New formulas derived from seismic interferometry to simulate phase velocity estimates from correlation methods using microtremor

I have developed two theoretical formulas on the basis of the power partition ratios among different modes of Rayleigh waves that are newly derived theoretically from seismic interferometry (SI). These formulas, one for the spatial auto-correlation method (SPAC) and another for the centerless circular array method (CCA), are used to simulate the estimates of the dispersion curves that can be obtained from the correlation methods using microtremor in situations when the higher normal modes are present along with the fundamental mode. The formulas can provide a way to overcome the problem caused by the assumption of the dominance of the fundamental mode, which is not always true. In addition, I have conducted a numerical validation check using the synthetic microtremor waveform data that were produced by the finite-difference method. I have found that the CCA can be an alternative to the SPAC to estimate the dispersion curves. The formula for the CCA can well simulate the dispersion curves estimated by the SPAC and CCA methods, and are better than the formula based on the above-mentioned assumption. Moreover, using the data mentioned above, I have discovered that the dual-mode inversion, which considers the presence of the fundamental and first higher modes on the basis of the formula for the CCA, performs better than the conventional single-mode inversion, which rests on the above-mentioned assumption. These positive results partially support the theoretical consequence of SI, i.e., the power partition ratio, and, further, SI itself on which the theoretically derived formulas fully rely.