Difference between revisions of "Born-approximate modeling formula"

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(Perturbation theory)
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Applying the forward [[Fourier transform]] in time to the scalar wave equation yields the scalar Helmholtz equation
 
Applying the forward [[Fourier transform]] in time to the scalar wave equation yields the scalar Helmholtz equation
  
<math> \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) </math>.
+
<math> \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) </math>.
  
We will consider the Helmholtz equation to be the governing equation of our problem. Because we are interested
 
in recording at a specific position, <math> \boldsymbol{x}_s </math> there is a second Helmholtz equation that
 
is of interest
 
 
<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
 
Green's function. In this case we have a [[self-adjoint]] problem.
 
  
 
=== Perturbation theory ===
 
=== Perturbation theory ===
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Substituting these items into the Helmholtz equation, we obtain
 
Substituting these items into the Helmholtz equation, we obtain
 +
 +
<math> \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) </math>.
 +
 +
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>
 +
 +
Give 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
 +
 +
<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>.
 +
 +
The scattered field <math> u_S(\boldsymbol{x},\boldsymbol{x}_s, \omega) </math> 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
 +
<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>.
 +
 +
 +
 +
 +
 +
We will consider the Helmholtz equation to be the governing equation of our problem. Because we are interested
 +
in recording at a specific position, <math> \boldsymbol{x}_s </math> there is a second Helmholtz equation that
 +
is of interest
 +
 +
<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
 +
Green's function. In this case we have a [[self-adjoint]] problem.

Revision as of 22:00, 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

.


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

Give 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

.

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

.



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.