Adjustment of Discharge Measurements Made at a Distance from the Gaging Station During Periods of Changing Stage and Discharge In Reply Refer To: June 17, 1992 WGS-Mail Stop 415 OFFICE OF SURFACE WATER TECHNICAL MEMORANDUM 92.09 SUBJECT: Adjustment of Discharge Measurements Made at a Distance from the Gaging Station During Periods of Changing Stage and Discharge Stage-discharge relations at streamflow gaging stations are defined by correlating recorded gage heights (stages) with measured discharges. Practical considerations frequently require that the discharge be measured at some place other than the gage. When the flow is steady (unchanging in time) and there are no lateral or tributary inflows, the measurement may be made at any convenient location, because the flow also is constant with respect to location along the stream course. When the flow or stage is changing with time, however, the flow also is varying from place to place along the stream. In this case, if the measurement cannot be made at the gage, an adjustment must be applied to make the measured flow and the recorded gage height hydraulically comparable. Procedures for determining and applying such adjustments are described in Techniques of Water-Resources Investigations (TWRI) Book 3, Chapter A8 (p.54-57), in Water Supply Paper (WSP) 2175 (p.177-179), and in a former stream-gaging manual, WSP 888 (p.79). The purpose of this memo is to explain the hydraulic basis for these procedures, to provide guidance for their practical application, and to correct an error in the description of one of the procedures. The fundamental hydrologic principle that governs this adjustment is the equation of continuity or hydrologic storage equation for a reach of stream channel between the measurement section and the gage: storage change = inflow - outflow. As explained in the TWRI and in WSP 2175, the change of storage in this reach during a measurement can be computed directly from knowledge of reach length, top width, and stage changes at the upstream and downstream ends of the reach. If there are no lateral or tributary inflows, the inflow and outflow are the measured discharge and the discharge at the gage; which one is which depends on whether the measurement is made above or below the gage. In either case, the equation can be solved for the flow at the gage as follows: Qg = Qm + WLdh/dt where Qg is the flow at the gage, Qm the measured flow, dh/dt the average rate of change of stage at the gage and measurement sections during the measurement, W the average channel top width, and L the distance from the gage to the measurement section, measured positive in the direction of flow. Both dh/dt and L have algebraic signs: dh/dt is negative if the stage is falling, positive if rising; L is negative if the measurement is made upstream from the gage, positive if downstream. This equation, computed with the rules for arithmetic with signed quantities, is equivalent to the sign rules given in the TWRI (p. 55-56) and in WSP 2175 (p.177) for the storage-change adjustment. The storage-change adjustment is based on the assumption that the storage change volume is adequately approximated by a rectangular slab of length L, average width W, and average height dh. Suitable average values of W and dh sometimes are not known with certainty. In this case, alternative methods of adjustment may be sought. An alternative adjustment method is based on the general observation that, in upland (non-tidal) stream channels of generally uniform geometry and not excessively affected by backwater or lateral or tributary inflows, streamflow peaks normally move downstream by pure translation, without appreciable change in shape, over moderate distances. That is, the discharge profile (plot of discharge versus distance along the channel) at a time t+dt can be found by translating the profile at time t a distance L downstream, where L = cdt, and c is the speed of propagation (celerity) of the flood wave. If the wave celerity is known, a discharge measured at a particular time and location can be related to the discharge (and gage height) that occurred at the gage at an earlier or later time. The speed of the flood wave, c, can be determined by direct measurement of travel times, but this is not usually feasible unless discharge or stage hydrographs are available at upstream and downstream sites. A theoretical estimate may be obtained by applying the continuity equation to a short segment of the rising limb of the flood wave. Because the segment is moving along with the wave, and the wave shape is not changing, the volume of water in the segment also