### Standard errors of annual discharge and change in reservoir content data from selected stations in the lower Colorado River streamflow-gaging station network, 1995-99

By: David W. Anning

                            https://doi.org/10.3133/wri014240

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#### Abstract

The Bureau of Reclamation is currently (1995–2001) testing the Lower Colorado River AccountingSystem as a method to estimate the consumptive use of Colorado River water by diverters from HooverDam to Mexico. Consumptive use is estimated in the Lower Colorado River Accounting System method,in part, on the basis of the annual discharge or annual change in reservoir contents, as well as the varianceof estimate of the annual discharge or the annual change in reservoir contents at several surface-watergaging stations in the lower Colorado River stream-gaging network. The standard error and the varianceof estimate were determined for the annual discharge at 14 streamflow-gaging stations and for the annualchange in content at 2 reservoir-content gaging stations used in the Lower Colorado River AccountingSystem for calendar years 1995–99.

The standard error of the annual discharge was determined by using modifications to an existingmethod that assumes that the uncertainty in the discharge-rating shift is the main source of uncertainty incomputed discharges and that the discharge-rating shift behaves as a first-order Markovian process. Themethod uses Kalman filtering of a first-order Markovian process as a statistical analogy to computingstreamflow with a shifted discharge rating. Temporally unbiased residuals from a discharge rating are usedas a surrogate for the actual shifts used to compute discharge. The standard error of the annual discharge isdetermined by using Kalman-filter theory and estimates of four parameters: (1) the measurement varianceof the discharge measurements used to determine the discharge-rating shift, (2) the process variance of thedischarge-rating residuals, (3) the serial correlation of the discharge-rating residuals, and (4) thefrequency of the discharge measurements. The existing methodology was improved by estimating themeasurement variance from a semivariogram of the discharge-rating residuals, rather than on the basis ofempirically derived error estimates for discharge measurements. The process variance and serialcorrelation of the discharge-rating residuals are estimated from the semivariogram, rather than avariogram, of the discharge-rating residuals. The empirically derived estimates are based oncharacteristics of the discharge measurements such as number of depth and velocity observation sections,type of current meter, and bed material composition and stability. Measurement variance determined fromthe semivariograms was site specific and is therefore considered a better estimate than measurementvariance determined from the empirically-derived estimates. The method of estimating the standard errorof the annual discharge requires the assumption of unbiased discharge-rating residuals, and for thisreason, the standard errors presented in this report only represent the random error in the annual dischargedata. Estimates of the standard error of the annual change in reservoir content were determined on thebasis of the reservoir-surface area and the standard error of reservoir-stage readings.

The standard error of the annual discharge, as a percentage, ranged from 0.11 percent for the All-American Canal near Imperial Dam in 1998 to 12.3 percent for the Colorado River below Imperial Damin 1996. The standard error of the annual discharge was less than 2 percent for all 5 years for 11 of the14 streamflow-gaging stations. In terms of flow volume, the standard error of the annual discharge rangedfrom 97 acre-feet for the Mittry Lake Diversions in 1995 to 77,000 acre-feet for the Colorado River at thenortherly international boundary with Mexico in 1998. In general, the standard error of the annualdischarge, as a percentage, was smallest at streamflow-gaging stations on the main stem of the ColoradoRiver; however, the standard error of the annual discharge in acre-feet was largest at these stationsbecause of the large annual discharge on the main stem. The standard error of the annual change incontent for the two reservoirs ranged from 1,590 acre-feet for Lake Havasu in 1996 to 2,790 acre-feet forLake Mohave in 1995.

The variance of estimate of the annual discharge for a streamflow-gaging station can be reduced bymaking additional discharge measurements; either by increasing the number of discharge measurementsmade per site visit, or by increasing the frequency of site visits. Measurement error can be reduced byusing the average shift for two or more discharge measurements made during a site visit. For astreamflow-gaging station where measurement error is much greater than process error and the serialcorrelation of the discharge-rating residuals is high, an improved gaging strategy would involve makingmultiple discharge measurements per site visit. In contrast, for a streamflow-gaging station where processerror is much greater than measurement error and the serial correlation of discharge-rating residuals islow, the gaging strategy would consist of several single discharge-measurement site visits. For a givenoperating cost or for a given variance of estimate of the annual discharge at a streamflow-gaging station,the optimal site-visit and discharge-measurement strategy can be determined, providing that the travelcosts as well as the measurement variance, process variance, and serial correlation of discharge-ratingresiduals are known.

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