Estimation under model uncertainty remains a practical concern in many scientific and engineering fields. A commonly encountered example in groundwater remediation is the contaminant source identification problem. Like many other inverse problems, contaminant source identification is inherently ill posed and is sensitive to both data and model uncertainties. Model uncertainties, which may be introduced at virtually any stage of a model building process, can adversely affect estimator performance if they are not accounted for properly. In this paper, a robust geostatistical approach (RGS), which is extended from the geostatistical inversion approach (GS), is used to solve linear estimation problems. The uncertainties in both model and covariance matrices are taken into account in the RGS formulation. The nominal correlation structural parameters are estimated using a structural analysis procedure. The resulting minimax optimization problem is solved using semidefinite programming techniques. The RGS is generic and can be applied to any problem for which the GS is suitable and the upper bound of uncertainty can be quantified. The RGS is illustrated for source release history identification in a two-dimensional aquifer where the model uncertainty is caused by variability in hydraulic conductivity. It is shown that when the model is perfectly known, the RGS solution coincides with that of the GS; when the model has uncertainty, the RGS is robust against unknown variations from a nominal model, and its overall performance is better than that of the GS using the same nominal model.