Misfit

The misfit is evaluated for all data samples. It is defined in the same way as for the dispersion curve inversion (equation (3.38) and Wathelet et al. (2004)), taking into account the standard deviation observed for each spatial auto-correlation sample :

$$misfit=\sqrt{\frac{1}{\sum_{k=1}^{n_R}{n_{Fk}}}\; \sum_{i=1}^{n_R}\sum_{j=1}^{n_{Fi}}\frac{(\rho_{dij}-\rho_{cij})^2}{\sigma_{ij}^2}}$$ (3.51)

where, $\rho_{dij}$ is the SPAC ratio of data curves at frequency $f_j$ and for ring $i$ which is defined by all inter-station distances between $r_{i1}$ and $r_{i2}$ , $\rho_{cij}$ is the SPAC ratio of calculated curves at frequency $f_j$ and for ring $i$ , $\sigma_{ij}$ is the observed variance for the sample at frequency $f_j$ and for ring $i$ , $n_R$ is the number of rings considered, and $n_{Fi}$ is the number of frequency samples for ring $i$ .

As for dispersion curves, the implemented algorithm can calculate a misfit for a set of modal curves by including the contributions of all modes in the sum of equation (3.51). The technique described in section 3.1.7 is also used for higher modes with a limited valid frequency band. Options exist to restrict the misfit computation to the first decreasing part of the auto-correlation curve (argument of Bessel's function less than 3.2) and to avoid the part of the curves close to 1 (argument of Bessel's function greater than 0.4). In this case, even the fundamental mode may have a restricted valid frequency interval for which the misfit is corrected in the same way as for higher modes. However, experience has proved that those options are generally useless and that the whole frequency range can be used for inversion (section 5.2).