Estimation of co-conditional moments of transmissivity, hydraulic head, and velocity fields Samuel Hanna & T.-C. Jim Yeh* Department of Hydrology and Water Resources, The University of Arizona, Tucson, AZ 85721, U.S.A. (Received 26 August 1996; accepted 21 March 1997) An iterative co-conditional Monte Carlo simulation (IMCS) approach is developed. This approach derives co-conditional means and variances of transmissivity (T), head (f), and Darcy’s velocity (q), based on sparse measurements of T and f in heterogeneous, confined aquifers under steady-state conditions. It employs the classical co-conditional Monte Carlo simulation technique (MCS) and a successive linear estimator that takes advantage of our prior knowledge of the covariances of T and f and their cross-covariance. In each co-conditional simulation, a linear estimate of T is improved by solving the governing steady-state flow equation, and by updating residual covariance functions iteratively. These residual covariance functions consist of the covariance of T and f and the cross-covariance function between T and f. As a result, the non-linear relationship between T and f is incorporated in the co-conditional realizations of T and f. Once the T and f fields are generated, a corresponding velocity field is also calculated. The average of the co-conditioned realizations of T, f, and q yields the co-conditional mean fields. In turn, the co-conditional variances of T, f, and q, which measure the reduction in uncertainty due to measurements of T and f, are derived. Results of our numerical experiments show that the co-conditional means from IMCS for T and f fields have smaller mean square errors (MSE) than those from a non-iterative Monte Carlo simulation (NIMCS). Finally, the co-conditional mean fields from IMCS are compared with the co-conditional effective fields from a direct approach developed by Yeh et al.[Water Resources Research, 32(1), 85–92, 1996].  1998 Elsevier Science Limited. Keywords: inverse problem, iterative approach, geostatistics, Monte Carlo simulation, heterogeneous aquifers, conditional means. 1 INTRODUCTION During the past few decades, numerous mathematical models have been developed to solve the inverse problem associated with groundwater systems, given scattered measurements of hydraulic head, f, and transmissivity, T (refer to Yeh 32 and Hanna 14 ). One popular method is the minimum-output-error-based approach (e.g. Yeh and Tauxe 31 ; Gavalas et al. 10 ; Neuman and Yakowitz 23 ; Neuman 24 ; Clifton and Neuman 4 ; Cooley 5 ; Carrera and Neuman 1,2 ; Willis and Yeh 27 ). A drawback of this approach is that the solution is non-unique, and the identity of the estimate is often undefined. In other words, using different initial guesses this approach can lead to different results. Subsequently, it is unclear whether the estimate is a conditional mean, an effective mean, a conditional realiza- tion or simply an estimate without any statistical meaning, and the uncertainty associated with the estimated field cannot be addressed properly. Uniquely identifying the spatial distribution of transmissivity in a heterogeneous aquifer under steady- state flow conditions is an impossible task, unless all the hydraulic heads are known and boundary fluxes are specified. For cases with scattered T and f measurements (or stochastic inverse problems referred to by Yeh et al. 30 ), a logical inverse approach should adopt the conditional stochastic concept. That is, one should attempt to obtain T and f fields that preserve their observed values at all sample locations and their underlying statistical properties (i.e. the mean and covariance). Furthermore, the estimated T and f Advances in Water Resources Vol. 22, No. 1, pp. 87–95, 1998  1998 Elsevier Science Ltd Printed in Great Britain. All rights reserved 0309-1708/98/$ - see front matter PII: S 0 3 0 9 - 1 7 0 8 ( 9 7 ) 0 0 0 3 3 - X 87 *Corresponding author. E-mail: ybiem@mac.hwr.arizona.edu