Likelihood

For the likelihood, we mostly follow the approach of Bodin et al. (2012). Summary of the definition is as follows:

Likelihood as multivariate Gaussian distribution
$P(\mathbf{d}|\mathbf{m})= \prod_{j=1}^{M}\frac{1}{\sqrt{(2\pi)^N|\mathbf{C}^{j}|}} \exp\left[\frac{1}{2}(\mathbf{d}_{calc}^{j}-\mathbf{d}_{obs}^{j})^T\mathbf{C}^{j}(\mathbf{d}_{calc}^{j}-\mathbf{d}_{obs}^{j})\right]$
Noise covariance matrix with r^2 decay
- σ_j is treated as a model parameter
- r_j is automatically determined from Gaussian low-pass filter using $r_j=\exp(-a_j^2\Delta t^2)$
- The inverse of C is calculated using singular value decomposition.
- Computing the determinant of C is not necessary becuase it is canceled out.

RF_INV: A Fortran Program for Trans-Dimensional Inversion of Receiver Functions