  We consider the computational challenges associated with uncertainty
quantification involved in parameter estimation such as seismic slowness and
hydraulic transmissivity fields. The reconstruction of these parameters can be
mathematically described as Inverse Problems which we tackle using the
Geostatistical approach. The quantification of uncertainty in the
Geostatistical approach involves computing the posterior covariance matrix
which is prohibitively expensive to fully compute and store. We consider an
efficient representation of the posterior covariance matrix at the maximum a
posteriori (MAP) point as the sum of the prior covariance matrix and a low-rank
update that contains information from the dominant generalized eigenmodes of
the data misfit part of the Hessian and the inverse covariance matrix. The rank
of the low-rank update is typically independent of the dimension of the unknown
parameter. The cost of our method scales as $\bigO(m\log m)$ where $m $
dimension of unknown parameter vector space. Furthermore, we show how to
efficiently compute measures of uncertainty that are based on scalar functions
of the posterior covariance matrix. The performance of our algorithms is
demonstrated by application to model problems in synthetic travel-time
tomography and steady-state hydraulic tomography. We explore the accuracy of
the posterior covariance on different experimental parameters and show that the
cost of approximating the posterior covariance matrix depends on the problem
size and is not sensitive to other experimental parameters.
