INTRODUCTION

The Delta region of California consists of a network of branching, interconnected channels which are strongly tidally affected . As these channels merge, the tidal flow increasingly dominates until the net flow of fresh water becomes undetectable. Four sites have been selected for measurement of the discharge (see figure 1). They are far enough downstream that uncertainties due to consumptive use are minimized, and far enough upstream that the tidal flow may not completely swamp the net discharge.

In recent years, two developments have made possible the accurate measurement of flow in tidally affected rivers and estuaries: (1) the acoustic Doppler current profiler (ADCP)and (2} improved methods for the in-situ measurement of a channel velocity "index."

As an illustration of the accuracy attainable, we will carry out a detailed analysis of measurement errors for the Three-Mile Slough measurement site, with an ultrasonic velocity meter (UVM) as the index device. (We do not consider in detail the effects of stratification in temperature or salinity, since at the four sites chosen the salinity is negligible.)

Accuracy Required

The Delta outflow Q_{D }is given in terms of the discharge at the four measurement sites by

Q_{D} = Q_{4}- Q_{5} + Q_{6} + Q_{7}, (1)

where the positive direction is north-to-south or east-to-west. For most purposes it is desirable to calculate an average value of Q_{D} with the effects of the tides removed. The Delta outflow varies dramatically throughout the year and from season to season. While monitoring the high-flow regime is useful, the most important flow figures are the lowest ones. This is because this part of the regime is more directly regulated, and because the competition between the various demands for water is strongest at times of low flow.

During a three-year period from 1994 to 1996, the full Delta outflow peaked at 60,000 m^{3}/s, with a minimum flow value of about 60 m^{3}/s. If we try to measure this minimum flow with a 10% error, and share it the error equally among the four sites involved in the measurement, we arrive at a goal for Three-Mile Slough of an error of 1.5 m^{3}/s. (For comparison, the peak tidal flow in TMS is about 1100 m^{3}/s.) While these values are specific to the chosen site, the methodology and analysis described below should be generally applicable at other estuarine sites.

THE INDEX VELOCITY, WITH THE UVM AS AN EXAMPLE

Determining the net flow in the presence of a large tidal flow requires continuous, rapid sampling of the flow. However, continuous, direct measurement of channel discharge is prohibitively time consuming. Thus, an average velocity such as that measured by a UVM is usually selected for continuous monitoring. In UVM measurements, ultrasonic transducers are mounted on pilings near the shore on opposite sides of the channel, as shown in figure 2. The signal propagation time is measured for each propagation direction, and the current velocity computed from the relation (Laenen 1985)

Here:

V_{L} = line velocity, or average water velocity parallel to the channel,

B = length of the acoustic path from A to C,

q = angle between the channel centerline and the acoustic path,

t_{AC} = traveltime from A to B,

and

t_{CA} = traveltime from C to A.

The same transducers are used to transmit and receive, to maintain reciprocity.

Random Error

The main sources of random error in the UVM line velocity are timing error and water variability (turbulence, speed-of-sound fluctuations). We have studied the random error, including all sources of noise, and find an rms error of about 0.005 m/s, including water variability. Averaged over the many individual measurements that contribute to a single measurement of the discharge, this leads to a negligible error in the daily discharge.

Principal UVM Bias Errors

There are many systematic errors in measuring the UVM line velocity. However, most of them cancel out in measuring net flow. Two examples are error in determining the path length, and error in determining the baseline angle theta. A *constant* 3% error in either of these quantities would lead to a negligible 3% error in the daily discharge. On the other hand, a 1% error occurring for one current direction but not the other would lead to a much larger error in the daily discharge. We will focus on systematic errors of this type (biases). In each case we will estimate the effect of a hypothetical bias on a daily discharge of 10 m^{3}/s, with a peak tidal flow of 1000 m^{3}/s and a peak tidal current velocity of 1.0 m/s.

**1. Timing errors leading to late detection of threshold crossing.** Possible sources: reduction in signal size, EM interference, and boat noise. To cause a bias, the errors must occur only for one sound propagation direction, and only for one direction of current flow, which seems very improbable. A one-cycle delay in 1% of the measurements, biased as described, would lead to an error in the daily discharge of 1.4 m^{3}/s. We have monitored the triggering conditions for a commercial UVM under typical operating conditions and observed no one-cycle errors in 138 triggers. For a well-designed system this error should be negligible.

