Regression Residuals and Mean Profiles in Uniform Open-Channel Flows

In spite of the frequent use of regression in hydraulics, the analysis of regression residuals is often not performed and yet may be important in assessing the performance of proposed models. The present work discusses residuals analysis with respect to (1) velocity profiles in clear-water channel flows; and (2) suspended-sediment concentration profiles in sediment-laden flows. Three models proposed for the mean velocity profile in turbulent uniform open-channel clear-water flows, differing only in the treatment of the wake component, are studied. The analysis of residuals from linear regression applied to velocity measurements suggests that there are statistical grounds for believing that these wake-function models are inadequate. The traditional Rouse suspended-sediment concentration profile provides the context for a comparison of a direct application of nonlinear regression for fitting purposes rather than the more traditional linear regression of a transformed problem. The importance of the uniform-variance assumption in least-squares fitting is emphasized. The analysis of residuals indicates that, in spite of smaller residual mean squares, direct nonlinear regression is inappropriate due to the nonuniform variance of the error term. Possibly misleading interpretations of high values of the coefficient of determination (R²) are pointed out, and implications for velocity profiles in sediment-laden flows and more complicated models of suspended sediment concentration are discussed.