An optimized routing model for flood forecasting

Transcription

1 WATER RESOURCES RESEARCH, VOL. 45,, doi:1.129/28wr713, 29 An optimized routing model for flood forecasting Roland K. Price 1 Received 19 April 28; revised 31 October 28; accepted 5 December 28; published 24 February 29. [1] A conceptual flow routing model for flood forecasting is adopted for a reach of river given only historic upstream and downstream stage hydrographs and the associated in-bank rating curves, local rainfall, and minimal additional information. The ungauged lateral inflows are generated by a unit hydrograph rainfall-runoff model. The model is calibrated by optimizing the parameters of the functional form for the average cross section along the reach, the base lateral inflow and the peak of the unit hydrograph, and the rating curve extensions downstream and upstream. The model is confirmed by applying it to a synthetic river reach and then to the forecasting of floods in a reach of the River Wye, UK. Citation: Price, R. K. (29), An optimized routing model for flood forecasting, Water Resour. Res., 45,, doi:1.129/28wr Introduction [2] Flood propagation in a natural river can be a highly nonlinear phenomenon, especially when there are significant floodplains along the channel [Price, 1973]. Forecasting a flood at the downstream end of a reach therefore benefits from having an accurate, volume conservative routing method that accommodates the nonlinear behavior of the flood. Given sufficient topographic data for the river, the time series for the upstream and downstream stage hydrographs, reliable rating curves and prescribed lateral inflows it is possible to develop a one or two dimensional computational model for the river to assist in forecasting downstream water levels during extreme flood events. But in some cases such detailed topographic data for the channel and floodplains is not available or can be collected only at considerable cost and delay. In addition, the rating curves often do not exist for above-bank flows, and the lateral inflows are more than likely to be ungauged, requiring them to be estimated or generated from rainfall. The question arises therefore, whether a simpler, conceptual flow routing model can be developed that adequately accommodates the nonlinear nature of the flow and that is calibrated in terms of an averaged cross section for the river reach, the ungauged lateral inflow, and the unknown rating curve extensions above bank upstream and downstream. [3] The purpose of this paper is to develop a method to produce such a calibrated conceptual model with which to forecast the downstream discharge and stage time series, given the time series for the upstream stage and the rainfall along the reach. It is assumed that the only data available to calibrate the model are historic time series for the upstream and downstream stage hydrographs and the rainfall, together with reliable in-bank rating curves; cross sections for the river are not generally available, and there are no reliable above-bank rating curves at the upstream and downstream gauging sites. 1 UNESCO-IHE, Delft, Netherlands. Copyright 29 by the American Geophysical Union /9/28WR713 [4] Developing a robust conceptual model for flood forecasting depends on addressing several issues, including (1) formulation of a flow routing model that is the uniform equivalent of what can be a very irregular natural river; (2) development of relationships between the model parameters and the physical attributes of the natural river; (3) calibration of these model parameters, taking into account any ungauged lateral inflows and any need to extend the rating curves upstream and downstream for above-bank flows; (4) modeling of ungauged lateral inflows along the river reach in terms of recorded rainfall; (5) forecasts of the primary input variables; and (6) treatment of uncertainty in parameter estimation and model forecasts. [5] The first issue is addressed in this paper by adopting a nonlinear, advection-diffusion equation whose parameters are derived for the averaged cross section of the modeled reach, which is regarded as being equivalent to the irregular natural reach. Such an equation was developed by Price [1985] in terms of the kinematic wave speed and attenuation parameter as functions of discharge. Others have subsequently developed similar models, such as Cappelaere [1997], who developed a diffusive wave equation in terms of the same parameters by neglecting the inertia terms in the momentum equation of the Saint Venant equations. Todini [27b] also developed a similar routing model to the one used in this paper. The second issue concerns the identification of these model parameters with a uniform cross section that is essentially an average of the natural cross sections. A specific functional relationship for the cross section based on the notions of Knight [26] is used as the basis of calculating the model parameters. The third issue may be addressed by solving the routing equation for a prescribed upstream discharge time series and optimizing the parameters for the cross section to minimize the error in the downstream discharge time series. However, unknown lateral inflows and uncertain rating curves complicate the optimization process. Given a time series for rainfall along the reach the lateral inflow may be modeled by a design unit hydrograph rainfall-runoff model. [6] The reliability of forecasts for the stage downstream given the stage upstream and downstream to the current 1of15

