Misfit

The misfit is a value that represents the distance between a calculated dispersion curve and an experimental curve. If the data curve is affected by an uncertainty estimate, the misfit is given by:

$$misfit=\sqrt{\sum_{i=1}^{n_F}\frac{(x_{di}-x_{ci})^2}{\sigma_{i}^2 n_F}}$$ (3.38)

where $x_{di}$ is the velocity of data curve at frequency $f_i$ , $x_{ci}$ is the velocity of calculated curve at frequency $f_i$ , $\sigma_i$ is the uncertainty of the frequency samples considered, $n_F$ is the number of frequency samples considered. If no uncertainty is provided, $\sigma_i$ is replaced by $x_{di}$ in equation (3.38).

When various modes are observed and clearly identified, the inversion of all modes requires a multi-modal misfit. The sum in equation (3.38) is extended to all samples available for all modes. For higher modes, the curves may be defined over a restricted frequency range. Hence, it is not always possible to calculate a theoretical dispersion curve for some experimental samples. If the calculated one-dimensional model is close to the real one, the valid ranges of higher modes are similar and the number of experimental samples is equal to the number of calculated samples. To force both curves to be defined in the same frequency range, the misfit is multiplied by a factor equal to

$$misfit=misfit*(1+n_{experimental}-n_{calculated})$$ (3.39)

$n_{experimental}$ being the number of available samples for each curve ( $n_{experimental} \geq n_{calculated}$ ).