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Theory

The theory of least-squares reverse time migration is well established (Symes and Carazzone, 1991; Mulder and Plessix, 2004; Dai et al., 2012). In this section, I will first review the theory of LSRTM assuming the constant density acoustic wave equation,

$\displaystyle \frac{1}{c(\textbf{x})^2}\frac{\partial^2 p(\textbf{x},t;\textbf{...
...partial t^2}-\bigtriangledown^2 p(\textbf{x},t;\textbf{x}_s)=s(t;\textbf{x}_s),$ (14)

where $ c(\textbf{x})$ is the velocity distribution, and $ p(\textbf{x},t;\textbf{x}_s)$ is the pressure field associated with the source term $ s(t;\textbf{x}_s)$ . A perturbation in the velocity model $ c(\textbf{x})\rightarrow c(\textbf{x})+\delta c(\textbf{x})$ will generate a wavefield $ p(\textbf{x},t;\textbf{x}_s)\rightarrow p(\textbf{x},t;\textbf{x}_s)+\delta p(\textbf{x},t;\textbf{x}_s)$ , which obeys the equation

$\displaystyle \frac{1}{(c(\textbf{x})+\delta c(\textbf{x}))^2}\frac{\partial^2 ...
...xtbf{x},t;\textbf{x}_s)+\delta p(\textbf{x},t;\textbf{x}_s))=s(t;\textbf{x}_s).$ (15)

Expanding the velocity term according to

$\displaystyle \frac{1}{(c(\textbf{x})+\delta c(\textbf{x}))^2} \approx \frac{1}{c(\textbf{x})^2}-\frac{2\delta c(\textbf{x})}{c(\textbf{x})^3},$ (16)

and subtracting equation [*] from equation [*] yields the wave equation for the wavefield perturbation $ \delta p(\textbf{x},t;\textbf{x}_s)$

$\displaystyle \frac{1}{c(\textbf{x})^2}\frac{\partial^2 \delta p(\textbf{x},t;\...
...al t^2}\frac{2\delta c(\textbf{x})}{c(\textbf{x})^3}+O(\delta c(\textbf{x})^2).$ (17)

Neglecting the higher order terms and defining the reflectivity model as $ m(\textbf{x})=\frac{2\delta c(\textbf{x})}{c(\textbf{x})}$ , the above equation becomes

$\displaystyle \frac{1}{c(\textbf{x})^2}\frac{\partial^2 \delta p(\textbf{x},t;\...
...},t;\textbf{x}_s)=m(\textbf{x})\bigtriangledown^2 p(\textbf{x},t;\textbf{x}_s).$ (18)

Equations [*] and [*] will be used to derive the Born modeling operator. Numerically, the calculation of the reflection data $ \delta p(\textbf{x},t;\textbf{x}_s)$ requires two finite-difference simulations: one to solve equation [*] to obtain the wavefield $ p(\textbf{x},t;\textbf{x}_s)$ , and one to solve equation [*] for the reflection data $ \delta p(\textbf{x},t;\textbf{x}_s)$ . The wavefield $ \delta p(\textbf{x},t;\textbf{x}_s)$ will be recorded at the receiver position $ \textbf{x}_g$ to give the shot gather $ d(\textbf{x}_g,t;\textbf{x}_s)$ . By the adjoint state method (Plessix, 2006), the migration operation of a shot gather $ d(\textbf{x}_g,t;\textbf{x}_s)$ requires two finite-difference simulations, one for the source-side wavefield and one for the receiver-side wavefield:

    $\displaystyle \frac{1}{c(\textbf{x})^2}\frac{\partial^2 p(\textbf{x},t;\textbf{...
...partial t^2}-\bigtriangledown^2 p(\textbf{x},t;\textbf{x}_s)=s(t;\textbf{x}_s),$ (19)
    $\displaystyle \frac{1}{c(\textbf{x})^2}\frac{\partial^2 q(\textbf{x},t;\textbf{...
...\bigtriangledown^2 q(\textbf{x},t;\textbf{x}_s)=d(\textbf{x}_g,t;\textbf{x}_s),$ (20)

where $ q(\textbf{x},t;\textbf{x}_s)$ is the receiver-side wavefield. Note that the source-side wavefield $ p(\textbf{x},t;\textbf{x}_s)$ propagates forward in time but the receiver-side wavefield $ q(\textbf{x},t;\textbf{x}_s)$ propagates backward in time. The migration image associated with the shot at $ \textbf{x}_s$ is produced by applying the imaging condition

$\displaystyle m(\textbf{x};\textbf{x}_s)=\sum_t \bigtriangledown^2 p(\textbf{x},t;\textbf{x}_s) \cdot q(\textbf{x},t;\textbf{x}_s).$ (21)

To simplify the formulas, matrix-vector notation will be used to represent the Born modeling operator

$\displaystyle \textbf{d}_i=\textbf{L}_i\textbf{m},$ (22)

where $ \textbf{d}_i$ is the reflection data vector for the $ ith$ shot, $ \textbf{m}$ is a reflectivity model, and $ \textbf{L}_i$ represents the Born modeling operator associated with the $ ith$ shot. Similarly, the reverse time migration operator can be expressed as

$\displaystyle \textbf{m}_{mig,i}=\textbf{L}^{T}_{i}\textbf{d}_{i}.$ (23)

with $ \textbf{m}_{mig,i}$ indicating the migration image for the $ ith$ shot and $ \textbf{L}^{T}_{i}$ representing the migration operator associated with the $ ith$ shot.



Subsections
next up previous contents
Next: Least-squares Migration Up: Plane-wave Least-squares Reverse Time Previous: Introduction   Contents
Wei Dai 2013-07-10