2 PRICE: OPTIMIZED FLOW ROUTING MODEL time is a function of the time taken for discharge information to propagate along the reach. This can evidently be a nonlinear function of the discharge because of the nonlinear nature of the kinematic wave speed. Forecasts downstream can be improved if there are reliable forecasts of the primary input variables, namely the upstream stage or discharge and the rainfall for the lateral inflow. [7] Finally, the acknowledgment and allowance for uncertainty in model parameters and forecasts in practical applications of a model are important for decision making, but this issue is not covered in this paper; see Abebe and Price [24] for the improvement of forecasts by modeling separately the errors in the predictions. [8] In what follows, the theoretical aspects of the model and its calibration are described first. The model is then applied to a synthetic river, and confirmed in the case of the River Wye, UK. 2. Theoretical Basis of the Method 2.1. Model Equations [9] Following Price [1985] and beginning with the Saint Venant equations, assuming that the flow surface gradient defined relative to the bed gradient is small compared to the bed gradient and the lateral inflow is orthogonal to the stream flow, then to first order Q v þ Z Q a ðqþ c 2 ðqþ ¼ Qq 2gAs c ðqþ ¼ 1 dq B sf ðqþ dy n dq c ðqþ ¼ AQ ð Þ ¼ Q vq ð Þ ( " a ðqþ ¼ Q 1 B sf c Q 2 Q2 2s B sf ga A A 2 1 B # ) c B sf See Appendix A. Here Q discharge, m 3 /s; v averaged velocity at a cross section, m/s; q lateral inflow, m 2 /s; g acceleration due to gravity, m/s 2 ; Awetted cross-sectional area, m 2 ; s bed gradient; x distance, m; t time, s; c (Q) kinematic wave speed, m/s; a (Q) attenuation parameter, m 2 /s; y n normal depth, m; B c surface flow width, m; B sf surface storage width, m. [1] A numerical solution of equation (1) can be based on a finite difference scheme that uses a minimum of four grid points in space-time such that the expressions for each term are centered in time and space: C 1 Q nþ1 jþ1 þ C 2Q nþ1 j þ C 3 Q n jþ1 þ C 4Q n j þ 1 2 C q nþ1 þ q n ¼ ð1þ ð2þ ð3þ ð4þ ð5þ where C ¼2Dt C 1 ¼ 1 vjþ1 nþ1 þ Dt Dx þ 2 a Dx c 2 C 2 ¼ 1 v n þ Dt jþ1 Dx 2 Dx C 3 ¼ 1 vj nþ1 Dt Dx 2 a Dx c 2 C 4 ¼ 1 Dt Dx þ 2 Dx vj nþ1 nþ1 þ jþð1=2þ a n c 2 jþð1=2þ nþ1 jþð1=2þ a n c 2 þ jþð1=2þ q nþ1 2gA nþ1 jþð1=2þ s q n 2gA n jþð1=2þ s q nþ1 2gA nþ1 jþð1=2þ s q n 2gA n jþð1=2þ s Such a scheme conserves the fluxes into and out of the cell ( jdx, ndt), (( j +1)Dx, ndt), ( jdx,(n +1)Dt), (( j +1)Dx, (n +1)Dt)) in space-time. Note that C 1 + C 2 + C 3 + C 4 =. Here Q n j Q defined at the point ( jdx, ndt), m 3 /s; Dx space increment, m; Dt time increment, s. [11] C 1, C 2, C 3 and C 4 are nonlinear functions of the discharges and of Q n+1 j and Q n+1 j+1 in particular. Equation (5) can be solved for Q n+1 j+1 knowing Q n+1 j, c (Q), a (Q) and q n+1 using Newton s iterative method [see Price, 1985]. The Courant condition c Dt Dx 1 should be approximately satisfied to ensure reasonable accuracy of the solution. Using a centered difference scheme ensures that any numerical diffusion is an order of magnitude smaller than the physical diffusion induced by the secondorder derivative term in equation (1); that is, provided Dx Le 1 = 2 and Dt Te 1 = 2, where L and T are the length and time scales of the flood event, respectively. The second derivative term also assures stability of the numerical scheme. It can be shown that the scheme is consistent with the analytical equation and converges to it as Dx and Dt tend to zero Functional Forms for c (Q) and a (Q) [12] The terms c (Q) and a (Q) can be derived from the uniform cross section for the river channel at a uniform slope s. We define the channel as shown in Figure 1. Figure 1 is based on a form of cross section originally proposed by Knight [26]. In particular, we adopt the concept of a varying flow width and storage volume on the associated floodplains, which are functions of the flow depth. We define the channel by channel semi bed width B b, channel side slope s s, semi flow width B f for in-bank flow, semi storage width B s for in-bank flow, semi flow width B f for above-bank flow, and semi storage width B s for abovebank flow. Semi flow width B f for in-bank flow is B f ¼ B b þ s s y where y is the depth above the bed. Semi storage width B s for in-bank flow is " B s ¼ B b þ ðs s y ch þ B ch Þ 1 1 y # k1 y 2 ð9þ y ch y ch ð6þ ð7þ ð8þ 2of15

3 PRICE: OPTIMIZED FLOW ROUTING MODEL Figure 1. Half channel for the calculation of c (Q) and a (Q). provided B s > B f ; otherwise B s = B f. Semi flow width B f for above-bank flow is unit hydrograph. For this reason, q p can be adjusted by the factor y k3 y B f ¼ k 4 B fl þ B ch 1 s s y ch 1 y ch y ch ð1þ X N M Q n XNM J Q n 1 q base!, NM=2 X ðq n q base Þ ð14þ M=2 Semi storage width B s for above-bank flow is y k2 B s ¼ B b þ s s y ch þ B ch þ B fl 1 ð11þ y ch Here k 1, k 2, k 3 and k 4 are dimensionless numbers such that k 1 <1,k 3 < k 2 < 1 and k 4 <1. [13] We calculate the flow from Q ¼ A chr 2=3 ch n ch þ 2 A flr 2=3 fl n fl! s 1=2 ð12þ where A ch cross-sectional area for the channel flow, m 2 ; A fl cross-sectional areas for the left and right overbank flows, m 2 ; R ch hydraulic radius for the channel flow, m; R fl hydraulic radius for the left and right overbank flows, m; n ch Manning s n value for the channel, s/m 1/3 ; n fl Manning s n value for the left and right overbank flows, s/m 1/3. The derived variables, such as A ch and A fl, are determined by analytical integration where possible Determination of the Lateral Inflow [14] The lateral inflow is estimated from the rainfall along the reach. Suppose we have the rainfall time series {P n+1 } for the observed flood event. We convolute this series with a design unit hydrograph given in Figure 2 such that where N total number of time steps for the event; M number of time steps for a signal to travel along the reach at the average of the maximum in-bank kinematic wave speed and the minimum above-bank speed Determination of Rating Curve Extensions [17] In many practical cases, the rating curves relating discharge to stage (or depth) at a section are inadequate for above-bank flows because of bypassing of the main channel and the difficulty in taking appropriate current meter measurements. Often the more reliable in-bank rating curve is extrapolated to cover above-bank flows. Such an assumption introduces an error into the optimizing method above. The error affects the values of the parameters for the uniform channel cross section. We address this problem as follows. [18] Suppose that the rated section does have bypassing and that the use of a single rating curve introduces considerable errors for the discharges above bank. If we continue to use the in-bank rating curve for above-bank flows, it is likely that we will generate a lower peak discharge downstream than actually occurred. This happens to distort the optimization procedure below to determine the parameter values for the cross section. q nþ1 ¼ : XI 1 P niþ2 U i þ q base ð13þ where P n+1 rainfall at time (n + 1)Dt, mm/h; U i ith component of the design unit hydrograph, m; IDt duration of the design unit hydrograph, h; q base base flow, m 2 /s. [15] In this paper the time to peak, T p, and duration, D, of the unit hydrograph and q base are determined from the UK flood studies report [Natural Environment Research Council (NERC), 1985]. The peak of the design unit hydrograph, q p, is given by 22/T p. [16] In practice, the actual volume of the cumulative lateral inflow may differ from the prediction of the design 3of15 Figure 2. Design unit hydrograph.

