# 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

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

We will consider the Helmholtz equation to be the governing equation of our problem. Because we are interested in recording at a specific position, ${\displaystyle {\boldsymbol {x}}_{s}}$ there is a second Helmholtz equation that is of interest

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

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 Green's function. In this case we have a self-adjoint problem.

### 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