The Upper Half and Lower Half methods split the data and search the specified half for a local minimum, or users can specify a range over which the algorithm will search for a minimum. The tool includes three options to constrain the search range: The global minimum is associated with a low flow, but there is a much higher flow inflection point that improves the overall fit substantially.īy constraining the inflection-point candidates to a particular flow range, users can force the model to select a local minimum and specify an inflection point that is more morphologically meaningful. For example, the following figure includes the RMSE computed from each candidate inflection point. In some cases, the data can have multiple, local, minimums, in the RMSE, including a potential inflection point in the higher flow data that is not the global RMSE minimum. But because these flows do not deliver much sediment, the total sediment load will not be very sensitive to this constraint. Users can request the biased result, but this is not recommended.īecause sediment data often over-represent the low flows, the flow-load inflection point that minimizes the RMSE can turn up in the lower flows (see Figure below). If you select this option, it will update the Piecewise correction factors for both Duan and Furguson (below) and will plot and report the unbiased result by default. The piecewise linear model also computes biased and unbiased regressions. Because most of the sediment moves in the moderate-to-large flow range, fitting a separate slope to these larger flows can affect the sediment budget and model dramatically. In the case pictured above, the model computed an inflection point at 6,490 cfs, and fit a steeper slope to the lower flows than the higher flows (which is typical of rivers in this region). The algorithm selects the inflection point with the lowest Root Mean Square Error (RMSE). "knot" in the statistical terminology) and fits separate-but-continuous power functions to the upstream and downstream data. The calculator evaluates every data point as a potential inflection point (i.e. The simplest form of the piecewise linear model is included in the figure above. HEC worked with UC Davis () to develop a piecewise linear algorithm that identifies the inflection point in an inflected or bent rating curve and fits a continuous model to the upper and lower halves of the data. But a two-slope, piece-wise linear, regression can capture some of this complexity. Future version of this tool will include local regression methods, that will allow users to develop more sophisticated rating curves. Additionally, "bent" or "inflected" rating curves are relatively common, particularly in supply limited systems. Sediment data tend to over-represent low flows, which can dominate the regression in the moderate-to-large flow range that is most morphologically active. Often, a single power function does not capture the complexity of the data.
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