(2) Wave theory tomography

(a) Fourier Diffraction Tomography for constant background Wu and Toksoz (1987) gave the plane-wave response in the direction $\vec{r}$ from an incident wave $\vec{i}$:  

$$P_{S}^{pl}\left(\vec{i}, \vec{r} \right) = -k^{2}\tilde{O}\left[k\left( \vec{r}-\vec{i}\right) \right]$$ (39)

where $\tilde{O}\left[k\left( \vec{r}-\vec{i}\right) \right]$ is the 3D Fourier transform of the object function $O\left( \vec{r}\right)$. $P_{S}^{pl}\left(\vec{i}, \vec{r} \right)$ is some kind of projection. Comparing this to linear Radon transform, we know that the object function can be accurately restored if the angles of the plane waves continuously change around the object. Fig.3 shows the projection from the real plane wave source and from the virtual plane wave source.

 

planewave_tomography Figure 3 The geometry of plane wave propagation.

(b) Inverse Generalized Radon Transform for variable background The scattered wavefield after Bron and WKBJ approximation is of the following form:

$$\begin{eqnarray} P_{S}\left( \vec{r}, \vec{s},t \right)& = &-\frac{\partial^{2}}... ...ec{x}, \vec{s}\right)\right] f\left( \vec{x} \right) d^{3}\vec{x},\end{eqnarray}$$ (40)

where $A\left( \vec{r}, \vec{x}, \vec{s}\right) = A\left( \vec{r}, \vec{x}\right) A\left( \vec{x}, \vec{s}\right)$ and $\tau\left( \vec{r}, \vec{x}, \vec{s}\right)=T\left( \vec{s}, \vec{x}\right) +T\left( \vec{r}, \vec{x}\right)$.It is known that the diffraction-time surface $R_{x}=\left\lbrace \textbf{d}:t=T\left( \vec{s}, \vec{x}\right) +T\left( \vec{r}, \vec{x}\right)\right\rbrace$ in the data space is a counterpart of the isochron surface $I_{d}=\left\lbrace \textbf{x}:t=T\left( \vec{s}, \vec{x}\right) +T\left( \vec{r}, \vec{x}\right)\right\rbrace$ in the model space. These dual geometric associations naturally give rise to a corresponding pair of projection operators. Equation (40) can be written as  

$$P_{S}\left( \textbf{d}\right) = -\frac{\partial^{2}}{\partia... ...ft( \vec{r}, \vec{x}, \vec{s}\right)f\left( \textbf{x} \right).$$ (41)

The diffraction curve in the data space is a projection of an isochron in the model space. This is a kind of Radon transform Hubral et al. (1996); Miller et al. (1987). The standard Radon transform and inverse Radon transform in three dimensions are given by  

$$f^{\triangle}\left( \vec{\xi} , p\right)=\int \delta \left(p... ...{\xi} \cdot \vec{x} \right) f\left(\vec{x} \right) d^{3}\vec{x}$$ (42)

and  

$$f\left(\vec{x}_{0}\right)=-\frac{1}{8\pi^{2}} \int \left[ \f... ...\right) \vert _{p=\vec{\xi} \cdot \vec{x}_{0}}\right] d^{2} \xi$$ (43)

respectively, where p is the distance from the origin to a plane which cuts through the object body, $\xi$ is the unity direction vector which is normal to the plane, and $\vec{x}$ is a point on the plane.

 

GRT_fig Figure 4 The geometry between the incident and scattering rays near the scattering point, or imaging point $\textbf{x}_{0}$.

Comparing this with the classical Radon transform and its inverse, the final 3D inversion formula can be given as  

$$f\left( \vec{x}_{0}\right) = \frac{1}{\pi^{2}}\int d^{2}\xi ... ...vec{s} \right)}P_{S}\left( \vec{r}, \vec{s},t=\tau_{0} \right).$$ (44)

In equation (44), the angle variable $\xi \left( \vec{r}, \vec{x}_{0},\vec{s} \right)$ near the imaging point $\vec{x}_{0}$ is used, rather than the measurement configuration at the surface. Fig.4 illustrates this. The angle variable is related to the measurement configuration and reflects the seismic wave illumination aperture. Only if the aperture is large can a high resolution image be obtained. The relative true-amplitude imaging is severely affected by the angle variable. Bleistein and Stockwell (2001); Zhang (2004) gave some similar true-amplitude migration/inversion formulas.