A Fast Method of Flood Discharge Estimation

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HYDROLOGICAL PROCESSES Hydrol. Process. 18, 1671–1684 (2004) Published online 8 March 2004 in Wiley InterScience (www.interscience.wiley.com). DOI: 10.1002/hyp.1476

A fast method of ﬂood discharge estimation

Yen-Chang Chen1* and Chao-Lin Chiu2 1 Department of Civil Engineering, National Taipei University of Technology, Taipei, Taiwan 2 Department of Civil and Environmental Engineering, University of Pittsburgh, Pittsburgh, PA 15261, USA

Abstract: Discharge, especially during flood periods, is among the most important information necessary for flood control, water resources planning and management. Owing to the high flood velocities, flood discharge usually cannot be measured efficiently by conventional methods, which explains why records of flood discharge are scarce or do not exist for the watersheds in Taiwan. A fast method of flood discharge estimation is presented. The greatest advantage of the proposed method is its application to estimate flood discharge that cannot be measured by conventional methods. It has as its basis the regularity of open-channel flows, i.e. that nature maintains a constant ratio of mean to maximum velocities at a given channel section by adjusting the velocity distribution and the channel geometry. The maximum velocity at a given section can be determined easily over a single vertical profile, which tends to remain invariant with time and discharge, and can be converted to the mean velocity of the entire cross-section by multying by the constant ratio. Therefore the mean velocity is a common multiple of maximum velocity and the mean/maximum velocity ratio. The channel cross-sectional area can be determined from the gauge height, the water depth at the y-axis or the product of the channel width multiplied by the water depth at the y-axis. Then the most commonly used method, i.e. the velocity–area method, which determines discharge as the product of the cross-sectional area multiplied by mean velocity, is applied to estimate the flood discharge. Only a few velocity measurements on the y-axis are necessary to estimate flood discharge. Moreover the location of the y-axis will not vary with time and water stage. Once the relationship of mean and maximum velocities is established, the flood estimation can be determined efficiently. This method avoids exposure to hazardous environments and sharply reduces the measurement time and cost. The method can be applied in both high and low flows in rivers. Available laboratory flume and stream-flow data are used to illustrate accuracy and reliability, and results show that this method can quickly and accurately estimate flood discharges. Copyright  2004 John Wiley & Sons, Ltd.

KEY WORDS discharge; entropy; ﬂood estimation; mean/maximum velocity ratio; probability

INTRODUCTION Discharge, the volume of water ﬂowing through a cross-section of a stream in a given amount of time provides useful information for understanding hydrological processes. These data are very useful for water resources planning, design of hydraulic structures, ﬂood control and decision making. A continuous record of discharge and stage can be used to construct a stage–discharge rating curve that is the most commonly used method to estimate discharge. A stage–discharge rating curve commonly obeys the equation of the form (Rantz, 1982b)

Q D pG eN 1 where Q is the discharge that is usually made by current meter, p and N are constants, G is the stage and e is the water stage of zero flow. Under uniform flow conditions, discharge is then obtained by the measured stage and stage–discharge rating curve. However, a large portion of the annual transportation of water and sediment occurs during floods. It is particularly important to make discharge measurements when stream

* Correspondence to: Yen-Chang Chen, Department of Civil Engineering, National Taipei University of Technology, Taipei, Taiwan. E-mail: [email protected]

