## 3 Earthquake Modelling
### Generalized Akaike's Bayesian Information Criterion (gABIC)  
A precise weighting of each data set is fundamental to realize the joint inversion. These weightings strongly affect the result of the inversion (as explained in Reference [28]). For instance, an over-weighting of one data set could induce an under-weighting of the other data sets and disturb the estimation of each parameter of the fault geometry. In that goal, the weight determination method  
Figure 6: Coseismic displacement vectors, black arrows represent SPOT offsets and red arrows GPS data. developed by [PERSON] ([29; 30]) is used: gABIC based on the entropy maximization principle [31] to determine the regularization parameter (a spatial constraint used to regulate the rupture process), and the Bayesian theorem to estimate the probability distribution of the slip for given observations (GPS, benchmarks, levelling, InSAR, SPOT offsets) and prior constraints (Laplacian constraint in our case). In this generalized version, the type of observations and prior constraints do not need to be specified; only diagonal covariance matrices have to be estimated before running gABIC script. As we have 8 sets of (pseudo-) observations (3 interferograms, GPS, benchmarks, levelling and SPOT offsets in addition to the Laplacian constraint), 7 hyperparameters need to be determined using the search-based gABIC method. To obtain optimal weight ratios that minimize the ABIC value, large enough ranges of these hyperparameters have to be considered. In practice, searching the seven data sets with a small enough interval is highly time consuming. In order to reduce the processing time, a first estimation of each relative weight was done by calculating it for each data set in respect with the GPS one, which covers the whole area with a low uncertainty. For this first step, the weighting range was determined using a power law method (\(10^{x}\)) and then refined using a linear method (x) for the interval of inputs (x representing the values to test). Once the interval of hyperparameter was reduced, gABIC could be used using several data sets at once. The inversion equations [30] are the following (to make the equations more manageable, only one interferogram is mentioned):  
\[\mathbf{d}=\mathbf{Ha}+\mathbf{e} \tag{1}\]  
\[\mathbf{0}=\mathbf{Sa}+\mathbf{e}, \tag{2}\]  
with:  
* \(\mathbf{d}=[\mathbf{d}_{GPS}\ \ \mathbf{d}_{Bench}\ \ \mathbf{d}_{lev}\ \ \mathbf{d}_{InSAR}\ \ \mathbf{d}_{SPOT}]_{Nx1}^{T}\): Nx1 vector containing 5 data types (total N observations);
* \(\mathbf{H}=[\mathbf{H}_{GPS}\ \ \mathbf{H}_{Bench}\ \ \mathbf{H}_{lev}\ \ \mathbf{H}_{InSAR}\ \ \mathbf{H}_{SPOT}]_{NxM}^{T}\): NxM coefficient matrix;
* \(\mathbf{S}=[\mathbf{S}_{S}]_{P_{xM}}^{T}\): \(PxM\) coefficient matrix containing Prior constraints
* \(\mathbf{a}\) is a M dimensional model parameter vector that varies with the time windows;
* \(\mathbf{e}=[\mathbf{e}_{GPS}\ \ \mathbf{e}_{Bench}\ \ \mathbf{e}_{lev}\ \ \mathbf{e}_{InSAR}\ \ \mathbf{e}_{SPOT}]_{Nx1}^{T}\)  &  \(\mathbf{\epsilon}=[\mathbf{\epsilon}_{S}]_{P_{x1}}^{T}\): are vectors of the Gaussian-distributed uncertainties of observations and constraints, respectively. Furthermore, \(\mathbf{e}\sim\mathbf{\tilde{Q}}(\sigma_{i}^{2})\) where \(\mathbf{\tilde{Q}}(\sigma_{i}^{2})=\sigma_{GPS}^{2}\) with \(\mathbf{Q}\) (i refers to the five data sets) covariance matrices of observations and a similar equation for the constraint).