State Space Identification of Linear Deterministic Rainfall-Runoff Models

Rainfall‐runoff models of the black box type abound in the water resources literature (i.e., transfer function, autoregressive moving average (ARMA), ARMAX, state space, etc.). The corresponding system identification algorithms for such models are known to be numerically efficient and accurate, leading in most cases to good parsimonious representations of the rainfall‐runoff process. Alternatively, every model in transfer function, ARMA, and ARMAX form has an equivalent state space representation. However, state space models do not necessarily have simple system identification algorithms, unless the system matrices are restricted to some canonical form. Furthermore, state space system identification algorithms that work with the rainfall/runoff data directly (i.e., covariance free), require initial conditions and are inherently iterative and nonlinear. In this paper we present a state space system identification theory which overcomes these limitations. One advantage of such a theory is that the corresponding algorithms are highly robust to additive noise in the data. They are referred to as “subspace algorithms” due to their ability to separate the signal subspace from the noise subspace. The main advantages of the subspace algorithms are the automatic structure identification (system order), geometrical insights (notions of angle between subspaces), and the fact that they rely on robust numerical procedures (singular value decomposition). In this paper, two algorithms are presented. The first one is a two‐step procedure, where the impulse response (unit hydrograph ordinates for the single‐input, single‐output case) are computed from the input/output data by solving a constrained deconvolution problem. These impulse response ordinates are then used as inputs for identifying the system matrices by means of a Hankel‐based realization algorithm. The second approach uses the data directly to identify the system matrices, bypassing the deconvolution step. The algorithms are tested with real data from the Voer catchment in Belgium