is not changing. Thus, the continuity equation for the moving segment is: 0 = (V + dV - c)(A + dA) - (V - c)(A) where c is the velocity of the wave (and the moving segment), V and A are the water velocity and area at the downstream face of the segment, and V+dV and A+dA are the area and velocity at the upstream face. This equation can be solved for c: c = dQ/dA = (dQ/dh)/W = f'/W where dQ = (V+dV)(A+dA) - VA, h = gage height, and W = mean top width. The quantity f' = dQ/dh is the slope, in cubic feet per second per foot of stage rise, of the stage-discharge rating curve drawn on arithmetic (rectangular) plotting paper. This derivation of the wave speed can be found in many standard open-channel hydraulics texts under the heading of monoclinal or uniformly progressive wave. The result is often called Seddon's (Trans. ASCE, 1900) principle. It also can be derived by similar analysis of the receding limb of the hydrograph. Although the quantity f' = dQ/dh is defined by usual gaging- station data, a suitable value for top width W, as previously noted, is not always readily determinable. However, under channel-control conditions, Manning's or Chezy's formula can be used with a suitable cross-sectional shape to approximate the stage-discharge rating curve. The quantity (dQ/dh)/W can be evaluated for such a theoretical rating, and the wave speed c can be expressed as a factor times the mean water velocity. As shown in WSP 2175, p.415, the ratio of wave speed to water speed ranges from about 1.3 to 1.7 depending on the cross section shape and choice of Manning or Chezy formula. The value of 1.3 seems to be the most probable (WSP 2175, p. 415, TWRI, p.57). If the wave velocity c is known, a discharge measurement made at a distance L downstream from the gage during a period of changing stage can be adjusted as follows. The segment of the flood wave that was measured at the measuring section at time t passed by the gage at time t-dt, where dt = L/c, and thus is hydraulically associated with the stage recorded at that time. It is immaterial whether the stage is rising or falling (i.e., whether the measurement is made on the rising or the falling limb of the flood wave). A similar analysis applies to measurements made upstream from the gage. All cases can be summarized by the formula: Qm(t) = f( h(t - L/c) ) in which Qm(t) is the discharge measured at the measurement section at time t, h(t-L/c) is the recorded gage height at time t- L/c, c is the wave speed, L is the distance from the gage to the measurement section, measured positive in the direction of flow, and f is the stage-discharge relation, which expresses the discharge at the gage at any time as a function of the corresponding stage h. The above equation implies that Qm(t), the discharge measured at time t, is to be plotted on the stage-discharge rating curve in association with h(t-L/c), the stage observed at time t-L/c. Under conditions of changing stage, the stage plotted on the rating is computed as a discharge-weighted average of stages observed at selected times ti' during the course of the measurement. For measurements made at a distance from the gage, the weighted stage can be computed as a discharge-weighted average of the time-adjusted recorded stages, h(ti'-L/c). Alternatively, a weighted gage height hw can be computed in the usual way without any time adjustment, such that hw = weighted average of the h(ti'); the time of occurrence, t", of hw then can be looked up in the stage record, adjusted by L/c, and used to read off the time- adjusted weighted gage height from the stage record as h(t"-L/c). This computation can be summarized by the formula ha(t) = h(t" - L/c) = hw - (L/c)h' in which ha(t) is the time-adjusted gage height associated with the measured discharge Qm(t), hw is the weighted mean gage height omputed without time adjustment, h' = dh/dt is the average time- rate of change of stage between times t" and t"-L/c, and L and c are as defined previously. The travel-time and storage-change adjustments are alternative methods for accounting for the single fact that the flow at the measurement site is not the same as the flow passing the gage at the time of measurement. A measurement may be adjusted by one method or the other, but not both. The two methods can be illustrated and compared by an example. A measurement was made on a falling stage at a site 3.3 miles upstream from the gage. During the approximately 2-hour measurement period, the stage dropped at a rate of 1.0 ft/hr at the measurement site and at 0.3 ft/hr at the gage. The measured discharge was 43,610 cfs, the weighted-mean gage height (without