4 PRICE: OPTIMIZED FLOW ROUTING MODEL Table 1. Results of the Optimization for the Synthetic River Parameter Units Prescribed Values Given Upstream and Downstream Rating Curves Given Upstream and Downstream in-bank Rating Curves Optimized Above-Bank Rating Curve Downstream Optimized Above-Bank Rating Curves Upstream and Downstream B b m y ch m s s B ch m B fl m k k k k b fl,up y fl,up m b fl,dn y fl,dn m O f [19] If this problem occurs the solution is to define separate above-bank rating curves: bab;up Q ¼ Q bf ;up y y bf ;up = yab;up þ y bf ;up þ 1 for y > ybf ;up ð15þ bab;dn Q ¼ Q bf ;dn y y bf ;dn = yab;dn þ y bf ;dn þ 1 for y > ybf ;dn ð16þ Here Q bf bank full discharge, m 3 /s; b ab above-bank rating curve index; y ab above-bank rating curve control depth, m; y = y bf transition between the in-bank and above-bank flows. The second subscript refers to upstream (up) or downstream (dn), respectively. Q bf is determined from the corresponding in-bank rating curve for y = y bf Calibration of the Model Parameters [2] We assume that the data for a given reach of river contains sufficient information such that we can calibrate the various unknown parameters by finding an optimal prediction of the discharge time series downstream. We do this by posing the optimization problem: Given the upstream stage (or depth) y up (t) and the downstream stage or depth y dn (t) find the dimensions of the parameterized uniform cross section (from which c (Q) and a (Q) are determined) and the parameters of the above-bank rating curve(s) (upstream and) downstream such that kq dn Q J k!min The objective function for the optimization can be identified with the root mean square error (RMSE) in the predicted downstream discharge time series, appropriately normalized: 2 O f O f ;1 RMSE ¼ 6 4 P N N L1 P N N L1 Q nþ1 2 J Q nþ1 dn Q nþ1 2 dn Q dn ð17þ The values of the resulting set of thirteen parameters {B b, y ch, s s, B ch, B fl, k 1, k 2, k 3, k 4, b fl,up, y fl,up, b fl,dn, y fl,dn } can be determined by minimizing O f in equation (17) over an extended time series that preferably includes significant high discharges. There can be, however, a significant connection between the parameters for the above-bank rating curves upstream and downstream in that additional volume introduced by arbitrarily adjusting the rating curve parameters upstream can lead to a corresponding change in the Figure 3. Wave speed and attenuation parameter curves for the synthetic river. 4of15

5 PRICE: OPTIMIZED FLOW ROUTING MODEL Figure 4. Test event for the synthetic river. parameters downstream. It may be necessary therefore to introduce a second constraint in the case of optimizing both of the above-bank rating curves. This could be defined as the difference between the above-bank volume for the event upstream as defined by the above-bank rating curve and the in-bank rating curve extended for stages above bank: O f ;2 ¼ XN " Qup;ab nþ1 Q # bf ;up Qup;ib nþ1 Q 1 bf ;up ð18þ n+1 Here Q up,ab upstream discharge determined by above-bank rating curve, m 3 n+1 /s; Q up,ib upstream discharge determined by in-bank rating curve, m 3 /s; Q bf,up upstream bankfull discharge, m 3 /s; The objective function could therefore take the form O f ¼ k 1 O f ;1 þ k 2 O f ;2 ð19þ where k 1 parameter for the RMSE; k 2 parameter for the difference in the above-bank discharges upstream. The introduction of the second constraint in equation (19) means that values have to be selected for k 1 and k 2. This is a wellknown Pareto problem (see V. Pareto, Cours d économie politique professé àl Université de Lausanne, two volumes, , available at [21] Alternatively, if the above-bank rating curve upstream is defined then the parameters for the above-bank rating curve downstream can be determined using the single objective function (k 2 = ). This is also the case for the optimization with both above-bank rating curves defined, or for the above-bank rating curve defined downstream but not upstream. We use the global optimization tool, GLOBE, and the adaptive cluster covering with local search (ACCOL) algorithm; see and Solomatine [1995, 1999] Forecasting Algorithm [22] When the routing model is calibrated it can be used straightforwardly to forecast flows and therefore stages downstream. But it is also necessary to provide some form Figure 5. Results for the test event with in-bank rating curves extended above bank both upstream and downstream. 5of15

