The K-Space Green’s Functions for Decoupled Constant-Q Wave Equation and its Adjoint Equation

Motivated by exactly explaining the numerical instability of attenuation compensation in lossy media, we analytically derive k-space Green’s functions for decoupled constant-Q (DCQ) wave equation and its adjoint equation, then we find that Green’s function for DCQ wave equation is exponentially decreasing, whereas that of adjoint DCQ wave equation is exponentially increasing. These two Green’s functions can be taken as theoretical explanations for the fact that attenuation and compensation are both a nonstationary process with energy exponentially attenuated or amplified over propagation time, especially for high-frequency components. It is this exponential amplification which eventually results in numerical instability of adjoint DCQ wave equation. For a more intuitive understanding about the k-space Green’s functions, we numerically compare them with that of acoustic wave equation.