Modification 1: Retransformation and Bias Correction Factors

The rating curve estimator of equation (5) is not statistically consistent [ Lane, 1975; DeLong, 1982; Thomas, 1985; Jansson, 1985; Ferguson, 1986; Koch and Smillie, 1986; Cohn et al., 1989]. Its results are biased, in general systematically underestimating loads. In studies with field data [ Walling et al., 1981; Fenn et al., 1985], this bias sometimes exceeded 50%. Thomas [1985], Ferguson [1986], and Koch and Smillie [1986] note that the bias arises when model results computed using the logarithm of C are ``retransformed'' into real units. Three methods for compensating for this bias are now commonly employed. Thomas [1985], Ferguson [1986] and Koch and Smillie [1986] describe an approximately-unbiased estimator:

where

s

is the estimated standard deviation of the residuals.

where

N

is the number of observations (measured concentrations) used to calibrate the regression model;

is the regression residual corresponding to the i-th observation

The three bias-correction methods yield nearly identical results under the following conditions:

1. the assumed linear model is approximately correct;

2. there is a reasonably large sample for calibrating the regression model (i.e. 30 observations or more);

3. the model is not being used to extrapolate beyond the range of data used for model calibration. For example, if only low-flow data are used to calibrate the regression model, then the methods will generate different load estimates for periods of high flow.

If condition 1 is satisfied, is best. Where conditions 2 and 3 are also satisfied, may be a good choice because it is relatively easy to compute and will closely approximate . Where regression residuals are not normally distributed, the nonparametric smearing estimator, , may be best. However, under such circumstances one must verify that use of a regression model is appropriate.