Rating Curves

Campbell and Bauder [1940] observed that the relation between the logarithm of sediment concentration and the logarithm of discharge was approximately linear. They suggested that this relation could be used as a ``rating curve.'' For periods when no sediment data had been collected, sediment concentrations could be estimated from water discharge. Although the rating curve remains an empirical result without physical justification, it has come into widespread use. It appears to be adequate for many purposes [ Miller, 1951; Colby, 1955]. The rating curve is simple, and, by including additional regressor variables (for example, see Cohn et al. [1992a]), can be easily modified to account for variability associated with non-linear flow dependence and time trends. One can stratify data (e.g. by season, discharge or other variable) or use multiple rating curves to describe more complicated concentration/discharge relations [ Colby, 1955]. The rating-curve's shortcomings, some of which are discussed below, are also well-documented [ Walling and Webb, 1981; Walling and Webb, 1988].

By making some assumptions about sediment transport functions, the rating curve can be converted into an intrinsically linear model [ Draper and Smith, 1981]:

where

is the natural logarithm function;

are model coefficients;

is residual error; the regression residuals are commonly assumed independent and identically distributed (iid) normal random variables, with a mean of zero and variance denoted by .

With coefficients estimated by linear regression, a continuous trace of concentrations can be estimated from:

where

are ordinary least squares regression coefficients.

Because this model provides a convenient statistical framework, much recent work has been devoted to correcting its deficiencies and expanding its applicability.