We establish an asymptotic approach for checking the appropriateness of an assumed multivariate spatial regression model by considering the set-indexed partial sums process of the least squares residuals of the vector of observations. In this work, we assume that the components of the observation, whose mean is generated by a certain basis, are correlated. By this reason we need more effort in deriving the results. To get the limit process we apply the multivariate analog of the well-known Prohorov's theorem. To test the hypothesis we define tests which are given by Kolmogorov-Smirnov (KS) and Cramér-von Mises (CvM) functionals of the partial sums processes. The calibration of the probability distribution of the tests is conducted by proposing bootstrap resampling technique based on the residuals. We studied the finite sample size performance of the KS and CvM tests by simulation. The application of the proposed test procedure to real data is also discussed.
Joint inversion codes of the effective Rayleigh wave and multimode Love wave dispersion curves based on modified GA have been developed. We focus on the attempt to improve the estimated shear‐wave velocity profile for a reversal subsurface structure. Two different profiles representing the high‐velocity layer (HVL) and low‐velocity layer (LVL) cases are used. Effective Rayleigh wave and multimode Love wave dispersion curves are synthesized from these profiles using full P‐SV waveform and full SH‐waveform reflectivity, respectively. The dispersion curves are then inverted using a modified genetic algorithm with two different inversion approaches, namely single inversion of Love/Rayleigh dispersion curves and joint inversion of Love and Rayleigh dispersion curves. To asses the accuracy of each inversion approach, differences between the true and inverted shear‐wave velocity profile are quantified in terms of shear‐wave velocity error, ES. Our numerical modelling showed that the shear‐wave velocity errors of the joint inversion approach are relatively smaller than the single inversion approach. These errors indicate that the accuracy of shear‐wave velocity reversal can be improved by jointly inverting Rayleigh wave and Love wave dispersion curves. All of these approaches have been applied on the real data acquired at a site in Niigata prefecture, Japan. In this field example, our inversion results show good agreement between the calculated and experimental dispersion curves and can well detect LVL targets.
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