6 PRICE: OPTIMIZED FLOW ROUTING MODEL Figure 6. Pareto front for the optimization of the above-bank rating curves both upstream and downstream. of forecast for the discharge upstream and the lateral inflow (or rainfall). [23] For simplicity, the forecast of the upstream discharge is done using a projected hydrograph of the form Q nþi up ¼ Q b þ Q p Q b ½:5ðm þ iþexpð1 :5ðm þ iþþš 8 ð2þ where Q b base flow of forecast discharge hydrograph upstream, m 3 /s; Q p peak flow of forecast discharge hydrograph upstream, m 3 /s; m number of forecast time steps to the current time; i forecast time steps. [24] The form of the expression for Q n+1 up in equation (2) is derived from data for significant flood events upstream. Q b, Q p and m are determined in terms of the current discharge upstream and the recent discharge values there by seeking a good fit to the data. Similarly, the forecast of the rainfall generating the lateral inflow is made such that P nþi ¼ P n maxð1 i=i; Þ ð21þ where I number of time steps for the decay of the rainfall. If rainfall is available for the upstream catchment then the predictions of an appropriate rainfall-runoff model could be used in place of equation (2). This is not pursued further in this paper. 3. Application to a Synthetic River [25] In that we do not claim to know the rating curve extensions for a real river, we cannot confirm the method above unless we resort to data generated for a synthetic river with prescribed above-bank rating curves. Therefore, we set up a synthetic reach, which has a typical uniform cross section as may be deduced for a real river. We assume that the synthetic river has a reach length of 1 km and a bed slope of.8. We define the upstream stages (or depths) by yð; tþ ¼ y base for t < t y yð; tþ ¼ y base þ y peak y base t t y exp 1 t t by y for t y < t ð22þ T y T y where y base base depth, m; y peak peak depth, m; t time, h; t y time of the start of the flow event, h; T y time to peak of the flow event, h; b y index. and the lateral inflow by q(t) =q base for t < t q qt ðþ¼q base þ q peak q base t t q exp 1 t t bq q for t q < t ð23þ T q T q Figure 7. River Wye, Erwood to Belmont. 6of15

7 PRICE: OPTIMIZED FLOW ROUTING MODEL Table 2. Event Data for the River Wye Event Number Start Date End Date Peak Discharge at Erwood With in-bank Rating Curve (m 3 /s) 1 2 Dec Jan Oct 2 11 Nov Nov 2 19 Dec Jan 22 3 Mar Here q base base lateral inflow, m 2 /s; q peak peak lateral inflow, m 2 /s; t q time of the start of the lateral inflow event, h; T q time to peak of the lateral inflow event, h; b q index. [26] It is assumed that the typical upstream flood event for the synthetic reach is defined by y base =.5m,y peak =5.m, T y = 24. h, and b y = 5.. Similarly, q base =.2 m 2 /s, q peak =.5m 2 /s, T q =6.h,t q = 36. h and b q = 1.. Table 1, third column, gives the parameters for the uniform channel with n ch =.35 and n fl =.8. At the upstream and downstream boundaries, we adopt in-bank rating curves, which are distinct from the normal depth curve for the uniform cross section along the reach: h. i bch;up Q ¼ Q bf ;up y y bf ;up y ch;up þ y bf ;up þ 1 ð24þ h. i bch;dn Q ¼ Q bf ;dn y y bf ;dn y ch;dn þ y bf ;dn þ 1 ð25þ Here Q bf,up = m 3 /s, b ch,up = 1.9, y bf,up = 4. m, and y ch,up =.3 m, and Q bf,dn = m 3 /s, b ch,dn = 1.5, y bf,dn = 5. m, and y ch,dn =.15 m. Similar expressions define the above-bank rating curves. The coefficients for the above-bank rating curves (see equations (15) and (16)) are selected such that the in-bank and above-bank curves are continuous at the bankfull depth. [27] Figure 3 displays the resulting wave speed and attenuation parameter curves. The discharges at the downstream boundary are generated using a four-point implicit and iterative (nonlinear) finite difference solution of the Saint Venant equations; see Price [1974]. The in-bank and above-bank rating curves are used at the downstream boundary, which are adjusted using the Jones formula; see Jones [1916]. The space and time increments in the calculations are 1 m and 225 s, respectively. [28] Figure 4 shows the test event with the lateral inflow timed to affect the peak discharge of the downstream hydrograph. The predictions of the Saint Venant model (without the corresponding lateral inflow time series) form the input to the routing model. The results from the routing model for the same channel dimensions, lateral inflows and the above-bank rating curves upstream and downstream are only marginally different from the predicted values, as shown in Figure 4 (the calculated downstream discharges are indistinguishable from the observed discharges at this level of detail), which confirms the accuracy of the routing equation. The Nash-Sutcliffe (NS) performance parameter, 2 given by NS =1 O f,1 [see Nash and Sutcliffe, 197]), takes the value.9998, which indicates an excellent performance by the routing model. [29] A number of experiments using the optimization procedure are carried out. The first is for the channel parameters, assuming that the upstream and downstream Figure 8. River Wye monitored and predicted hydrographs for in-bank rating curves extended for above-bank stages at Erwood and Belmont. 7of15

