Stability and Accuracy Analysis of Coarse-Grain Viscoelastic Simulations

We analyze the stability and accuracy of the coarse-grain memory variable technique used for viscoelastic wave-field simulations. Our analysis shows that the general behavior of the coarsegrain system is well described by effective parameters (ME and QE) that are derived from the harmonic average of the moduli over the volume of the coarse-grain cell. The use of these effective parameters proves essential for analyzing the performance and accuracy of the coarse-grain system for Q values less than about 20. A necessary stability condition for the coarse-grain system requires that the weighting coefficients be bounded between 0 and 1. Specifying the weights using the approach of Day and Bradley (2001) satisfies this condition for Q values of about 3 and larger; however, using unconstrained optimization techniques will often produce weights that violate this condition at much higher Q values. We also derive a variation of the original coarse-grain methodology called the element-specific modulus (ESM) formulation in which each element of the coarse-grain cell uses a different unrelaxed modulus. We demonstrate that the accuracy of the coarse-grain system for Q values lower than about 20 is significantly improved with the ESM formulation without increasing the computational cost. Finally, we present a technique for optimizing the accuracy of the coarse-grain system for very low Q based on the use of the effective quality factor (QE). We demonstrate that using conventional optimization techniques that do not employ the effective parameter QE will actually degrade the accuracy of the coarse-grain system.