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Mechanistic simulation models are inverted against observations in order to gain inference on modeled processes. However, with the increasing ability to collect high resolution observations, these observations represent more patterns of detailed processes that are not part of a modeling purpose. This mismatch results in model discrepancies, i.e. systematic differences between observations and model predictions. When discrepancies are not accounted for properly, posterior uncertainty is underestimated. Furthermore parameters are inferred so that model discrepancies appear with observation data stream with few records instead of data streams corresponding to the weak model parts. This impedes the identification of weak process formulations that need to be improved. Therefore, we developed an efficient formulation to account for model discrepancy by the statistical model of Gaussian processes (GP). This paper presents a new Bayesian sampling scheme for model parameters and discrepancies, explains the effects of its application on inference by a basic example, and demonstrates applicability to a real world model-data integration study. The GP approach correctly identified model discrepancy in rich data streams. Innovations in sampling allowed successful application to observation data streams of several thousand records. Moreover, the proposed new formulation could be combined with gradient-based optimization. As a consequence, model inversion studies should acknowledge model discrepancies, especially when using multiple imbalanced data streams. To this end, studies can use the proposed GP approach to improve inference on model parameters and modeled processes., Comment: 25 pages, 10 figures