travel-time adjustment) was 8.76 ft, and the mean velocity (Q/A) was 8.8 ft/s. The average top width of the channel, scaled at 10 points on a topographic map, was 340 ft. The slope of the stage-discharge relation, which plots as a straight line between 8 and 10 ft on rectangular coordinates, is 8500 cfs/ft. The adjustment by the storage-change method is as follows: Qg = Qm + WLdh/dt = 43,610 + (340)(-3.3x5280)(-(1+.3)/2)/(1x60x60) = 43,610 + 1070 = 44,680 cfs (GH = 8.76 ft) For adjustment by the travel-time method, two estimates of the flood wave speed are available: c = 1.3V = 1.3x8.8 = 11.4 ft/s = 7.8 mph, and c = f'/W = 8500/340 = 25.0 ft/s = 17.0 mph. The corresponding calculated gage-height adjustments are: ha = hw - h'L/c = (8.76) - (-0.3)(-3.3)/(7.8) = 8.76 - 0.13 = 8.63 ft ha = hw - h'L/c = (8.76) - (-0.3)(-3.3)/(17.0) = 8.76 - 0.06 = 8.70 ft For wave speed c = 1.3V (travel time 25 min), the discharge- weighted mean of time-adjusted recorded stages is 8.67 ft; the recorded gage height 25 min after the time of occurrence of the unadjusted weighted gage height was 8.68 ft. For wave speed c = f'/W (travel time 12 min), the corresponding gage heights were 8.72 and 8.70 ft. These values differ from the above ha-values because the rate of change of stage was not perfectly uniform. The adjusted gage heights are plotted against the measured discharge (Qm = 43,610 cfs) on the rating curve. The results may be compared with the rating curve as follows: Method GH (ft) Q (cfs) Qr (cfs) dQ/Qr (pct) Storage 8.76 44,680 42,540 5.0 Travel (1.3V) 8.63 43,610 41,440 5.2 Travel (f'/W) 8.70 43,610 42,030 3.8 Travel (wtd 25) 8.67 43,610 41,780 4.4 Travel (t"+25) 8.68 43,610 41,860 4.2 Travel (wtd 12) 8.72 43,610 42,200 3.3 Travel (t"+12) 8.70 43,610 42,030 3.8 These results illustrate that the travel-time and storage-change adjustments give generally comparable results. For a perfect translatory wave in a uniform channel the travel-time adjustment (with c = f'/W) and the storage-change adjustment yield the same difference between the plotted measurement and the rating curve. The results also illustrate, however, that the travel-time adjustment is sensitive to judgements and assumptions that may be hard to verify and subject to dispute. The difference in rate of change of stage at the gage and measurement sections in this case introduces additional uncertainty. The storage-change adjustment, on the other hand, involves only straightforward calculations on measurable data. Therefore, as stated in the TWRI, p.57, the storage-change method generally is preferred; WSP 2175 does not even mention the travel-time adjustment for discharge measurements. The formulas given in this memo for adjustment of measured discharge, time, and stage are computed using the normal rules of algebraic signs, where L is a signed quantity, positive if the measurement is made downstream from the gage, negative if upstream. The computation of corrected times does not depend on whether the stage is rising or falling. The computation of corrected gage height, ha(t), involves in addition the signed quantity h', which is negative if the stage is falling and positive if the stage is rising. The standard rules of arithmetic with signed quantities yield the proper signs for the adjustments. The discussion of the travel-time adjustment in the TWRI and WSP 888 includes a verbal rule for determining the algebraic sign of the correction to times observed at the stage recorder. This rule, which includes consideration of whether the stage is rising or falling, is incorrect. (The rule would have given the correct sign for adjustment to gage height, but does not give the correct sign for time.) To prevent any future confusion or error, the rule for application of the travel time adjustment should be corrected on page 57 of the TWRI to read as follows: In applying the time adjustment, the time of travel is subtracted from the observed time at the measurement section if the measurement is made below the gage and is added if the measurement is made above the gage. The resulting adjusted time is used to look up the corresponding gage height in the gage-height record. (Corrected per OSW Technical Memorandum 92.09, June 17, 1992.) A similar correction should be made to the corresponding statement on page 79 of any copies of WSP 888 that may still be in use. In addition, necessary steps should be taken to ensure that all personnel who make, use, or review discharge measurements are familiar with the priciples and computations summarized in this memo. Charles W. Boning Chief, Office of Surface Water WRD DISTRIBUTION: A, B, FO, PO