On the art of weighting an objective function with heterogeneous datasets

(Sub)surface hydrological models are more and more integrated, coupling multiple physical, biological and chemical processes. Such models are highly parameterized and most often, prior knowledge on these parameters is loose. Hopefully, such complex models may assimilate rich observation datasets, which constrain model parameters and reduce forecast uncertainties. The inclusion of diverse data types (aka “calibration targets”) within the so called “objective function” deserves particular attention to avoid bias in estimated parameters and forecasts of interest. In the most common approach, the fit between model outputs and data is described with a single objective function composed of the sum of weighted squared residuals between simulated values and their observed counterparts. When the residuals are statistically independent, homoscedastic and can be described with a gaussian probability distribution, the least square estimates obtained through the minimization of the objective function presents numerous advantages. However, when assimilating diverse data types with a model presenting structural error, the above-mentioned hypotheses on model residuals are at best very unlikely and in practice, never matched. Numerous studies investigated the interest of error modeling and data-transformation. Less attention has been paid to the integration of various data types (flows, heads, concentrations, soft data, ...) potentially spanning over several orders of magnitudes and originating from spatially distributed locations (wells, gaging stations, ...) each with contrasting sampling frequency (years, days, hours, ...). A purely formal statistical approach is challenging to put in practice, but the integration of such dataset into a single objective function deserves a relevant weighting strategy. Based on a synthetic model, different weighting strategies are compared based on their ability to reduce predictive bias and uncertainty. We propose an informal but practical formulation of the objective function that may be used for operational groundwater modeling case studies. The approach is eventually illustrated on a real-world case study.