Copyright  2004 John Wiley & Sons, Ltd. 1672 Y.-C. CHEN AND C.-L. CHIU stages are high. Such measurements form the basis for flood warning, flood forecasting, reservoir operation and estimates of annual flow volumes. The stage–discharge relationship is subject to shifts as a result of flood wave during high water. Owing to the energy slope varying with time, the stage–discharge rating curve cannot be used to estimate discharge when backwater or highly unsteady flow exits. During a flood period, which is invariably unsteady, the stage–discharge relation is a loop as described in Figure 1, which shows that at a given water level the stage–discharge relationship obtained during steady flow periods tends to underestimate the discharge when the flow is rising, and overestimate it when the flow is subsiding. The individual conditions of an unsteady flow have individual energy slopes. Each individual flood should have an individual loop. Each loop therefore can only display only an individual flood. Consequently, it would be difficult to estimate flood discharge accurately by using a stage–discharge rating curve. Many indirect methods (Rantz, 1982b; Bureau of Reclamation, 1997) are used to estimate flood discharge. The slope–area method, which makes use of empirical equations along with the energy slope, is the most commonly used technique to make determinations of peak discharge. The channel-geometry method, which relies on discharge and channel dimensions, is used to estimate mean annual flood (Wharton and Tomlinson, 1999). However, it is difficult to measure real-time flood discharge accurately. The indirect methods cannot be of sufficient accuracy even when the flood discharge is estimated by floats (Chow, 1964). Floats are used under conditions for which no conventional method is available. Moreover flood discharge estimated by indirect methods is unreliable (Quick, 1991) and usually cannot be verified. The current-meter method is the most popular, convenient and widely used direct method of discharge measurement. The concept of the current-meter method is to divide the cross-section of an open channel into several elemental strips, and measure the velocities and depths by current meter and cable at the centre of each strip. The discharge of the stream cross-section is computed by using the mid-section or mean-section methods. The disadvantages of the current-meter method are that they are costly, labour intensive and tedious. Thus flood discharge measurement is difficult using the current-meter method. The moving-boat method is usually used to measure discharge rapidly on large steams and estuaries. However, it is frequently impractical and dangerous to use the moving-boat method during floods. Accurate

30 Unsteady flow in flume with unvegetated flood plain (Tu et al. 1995) Observed data (5) 25 Variation with time (4) t(s) (3) (1) 99 (2) (2) 100 (1) 20 (3) 105 (4) 120 (5) 150 Steady (6) (6) 200 (7) 220 15 (8) 250 (7) Q (l/s) (9) 254 (8) 10 (9)

0 45678910 D (cm) Figure 1. Stage–discharge loop during a ﬂood simulated in the laboratory. Reproduced from Chiu and Chen, 2003

Copyright  2004 John Wiley & Sons, Ltd. Hydrol. Process. 18, 1671–1684 (2004) FLOOD DISCHARGE ESTIMATION 1673 estimation of flood discharge requires velocity sampling and measurement of water depths across the channel. Measuring flood discharge during highly unsteady flow has to be completed as rapidly as possible, owing to the fact that flow conditions change rapidly. Flood discharge measurement can be hazardous to personnel and instruments. In order to avoid exposure to the hazardous environment too long, some non-contact methods (Spicer et al., 1997; Herschy, 1999; Costa et al., 2000) that do not involve immersing equipment in the stream have been developed to measure flood discharge. They measure only water surface velocities. A surface-velocity coefficient of 0Ð85 or 0Ð86 is preferred to convert water surface velocity to mean velocity when a surface-velocity method is used (Rantz, 1982a). However, the maximum velocity usually occurs beneath the water surface during flood periods. The higher the water stage is, the deeper the location of maximum velocity. It is, thus, easy to underestimate flood discharges by using non-contact methods. Figure 2 shows the velocity distribution of unsteady flow. The mean velocity is 54Ð1 cm/s, and the maximum velocity occurs beneath the water surface at around 0Ð6 depth. If the mean velocity is estimated by non-contact methods, and the surface-velocity coefficient 0Ð85 is used, it will be underestimated and be only 41Ð1 cm/s. Consequently, flood discharge cannot be determined easily and accurately using conventional methods, particularly in wide open-channels. Until recently, a key limitation to understanding flood flow has been the difficulty and expense associated with using conventional methods. Owing to rapid change of flow conditions allowing very little time for measurements, new techniques have been applied to measure velocity quickly (Klein et al., 1993; Schultz, 1996; Callede et al., 2000; Sulzer et al., 2002). However, a new method to determine flood discharge efficiently is still required. A fast method of flood discharge measurement that is efficient and requires only a small number of velocity samples is preferred. (Chen and Chiu, 2002) have successfully used an efficient method to estimate discharge in tidal streams with the strong effects of turbulent flow. The concept of the method presented in this study is to attain efficiency of discharge measurements by observing and understanding the physical process of flood flow and by taking advantage of regularities recognizable in the flow. Discharge through a cross-section in a natural channel is estimated from the mean velocity in the section and the cross- sectional area. The method proposed is based on the constant ratio of mean to maximum velocities (Chiu and

