  We present a method for obtaining efficient probabilistic solutions to
geostatistical and linear inverse problems in spherical geometry. Our Spherical
Direct Sequential Simulation (SDSSIM) framework combines information from
possibly noisy observations, that provide either point information on the model
or are related to the model by a linear averaging kernel, and statistics
derived from a-priori training models. It generates realizations from marginal
posterior probability distributions of model parameters that are not limited to
be Gaussian. We avoid the restriction to Cartesian geometry built into many
existing geostatistical simulation codes, and work instead with grids in
spherical geometry relevant to problems in Earth and Space sciences.
  We demonstrate our scheme using a synthetic example, showing that it produces
realistic posterior realizations consistent with the known solution while
fitting observations within their uncertainty and reproducing the model
parameter distribution and covariance statistics of a-priori training models.
Secondly, we present an application to real satellite observations, estimating
the posterior probability distribution for the geomagnetic field at the
core-mantle boundary. Our results reproduce well-known features of the
core-mantle boundary magnetic field, and also allow probabilistic
investigations of the magnetic field morphology. Small-length scale features in
the posterior realizations are not determined by the observations but match the
covariance statistics extracted from geodynamo simulation training models. The
framework presented here represents a step towards more general approaches to
probabilistic inversion in spherical geometry.
