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

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We can make sense of this by rearranging the terms | We can make sense of this by rearranging the terms | ||

− | <math> \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] </math> | + | <math> \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] </math>. |

− | + | === The incident (or background) field === | |

+ | Given the way that we have stated the problem, the incident field <math> u_I(\boldsymbol{x},\boldsymbol{x}_s, \omega) </math> is described by the Helmholtz equation written in terms of the background wavespeed profile, | ||

with a source term | with a source term | ||

<math>\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) </math>. | <math>\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) </math>. | ||

+ | |||

+ | 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 | ||

+ | <math> F(\omega) </math> such that | ||

+ | |||

+ | <math> u_I(\boldsymbol{x},\boldsymbol{x}_s, \omega) = -F(\omega) g(\boldsymbol{x},\boldsymbol{x}_s, \omega). </math> | ||

+ | |||

+ | === The scattered (or perturbation) field === | ||

The scattered field <math> u_S(\boldsymbol{x},\boldsymbol{x}_s, \omega) </math> is represented by the | The scattered field <math> u_S(\boldsymbol{x},\boldsymbol{x}_s, \omega) </math> is represented by the | ||

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<math> \alpha(\boldsymbol{x}) </math> | <math> \alpha(\boldsymbol{x}) </math> | ||

− | <math> \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] </math>. | + | <math> {\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] </math>. |

+ | |||

+ | The problem that we want to solve, is to create an integral equation representation of the scattered field | ||

+ | for a specific field location <math> \boldsymbol{x} \equiv \boldsymbol{x}_g </math>. 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 | ||

+ | |||

+ | <math> {\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) </math>. | ||

+ | |||

+ | Here <math> g^\star(\boldsymbol{x}, \boldsymbol{x}_g; \omega ) </math> is the Green's function of the medium. | ||

+ | The superscript <math> \star </math> indicates that these are formally the [[adjoint]] operator and respective | ||

+ | adjoint Green's function. In this simple problem, called ''self-adjoint'' <math> {\cal L}^\star = {\cal L} </math>. | ||

+ | |||

+ | === 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 | ||

+ | |||

+ | <math> {\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] </math> | ||

+ | |||

+ | and | ||

+ | |||

+ | <math> {\cal L}^\star g^\star(\boldsymbol{x}, \boldsymbol{x}_g; \omega ) = -\delta(\boldsymbol{x} - \boldsymbol{x}_g) </math>. | ||

+ | |||

+ | In this case, [[Green's theorem]] may be stated as | ||

+ | |||

+ | <math> \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]</math>. | ||

+ | The problem is further simplified by considering it to be ''unbounded,'' which means that the boundary | ||

+ | <math> S </math> 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 | ||

+ | <math> 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 </math>. | ||

+ | === 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 <math> u_S </math> 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 <math> u_S </math> and the perturbation <math> \alpha </math>, will then be lower order | |

− | + | than the incident and scattered field. This is called the Born approximation. | |

+ | |||

+ | Making the Born approximation and recognizing that <math> u_I = F(\omega) g </math>, and recognizing | ||

+ | that for the unbounded media problem, the [[Reciprocity theorem]] takes the form <math> g^\star(\boldsymbol{x}, \boldsymbol{x}_g, \omega ) \equiv g(\boldsymbol{x}_g, \boldsymbol{x},\omega ) </math> | ||

+ | |||

+ | <math> 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 </math>. |

## Revision as of 22:41, 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.

## Contents

## 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

.

### 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

.

We can make sense of this by rearranging the terms

.

### The incident (or background) field

Given the way that we have stated the problem, the incident field is described by the Helmholtz equation written in terms of the background wavespeed profile, with a source term

.

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 such that

### The scattered (or perturbation) field

The scattered field 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

.

The problem that we want to solve, is to create an integral equation representation of the scattered field
for a specific field location . 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

.

Here is the Green's function of the medium.
The superscript indicates that these are formally the adjoint operator and respective
adjoint Green's function. In this simple problem, called *self-adjoint* .

### 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

and

.

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

.

The problem is further simplified by considering it to be *unbounded,* which means that the boundary
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

.

### 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 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 and the perturbation , will then be lower order than the incident and scattered field. This is called the Born approximation.

Making the Born approximation and recognizing that , and recognizing that for the unbounded media problem, the Reciprocity theorem takes the form

.