Discussion on the rank deficiency of the representation matrix of the smoothing constraint in inversion methods using a Bayesian Information Criterion

Akaike's Bayesian Information Criterion (ABIC) has been used in a number of studies to optimize the weights of constraint conditions to estimate the distributions of displacements on seismic faults and of slip deficits on plate boundaries from geodetic data such as displacements, tilts, and strains based on Bayesian models. The treatment of the prior probability density function (PDF) for the case in which a matrix, which represents the spatial derivatives in the existing inversion methods to include the smoothness of the distribution of slips or slip deficits into the inversion, is rank deficient is discussed. If the matrix consists only of spatial derivatives and their linear combinations and if the effects of boundary conditions are not taken into account, the matrix must be rank deficient and singular. On the other hand, the prior PDF cannot be uniquely defined using such a rank deficient matrix for the full space of model parameters, and, therefore, the marginal likelihood that is necessary to obtain ABIC cannot be uniquely calculated. In addition, an inversion method to avoid the rank deficiency of the matrix in the prior PDF is introduced.