The Missing Sediment: Retransformation Bias in Log-Log Rating Curves, Tested on Daily Loads of the Mississippi at St. Louis and Thebes

Sediment loads are often estimated from a few samples through a rating curve fitted in logarithms and then exponentiated. Exponentiating a fitted log gives a median, not a mean, so the load is underestimated. I prove that the naive estimate is biased low for any error distribution, derive the exact small-sample expectations of the two standard corrections, the lognormal factor exp(s squared over 2) and Duan's smearing estimator, and test them on daily records of suspended-sediment discharge from the US Geological Survey. At St. Louis, over 41 complete water years from 1981 to 2022, the log-log rating has slope 2.0965 and residual standard deviation 0.5940, and the naive curve recovers only 85.54% of the measured load. The lognormal correction gives 102.05% and smearing 103.60%. In a sampling experiment that draws two days a month and estimates each year's load, the naive estimate is low by 10.73% on average, with a root mean square error of 0.1692; both corrections remove the bias and lower the error to about 0.16. Thebes gives the same picture with a naive shortfall of 11.19% over the record. The rating curve itself has shifted: the load predicted at the median flow at St. Louis fell from 176,635 tonnes a day in the 1980s to 88,372 in the 2010s, so a curve from one decade misstates loads in another by far more than the retransformation bias.