8 PRICE: OPTIMIZED FLOW ROUTING MODEL Table 3. River Wye Parameter Values for the Channel and the Rating Curve Extension Parameter Units Knight [26] Given Upstream and Downstream Rating Curves Optimized Above-Bank Rating Curve Downstream Optimized Above-Bank Rating Curve Downstream Volume Conserved Optimized Above-Bank Rating Curve Upstream B b m y ch m s s B ch m B fl m k k k k q base q p b fl,up y fl,up b fl,dn y fl,dn O f, rating curves, both in bank and above bank, are given. Table 1, fourth column, gives the parameter values for the optimized cross section. There are some differences between the original and optimized values due to the approximations in the routing model, but the values are acceptably close. Note that the optimized values are not unique. In general, there are (small) ranges of values that are acceptable for each parameter. [3] The second experiment involves the assumption that the in-bank rating curves can be extended above-bank both upstream and downstream. The results are shown in Table 1 column (3). Evidently, the optimization is seeking to increase the attenuation of the peak by restricting the width of the channel and increasing the bankfull depth; see Figure 5. This trend is evidence that the in-bank rating curves are inadequate for above-bank stages. [31] Next, the optimization is carried out for the channel and the above-bank rating curve downstream with the separate in-bank and above-bank rating curves upstream. The results are shown in Table 1, sixth column. These confirm that the optimization generates realistic values for the above-bank rating curve downstream. [32] Last, we carry out the optimization when the parameters for the channel and both above-bank rating curves are unknown. This time a second objective function is introduced as explained above. The Pareto front is shown in Figure 6. What is interesting is that the Pareto front indicates that the optimum is achieved when O f,2 =, and the single objective function is sufficient; see Table 1, seventh column. Note that the constraint given by O f,2 is still needed to contain any arbitrary growth in the upstream hydrograph above bank. [33] Having demonstrated that the optimization procedure applied to a synthetic river is effective for several cases when parameters for the channel cross section and the abovebank rating curves are unknown, we now consider data for Figure 9. River Wye Pareto front for the optimization of the above-bank rating curves simultaneously at Erwood and Belmont. 8of15

9 PRICE: OPTIMIZED FLOW ROUTING MODEL Figure 1. River Wye revised rating curves for Erwood and Belmont. a real river where, in addition, we can only estimate the lateral inflow. 4. Application to the River Wye [34] The River Wye has its source above Builth Wells (longitude W, latitude N) in Wales, and flows eastward into England; see Figure 7. The 7 km reach between Erwood and Belmont near Hereford has an average slope of.88, and has extensive floodplains, for example in the region of Bredwardine, that become inundated for flows above 4 m 3 /s. [35] The gauging stations at Erwood and Belmont (which have catchment areas km 2 and km 2, respectively) have well established rating curves for in-bank flows, though there are no reliable curves for overbank flows. A review of the rating curve data at the upstream gauging station, Erwood, shows that the in-bank rating curve is valid up to about 561 m 3 /s for a depth of 4. m. Q ¼ 56:82½ðy 4:Þ= ð:324 þ 4:Þþ1Š 2:336 ð26þ The river section at Erwood is reasonably well confined, and there are current meter readings for higher discharges. However, these are inconsistent, and confidence in the extension of the in-bank rating curve to deal with out-ofbank flows is somewhat limited. [36] At Belmont the rating curve for in-bank flows is Q ¼ 417:485½ðy 5:Þ= ð:15 þ 5:Þþ1Š 1:53263 ð27þ and this is recommended as being valid up to y 5. m. Bypassing of the main channel by the out-of-bank flows at Figure 11. River Wye optimized forms for c (Q) and a (Q) deduced from event 2. 9of15

10 PRICE: OPTIMIZED FLOW ROUTING MODEL Figure 12. River Wye model applied to event 4. Belmont is more significant than at Erwood. It is therefore reasonable to suppose that discharges calculated with equation (25) for larger values of y will be significantly in error. [37] The Environment Agency in the UK made several time series events available for the River Wye. Table 2 lists the more recent events used in this study. In each case, the data for water levels and discharges are at 15 min intervals. Hourly records from two rainfall gauges are available for the duration of the four events at Broomy Hill and Tregoyd; see Figure 7. Using the flood studies report [NERC, 1985], q base =.35 m 2 /s for the reach between Erwood and Belmont. [38] We apply the routing method and optimizing procedure to the River Wye. Taking into account the Courant condition (see equation (7)) and the need of the optimization procedure for computational speed, we select Dt = 36 s, which implies that Dx = 5 m is acceptable. [39] We optimize the channel parameters for the River Wye first for single (in-bank) rating curves upstream and downstream with event 2, which has a maximum upstream discharge of 855 m 3 /s using the in-bank rating curve. Table 3, fourth column, gives the values of the parameters for the uniform cross section, and Figure 8 shows the observed and predicted discharge hydrographs. The channel depth and semi bed width are m and m, respectively. The former value is significantly larger and the latter value is significantly smaller than the estimates made by Knight [26] from surveyed cross sections for the reach. There are also correspondingly sizable errors around the peak of the flood at Belmont. According to the guidelines developed above, there is therefore reasonable doubt that the single rating curve downstream is sufficient. For this reason, the optimization procedure is applied to event 2, again assuming that the in-bank rating curve upstream can be extended for above-bank stages, but this time the abovebank rating curve downstream is to be determined along with the channel parameters. In contrast to the first optimization for single rating curves only, the channel depth and semi bed width are now m and m, respectively; see Table 3, fifth column. This is in much better agreement with Knight [26]. However, there is an imbalance in the volume. This can be dealt with by adjusting the base lateral inflow and/or the peak of the unit hydrograph. It was found that the RMSE value is reduced when the base flow of the lateral inflow is included in the optimization and the peak of the unit hydrograph, q p, is adjusted to conserve volume at the downstream boundary (see equation (13)); the parameter values for the channel cross section and the above-bank rating curve downstream are given in Table 3, sixth column. [4] At this stage we can proceed to optimize the parameters for both the above-bank rating curves and the channel cross section using both objective functions. However, the sizable uncertainty in the lateral inflow as well as the abovebank rating curves appears to contribute to an unacceptable increase in the (peak) discharges upstream and downstream while reducing the objective function RMSE value; see the Pareto front given in Figure 9. Therefore a different approach is adopted in order to determine the above-bank rating curve upstream. Assuming that the parameters derived for the above-bank rating curve downstream are correct, the optimization is carried out for the above-bank rating curve upstream, again including the cross section parameters and the base lateral inflow while adjusting q p to conserve the volume at the downstream boundary. The results of this analysis are given in Table 3, seventh column. The peak discharge upstream for event 2 is m 3 /s, which is a Table 4. River Wye Performance for Four Events Parameter Event 1 Discharge Event 2 Discharge Event 3 Discharge Event 4 Discharge NS of 15

