An improved affine mixed-grid method for frequency-domain finite-difference elastic modelling

In seismic frequency-domain finite-difference modelling, the affine mixed-grid method effectively eliminates the spatial sampling restriction associated with square meshes of the rotated mixed-grid method. Nevertheless, the affine mixed-grid method makes a weighted average of the entire elastic wave equations, resulting in reduced accuracy compared to the average-derivative method in the case of rectangular meshes. It is worth noting, however, that the average-derivative method is presently inapplicable to free-surface scenarios, whereas the affine mixed-grid method is applicable. By performing weighted averages of the derivative terms instead of the entire elastic wave equations in Cartesian and affine rotated coordinate systems, we have developed an improved affine mixed-grid method for elastic-wave frequency-domain finite-difference modelling. The proposed improved affine mixed-grid method 9-point scheme overcomes the drawback that the accuracy of affine mixed-grid method is lower than that of average-derivative method for unequal directional grid intervals. Moreover, the improved affine mixed-grid method 6-point scheme provides much higher numerical accuracy than the affine mixed-grid method 6-point scheme at either equal or unequal directional grid intervals. On the other hand, the proposed improved affine mixed-grid method simplifies the coding complexity for implementing free-surface condition in elastic-wave frequency-domain finite-difference modelling by modifying the elastic parameters of the free-surface layer and thus constructing the impedance matrix containing the free-surface condition directly.