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[1] Acoustic wave equation

Based on inverse theory, the characterization of seismic wave propagation is important for parameter estimation. Here I use the acoustic wave equation with two elastic parameters -- bulk modulus and density -- to model seismic wave propagation in a geological medium, though we know that this is a simplification.  
LP\equiv \left( \nabla \cdot \frac{1}{\rho\left( \vec{x}\rig...
 ...\delta\left(\vec{x}-\vec{x}_{s} \right) S\left( \omega \right),\end{displaymath} (1)
where $\kappa$ is the bulk modulus and $\rho$ is the density. Both parameters vary horizontally as well as vertically. $P\left(\vec{x},\vec{x}_{s},\omega \right) $ is the acoustic pressure wave field, and $S\left( \omega \right) $ is the monochromatic source function. We can carry out the full waveform inversion with equation (1). Tarantola (1984) gave a detailed theoretical framework. Pratt and Hicks (1998) discussed in detail how to implement seismic waveform inversion in the frequency domain. Now I introduce a background model which is so close to the true model that we can neglect the second and higher-order reflection and transmission effects caused by the interaction between the incident wave and the scattering potential. The background wavefield obeys the following equation:  
L_{0}P\equiv \left( \nabla \cdot \frac{1}{\rho_{0}} \nabla +...
 ...\delta\left(\vec{x}-\vec{x}^{s} \right) S\left( \omega \right).\end{displaymath} (2)
With the definition V=L-L0, the identity $A=B+B\left(B^{-1}-A^{-1} \right)A$ becomes

G=G0+G0VG, (3)

if we associate G with A and G0 with B. And equation (3) is further rearranged to  
G=\left( I-G_{0}V\right)^{-1}G_{0}.\end{displaymath} (4)
Performing a Taylor expansion on the right term of equation (4) yields  
G=\left[\sum\limits_{j=0}^{\infty}\left( G_{0}V\right)^{j}\right] G_{0} .\end{displaymath} (5)
Equation (3) is called the Lippmann-Schwinger equationClayton and Stolt (1981). Clearly, if $j\geq 2$, equation (5) depicts second and higher-order scattering terms of wave propagation, which are neglected. The linearized propagator characterizes only the first scattering of wave propagation. That is,

G=G0+G0VG0. (6)

This is the Born approximation, the physical meaning of which is clearly demonstrated by equations (5) and (6). From $L=-\left( \nabla \cdot \frac{1}{\rho} \nabla +\frac{\omega^{2}}{\kappa}\right)$ and $L_{0}=-\left( \nabla \cdot \frac{1}{\rho_{0}} \nabla +\frac{\omega^{2}}{\kappa_{0}}\right)$, the scattering potential V is defined as follows:
V&=&\left( \nabla \cdot \frac{1}{\rho} \nabla +\frac{\omega^{2}...
 ...t \frac{a_{1}}{\rho_{0}}\nabla +\omega^{2}\frac{a_{2}}{\kappa_{0}}\end{eqnarray}
where $a_{1}=\frac{\rho_{0}}{\rho} -1=\frac{\triangle \rho}{\rho}$ and $\triangle \rho=\rho_{0}-\rho$; $a_{2}=\frac{\kappa_{0}}{\kappa} -1=\frac{\triangle \kappa_{0}}{\kappa}$ and $\triangle \kappa=\kappa_{0}-\kappa$.Therefore the linearized synthetic wave field is composed of two parts: one is the background wave field described by the background Green's function; the other is the scattering wavefield caused by the scattering potential V. According to equation (3), the total wave field is written as  
P\left(\vec{x}_{r}, \vec{x}_{s}, \omega \right)=G_{0}\left(\...
 ...left(\vec{x}\right)P\left(\vec{x}, \vec{x}_{s}, \omega \right),\end{displaymath} (8)
and the scattering wavefield after the Born approximation from equation (6) is  
P_{s}\left(\vec{x}_{r}, \vec{x}_{s}, \omega \right)=\omega^{...
 ...(\vec{x}\right)G_{0}\left(\vec{x}, \vec{x}_{s}, \omega \right).\end{displaymath} (9)

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