# 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.[1]

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

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

"A volume ${\displaystyle V}$ bounded by a surface ${\displaystyle S}$, with outward pointing unit normal vector ${\displaystyle {\boldsymbol {\hat {n}}}}$. General position in the volume is given by ${\displaystyle {\boldsymbol {x}}\equiv (x_{1},x_{2},x_{3})}$. The source position is ${\displaystyle {\boldsymbol {x}}_{s}\equiv (x_{s1},x_{s2},x_{s3})}$. The receiver group (or geophone) location is ${\displaystyle {\boldsymbol {x}}_{g}\equiv (x_{g1},x_{g2},x_{g3})}$."

### 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

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

Here, the normal derivative is given by ${\displaystyle {\frac {\partial }{\partial {\boldsymbol {\hat {n}}}}}\equiv {\boldsymbol {\hat {n}}}\cdot \nabla }$, where ${\displaystyle {\boldsymbol {\hat {n}}}}$ 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 ${\displaystyle S}$ 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 ${\displaystyle g^{\star }({\boldsymbol {x}},{\boldsymbol {x}}_{g},\omega )\equiv g({\boldsymbol {x}}_{g},{\boldsymbol {x}},\omega )}$

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

## 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 ${\displaystyle c({\boldsymbol {x}})=c_{0}}$ then we have a closed form representation of the Green's functions. In three dimensions these are

${\displaystyle g({\boldsymbol {x}},{\boldsymbol {x}}_{s},\omega )={\frac {e^{i\omega r_{s}/c_{0}}}{4\pi r_{s}}}}$

where ${\displaystyle r_{s}=|{\boldsymbol {x}}-{\boldsymbol {x}}_{s}|={\sqrt {(x_{1}-x_{s1})^{2}+(x_{2}-x_{s2})^{2}+(x-x_{s3})^{2}}}}$

and

${\displaystyle g({\boldsymbol {x}}_{g},{\boldsymbol {x}},\omega )={\frac {e^{i\omega r_{g}/c_{0}}}{4\pi r_{g}}}}$

where ${\displaystyle r_{g}=|{\boldsymbol {x}}_{g}-{\boldsymbol {x}}|={\sqrt {(x_{g1}-x_{1})^{2}+(x_{g2}-x_{2})^{2}+(x_{g3}-x_{3})^{2}}}}$.

Substituting these into the Born-approximate modeling formula yields

${\displaystyle u_{S}({\boldsymbol {x}}_{g},{\boldsymbol {x}}_{s},\omega )\approx F(\omega ){\frac {\omega ^{2}}{16\pi ^{2}}}\int _{V}{\frac {\alpha ({\boldsymbol {x}})}{c^{2}({\boldsymbol {x}})}}{\frac {e^{i\omega (r_{s}+r_{g})/c_{0}}}{r_{s}r_{g}}}\;dV}$

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 ${\displaystyle c({\boldsymbol {x}})}$. Under this high frequency or large wavenumber approximation, the WKBJ or ray theoretic Green's functions may be assumed. These are

${\displaystyle g_{s}^{WKBJ}({\boldsymbol {x}},{\boldsymbol {x}}_{s},\omega )\sim A({\boldsymbol {x}},{\boldsymbol {x}}_{s})e^{i\omega \tau ({\boldsymbol {x}},{\boldsymbol {x}}_{s})}+O(\omega ^{-1})}$

as ${\displaystyle \omega \rightarrow \infty }$

${\displaystyle g_{g}^{WKBJ}({\boldsymbol {x}}_{g},{\boldsymbol {x}},\omega )\sim A({\boldsymbol {x}}_{g},{\boldsymbol {x}})e^{i\omega \tau ({\boldsymbol {x}}_{g},{\boldsymbol {x}})}+O(\omega ^{-1})}$

as ${\displaystyle \omega \rightarrow \infty }$ .

Here ${\displaystyle \tau ({\boldsymbol {x}},{\boldsymbol {x}}_{s})}$ and ${\displaystyle \tau ({\boldsymbol {x}}_{g},{\boldsymbol {x}})}$ are the respective travel times from the source and receiver positions to a general position in the medium. The ${\displaystyle A({\boldsymbol {x}},{\boldsymbol {x}}_{s})}$ and ${\displaystyle A({\boldsymbol {x}}_{g},{\boldsymbol {x}})}$ are the respective ray-theoretic amplitudes.

Substituting these quantities into the Born-approximate modeling formula yields

${\displaystyle u_{S}({\boldsymbol {x}}_{g},{\boldsymbol {x}}_{s},\omega )\approx F(\omega )\omega ^{2}\int _{V}{\frac {\alpha ({\boldsymbol {x}})}{c^{2}({\boldsymbol {x}})}}A({\boldsymbol {x}},{\boldsymbol {x}}_{s})A({\boldsymbol {x}}_{g},{\boldsymbol {x}})e^{i\omega (\tau ({\boldsymbol {x}},{\boldsymbol {x}}_{s})+\tau ({\boldsymbol {x}}_{g},{\boldsymbol {x}}))}\;dV+O(\omega )}$

as ${\displaystyle \omega \rightarrow \infty }$, which is the WKBJ Born-approximate modeling formula.

## Other approximations

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).[2]

## References

1. Bleistein, N. (1984). Mathematical methods for wave phenomena. Academic Press.
2. Goodman, J. W. (2005). Introduction to Fourier optics. Roberts and Company Publishers.
3. Whaley, J., 2017, Oil in the Heart of South America, https://www.geoexpro.com/articles/2017/10/oil-in-the-heart-of-south-america], accessed November 15, 2021.
4. Wiens, F., 1995, Phanerozoic Tectonics and Sedimentation of The Chaco Basin, Paraguay. Its Hydrocarbon Potential: Geoconsultores, 2-27, accessed November 15, 2021; https://www.researchgate.net/publication/281348744_Phanerozoic_tectonics_and_sedimentation_in_the_Chaco_Basin_of_Paraguay_with_comments_on_hydrocarbon_potential
5. Alfredo, Carlos, and Clebsch Kuhn. “The Geological Evolution of the Paraguayan Chaco.” TTU DSpace Home. Texas Tech University, August 1, 1991. https://ttu-ir.tdl.org/handle/2346/9214?show=full.