8 Flume Data (Tu et al., 1995) Water in unvegetated flood plain Surface velocity = 48.4 cm/s 7 Unsteady flow, t=120 s Φ=0.78 umax=59.1 cm/s 6 D=8.4 cm h=5.4 cm h/D=0.645 5 Mean velocity = 54.1 cm/s Mean velocity by surface-velocity method = 41.1 cm/s 4 y (cm)

0 0 10203040506070 u (cm/s) Figure 2. Velocity distribution of unsteady ﬂow

Chen, 1998). Fundamentally, this method is based on the velocity–area principle

Q D uA 2 where u is the mean velocity of the cross-section and A is the cross-sectional area. The mean velocity can be quickly determined using maximum velocity data, and the maximum velocity can be estimated by sampling only a few velocity measurements. The cross-section area can be estimated easily by gauge height, water depth or channel width. The difference between the conventional methods and the method proposed is the determination of the mean velocity and cross-section area. The mean velocity and cross-section area determined by the proposed method are more efﬁcient and accurate. This paper presents and illustrates the details of this method, and the results of testing and the various steps of this method, by using laboratory and ﬁeld data.

DETERMINATION OF MEAN VELOCITY Mean velocity data for a channel cross-section are useful for human applications. In order to measure flood flows, the mean velocities have to be determined quickly. Fortunately, the ratio of the mean to maximum velocities of a given cross-section that approaches a constant can be used to rapidly determine mean velocities. A velocity distribution equation (Chiu, 1989) that can describe the maximum velocity occurring beneath the water surface is given as u u D max ln 1 C eM 1 0 3 M max 0 where u is velocity on an isovel of a , is the isovel in Figure 3 (Chiu and Chiou, 1988), umax is maximum velocity in a channel section, M is a parameter, 0 is the minimum value of (which occurs at the channel bed where u D 0) and max is the value of when u is the maximum velocity. Isovel can be expressed as function of y on the y-axis, which is the vertical profile containing the maximum velocity occurring in the cross-section y y D exp 1 4 D h D h where y is distance from the channel bed, D is water depth and h is a parameter that indicates the location of maximum velocity. If umax occurs on the water surface, h 0. If umax occurs below the water surface, h>0 and h is the actual depth of umax below the water surface. Probabilistically, 0/max 0 in Equation (3) is the area of the isovel between and 0 divided by the total area that is the probability of velocity less than u. Therefore, u 0 D pudu5 max 0 0 in which pu is the density function of u. For a wide open-channel, 0/max 0 D y/D.Forflow in a circular pipe, umax occurs in the centre of the pipe and the minimum velocity occurs at the pipe wall, 2 therefore 0/max 0 D 1 r/R ,inwhichr is the radial distance from the centre of the pipe and R is the radius of the pipe (Chiu and Chen, 1999). Mathematically, information entropy (Shannon, 1948), which is a measure of information content depending on the current level of uncertainty, is expressed as umax Hu D pu ln pudu6 o

B ∆ i CL

|h|

Bi ∆ CL

D ξ ξ y y η η

Channel bed (ξ = ξ ) ξ = ξ 0 Channel bed ( 0) (a) Pattern I (h≤0) (b) Pattern II (h>0) Figure 3. Patterns of velocity distribution and isovels; (a) h<0; (b) h ½ 0

Equation (6) presents the average information content. Variable pu must satisfy the following conditions umax pudu D 1 7 0 umax upudu D u8 0

If the entropy is maximized subject to the constraints in Equations (7) and (8) by the method of Lagrange multipliers, the probability density function can be obtained as

pu D e11e2u 9 where 1 and 2 are coefﬁcients. Substitution of Equations (9) into (7) yields

11 2 e D umax 10 e2 1

By deﬁning M D 2umax and substituting Equation (10) with (9), pu becomes

M M u umax pu D M e 11 umaxe 1

Substituting Equation (11) into (8) gives

u eM 1 D M D 12 umax e 1 M

3.5 Ohio River at Sewickley USGS Data, 10/1938-8/1974 3.0 u = Q/A φ=0.76 (M = 3.70)

2.5

2.0

u (m/s) 1.5

1.0

0.5

0.0 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5

umax (m/s)

