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

(→Perturbation theory) |
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One way of representing this that preserves the form of the Helmholtz equation is | One way of representing this that preserves the form of the Helmholtz equation is | ||

− | <math> \frac{1}{v^2(\boldsymbol{x}) } \equiv \frac{1}{c^2(\boldsymbol{x}) }\left[ 1 + \alpha(\ | + | <math> \frac{1}{v^2(\boldsymbol{x}) } \equiv \frac{1}{c^2(\boldsymbol{x}) }\left[ 1 + \alpha(\boldsymbol{x}) \right] </math>. |

Correspondingly, we consider that the wavefield is similarly decomposable into an ''incident wavefield'' | Correspondingly, we consider that the wavefield is similarly decomposable into an ''incident wavefield'' | ||

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<math> u_S </math>, such that | <math> u_S </math>, such that | ||

− | <math> u(\boldsymbol{x},\boldsymbol{x}_s, \omega) \equiv | + | <math> u(\boldsymbol{x},\boldsymbol{x}_s, \omega) \equiv u_I(\boldsymbol{x},\boldsymbol{x}_s, \omega) + u_S(\boldsymbol{x},\boldsymbol{x}_s, \omega) </math>. |

Substituting these items into the Helmholtz equation, we obtain | Substituting these items into the Helmholtz equation, we obtain |

## Revision as of 21:41, 30 December 2020

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

.

We will consider the Helmholtz equation to be the governing equation of our problem. Because we are interested in recording at a specific position, there is a second Helmholtz equation that is of interest

.

Here is the Green's function of the medium. The superscript 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 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