River Quality Modeling: Time Domain Approach

River quality and quantity variables such as specific conductance (SPC, μs/cm), total dissolved solids (TDS, mg/L), and daily mean discharge (DDM, $m^3/s$) reported by others for Battle River in Saskatchewan, Canada are analyzed by state space methods. The variables are observed over a period of 72 months at one month intervals and there are a total of 12 missing observations. A log transformation is used on DDM data before carrying out the analysis because of orderof‐magnitude changes in the data. The SPC and TDS series are divided by 100 for numerical stability. The scatter diagram of the variables indicate a deterministic inversely varying relationship between DDM, SPC, and TDS. There is a positive correlation between SPC and TDS. A trivariate second‐order state space model is fitted to the observations. This model is selected because it provides the smallest value of the Akaike Information Criterion (AIC) among first‐, second‐, and thirdorder models. This model is used to estimate the missing observations and to forecast future values of the variables along with their standard deviation of estimation. The expectation‐maximization (EM) algorithm and Kalman smoothed estimators are used to estimate the model parameters by the maximum likelihood. Dynamic multivariate models such as this can model the stochastic relationship between variables that are measured in time or space. Such models can improve our understanding of complex systems such as Battle River in terms of irrigation, policymaking decisions, and environmental considerations.