[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.  

$$LP\equiv \left( \nabla \cdot \frac{1}{\rho\left( \vec{x}\rig... ...\delta\left(\vec{x}-\vec{x}_{s} \right) S\left( \omega \right),$$ (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:  

$$L_{0}P\equiv \left( \nabla \cdot \frac{1}{\rho_{0}} \nabla +... ...\delta\left(\vec{x}-\vec{x}^{s} \right) S\left( \omega \right).$$ (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  

$$G=\left( I-G_{0}V\right)^{-1}G_{0}.$$ (4)

Performing a Taylor expansion on the right term of equation (4) yields  

$$G=\left[\sum\limits_{j=0}^{\infty}\left( G_{0}V\right)^{j}\right] G_{0} .$$ (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:

$$\begin{eqnarray} V&=&\left( \nabla \cdot \frac{1}{\rho} \nabla +\frac{\omega^{2}... ...t \frac{a_{1}}{\rho_{0}}\nabla +\omega^{2}\frac{a_{2}}{\kappa_{0}}\end{eqnarray}$$ (7)

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  

$$P\left(\vec{x}_{r}, \vec{x}_{s}, \omega \right)=G_{0}\left(\... ...left(\vec{x}\right)P\left(\vec{x}, \vec{x}_{s}, \omega \right),$$ (8)

and the scattering wavefield after the Born approximation from equation (6) is  

$$P_{s}\left(\vec{x}_{r}, \vec{x}_{s}, \omega \right)=\omega^{... ...(\vec{x}\right)G_{0}\left(\vec{x}, \vec{x}_{s}, \omega \right).$$ (9)