Figure 4. u/umax relationship for the Ohio River at Sewickley

Equation (12) shows the relationship between mean and maximum velocities. The ratio of u to umax of a cross-section, , approaches to a constant. It is a linear relationship passing through the origin and is a law of nature that a channel section maintains a constant ratio of mean to maximum velocities (Chiu, 1996). Different cross-sections have different ratios. Figure 4 presents the excellent linear relationship between u and umax. It indicates that  for the Ohio River at Sewickley between 1938 and 1974 was quite stable, i.e. that  does not vary with time and discharge. Thus, u of a cross-section can be easily and quickly determined by the product of umax multiplied by .

DETERMINATION OF THE Y-AXIS It is difficult to find the location of the maximum velocity in an open channel. Fortunately, the velocity data applied to determine the discharge that can be used to plot the isovels in the cross-section reveals the location of the y-axis Zy. The position of Zy in a natural channel can be located anywhere in the cross-section. For a straight and regular artificial channel, the y-axis usually occurs at the centre of the cross-section. Figure 5, plotted using conventional velocity data, shows isovel patterns of an open channel and indicates the position of Zy, which is very stable and invariant with time, discharge and gauge height if the channel bed does not change drastically (Chen, 1998). However, once the water exceeds a certain level, e.g. banks are overtopped and water spreads over the floodplain, Zy in certain situations may shift to a new location. Figure 6 illustrates such a situation in the Kaoping River at the Leeling Bridge. The maximum velocity in the high-velocity region on the right-hand side is greater than that on the left-hand side when the water level is above 29 m (gauge height) but Zy is found about 47Ð5 m from a reference point on the left bank when water flows in the main channel. Figure 6 also shows that there are five anomalies between 25 and 26 m. The maximum velocities of the cross-section of the five flood events are very close to the maximum velocities on Zy. Therefore the discharge of the five anomalies still can be estimated accurately on Zy. Without records of data, Zy may be estimated by using floats thrown on to the water surface to determine the velocity profile on the water

surface. During the ﬂood period, the maximum velocity on the water surface can fairly indicate Zy. A slight shift of Zy will not have much effect on the estimation of the maximum velocity (Chen, 1998). Therefore the maximum velocities can be estimated at the mean location of the y-axis Zy.

DETERMINATION OF MAXIMUM VELOCITY

Measurement of discharge is by current-meter using the two-point method. Once Zy is determined, umax can be determined by u0Ð2 and u0Ð8 sampled at Zy,whereu0Ð2 and u0Ð8 are the velocities at 0Ð2and0Ð8ofthe depth below the water surface. The equations for computing u from u Ð and u Ð are given by max 0 2 0 8 umax 0Ð2 u D ln 1 C eMl 1 13 0Ð2 M l max u max Ml 0Ð8 u0Ð8 D ln 1 C e 1 14 Ml max

Zy = 45 m 24

G (m) 22

21 Kaoping River at the Leeling Bridge Taiwan Water Conservancy Agency Data, 07/11/1992 Q=680 cms, G=24.01 m 0 50 100 150 200 250 300 Distance from relative point (m)

Figure 5. Typical isovel pattern indicating location Zy

700 Kaoping River at the Leeling Bridge Taiwan Water Conservancy Agency Data, 1/1991-8/1996 600 obtained at G>24 m Zy = 47.5 m 500

400 (m) y

Z 300

200

100

0 23 24 25 26 27 28 29 30 31 32 G (m) Figure 6. Two locations of the y-axis in the Kaoping River at Leeling Bridge

D u0Ð2 C u0Ð8 umax u D D ln 1 C eMl 1 dy15 y 2 MD 0 max 0Ð8D 0Ð8D D exp 1 16 0Ð2 D h D h 0Ð2D 0Ð2D D exp 1 17 0Ð8 D h D h D 1ifh ½ 0 18 max D D D exp 1 if h<0 19 max D h D h where Ml is the local M of every flood on Zy.Valuesumax, Ml and h can be approached by solving the above equations. However, only umax is needed to establish the relationship of the mean and maximum velocities of the given cross-section. For some purposes, a detailed velocity profile at Zy could be made. In such a condition, Equation (3) can be fitted to the velocity profile data with parameter estimates given by a non-linear regression method to determine umax.

