I am trying to get an overview of different approaches hydrologists have developed to deal with uncertain boundary conditions in inversions for subsurface flow such as hydraulic tomography.
This may help: A critical review of the properties of forward and inverse problems in groundwater hydrology / M. Giudici, T.R. Ginn, C. Vassena, H. Haeri, L. Foglia (IAHS Publications; 320). - In: Calibration and Reliability in Groundwater Modelling: Credibility of Modelling