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Water Resources Research

Stochastic Modeling of Groundwater Flow by Unconditional and Conditional Probabilities: The Inverse Problem


  • Gedeon Dagan


The inverse problem is defined here as follows: determine the transmissivity at varius points, given the shape and boundary of the aquifer and recharge intensity and given a set of measured log-transmissivity Y and head H values at a few points. The log-transmissivity distribution is regarded as a realization of a random function of normal and stationary unconditional probability density function (pdf). The solution of the inverse problem is the conditional normal pdf of Y, conditioned on measured H and Y, which is expressed in terms of the unconditional joint pdf of Y and H. The problem is reduced to determining the unconditional head-log-transmissivity covariance and head variogram for a selected Y covariance which depends on a few unknown parameters. This is achieved by solving a first-order approximation of the flow equations. The method is illustrated for an exponential Y covariance, and the effect of head and transmissivity measurements upon the reduction of uncertainty of Y is investigated systematically. It is shown that measurement of H has a lesser impact than those of Y, but a judicious combination may lead to significant reduction of the predicted variance of Y. Possible applications to real aquifers are outlined.

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