DETERMINATION OF CROSS-SECTIONAL AREA The cross-sectional area may be estimated by gauge height, depth at the y-axis and the product of channel width multiplied by depth at the y-axis. If the streambed is free of scour and deposits and is stable, it is most convenient to use the gauge height to estimate the cross-sectional area. The relationship of the cross-sectional area to the gauge height can be expressed as

c1 A D a1G b1 20 where G is gauge height, and a1, b1 and c1 are coefﬁcients. For a highly erodible channel under the effects of scour or sediment deposition, the cross-sectional area can be evaluated by

c2 A D a2d b2 21 where d is water depth at y-axis, and a2, b2 and c2 are coefﬁcients that can determined from data. If the main channel changes frequently, the following equation may be more suitable than Equations (20) and (21)

c3 A D a3Bd b3 22 where B is channel width at the water surface, and a3, b3 and c3 are coefﬁcients. The parameters b1, b2 and b3 are used to adjust the zero of the staff gauge in low ﬂow. Figure 7 compares the performances of Equations (20), (21) and (22) and shows the superiority of Equation (22) (Chiu and Chen, 1999).

DESCRIPTION OF DATA The laboratory flume data (Jayaratne et al., 1995; Tu et al., 1995) and the flood discharge data of the Kaoping River at the Leeling Bridge are used to evaluate and verify the applicability of this method. The flume used to simulate unsteady flows was 25 m long, 0Ð6 m high and 1 m wide. The width of the main channel was 40 cm and the floodplain was 60 cm wide. Flow velocities were measured on 16 vertical profiles at every 5 mm along each profile. The first measuring point on each profile was 10 mm from the channel bed. The floodplain had vegetation 6Ð45 cm high. The diameter and height of the model plants were 2 mm and 50 mm, respectively. The space between plants was 2Ð5 cm. The slope of the tilting flume was 1/2000. Figure 8 shows the time

10000 Erh-jen Creek at Nan-Hsiung Bridge Taiwan Water Conservation Bureau Data, 3/91-8/96 2.25 Aest = 5.22(D+0.31) 1.11 1000 Aest = 37.2(G-3.60) , G<5.49 m 2.73 ≥ Aest = 1.12(G-1.57) , G 5.49 m 0.87 Aest = 1.61(BD-2.27) 100 ) 2 (m est A 10

0.1 0.1 1 10 100 1000 1000 2 Aobs (m ) Figure 7. Comparison of models of cross-sectional area estimation variation of velocity profiles of the flume data. The numerical numbers above the water surface indicate the time sequence during the flood process. The velocity profile numbers 1 to 5 were measured during the rising stage, and the profile numbers 6 to 9 were taken during the receding stage. The velocity profile of steady flow is also shown in Figure 8. Velocity profiles 3, 7 and steady flow describe the fundamentally different flow structures at the water depth of 7Ð8cm. The Kaoping River, as shown in Figure 9, has the largest catchment in Taiwan, which flows south- westward and drains an area of 3257 km2 into the Taiwan Strait. It is 171 km long and the annual runoff is 8455 M m3. Owing to the tropical location, the catchment receives 3046 mm of rainfall each year; however, the rainfall is uneven. Typhoons accompanied by torrential rainfall, which usually occur in the summer, are principally responsible for the floods. The catchment extends from the highest mountain in Taiwan (elevation 3952 m) to sea-level. The Kaoping River above the Leeling Bridge is a mountain river with a steep slope and a catchment area covered by mature forests. The streambed at the Leeling Bridge is sandy. Several channels exist in the regular period and the streambed of main channel is considerably unstable during the flood period. However, only one main channel is found when the water is above 24 m (gauge

Table I. Summary description of the ﬂood data for the Kaoping River at Leeling Bridge

