# 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

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

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

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

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

### Perturbation theory

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

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

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

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

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

Substituting these items into the Helmholtz equation, we obtain

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

We can make sense of this by rearranging the terms

$\left[\nabla ^{2}+{\frac {\omega ^{2}}{c^{2}({\boldsymbol {x}})}}\right]\left[u_{I}({\boldsymbol {x}},{\boldsymbol {x}}_{s},\omega )+u_{S}({\boldsymbol {x}},{\boldsymbol {x}}_{s},\omega )\right]=-F(\omega )\delta ({\boldsymbol {x}}-{\boldsymbol {x}}_{s})-{\frac {\omega ^{2}\alpha ({\boldsymbol {x}})}{c^{2}({\boldsymbol {x}})}}\left[u_{I}({\boldsymbol {x}},{\boldsymbol {x}}_{s},\omega )+u_{S}({\boldsymbol {x}},{\boldsymbol {x}}_{s},\omega )\right]$ Give the way that we have stated the problem, the incident field $u_{I}({\boldsymbol {x}},{\boldsymbol {x}}_{s},\omega )$ is described by the Helmholtz equation written in terms of the background wavespeed profile, with a source term

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

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

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

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

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