Estimation of Discharge Error

Uncertainty in estimating instantaneous discharge is produced by random and systematic errors. Random discharge errors can be removed by data averaging, whereas systematic discharge errors cannot be removed by averaging. Two principal sources of error in the estimate of instantaneous discharge are instrument errors associated with measurement of water depth and index velocity, and errors associated with the cross-sectional area and mean-velocity equations (Sloat and Gain, 1995). Two methods of estimating discharge error are discussed in this report. The first method uses the standard error of the estimate of the regression equations for estimating mean velocity in the river; the second method compares the difference of computed discharge (estimated using regression equations) to measured discharge and then uses the Wilcoxon Signed-Ranks test to determine if the median difference between the values is different from zero. A difference from zero indicates a bias in the computed discharge.

The first method estimates instantaneous discharge error by multiplying the standard error estimate for each index-to-mean velocity regression equation by the corresponding median cross-sectional area. The instantaneous discharge errors using the standard error and median cross-sectional areas are listed in table 3 along with basic regression statistics for the velocity regression equations. Residual plots of the index-to-mean velocity regression equations were analyzed and appear to be random, indicating that biases are unlikely. Maximum instantaneous discharge errors were approximately 213, 65, 326, 453, and 193 ft³/s for Lostmans Creek, Broad River, Harney River, Shark River, and North River, respectively (table 3). The maximum discharge errors as a percentage of maximum instantaneous discharge are 9, 2, 2, 4, and 18 percent for Lostmans Creek, Broad River, Harney River, Shark River, and North River, respectively. Random errors associated with individual acoustic discharge measurements were estimated to be less than 3 percent of the true discharge based on a simple random error model for acoustic discharge measurement systems (Simpson and Bland, 2000). Simpson and Bland (2000) also noted that the measurement of less than the entire river cross section for an index velocity may result in sampling bias (systematic error). The index-velocity methods used in this study sampled a relatively small horizontal cross section of the rivers and could be biased. Other methods of index-velocity measurement such as horizontal-averaging velocity systems can also be biased, so care must be used when determining any index-to-mean velocity relation.

The second method for estimating instantaneous discharge error provides information about discharge random error and bias. Discharge errors were calculated by subtracting the computed discharge from the measured discharge, and then basic statistics (minimum, maximum, mean, and standard deviation) were calculated for the discharge errors. The standard deviations of the discharge error are similar to the errors computed using the first method (table 3). Additional discharge measurements that were not used to develop the index-velocity relations were used in this method for three of the five stations (Broad, Harney, and Shark Rivers which had data-collection periods that were 2 years longer).

The Wilcoxon-Signed-Ranks test (WSR) was used to compare the difference between the medians of computed discharges and measured discharges. The WSR test can provide information relating to biases between the computed and measured discharges. The median values for discharge error are not significantly different from zero for four of the five stations based on the WSR probabilities (table 4).

The median values not significantly different from zero (probabilities 0.29 to 0.85) indicate that bias in the discharge is unlikely. However, the Shark River WSR probability less than 0.002 indicates that the median error for discharges is significantly different from zero, which could indicate a positive bias for computed discharge.

The Shark River instantaneous discharge bias based on the WSR is between 74 and 125 ft³/s (table 4), indicating that the true discharges for the Shark River may be slightly lower than the values in this report. This bias may be attributed in part to the confluence of a small creek that enters the Shark River upstream of the discharge measurement section that was not well represented using the index-velocity system.

The North River station had the greatest percentage error in instantaneous discharge, as expected based on the velocity regression model (R² = 0.60 compared to 0.92 to 0.98 for the other stations). However, the long-term North River mean discharge should be close to the actual discharge because residual plots (figs. 4A-E) of the index-to-mean velocity regression data and Wilcoxon-Signed-Ranks test (table 4) indicate the uncertainty is random and the mean and median WSR errors are not significantly different from zero. The poor index-to-mean velocity regression model for the North River indicates the velocity sensor was located in a section of the river that did not represent the mean channel velocity all of the time. In addition, a relatively low tidal range (typically less than 1/2 ft) measured at this station could allow the flow distribution across the channel to be easily changed by wind forcing. Additionally, the channel geometry (wide and relatively shallow with no distinct center channel and confluence of two river sections upstream of the station) also could have contributed to a poor index-to-mean velocity relation.

Figure 4. Residual plots of index-to-mean velocity regression relations for Lostmans Creek, Broad River, Harney River, Shark River, and North River stations. [click on images above for larger versions]

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Table 4. Computed discharge error estimation based on measured discharge and Wilcoxon Signed Ranks probability for determining median error is different from zero. [Error = computed discharge minus measured discharge; ft3/s, cubic feet per second; WSR, Wilcoxon Signed Ranks test]
Station name	Number of discharge measurements	Minimum error (ft3/s)	Maximum error (ft3/s)	Mean (ft3/s)	Median (ft3/s)	WSR probability that median difference is zero	Standard deviation (ft3/s)
Lostmans Creek	69	-334	490	16	6	0.85	209
Broad River	227	-350	335	-9	-5	0.29	98
Harney River	196	-1,268	900	-10	-31	0.67	337
Shark River	205	-1,208	1,055	74	125	0.002	398
North River	131	-378	510	12	3	0.62	192

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