11 PRICE: OPTIMIZED FLOW ROUTING MODEL Figure 13. River Wye 1 h forecasts at Belmont for event % increase over the value predicted using the in-bank rating curve; see Figure 1 for the full rating curves at Erwood and Belmont. Figure 11 presents the final optimized forms for c (Q) and a (Q), compared with those estimated by NERC [1975]. The calculated curve for c (Q) is similar to the curve given in NERC [1975], though the in-bank values are marginally smaller. The curve for a (Q) is now considerably lower. (Note that here a (Q) =aq/l as defined in NERC [1975]). Also of significance are a sizable increase in q base and a corresponding decrease in q p. These adjustments appear to favor a larger base lateral inflow than actually occurs to compensate for errors in the short-term generation of lateral inflow from rainfall using the design unit hydrograph. [41] The resulting instantiated model is applied to events 1, 3 and 4. It is evident from Table 4 that the values of the NS performance criterion are of the order of.9, and therefore the discharges are reasonably well predicted; see Figure 12 for event 4. Obviously a more accurate rainfallrunoff model for the lateral inflow would help in improving the routing model, but this is generally not available. [42] Forecasts are made for times up to 24 h ahead for each of the events. The values of NS for the stage and discharge forecasts are given in Table 5, while Figure 13 and Figure 14 show the forecasts for event 3 and event 4, respectively, 1 h ahead. It is evident that the forecasts are reliable up to 1 h ahead and can be used up to 14 h ahead for the larger events. Note that the forecasts of discharge perform better than the forecasts of the stage for event Discussion [43] The method above to calibrate a routing model for flow forecasting on the basis of knowledge of the upstream and downstream discharge time series uses a simplified flow routing equation derived from the 1-D Saint Venant equations and correct to the first order in the surface gradient number. It is theoretically possible to use the full 1-D Saint Figure 14. River Wye 1 h forecasts at Belmont for event of 15

12 PRICE: OPTIMIZED FLOW ROUTING MODEL Table 5. River Wye Forecast Performance for the Four Events Forecast Time Ahead (h) NS Event 2 Stage NS Event 2 Discharge NS Event 1 Stage NS Event 3 Stage NS Event 4 Stage Venant equations in place of the flow routing equation, although the computation time in the optimization of the uniform cross section and rating curve extension parameters would be greater. [44] Obviously there should be no impediment for any water on the floodplain returning to the channel during the recession of a flood. This prevents any application of the present model to rivers with embankments that are overtopped; a full 2-D hydrodynamic model would be needed. The assumption of one dimensional flow is also strained as the width of the inundated floodplain increases. This is because the form of the kinematic wave speed curve for above-bank flows can lead to a steepening of the recession of the hydrograph such that the drainage of water back to the channel cannot occur quickly enough and the one dimensional assumption is violated. The application to the reach of the River Wye in this paper presents a considerable test of this assumption. [45] The modeling approach adopted in this paper requires a precise treatment of the flow routing process, and this could not be achieved without the use of the nonlinear routing equation [see also Todini, 27b]. It is apparent that for rivers with extensive floodplains, the kinematic wave speed can vary significantly over the range of discharges occurring, and a reliable prediction of flood discharges would therefore require the full nonlinearity of the flow routing equation. [46] The result of the optimization in terms of a set of specific values for the model parameters is not unique. A set of values therefore must be seen as instantiating a particular model for the river. The parameters for the above-bank rating curves do not guarantee an actual discharge related to a given stage; they are however, integral components of the resulting model. [47] The forecasts for the River Wye decay significantly after 1 h. Such a threshold could possibly be extended by better forecasts of the rainfall used in predicting the lateral inflow and the upstream discharge hydrograph, possibly generated by a catchment model from rainfall. Because the forecasts are derived using a conceptual model based on physical principles there is a greater measure of confidence in the predictions of extreme events than with a purely data driven model, such as one based on a trained artificial neural network model. For actual application in forecasting some form of error correction using a data driven approach, such as an artificial neural network [see Abebe and Price, 24] or Kalman filter [see Todini, 27a], should be made to improve the discharge, and therefore the stage, prediction in general. 6. Conclusions [48] A flow routing model for flood forecasting is calibrated for a reach of river. The model is dependent primarily on knowledge of the time series for the stage hydrographs at the upstream and downstream gauging stations of the reach together with the corresponding in-bank rating curves, the rainfall along the reach, and a minimal amount of other information. The hydrographs are linked through a physically based flow routing equation that is dependent on the kinematic wave speed and attenuation parameter for the reach, which are nonlinear functions of discharge. The model assumes a parametric form for the cross section of the uniform channel (with floodplains) along the reach, from which the kinematic wave speed and attenuation parameter are derived. The lateral inflow is generated by a calibrated rainfall-runoff model based in this paper on the UK design unit hydrograph. The resulting model for the parametric cross section is used to predict the downstream discharge time series. The root mean square error of the predicted discharge time series forms the objective function in an optimization process devised to refine the uniform cross-section parameters. Where the mismatch remains poor for higher discharges there are possibly errors in the rating curves for the above-bank flows upstream and downstream. Parameters for the extended rating curves can also be included in the optimization process to generate better rating curve extensions. A numerical solution of the full 1-D Saint Venant equations for flow in a synthetic river reach is used to confirm the model, which is then applied to the River Wye, UK. The resulting model is calibrated for a given time series event and confirmed successfully for three other time series events. The forecast results show that the model has an effective forecast threshold of 1 h for the reach between Erwood and Belmont on the River Wye. Appendix A [49] The one-dimensional Saint Venant equations can be normalized to give eg Q A s þ s ¼ dq ða1þ ða2þ 12 of 15