Minimum Maximum Mean SD

Q (m3/s) 54 19 300 4127 4659 Am2 66 5560 1716 1309 G (m) 24Ð04 30Ð72 25Ð91 1Ð73 B (m) 320 690 471 95 u (m/s) 0Ð18 3Ð46 2Ð01 0Ð82 umax (m/s) 0Ð15 5Ð50 2Ð86 1Ð18

11 Flume Data (Tu, et al. 1995) 10 Water in unvegetated flood plain

Φ (5) =0.78 (M=4.31) ∆

9 (6) (4) ∆ t(s) ∆

(7) (3) ∆ ∆ (1) 99 Steady∆

(2)

8 (2) 100 (8) ∆

(9) ∆ (2)(1) ∆ (3) 105 ∆ 7 (4) 120 (5) 150 6 (6) 200 (7) 220 (8) 250 y (cm) 5 (9) 254 Steady 4

0 0 10203040506070 u (cm/s) Figure 8. Time variation of the velocity profile. Reproduced from Chiu and Chen, 2003 height). The Leeling Bridge Station is operated by the Taiwan Water Conservancy Agency from 1991 to the current year. Flood discharge measurement was made from the bridge by current-meter using a two-point method. The Price meter is applied to measure the velocity under 3 m/s. When velocity exceeds 3 m/s, the propeller current meter with a measuring range from 4 cm/s to 6 m/s is used. Thirty-one flood discharge measurements are available for the period of 1991 to 1996. They are applied to develop the measurement method. General information on the flood data of the Kaoping River at the Leeling Bridge is summarized in Table I.

DETERMINATION OF FLOOD DISCHARGE The fast method presented herein is for estimating flood discharge, for which data are extremely scarce and hence badly needed. The estimation performance of this method is evaluated by correlation coefficient as Qobs Qobs Qest Qest D 23 2 2 Qobs Qobs Qest Qest where Qobs and Qest are observed and estimated discharges, Qobs is the mean of the observed discharge and Qest is the mean of the estimated discharge. The closeness of the estimated discharge to observed discharge is indicated by , implies perfect matching when is unity. Figure 10 displays the linear relationship between u and umax. It shows the ratio of the mean velocity to the maximum velocity of the laboratory flume and the Kaoping River at the Leeling Bridge are about 0Ð78 and 0Ð71, respectively. The maximum velocities of the laboratory flume are estimated at the centre of the main channel. For the Kaoping River at the Leeling Bridge umax values are estimated at Zy D 47Ð5mand 433 m from the relative point when water stage is below and above 29 m (gauge height) respectively. All u

Figure 9. Location of the area under study

90 6 Flume Data (Tu et al. 1995 & Jayaratne et al. 1995) Kaoping River at the Leeling Bridge φ φ Taiwan Water Conservancy Agency Data, 1/1991-8/1996 80 U= Umax, =0.78 (M=4.27) Unsteady flow; flood plain vegetated umax obtained at G<29 m & Zy = 47.5 m 5 Unsteady flow; flood plain unvegetated umax obtained at G>29 m & Zy = 433 m 70 Steady flow, flood plain vegetated Φ=0.71(M = 2.84) for G>24 m Steady flow, flood plain unvegetated 60 4

50 3 (m/s)

40 u U (cm/s)

30 2

20 1 10

0 0 0 102030405060708090 0123456

Umax (cm/s) Umax (m/s) (a) (b)

Figure 10. Relationship between u and umax; (a) unsteady ﬂow (reproduced from Chiu and Chen, 2003); (b) the Kaoping River at Leeling Bridge

values are estimated by Qobs/A. The relationship between u and umax is very stable and does not vary with space and time. The relationship of A to G of the Kaoping River at the Leeling Bridge during 1991 and 1996 can be seen in Figure 11. Thus the cross-sectional area can be estimated by 8Ð49G 21Ð21Ð87 when

12 11 Kaoping River at the Leeling Bridge Taiwan Water Conservancy Agency Data, 1/1990-9/1996 10 obtained at G>24 m 9 A=84.9(G-21.2)1.87 8 7

5 G-21.2 (m)

2 50250 500 1000 2500 7500 A (m2) Figure 11. A/G relationship in the Kaoping River at Leeling Bridge

