Mathematically speaking, the compensated viscoacoustic wave equations 2 and 3 are severely ill-posed due to the presence of the compensating term . That is to say, amplitude compensation is a nonstationary process with energy exponentially amplified over travel time, which boosts high-frequency ambient noise and can even result in numerical instability. In the package, we apply an adaptive stabilization scheme for -RTM to suppress unwanted high-frequency artifacts, which is discussed in my previous work (Wang et al., 2018c). Here we briefly summarize the process.

I derive a k-space Green's function of equation 1 by enforcing a point source at time and . The time-space harmonic Green's function is the solution of the following Helmholtz equation

Solving for Green's function , applying dimensional inverse Fourier transformation, and then integrating the kernel function with respect to based on Cauchy's residue theorem, we have the following time-domain attenuated Green's function

The compensated Green's function can be obtained by reversing the absorption-related term in sign but leaving the other term unchanged:

where and represent attenuated and compensated Green's functions, respectively. These two Green's functions lay the foundation for designing an adaptive stabilization operator. Inspired by stabilization in inverse filtering (Irving and Knight, 2003; Wang, 2006,2002), we proposed a similar adaptive stabilization for -RTM, which can be defined as

where the amplitude-attenuated operator . The final form of the proposed stabilization operator can be given by

2020-04-03