Estimating longitudinal dispersion coefficient in rivers using non-linear regression model

Investigating the process of pollutant dispersion in natural rivers is important in pollution control and its distribution and management of aquatic environments. The process of pollution distribution is a function of dispersion coefficient that has been used in Advection-Diffusion equation. Since the one-dimensional models are applied for river water quality modeling, the first step in modeling the quality of river water is determining the Longitudinal Dispersion Coefficient (LDC). Many methods such as empirical, analytical, statistical, field measurements, and more sophisticated approaches, i.e., artificial intelligence techniques have been suggested for LDC estimation. By now, Laboratory methods are associated with many limitations. These methods are along with high costs and detrimental effects on the aquatic environment due to the application of some specific and harmful tracers. Artificial Intelligence Models have had a good prediction for LDC in recent years, but uncertainty in these models, as an annoying factor, always limited their results for practical purposes. These techniques act exactly like a black box model. Also, although different studies have been carried out for LDC estimation in natural rivers, the absence of a comprehensive study to investigate the effects of different patterns of dimensionless hydraulic and geometric parameters on LDC is still felt. Another challenge is the inaccuracy and less user-friendliness of the provided models to predict LDC. Thus, the main objective of this study is to investigate different types of LDC prediction by application of regression analysis.
The data used in this study for estimation of LDC contains 61 datasets of hydraulic parameters such as velocity U (m/s), shear velocity u*(m/s) and flow rate (Q), and also the geometric specifications measuring in cross-section of rivers like water depth H(m), stream width (W) and curvature of the river σ. This dataset was taken from different sections of 31 rivers in the United States of America. This information was used as input parameters (about 70 percent, i.e 43 data for calibration and 30 percent, i.e. 18 data for testing) for the non-linear regression model.
Many of researchers have used equations based on establishing a relationship between LDC and hydraulic-geometric specifications of rivers. Some researchers have used other parameters such as Q and σ to improve LDC estimation. So, in this research, all combination of input parameters for application of non-linear equations have been investigated. Non-linear equations could be changed to linear equations by a logarithmic method. Thus, all of the unknown coefficients of equations could be known by a linear Least Squares Regression.
Calibration and test steps were carried out using various patterns of hydraulic and geometric data of the rivers in the United States of America. The obtained equations have been compared by the statistics of regression method such as coefficient of determination (R2), t-value, p-value and Variance Inflation Factor (VIF). Higher values of t-value led to the lower value of p-value that indicated the importance of parameter in equation. VIF shows the existence of multi-collinearity in equation. Results showed that the pattern including the river curvature parameter had the best performance for LDC prediction model, while discharge parameter had the least effects on the LDC prediction model. Also, if the constant number is eliminated from the equation, the model’s performance is increased. The R2 of calibration and test stages of the best tuned model were 0.993 and 0.938, respectively. Selected equations in this research (with the best performance) compared with the models suggested by other researchers based on Root Mean Square Error (RMSE), Mean Related Error (MRE), Developed Discrepancy Ratio (DDR), and Threshold Statistics (TS) indices. The comparison of the developed model with the past studies revealed that it had a better performance in LDC estimation. Although the model proposed by Zeng and Huai (2014) had the second best performance, its statistical indices (RMSE and MRE)were about 2 and 83 times greater than that those obtained for the developed model. The best model based on the DDR graphic statistic has the largest crest and lowest width. So the comparison of models based on DDR indicated that the developed model in this research had the best performance. Also, the graph of the TS showed that the model developed in this research had the lowest threshold of errors in 100 percent of datasets. Thus, this research provides an LDC estimator model that outperforms the other existence equations.