2. Temperature variations. Since the velocity as determined from equation (1) does not depend on the speed of sound, temperature variations from ebb to flood have no effect.

3. Temperature gradients. A temperature gradient of 0.1 C^{o}/m (in fresh water at 12°
C) causes curvature of the sound path, deflecting it upwards at the center by 1.3 m and increasing the distance traveled by 0.022 m. If such a temperature gradient is present during flood tide (say) but not during ebb, a bias error occurs. The increase in length would cause a fractional velocity error of 0.01%, leading to a negligible error in the daily discharge of 0.03 m^{3}/s. The deflection of the path has the effect of causing the UVM to sample the velocity at a different depth. As an example, let us assume a y^{1/6} velocity profile, a water depth of 6.2 m and an acoustic path 2.1 m below the surface (numbers typical of TMS). Then raising the acoustic path by 1.3 m increases the measured velocity by 4.7%! While a gradient this large may be rare, and would probably be present to some extent at both ebb and flood tide, the magnitude of the effect makes it a significant concern.

4. Salinity gradients. A salinity gradient of 1 ppt per meter would have the same effect as the thermal gradient discussed just above.

5. Motion of the transducer mounts (pilings). Suppose that one piling deflects by 0.1 m, preferentially in one current direction (implausible?) This would change the baseline by one part in 2000, leading to an error in the daily discharge of 0.16 m^{3}/s.

6. Variation in the flow angle. Suppose that the angle of flow varies by one degree from an assumed value of 45^{o}, for one flow direction relative to the other. The calculated line velocity would be in error by one part in 45, leading to an error in the daily discharge of 8 m^{3}/s. The transverse motion at the level of the acoustic path would have to be balanced by an opposite motion at a different level, but this is not a far-fetched possibility. Differences between flood and ebb are also plausible, if the channel curves or has bottom irregularities near the measuring site. Such effects might depend on variables such as tidal amplitude and stage, making them difficult to calibrate out.

THE USE OF THE BOAT-MOUNTED ADCP TO MEASURE DISCHARGE

The ADCP discharge measurement system described in this report uses the acoustic Doppler shift technique to measure vertical profiles of horizontal water velocities at 25cm(centimeter) vertical intervals from a moving vessel. Figure 3 illustrates the operation of the ADCP. Four pulsed acoustic beams scatter off debris or inhomogeneities in the water, returning enough information to determine the average vector velocity of the region of water sampled by the beams. The use of this instrument mounted on a boat to measure discharge has been described by Simpson and Oltman (1990, 1993). Their tests showed that velocities measured by the ADCP agree to within 3.0 percent with velocities measured using conventional current meters (Simpson, 1986). A complete description of an ADCP system is beyond the scope of this report. Readers interested in details about the operation and accuracy of ADCP systems are referred to Simpson (1986).

One of the features of this system of special concern to us is the unmeasured parts of the water column. A blanking time after pulse transmission leaves a surface layer about 1 m thick unmeasured. And side-lobe reflection off the bottom prevents measurement of the lower 15% to 20% of the water column. To determine the discharge, the measured velocity profile must be extrapolated upwards and downwards. The channel at Three-Mile Slough is about 150 m across and 6 meters deep, with a fairly flat bottom, rising abruptly at the edges. The measuring station is situated in a three-km-long straight section which ends about 500 m from its confluence with the San Joaquin River. Chu (1995) has discussed vertical velocity profiles in channels such as that at Three-Mile Slough. He concludes that the distribution is reasonably well approximated by the relation

Here *y* is the distance from the bottom, and y_{max} is the water depth. We use this vertical profile to correct for the unmeasured top and bottom water layers. (For details, see Simpson and Oltman, 1990 and 1993.)

THE INDEX AND THE APPLICATION OF THE RATING CURVE

The measurement of flow in a tidal affected channel requires that the index velocity be accurately related to the mean channel velocity. This is done by collecting a data set of concurrent measurements of mean channel velocity using an ADCP discharge measurement system synchronized with index velocity measurements. The resulting data set is used to develop a relationship between mean channel velocity and index velocity. Here we consider the relationship between v_{ADCP} and the index velocity, the stage height, and other possible parameters. Our goal is to identify uncertainties in the indexing process, and to propagate these uncertainties to the measurement of discharge. We will discuss these issues in the context of the TMS case study, but with an eye towards application in any estuarine situation.