13 PRICE: OPTIMIZED FLOW ROUTING MODEL where e ¼ y x. s ða3þ In addition, we note that from the conservation of mass B sf ða9þ g ¼ Q 2 ga 2 y ða4þ Differentiating equation (A8) with respect to t and x and using equation (A9) we obtain to zeroth order in ¼ cq; ð y Þ þ OðÞ e ða1þ d ¼ x Q q ða5þ Here Q(x, t) discharge, m 3 /s; A(x, t) wetted cross-sectional area, m 2 ; g acceleration due to gravity, m/s 2 ; y(x, t) depth, m; q(x, t) lateral inflow (which is assumed to enter the channel normal to the direction of flow), m 2 /s; s (x) bed gradient of the river channel; s f (Q, A, y, x) friction slope; t time, s; x distance, m. [5] These variables are normalized by their scales, which are indicated by the same letters but with the superposition of a bar. The scale of t is given by xa=q from equation (A2). The three dimensionless numbers are: e slope number that expresses the ratio of the surface gradient (defined relative to the bed gradient) to the bed gradient; g Froude number squared; d lateral inflow number. e is essentially an event-based number in that it depends on the length scale for the (flood) event, whereas g depends primarily on the nature of the channel and the flow it conveys. d is typically less than unity, but it can still be significant. [51] Generally, for lowland rivers g e 1. We focus on generating a consistent approximation to the Saint Venant equations of the order of e; that is, in mathematical notation, O(e). We assume that the uniform cross section has both an active part that conveys the flow and a passive part that acts as storage; see Figure 2 above. Both the active and passive parts of the cross section are dependent on the depth of flow such that we can define a surface width B c (y) for the flow and a storage width B sf (y) (including the flow width). In addition, we define the friction slope as s f ¼ Q2 K 2 ða6þ where K(y) conveyance of the channel, m 3 /s; and we assume that the time-dependent lateral inflow is uniformly distributed along the reach. [52] We write the momentum equation equation (A1) as Q ¼ KðyÞs 1 = 2 1 eg Q 2 1 = 2 A We assume that e 1. It follows that Q ¼ KðyÞs 1 = 2 1 eg 2gAs þ O e þ Q 2 A ða8þ ¼ cq; ð y ÞB sf þ OðÞ e cq; ð yþ ¼ Q 1 dk ðyþ K dy This means we can rewrite equation (A8) as Q ¼ Ks 1 = e 2s B sf c gc þ gq 2gAs 2gA 2 s 2 QB c egdcq þ O e 2 AcB 2gAs We now define aq; ð yþ ¼ Q 2s B sf Then equation (A13) becomes Q ¼ Ks 1 = 2 1 ea egdcq 2gAs ða11þ ða12þ ða13þ ( " 1 gb sf c Q 2 Q2 ga A A 2 1 B # ) c B sf þ O e 2 ða14þ ða15þ Differentiating this equation with respect to t and noting equation ¼ cq; ð y Þ dq a cq cq þ O e ða16þ The problem with equation (A16) is that c is a function of y as well as Q. We need to replace y as a function of Q. From equation (A12) where cq; ð yþ ¼ QFðyÞ FðyÞ ¼ 1 1 dk ðyþ K dy B sf ða17þ ða18þ 13 of 15

14 PRICE: OPTIMIZED FLOW ROUTING MODEL Now from equation (A15) y ¼ f Qþ e a þ egdcqq 2gAs þ O e 2 where f(.) is a (undetermined) function. We define ða19þ c ðqþ ¼ QFð f ðqþþ ða2þ or dq c ðq Þ þ a ðqþ c 2 ðqþ ¼ Qq 2gAs ða27þ to O(e) (and ignoring e and g). The first term in equation (A27) can be replaced where Similarly from equation (A14) a ðqþ ¼ QGðQÞ ða21þ 2 v ¼ 4 1 Q Z Q 3 dq 5 c ðq Þ 1 ða28þ where ( " 1 GQ ð Þ ¼ 1 gb sf ðqþ c Q 2 2s B sf ðqþ gaðqþ AQ ð Þ Q2 AQ ð Þ 2 1 B #) cðqþ ða22þ B sf ðqþ according to the zeroth-order relationship between y and Q from equation (A19). [53] Note that c (Q) is determined from the rating curve defined by the normal depth flow at the averaged cross section: c ðqþ ¼ 1 dq B sf ðqþ dy n where y n is the normal depth. So cq; ð yþ ¼ c ðqþþec Q d c dq Q a c þ gdqq 2gAs ða23þ þ O e 2 ða24þ where a refers to a as a function of Q only. Substituting for c in equation (A16), and noting the change in the definition of a, we ¼ egd þ ec Q d c a dq Q c þ gdqq a c 2 Qq q c 2gAs þ Rearranging this equation þ a Qq ¼dc q 2gAs þ O e 2 þ O e 2 ða25þ ða26þ Notation A(x, t) wetted cross-sectional area, m 2. A ch cross-sectional area for the channel flow, m 2. A fl cross-sectional areas for the left and right overbank flows, m 2. a (Q) attenuation parameter. ab subscript above bank. B b semi bed width, m. B c (y) surface width of flow, m. B ch additional semi storage width for in-bank flow, m. B fl semi flow width for in-bank or above-bank flow, m. B sf (y) storage width (including flow width), m. B s semi storage width for in-bank or above-bank flow, m. C, C 1, C 2, C 3, C 4 nonlinear functions in finite difference equation. c (Q) kinematic wave speed, m/s. D duration of design unit hydrograph, h.,dn subscript referring to downstream. f(.) function. g acceleration due to gravity, m/s 2. I number of points in the design unit hydrograph. ib subscript in bank. j number of space increments. J number of space increments along the reach. K(y) conveyance of the channel, m 3 /s. k 1, k 2, k 3, and k 4 dimensionless numbers for channel cross section. L length of reach, m. m initial forecast time step. n number of time increments. N L number of time steps along reach at average above-bank wave speed. n ch Manning s n value for the channel. n fl Manning s n values for the left and right overbank flows. O f objective function for optimization. ith rainfall point, mm/s. P i 14 of 15

