Optimized High-order Finite-difference Modeling of Second-order Strain Gradient Wave Field Effects

The second-order strain gradient wave equation, which is based on the generalized continuum mechanics theory, can enrich the content of classical continuum mechanics theory with the incorporation of the second-order spatial derivative term of displacement. Furthermore, it incorporates scale parameters of the media characteristics to bridge the gap between a micromodel and classical continuum mechanics, thus reflecting the microstructure characteristics within the media. Wang derived a constitutive equation of the single-parameter second-order strain gradient theory using a nonlocal theory and provided a mathematical expression for the second-order strain gradient asymmetric elastic wave equation combined with the geometric equation and differential equation of motion. The second-order strain gradient theory can reflect the smaller-scale effect of seismic wave propagation within the neighborhood of average particle diameter l of the medium. Considering the relatively weak spatial scale effect, the numerical dispersion generated when the difference operator approximates the differential operator can suppress the scale effect. Thus, to accurately describe and analyze the scale effects, the finite-difference operator must be optimized. This paper proposes an improved black window optimization algorithm to obtain optimized finite-difference coefficients and perform numerical modelling based on the second-order strain gradient wave equation. Numerical modelling results reveal that the numerical dispersion is well suppressed, and the smaller spatial scale effects can be clearly observed and extracted from the seismograms.