The Number of Required Observations in Data Assimilation for a Shallow-Water Flow

The authors consider statistical ensemble data assimilation for a one-layer shallow-water equation in a twin experiment: data are generated by an N × N enstrophy-conserving grid integration scheme along with an Ekman vertical velocity at the bottom of an Ekman layer driving the flow and Rayleigh and eddy viscosity dissipation damping the flow. Data are generated for N = 16 and the chaotic flow that results is analyzed. This analysis is performed in a path-integral formulation of the data assimilation problem. These path integrals are estimated by a Monte Carlo method using a Metropolis Hastings algorithm. The authors' concentration is on the number of measurements Lc that must be assimilated by the model to allow accurate estimation of the full state of the model at the end of an observation window. It is found that for this shallow-water flow approximately 70% of the full set of state variables must be observed to accomplish either goal. The number of required observations is determined by examining the number needed to synchronize the observed data Lc and the model output when L data streams are assimilated by the model. Synchronization occurs when and the correct selection of which data are observed is made. If the number of observations is too small, so synchronization does not occur, or the selection of observations does not lead to synchronization of the data with the model output, state estimates during and at the end of the observation window and predictions beyond the observation window are inaccurate.