15 PRICE: OPTIMIZED FLOW ROUTING MODEL Q(x, t) Q base, Q b Q peak, Q p discharge, m 3 /s. base flow, m 3 /s. peak flow, m 3 /s. Q bf bankfull discharge, m 3 /s. Q dn (t) downstream discharge, m 3 /s. Q dn average downstream discharge, m 3 /s. Q J (t) calculated downstream discharge, m 3 /s. Q up (t) upstream discharge, m 3 /s. n Q j Q defined at the point (jdx, ndt), m 3 /s. q(x, t) lateral inflow (which is assumed to enter the channel normal to the direction of flow), m 2 /s. base lateral inflow, m 2 /s. peak lateral inflow, m 2 /s. q defined at the time (n + (1/2))Dt, m 2 /s. R ch hydraulic radius for the channel flow, m. R fl hydraulic radii for the left and right overbank flows, m. s (x) bed gradient of the river channel. s s channel side slope. T p time to peak of design unit hydrograph, h. T y time to peak of upstream discharge event, h. T q time to peak of lateral inflow event, h. t time, s. t Q start of upstream discharge event, h. q base q peak q n+(1/2) t q start of the lateral inflow event, h. U component of design unit hydrograph, m.,up subscript referring to upstream. v(q) averaged velocity across a wetted cross section, m/s. x distance, m. y(x, t) depth or stage, m. y base base flow stage, m. y bf bankfull depth, m. y ch channel depth, m. incremental depth for in-bank rating curve equation, m. incremental depth for above-bank rating curve equation, m. y n normal depth, m. y ch y fl b ch b fl b y b q g in-bank rating curve index. above-bank rating curve index. index for upstream discharge event. index for lateral inflow event. square of the Froude number. d lateral inflow number. e slope number that expresses the ratio of the surface gradient (defined relative to the bed gradient) to the bed gradient. k 1 first Pareto coefficient. k 2 second Pareto coefficient. Dx space increment, m. Dt time increment, s. {.} time series for the specified variable. v scale of the variable v. [54] Acknowledgments. The author is grateful to the Environment Agency, UK, for providing the data for Erwood and Belmont on the River Wye, to colleagues and students at UNESCO-IHE for their comments and insights, and to the reviewers for their constructive suggestions made on earlier versions of this paper. References Abebe, A. J., and R. K. Price (24), Information theory and neural networks for managing uncertainty in flood routing, J. Comput. Civ. Eng., 18(4), , doi:1.161/(asce) (24)18:4(373). Cappelaere, B. (1997), Accurate diffusive wave routing, J. Hydrol. Eng., 123(3), , doi:1.161/(asce) (1997)123:3(174). Jones, B. E. (1916), A method for correcting river discharge for a changing stage, U.S. Geol. Surv. Water Supply Pap., 375-E. Knight, D. W. (26), River flood hydraulics: Validation issues in onedimensional flood routing models, in River Basin Modelling for Flood Risk Mitigation, edited by D. W. Knight and A. Y. Shamseldin, chap 18., pp , Taylor and Francis, Leiden, Netherlands. Nash, J. E., and J. V. Sutcliffe (197), River flow forecasting through conceptual models part I A discussion of principles, J. Hydrol., 1(3), , doi:1.116/ (7) Natural Environment Research Council (NERC) (1975), Flood routing, Flood Stud. Rep. 4, 76 pp., Wallingford, U.K. Natural Environment Research Council (NERC) (1985), Flood studies, Suppl. Rep. 16, Inst. of Hydrol., Wallingford, U.K. Price, R. K. (1973), Flood routing methods for British rivers, Proc. Inst. Civ. Eng., 55(4), , doi:1.168/iicep Price, R. K. (1974), Comparison of four numerical methods for flood routing, J. Hydraul. Div. Am. Soc. Civ. Eng., 1(7), Price, R. K. (1985), Flood routing in rivers, in Developments in Hydraulic Engineering, edited by P. Novak, chap. 4, pp , Appl. Sci., London. Solomatine, D. P. (1995), The use of global random search methods for models calibration, paper presented at XXVIth Congress, Int. Assoc. for Hydraul. Res., London. Solomatine, D. P. (1999), Two strategies of adaptive cluster covering with descent and their comparison to other algorithms, J. Global Optim., 14(1), 55 78, doi:1.123/a: Todini, E. (27a), A mass conservative and water storage consistent variable parameter Muskingum-Cunge approach, Hydrol. Earth Syst. Sci., 11, Todini, E. (27b), Hydrological catchment modeling Past, present and future, Hydrol. Earth Syst. Sci., 11, R. K. Price, UNESCO-IHE, P.O. Box 315, Delft NL-261 DA, Netherlands. (r.price@unesco-ihe.org) 15 of 15