\left[1+\alpha ({\boldsymbol {x}})\right]\right]\left[u_{I}({\boldsymbol {x}},{\boldsymbol {x}}_{s},\omega )+u_{S}({\boldsymbol {x}},{\boldsymbol {x}}_{s},\omega )\right]=-F(\omega )\delta ({\boldsymbol {x}}-{\boldsymbol {x}}_{s})}$.

We can make sense of this by rearranging the terms

${\displaystyle \left[\nabla ^{2}+{\frac {\omega ^{2}}{c^{2}({\boldsymbol {x}})}}\right]\left[u_{I}({\boldsymbol {x}},{\boldsymbol {x}}_{s},\omega )+u_{S}({\boldsymbol {x}},{\boldsymbol {x}}_{s},\omega )\right]=-F(\omega )\delta ({\boldsymbol {x}}-{\boldsymbol {x}}_{s})-{\frac {\omega ^{2}\alpha ({\boldsymbol {x}})}{c^{2}({\boldsymbol {x}})}}\left[u_{I}({\boldsymbol {x}},{\boldsymbol {x}}_{s},\omega )+u_{S}({\boldsymbol {x}},{\boldsymbol {x}}_{s},\omega )\right]}$.

### The incident (or background) field

Given the way that we have stated the problem, the incident field ${\displaystyle u_{I}({\boldsymbol {x}},{\boldsymbol {x}}_{s},\omega )}$ is described by the Helmholtz equation written in terms of the background wavespeed profile, with a source term

${\displaystyle \left[\nabla ^{2}+{\frac {\omega ^{2}}{c^{2}({\boldsymbol {x}})}}\right]u_{I}({\boldsymbol {x}},{\boldsymbol {x}}_{s},\omega )=-F(\omega )\delta ({\boldsymbol {x}}-{\boldsymbol {x}}_{s})}$.

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 ${\displaystyle F(\omega )}$ such that

${\displaystyle u_{I}({\boldsymbol {x}},{\boldsymbol {x}}_{s},\omega )=-F(\omega )g({\boldsymbol {x}},{\boldsymbol {x}}_{s},\omega ).}$

### The scattered (or perturbation) field

The scattered field ${\displaystyle u_{S}({\boldsymbol {x}},{\boldsymbol {x}}_{s},\omega )}$ 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 ${\displaystyle \alpha ({\boldsymbol {x}})}$

${\displaystyle {\cal {L}}u_{S}({\boldsymbol {x}},{\boldsymbol {x}}_{s},\omega )\equiv \left[\nabla ^{2}+{\frac {\omega ^{2}}{c^{2}({\boldsymbol {x}})}}\right]u_{S}({\boldsymbol {x}},{\boldsymbol {x}}_{s},\omega )=-{\frac {\omega ^{2}\alpha ({\boldsymbol {x}})}{c^{2}({\boldsymbol {x}})}}\left[u_{I}({\boldsymbol {x}},{\boldsymbol {x}}_{s},\omega )+u_{S}({\boldsymbol {x}},{\boldsymbol {x}}_{s},\omega )\right]}$.

The problem that we want to solve, is to create an integral equation representation of the scattered field for a specific field location ${\displaystyle {\boldsymbol {x}}\equiv {\boldsymbol {x}}_{g}}$. 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

${\displaystyle {\cal {L}}^{\star }g^{\star }({\boldsymbol {x}},{\boldsymbol {x}}_{g};\omega )\equiv \left[\nabla ^{2}+{\frac {\omega ^{2}}{v^{2}({\boldsymbol {x}})}}\right]g^{\star }({\boldsymbol {x}},{\boldsymbol {x}}_{g};\omega )=-\delta ({\boldsymbol {x}}-{\boldsymbol {x}}_{g})}$.

Here ${\displaystyle g^{\star }({\boldsymbol {x}},{\boldsymbol {x}}_{g};\omega )}$ is the Green's function of the medium. The superscript ${\displaystyle \star }$ indicates that these are formally the adjoint operator and respective adjoint Green's function. In this simple problem, called self-adjoint ${\displaystyle {\cal {L}}^{\star }={\cal {L}}}$.

### Applying Green's theorem

Our formal problem is to create an integral equation for the scattered field, given the two forms of the Helmholtz equation given by

${\displaystyle {\cal {L}}u_{S}({\boldsymbol {x}},{\boldsymbol {x}}_{s},\omega )=-{\frac {\omega ^{2}\alpha ({\boldsymbol {x}})}{c^{2}({\boldsymbol {x}})}}\left[u_{I}({\boldsymbol {x}},{\boldsymbol {x}}_{s},\omega )+u_{S}({\boldsymbol {x}},{\boldsymbol {x}}_{s},\omega )\right]}$

and

${\displaystyle {\cal {L}}^{\star }g^{\star }({\boldsymbol {x}},{\boldsymbol {x}}_{g};\omega )=-\delta ({\boldsymbol {x}}-{\boldsymbol {x}}_{g})}$.

In this case, Green's theorem may be stated as

${\displaystyle \int _{V}\left[g^{\star }({\boldsymbol {x}},{\boldsymbol {x}}_{g};\omega ){\cal {L}}u_{S}({\boldsymbol {x}},{\boldsymbol {x}}_{s},\omega )-u_{S}({\boldsymbol {x}},{\boldsymbol {x}}_{s},\omega ){\cal {L}}^{\star }g^{\star }({\boldsymbol {x}},{\boldsymbol {x}}_{g};\omega )\right]\;dV=\int _{S}\left[g^{\star }{\frac {\partial u_{S}}{\partial {\boldsymbol {\hat {n}}}}}-u_{S}{\frac {\partial g^{\star }}{\partial {\boldsymbol {\hat {n}}}}}\right]}$.

The problem is further simplified by considering it to be unbounded, which means that the boundary ${\displaystyle S}$ 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 made

${\displaystyle u_{S}({\boldsymbol {x}}_{g},{\boldsymbol {x}}_{s},\omega )=\int _{V}g^{\star }({\boldsymbol {x}},{\boldsymbol {x}}_{g};\omega ){\frac {\omega ^{2}\alpha ({\boldsymbol {x}})}{c^{2}({\boldsymbol {x}})}}\left[u_{I}({\boldsymbol {x}},{\boldsymbol {x}}_{s},\omega )+u_{S}({\boldsymbol {x}},{\boldsymbol {x}}_{s},\omega )\right]\;dV}$.

### 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 ${\displaystyle u_{S}}$ 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 ${\displaystyle u_{S}}$ and the perturbation ${\displaystyle \alpha }$, will then be lower order than the incident and scattered field. This is called the Born approximation.

Making the Born approximation and recognizing that ${\displaystyle u_{I}=F(\omega )g}$, and recognizing that for the unbounded media problem, the Reciprocity theorem takes the form ${\displaystyle g^{\star }({\boldsymbol {x}},{\boldsymbol {x}}_{g},\omega )\equiv g({\boldsymbol {x}}_{g},{\boldsymbol {x}},\omega )}$

${\displaystyle u_{S}({\boldsymbol {x}}_{g},{\boldsymbol {x}}_{s},\omega )\approx F(\omega )\omega ^{2}\int _{V}{\frac {\alpha ({\boldsymbol {x}})}{c^{2}({\boldsymbol {x}})}}g({\boldsymbol {x}}_{g},{\boldsymbol {x}},\omega )g({\boldsymbol {x}},{\boldsymbol {x}}_{s},\omega )\;dV}$.