Calibration of transfer function-noise models to sparsely or irregularly observed time series

Abstract

A method is presented to calibrate transfer function-noise (TFN) models, operating at the same frequency as the input (auxiliary) variables, to sparsely or irregularly observed time series of the output (target) variable. Once calibrated, the TFN models can be used to predict or simulate the output variable at the same frequency as the input variable. Consequently, the method provides a useful tool for filling in gaps of irregularly or sparsely observed hydrological time series. Although generic and suitable for any type of time series, the method is described through the modeling of a time series of groundwater head data with precipitation surplus (precipitation minus potential evapotranspiration) as input variable. First, the TFN model is written in vector notation, yielding the state equation of a linear discrete stochastic system. Subsequently, the state equation is embedded in a Kalman filter algorithm. The Kalman filter is then combined with a maximum likelihood criterion to obtain estimates of the parameters of the TFN model for small time steps (e.g., 1 day) while using sparsely (e.g., two times a month) or even irregularly observed time series of groundwater head data. The method is illustrated using (subsets of) time series of groundwater head data with varying regular and irregular observation intervals.