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

The Bureau of Reclamation is currently (1995–2001) testing the Lower Colorado River Accounting
System as a method to estimate the consumptive use of Colorado River water by diverters from Hoover
Dam 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 variance
of estimate of the annual discharge or the annual change in reservoir contents at several surface-water
gaging stations in the lower Colorado River stream-gaging network. The standard error and the variance
of estimate were determined for the annual discharge at 14 streamflow-gaging stations and for the annual
change in content at 2 reservoir-content gaging stations used in the Lower Colorado River Accounting
System for calendar years 1995–99.

The standard error of the annual discharge was determined by using modifications to an existing
method that assumes that the uncertainty in the discharge-rating shift is the main source of uncertainty in
computed discharges and that the discharge-rating shift behaves as a first-order Markovian process. The
method uses Kalman filtering of a first-order Markovian process as a statistical analogy to computing
streamflow with a shifted discharge rating. Temporally unbiased residuals from a discharge rating are used
as a surrogate for the actual shifts used to compute discharge. The standard error of the annual discharge is
determined by using Kalman-filter theory and estimates of four parameters: (1) the measurement variance
of the discharge measurements used to determine the discharge-rating shift, (2) the process variance of the
discharge-rating residuals, (3) the serial correlation of the discharge-rating residuals, and (4) the
frequency of the discharge measurements. The existing methodology was improved by estimating the
measurement variance from a semivariogram of the discharge-rating residuals, rather than on the basis of
empirically derived error estimates for discharge measurements. The process variance and serial
correlation of the discharge-rating residuals are estimated from the semivariogram, rather than a
variogram, of the discharge-rating residuals. The empirically derived estimates are based on
characteristics 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 from
the semivariograms was site specific and is therefore considered a better estimate than measurement
variance determined from the empirically-derived estimates. The method of estimating the standard error
of the annual discharge requires the assumption of unbiased discharge-rating residuals, and for this
reason, the standard errors presented in this report only represent the random error in the annual discharge
data. Estimates of the standard error of the annual change in reservoir content were determined on the
basis 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 Dam
in 1996. The standard error of the annual discharge was less than 2 percent for all 5 years for 11 of the
14 streamflow-gaging stations. In terms of flow volume, the standard error of the annual discharge ranged
from 97 acre-feet for the Mittry Lake Diversions in 1995 to 77,000 acre-feet for the Colorado River at the
northerly international boundary with Mexico in 1998. In general, the standard error of the annual
discharge, as a percentage, was smallest at streamflow-gaging stations on the main stem of the Colorado
River; however, the standard error of the annual discharge in acre-feet was largest at these stations
because of the large annual discharge on the main stem. The standard error of the annual change in
content for the two reservoirs ranged from 1,590 acre-feet for Lake Havasu in 1996 to 2,790 acre-feet for
Lake Mohave in 1995.

The variance of estimate of the annual discharge for a streamflow-gaging station can be reduced by
making additional discharge measurements; either by increasing the number of discharge measurements
made per site visit, or by increasing the frequency of site visits. Measurement error can be reduced by
using the average shift for two or more discharge measurements made during a site visit. For a
streamflow-gaging station where measurement error is much greater than process error and the serial
correlation of the discharge-rating residuals is high, an improved gaging strategy would involve making
multiple discharge measurements per site visit. In contrast, for a streamflow-gaging station where process
error is much greater than measurement error and the serial correlation of discharge-rating residuals is
low, the gaging strategy would consist of several single discharge-measurement site visits. For a given
operating 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 travel
costs as well as the measurement variance, process variance, and serial correlation of discharge-rating
residuals are known.