Estimation Theory Applied to River Water Quality Modeling

Estimation Theory Applied to River Water Quality Modeling by David S. Bowles, Doctor of Philosophy Utah State University, 1977 Major Professor: Dr. J. Paul Riley Dissertation Director: Dr. William J. Grenney Department: Civil and Environmental Engineering The extended Kalman filter (EKF) is used to represent BOD, DO, and nitrogen cycling in a 36.4 mile (58.6 km) stretch in the Jordan River, Utah, under the assumption of steady-state conditions. Approximate minimum variance estimates of the water quality parameters are provided by the EKF filter. These estimates are obtained through a combination of two independent estimates of the state of the river water quality system: (1) predictions of the system state from a "phenomenologically meaningful" process model of the biochemical and stream transport processes; and (2) measurements of the water quality parameters. These two estimates are combined by a weighting procedure based on the uncertainties associated with the process model predictions and the measurements. The EKF also yields an estimation error covariance matrix from which confidence limits for the accuracy of the parameter estimates are obtained. A sequential arrangement of extended Kalman filters is utilized. Each EKF in the sequence represents a river reach for which hydraulic and water quality characteristics are fairly uniform. Initial conditions for each EKF are based on the final conditions of the previous EKF adjusted to xiii represent the effect of point loads or tributaries discharging into the main river between the two reaches. A trial-and-error calibration procedure is used to obtain values for the model coefficients in the process model operated as a deterministic model independent of the filter. Determination of values for the Q matrix by a trial-and-error procedure is described. The approach is based on the requirement that the mean square error of the differences between the filter estimates and the measurements be not less than the measurement noise variance. A property of the EKF is that information contained in the measurements is used only in subsequent estimates of the system state. Therefore, information in the measurements is used only downstream of the sampling point at which the measurement was taken. To make use of the measurements in both the upand downstream directions a fixed-interval smoothing algorithm (FIS) is implemented. This technique combines state estimates from filter passes in the upand downstream directions. Sequential linearized Kalman filters are used for the pass in the upstream, or backward direction. Unlike the forward and backward estimates of the estimation error (P), the smoothed values are not characterized by discrete jumps at sampling points. Instead, the estimation error rises to a peak approximately midway between sampling locations. This characteristic indicates that when information from all the measurements is used confidence in the estimates decreases with distance from the adjacent sampling points. Smoothed values of P are less than the values obtained from applying the forward EKF alone; thus indicating that the estimates obtained from the FIS are better, in a minimum variance sense, than the estimates obtained from the forward EKF.