35 25000 Unsteady flow in flume Kaoping River at the Leeling Bridge Taiwan Water Conservancy Agency Data, with unvegetated flood plain (Tu et al. 1995) Φ Φ 1/1991-8/1996 30 Qest= Aumax, =0.78 Φ=0.71 (M=2.84) for G>24 m Unsteady flow 20000 Φ 1.87 Qest= umaxAest, Aest=84.9(G-21.2) Steady flow U obtained at Zy=47.5 m 25 max Umax obtained at Zy=433 m 15000 /s)

20 3 (l/s) (m est obs Q 15 Q 10000

10 5000 5

0 0 0 5 10 15 20 25 30 35 0 5000 10000 15000 20000 25000 3 Qobs (l/s) Qest (m /s) (a) (b) Figure 12. Accuracy of estimated discharges; (a) unsteady flow; (b) the Kaoping River at Leeling Bridge water stage is above 24 m (gauge height). The observed and estimated discharges at the laboratory flume and the Kaoping River at the Leeling Bridge are compared in Figure 12. The discharge measurements of the laboratory flume with a vegetated floodplain are unavailable; therefore, only the discharge measurements of the flume without vegetated flood plain are shown in Figure 12a (B. L. Jayaratne, University of Tokyo,

Copyright  2004 John Wiley & Sons, Ltd. Hydrol. Process. 18, 1671–1684 (2004) FLOOD DISCHARGE ESTIMATION 1683 personal communication, 1997). The flood discharges of the flume and the Kaoping River at the Leeling Bridge are estimated by 0Ð78Aumax and 0Ð71Aumax, respectively. All the data in Figure 11 fall on the line of agreement, which indicates that the flood discharges estimated by this method agree quite well with the observed discharges. The correlation coefficients of the flume and the Kaoping River at the Leeling Bridge are 0Ð97 and 0Ð98, respectively. The results show that the performances of this method are accurate and reliable. Both correlation coefficients are very close to unity. It demonstrates that it is possible to successfully measure flood discharges by using this method.

SUMMARY AND CONCLUSION In order to accurately measure flood discharge, many methods of flood discharge measurement have been developed. Owing to inpracticality and difficulty in measuring flood discharge, the application of conventional methods in Taiwan, which has a tropical climate and high mountain watersheds all over the island, has so far been unsuccessful. In this study, a fast method that takes advantage of the constant ratio of mean to maximum velocities is proposed to estimate flood discharge. The elements of this method are:

1. The ﬂood discharge is a common multiple of mean velocity and cross-sectional area. 2. Mean velocity can be estimated by the production of the maximum velocity multiplied by the mean/maximum velocity ratio of the given cross-section. 3. In a given channel cross-section, the ratio of mean to maximum velocities is constant and will not vary with time and water stage. 4. The maximum velocity can be estimated at the y-axis by taking a few velocity samples using new techniques such as acoustic doppler current proﬁlers (ADCP). 5. The location of the y-axis is stable. 6. The cross-sectional area can be determined simply from G, D or BD.

Laboratory flume data and data from the Kaoping River at the Leeling Bridge are used for developing this method. The flood discharge is determined by only two velocity measurements at the y-axis. The results demonstrate that this method is efficient and can successfully and accurately estimate flood discharges. The method can be applied to both high and low flows. Moreover the most important utility of this method is in estimating flood discharges for which data are scarce and cannot collected by the conventional methods, and hence essentially needed for flood forecasting and design of flood control structures. It can drastically reduce the time and cost of flood discharge measurement, and avoids exposure to dangerous environments. In addition, this method can be coupled with modern devices, e.g. ADCP, to automate flood discharge measurements, and can be applied in real-time flood forecasting. This research is limited to an initial study of the application of the constant ratio of mean to maximum velocity in estimating flood discharge. Further studies could be extended to study modern instruments for potential use with the proposed method and explore the possibility of automating flood discharge measurements. This will not only provide the hydrographers with a better method to estimate flood discharge, but also make possible automatic flood discharge measurement.

ACKNOWLEDGEMENTS This work is based on work supported by the Water Resources Bureau of Taiwan. The authors are grateful to the Taiwan Water Conservation Agency and Taiwan Power Company for their generosity in providing the valuable data.

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