The calibration discharge measurements discussed here were taken with a boat-mounted acoustic Doppler discharge measuring system (ADDMS) (Simpson and Oltmann, 1990, 1993), while the UVM velocity and stage were logged at 5-minute intervals. In two calibrations, TMS3 and TMS4, the boat crossed the Slough about once every four minutes for twelve hours, thus covering a complete (semi-diurnal) tidal cycle. A third calibration (TMS1) was composed of measurements taken during several intervals of time over a 2-week period. In figure 4 we show the calibration data taken on June 2, 1998 (TMS3). The figure shows the tidal stage, the ADCP discharge, and the index velocity v_{UVM}. Together the three calibrations cover a wide range of current velocity and tidal stage.

In figure 6 we show the corresponding rating curves, giving ADCP velocity as a function of UVM velocity. Here we have corrected the UVM velocity for the effect of the current profile, assuming a y^{1/6} law:

where

v_{UVM } is the line velocity at the height of the acoustic path,

h is the instantaneous stage height above mllw (mean lower low water),

d_{bot} is the depth of the channel below mllw; here d_{bot} = 5.9 m,

d_{tran} is the depth of the transducers below mllw; here d_{tran} = 1.8 m,

vbar_{UVM} is the depth-averaged line velocity.

The basis for this correction was discussed in the previous section, and its validity has already been assumed in calculating discharge values. To calculate the ADCP velocity, first the channel cross-sectional area is calculated from survey data and the stage height. Then the ADCP discharge is divided by the channel cross-section to get an average channel velocity, v_{ADCP}.

If the index is a good estimator of channel velocity, the rating curve should be a straight line 45° from the x axis. All four calibrations are fairly close to this simple straight line. However, very slight departures have a significant effect on the net discharge. The curves of figure 6 show the following departures from the simple 45° line:

1. TMS1 and TMS3 both have data on the "turning in" (from maximum ebb to maximum flood) and "turning out" (from maximum flood to maximum ebb) branches of the curve. (TMS4 does not.) And they both show a hysteresis-like effect which we refer to as a "loop rating:" near zero velocity, the ADCP velocity "turns" before the UVM velocity does. This effect is what would be expected if the velocity turned sooner near shore, since the UVM measurement misses more water at the edges of the channel than does an ADCP transect. This effect can be seen in figure 7, where we show a 2-d plot of the ADCP velocity, projected on the channel centerline. This current distribution corresponds to a UVM velocity near zero, and shows the edges of the channel flowing in the opposite direction from the center. This effect is observed in all of the calibrations, during both the turning-in branch and the turning-out branch.

2. The slopes of the curves are not exactly equal to one.

3. TMS3 and TMS4 show a bend or kink in the middle, while TMS1 does not.

4. All three of the curves have a lower calibration for the positive velocities than for the negative velocities.

In order to use these calibration data to calculate discharge throughout the year, we need to fit the points to a smooth curve. This serves various functions. If the calibration data do not cover the full range of velocities (gaps, neap tide, etc.) the smooth curve interpolates and extrapolates as necessary. If a channel is well understood, a rating curve can be generated from a relatively small number of measurements taken at strategic places. Furthermore, if the curve is parametrized well, the values of the fitting parameters may shed light on important hydrodynamical effects taking place.

The rating-curve function we used is shown in figure 8. The corresponding function is

The parameters, illustrated in the figure, are:

x = index velocity

y = true average channel velocity

tan Q = slope of baseline

a = offset

b = ‘kink parameter’ - positive *b* raises both ends of the curve, relative to the middle

c = loop amplitude

d = loop width

Each of the five parameters above describes a type of deviation from the ideal 45° straight-line rating. Their interpretation is as follows:

slope: A slope less than 1 (theta = 45 deg) means that the UVM line average velocity over-estimates the channel average velocity. This might be expected because the UVM baseline preferentially samples the center of the channel and the top of the channel. The slope might also be different from 45° if the angle between the UVM acoustic path and the channel centerline is different from that used by the UVM software.

Offset *a* and kink parameter *b*: A non-zero value for either of these parameters has a direct effect on the average flow, boosting discharge in one direction while reducing it in the other direction.

Loop amplitude *c* and loop width *d:* The simple Gaussian model used here seems to fit the observed loop ratings. As long as the loop is symmetrical, as we assume here, it has a minimal effect on the discharge.

