@article{fdi:010074389,
  title = {{S}tochastic static fault slip inversion from geodetic data with non-negativity and bound constraints},
  author = {{N}ocquet, {J}ean-{M}athieu},
  editor = {},
  language = {{ENG}},
  abstract = {{D}espite surface displacements observed by geodesy are linear combinations of slip at faults in an elastic medium, determining the spatial distribution of fault slip remains a ill-posed inverse problem. {A} widely used approach to circumvent the illness of the inversion is to add regularization constraints in terms of smoothing and/or damping so that the linear system becomes invertible. {H}owever, the choice of regularization parameters is often arbitrary, and sometimes leads to significantly different results. {F}urthermore, the resolution analysis is usually empirical and cannot be made independently of the regularization. {T}he stochastic approach of inverse problems provides a rigorous framework where the a priori information about the searched parameters is combined with the observations in order to derive posterior probabilities of the unkown parameters. {H}ere, {I} investigate an approach where the prior probability density function (pdf) is a multivariate {G}aussian function, with single truncation to impose positivity of slip or double truncation to impose positivity and upper bounds on slip for interseismic modelling. {I} show that the joint posterior pdf is similar to the linear untruncated {G}aussian case and can be expressed as a truncated multivariate normal ({TMVN}) distribution. {T}he {TMVN} form can then be used to obtain semi-analytical formulae for the single, 2-{D} or n-{D} marginal pdf. {T}he semi-analytical formula involves the product of a {G}aussian by an integral term that can be evaluated using recent developments in {TMVN} probabilities calculations. {P}osterior mean and covariance can also be efficiently derived. {I} show that the maximum posterior ({MAP}) can be obtained using a non-negative least-squares algorithm for the single truncated case or using the bounded-variable least-squares algorithm for the double truncated case. {I} show that the case of independent uniform priors can be approximated using {TMVN}. {T}he numerical equivalence to {B}ayesian inversions using {M}onte {C}arlo {M}arkov chain ({MCMC}) sampling is shown for a synthetic example and a real case for interseismic modelling in {C}entral {P}eru. {T}he {TMVN} method overcomes several limitations of the {B}ayesian approach using {MCMC} sampling. {F}irst, the need of computer power is largely reduced. {S}econd, unlike {B}ayesian {MCMC}-based approach, marginal pdf, mean, variance or covariance are obtained independently one from each other. {T}hird, the probability and cumulative density functions can be obtained with any density of points. {F}inally, determining the {MAP} is extremely fast.},
  keywords = {{I}nverse theory ; {E}arthquake source observation ; {S}atellite {G}eodesy ; {PEROU}},
  booktitle = {},
  journal = {{G}eophysical {J}ournal {I}nternational},
  volume = {214},
  numero = {1},
  pages = {366--385},
  ISSN = {0956-540{X}},
  year = {2018},
  DOI = {10.1093/gji/ggy146},
  URL = {https://www.documentation.ird.fr/hor/fdi:010074389},
}