The combination of numerical model simulations with observations is called “data-model fusion” or “Data Assimilation (DA)”. Measurements are introduced into the model to improve the model states that exhibit uncertainties due to imperfect input data, parameters and errors in the model structure. DA can be defined in different ways, e.g., as weighted mean between model outputs and observations based on their uncertainties, or active integration of data into the model, i.e., modification of model states, which are used in the next model forward integration step. To define the “best” combination of model simulations and measurements, a so-called cost function must be formulated. This function defines how simulation and observation are weighted by their uncertainties within the data assimilation process. In the case that an observation is very accurate and large uncertainties for the model simulation exist, the weight for the observation should be high, while the weight for the model should be small. In contrast, when the model simulation is accurate but large uncertainties are associated with observations, the model should be weighted high, and the merged value should be closer to the model value.
DA has a longer history in the atmosphere and ocean sciences. In these fields, the dynamics are considered to be chaotic. As a result, small errors in the initial conditions result in large differences in the temporal evolution of model states. This is critical and a limiting factor, e.g., in numerical weather prediction. Data assimilation methods have been developed in order to estimate optimal initial condition, e.g., three- and four-dimensional variational methods (3D-Var and 4D-Var). In contrast to atmospheric and oceanic dynamics, the accuracy of hydrological simulations is predominantly influenced by uncertain meteorological forcing conditions and model parameterisation. Thus, an integration of observations whenever they become available helps to adjust the model simulations to reality.
Therefore, in sequential ensemble-based data assimilation (Evensen, 2007), observations are used to correct the present states of a model as soon as they become available.
Fig.1: Scheme of sequential data assimilation: Observations (black triangles) are used as soon as they are available to improve the current model states x−k (black points and solid line). The updated model states x+k (white points) give the best fit to the observations at the current time step. Schumacher (2016, Fig. 3.2)
In Fig. 1, the concept of sequential data assimilation is illustrated. The model run starts at time t0 with initial conditions x0 and is integrated forward until time tk, for which observations yk are available (black triangles in Fig. 1). The measurements are directly incorporated to correct the model prediction x−k (black points in Fig. 1). The corrected values x+k (white points in Fig. 1) are subsequently used to start the next model forward integration. This procedure is repeated sequentially for each time step, at which observations are available. The approach is suitable when systems are driven by forcing fields, e.g., precipitation, temperature, and other meteorological variables in hydrological models.
In our research group, we work on the implementation and extension of the ensemble based DA approaches, for example, in the form of Kalman filter processes (e.g., Khaki et al., 2017a,b), Bayesian averaging (Mehrnegar et al., 2020), and Gaussian/non-Gaussian ensemble average updating technique (by Retegui Schiettekatte et al., coming soon).
Related Publications:
Evensen, G. (2007). Data assimilation. The Ensemble Kalman Filter. Springer, Berlin, Heidelberg.
Schumacher, M. (2016) Methods for assimilating remotely-sensed water storage changes into hydrological models. PhD dissertation. University of Bonn, Germany. http://hss.ulb.uni-bonn.de/2016/4508/4508.pdf
Khaki, M., Ait-El-Fquih, B., Hoteit, I., Forootan, E., Kuhn, M., Awange, J. (2017a), A two-update ensemble Kalman _lter for land hydrological data assimilation with an uncertain constraint. Journal of Hydrology, 555, pages 447-462, doi:10.1016/j.jhydrol.2017.10.032
Khaki, M., Hoteit, I., Kuhn, M., Awange, J., Forootan, E., van Dijk, A., Schumacher, M., Pattiaratchi, C. (2017b), Assessing sequential data assimilation techniques for integrating GRACE data into a hydrological model. Advances in Water Resources, 107, pages 301-316, doi:10.1016/j.advwatres.2017.07.001
Mehrnegar, N., Jones, O., Singer, M.B., Schumacher, M., Bates, P., Forootan, E. (2020), Comparing global hydrological models and combining them with GRACE by Dynamic Model Data Averaging (DMDA). Advances in Water Resources, 138, pages 103528, doi:10.1016/j.advwatres.2020.103528