We have fit the three calibration data sets shown to this Gaussian-bulge model. The parameters of the fit are shown in Table 1. TMS1 shows a negative offset, and TMS3 and TMS4 show negative kink parameters. A rough estimate of the effect on the daily discharge can be made from the parameters *a* and *b,* along with a mean value v_{hat} of |v|, and assuming that theta is about 45°
:

Here D
Q gives the difference in the daily discharge calculated with a simple straight line through the origin (a = b = 0) and with the best-fit straight line. For an area of 1000 m^{2} and a value of v_{hat} of 0.7 m/s, the values of D
Q for TMS1, TMS3, and TMS4 are -12 m^{3}/s, -24 m^{3}/s, and -25 m^{3}/s. This is not an effect to be ignored!

The values of the loop parameters *c* and *d* are fairly close for TMS1 and TMS3. (The TMS4 data is not complete enough to determine these parameters.)** **Positive c corresponds to early turning of the edge of the channel, as explained above

The rating curve of figure 6 can be applied to time series of stage and UVM velocity to calculate discharge. We have used data from TMS between Oct. 10 and Nov. 21, 1996, to illustrate the procedure. During this period the stage height near one end of the UVM baseline and the UVM velocity were recorded every 15 minutes. The value recorded was an average over the preceding 15 minutes. The calculation proceeds as follows:

1) Scan the data to determine for each point whether it is on the "turning in" (from maximum ebb to maximum flood) or "turning out" (from maximum flood to maximum ebb) branch of the current cycle. Set the "branch flag" to +1 (turning in) or -1 (turning out).

2) For each sample, use the stage height to calculate the cross-sectional area.

3) For each sample, use the profile-corrected UVM velocity and branch flag to find the ADCP velocity on the rating curve.

4) The instantaneous discharge is the product of the area and the ADCP velocity.

5) To remove the tidal variation a running 25-hour boxcar-average filter is applied: for each sample time the average of 100 points (50 before, 49 after) is calculated.

6) The points are smoothed by applying a digital single-pole low-pass filter with a 50-hour time constant, run forwards and backwards to cancel out the phase shift.

The resulting smoothed daily discharge for the given period is shown in figure 9.

The method just described makes many assumptions. We will discuss those which we think are the most important, and estimate the error in the smoothed daily discharge which each produces.

**The Rating Curve.** The procedure above uses a single rating curve all the time. This assumes that the ADCP velocity depends only on the UVM velocity and the branch index. To test this, we compared the TMS3 calibration with two others. The TMS3 data were taken at neap tide, with a variation of only 0.5 m in stage. The TMS4 calibration (June 11, 1998) was at spring tide, with a variation of 0.9 m during the calibration; while the variation for TMS1 was 0.8 m. (See figures 4 and 5.)

The discharge as calculated from all three calibrations is shown in figure 9. Two of the calibrations (TMS3 and TMS4) agree to within less than 1 m^{3}/s, while TMS1 differs from the other two by about 5 m^{3}/s. Since we have no a priori reason to prefer one of these calibrations over another, this indicates a standard error on the discharge of about 4.0 m^{3}/s.

This error on the discharge far exceeds that expected from random errors on ADCP or UVM measurements. This suggests systematic errors. Furthermore, the three rating curves look different. This suggests the presence of additional variables. If these could be included in the rating process, the error on the discharge might be reduced.

CONCLUSIONS

We have studied the process of determining net flow in a tidally dominated estuary, using the path-averaged velocity measured by an ultrasonic velocity meter as an index or proxy, and calibrating with velocity distributions obtained with an acoustic Doppler current profiler. Most of our conclusions apply in some degree to other types of index measurements, such as fixed ADCP velocities or conventional point velocities.

We have studied in detail the process of establishing an index to channel discharge, with particular reference to the case of determining net flow in a dominantly tidal regime. We have presented an analytic form for the rating curve which can be used to represent a set of data (complete or partial) in the application of the rating.

The rather large random single-ping errors of ADCP measurements are reduced by averaging to an insignificant level in the calibration process. Random errors in single discharge measurements appear to be dominated by water variability. Some known ADCP systematic errors (bottom-tracking errors, for example) may contribute to the variation observed in multiple calibrations. We present guidelines for ADCP data collection aimed at producing consistent and repeatable discharge measurements.

We find that the accuracy of the index velocity from commercial UVM’s is not a limiting factor, provided that the signal is large enough and that firmware and software deal adequately with bad data. The main weakness of this index is that it does not sample the edges of the channel. Study of the ADCP velocity profiles show systematic differences between the edge flow and the flow in the center of the channel. These differences are related to features observed in ADCP-UVM rating curves. The use of an index velocity measured at a single point in the channel would be more strongly affected by such sampling biases.

