Viscoacoustic RTM and RTDM

To obtain a seismic image with an attenuated record from the $i$-th shot $d_i(\mathbf{x}_r,t)$, where $\mathbf{x}_r$ denotes the receiver location, viscoacoustic RTM can be carried out in the following three steps:

1. Forward propagate the source wavefield $S_i(\mathbf{x},t)$ by solving
   

   $${\frac{1}{c^2}} {\frac{\partial^2 S_i(\mathbf{x},t)}{\part... ...i(\mathbf{x},t) = \delta(\mathbf{x}-\mathbf{x}_i) f(t) \; .$$ (13)

   
   

2. Backward propagate the receiver wavefield $R_i(\mathbf{x},t)$ by injecting the observed seismic record as the boundary condition $R_i(\mathbf{x}_r,t) = d_i(\mathbf{x}_r,t)$ and solving
   

   $${\frac{1}{c^2}} {\frac{\partial^2 R_i(\mathbf{x},t)}{\part... ...{\partial}{\partial t} \mathbf{H} R_i(\mathbf{x},t) = 0 \;.$$ (14)

   
   

3. Apply the cross-correlation imaging condition (Claerbout, 1985):
   

   $$I(\mathbf{x}) = \sum\limits_i \sum\limits_t S_i(\mathbf{x},t) \bar{R}_i(\mathbf{x},t) \;,$$ (15)

   
   
   where $\bar{R}$ denotes the complex conjugate of $R$ (Sun and Fomel, 2013).

Reverse-time demigration (RTDM) in viscoacoustic media can be formulated as the adjoint of the RTM process:

1. Calculate the source wavefield $S_i(\mathbf{x},t)$ in the background velocity model in the same manner as RTM by solving equation 13.
2. Generate the receiver wavefield $R_i(\mathbf{x},t)$ by using the stacked image $I(\mathbf{x})$ as a secondary source and solving:
   

   $${\frac{1}{c^2}} {\frac{\partial^2 R_i(\mathbf{x},t)}{\part... ...{H} R_i(\mathbf{x},t) = I(\mathbf{x})S_i(\mathbf{x},t) \; .$$ (16)

   
   

3. Extract the predicted seismic record (denoted by the hat) at receiver locations $\mathbf{x}_r$:
   

   $$\widehat{d}_i(\mathbf{x}_r,t) = R_i(\mathbf{x}_r,t) \;.$$ (17)

   
   

Note that, in order to make the RTDM adjoint to RTM, the wave extrapolation operator used to solve equation 16 needs to be the adjoint of the operator used to solve equation 14. For example, if lowrank PSPI is used to solve equation 14, then lowrank NSPS (derived in the appendix) needs to be used to solve equation 16. If we write the RTM process symbolically as $\widehat{\mathbf{m}} = \mathbf{A}^* \mathbf{d}$, where $\widehat{\mathbf{m}}$ is the stacked image, $\mathbf{A}^*$ is the viscoacoustic RTM operator and $*$ denotes the adjoint, then the RTDM process corresponds to $\widehat{\mathbf{d}} = \mathbf{A} \mathbf{m}$, where $\widehat{\mathbf{d}}$ represents the predicted data and $\mathbf{A}$ is the viscoacoustic RTDM operator.