The extended Kalman filter (EKF) is used to represent BOD, DO, and nitrogen cycling in a 36.4 miles (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 mdoel 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 extened Kalman filters is utilized. Each EKF in the sequence represents a river research 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 represent the effect of the 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. Determinatino of values for the Q matrix by a trail-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 that the measurement noise variance. A property of the EKF is that information contained in the measurement 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. The make use of the measurement in both the up- and 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 estimations 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 points. Instead, the estimation error rises to a peak approximately midway between sampling locations. This characteristic indicates taht when inforamtion 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 along; thus indicating that the estimates obtained from the FIS are better, in a minimum variance sense, than the estimates obtained from the forward EKF. To assist in gaining familiarity with the filtering technique, several sensitivity studies are perfomred. The sensitivity of filter estimates to changes in the following statistics were investigated: the process model noise variance, the measurement noise variance, the initial estimation error variance, and the point load estimatino error variance. A large value for the process model noise variance has the effect of: (1) increasing the rate of growth of the estimation error, and (2) placing additional weighing on the measurements because the larger estimation error implies less confidence in the process model predicitions. At sampling points the measurement update procedure always results in an estimation error less than the measurement noise regardless of the values used for the process model noise variances. Changing the values of the measurement noise affects (1) the level of the estimatino error after measurement updates, and (2) the weighting given to measurement. A larger initial estimation error variance gives relatively more weighting to the measurements but this effect decreases with distance from the upstream boundary. The sensitivity study on the point load estimation error variances indicates a snall, but noticeable, effect on the estimation errors, and therefore, a slight effect on the state estimates via the weighting procedure. The capability of estimating model coefficients and lateral inflow concentraions simultaneously with the water quality prameters is demonstrated. In one run five coefficients in the equations descriving nitrogen cycling are estimated. This run also provides an example of filter divergence.
Bowles, David S.; Grenney, William J.; and Riley, J. Paul, "Estimation Theory Applied to River Water Quality Modeling" (1977). Reports. Paper 588.