We find the main source of error in determining the tidally averaged discharge to lie in the calibration of the index. In our case study at Three-Mile Slough, three different calibrations yielded significantly different values for the discharge, indicating the importance of variables besides the index variable. We feel that the best way to estimate the effect of such unidentified variables is through redundant calibrations under as wide a range of conditions as possible.

At Three-Mile Slough, using the UVM velocity as the index variable, uncertainties in the calibration limited the accuracy of determination of the tidally averaged discharge to 0.4 % of the peak (tidal) flow.

ACKNOWLEDGMENTS

We would like to acknowledge the contributions of Richard Oltman, Jim Derose and Adam Bolton to measurements at Three-Mile Slough, and the advice and encouragement of Jon Burau.

BIBLIOGRAPHY

**Chiu,** C-L and Chairil **[1995]**, A., "Maximum and Mean Velocities and Entropy in Open-Channel Flow,’ J. Hydraulic engineering **121,** 26.

**Hoffard,** Stuart H. **[1980],** "Feasibility of Using an Acoustic Velocity Meter to Measure Flow in the Chipps Island Channel, Suisun Bay, California," US Geological Survey Open-File Report 80-697.

**Laenen**, Antonius, and R.E. Curtis, Jr. **[1989]**, "Accuracy of Acoustic Velocity Metering Systems for Measurement of Low Velocity in Open Channels," US Geological Survey Water-Resources Investigations Report 89-4090.

**Laenen**, Antonius **[1985]**, "Acoustic Velocity Meter Systems," in *Techniques of Water-Resources Investigations of the United States Geological Survey,* Book 3, Chapter A17 (U.S. Government Printing Office, Washington).

**Lane**, A., D. Prandle, A.J. Harrison, P.D. Jones, and C.J. Jarvis **[1997],** "Measuring Fluxes in Tidal Estuaries: Sensitivity to Instrumentation and Associated Data Analyses," *Estuarine, Coastal and Shelf Science* 45, 433.

**Simpson,** Michael R., and Richard N. Oltmann **[1990]**, "An Acoustic Doppler Discharge-Measurement System," *Hydraulic Engineering Proceedings 1990 National Conference*, HY Div/ASCE, San Diego, CA/July 30-Aug. 3, 1990, p. 903.

**Simpson,** Michael R., and Oltmann, Richard N. **[1993],** "Discharge-Measurement System Using an Acoustic Doppler Current Profiler with Applications to Large Rivers and Estuaries," US Geological Survey Water-Supply Paper 2395

**Smith,** Winchell **[1971],** "Techniques and Equipment Required for Precise Stream Gaging in Tide-Affected Fresh-Water Reaches of the Sacramento River, California," Biological Survey Water-Supply Paper 1869-G, US Government Printing Office.

**Worcester**, Peter F. **(1977),** "Reciprocal acoustic transmission in a midocean environment," *J. Acoust. Soc. Am.,* **62**.

FIGURE CAPTIONS

Figure 1. Map of the San Francisco Bay-Delta. The sites labeled #4, #5, #6, and #7 are used in the measurement of the Delta outflow.

Figure 2. Diagram showing the layout of a UVM site.

Figure 3. Diagram illustrating the principles of operation of the ADCP.

Figure 4. Variables measured during the TMS-3 calibration. The measured stage is in the upper plot, the ADCP discharge in the middle plot, and the line velocity measured by the UVM in the lower plot.

Figure 5. Measured stage and UVM velocity for the TMS1 and TMS4 calibrations. The TMS1 calibration consists of five non-contiguous time series, recorded on Nov. 13, 14, 20, and 25, 1996.

Figure 6. Rating curves for the three calibrations at Three-Mile Slough.

Figure 7. A current profile taken at 6:30 PST on June 2, 1998, as part of the TMS3 ADCP calibration data. The red and yellow colors in the center indicate positive flow (south to north), while near the edge the blue indicates that the flow direction has reversed.

Figure 8. Parametrized model for the rating curve, including variable slope, offset, kink, and bulge. The parameters for this curve are: Q = 40° , a=0.1, b = 0.1, c = 0.071, and d = 0.4.

Figure 9. Smoothed daily discharge at Three Mile Slough, calculated from three different independent calibrations.