(1)School of Civil and Environmental Engineering, The University
of New South Wales, Sydney NSW 2052, Australia
Phone: 61 2 9385 5040, Fax: 61 2 9385 6139, Email: shangyou@civeng.unsw.edu.au
Abstract: Mitigation of flood-related disasters relies on accurate forecasting of flood hydrographs. A linear storage routing model, which attempts to provide a physically realistic distribution of storage effects, has recently been developed (Zhang et al, 2000a, 2000b). This model is particularly useful for estimation of large floods. It has been shown that for large floods, those with peak discharges greater than bankfull flow, catchments tend to operate as linear systems. The model with its physically realistic storage effects has been shown to provide consistently good reproduction of observed large floods without the need for direct parameter fitting or calibration. In the model framework, storage distribution is determined from a volume law that accounts for the distribution of storage in natural catchments. The model is briefly described, but the emphasis of the paper is on demonstrating its ability to reproduce large observed floods. Application results, including the estimation of probable maximum floods, are compared with those estimated using two other common storage routing models.
Keywords: flood forecasting; storage routing, rainfall runoff model
The reliable estimation of large floods is crucial for many engineering applications, including mitigation of flood related disasters. Various hydrological and hydraulic models have been developed for flood estimation [Pilgrim and Cordery, 1993, Fread, 1993]. While hydraulic models are suitable in dealing with complex flow regimes in large rivers, hydrological models are known for their simplicity and reasonable accuracy for small to medium sized catchments, where kinematic wave conditions may prevail. Lack of data and expertise on model application also prevents the use of many sophisticated hydraulic models for routine flood studies. The boundary and initial conditions of these models, which are crucial to the model success, are difficult to meet in such cases. Among numerous hydrological models, RORB (Laurenson and Mein, 1995) and WBNM (Boyd et al, 1987) have been recommended for estimation of large floods in Australia (Institution of Engineers, 1987). An important component in these models is the assumed nonlinear storage-discharge relationship. However, the question as to whether the nonlinear storage-discharge relationship is appropriate for estimation of large floods has never been answered. Zhang and Cordery (1999a, 1999b) challenged the nonlinear storage-discharge relationship. They illustrated marked linear behaviour at high flows, and suggested the use of the linear storage-discharge for estimation of large floods.
A large flood here refers to any flood with ARI equal to or greater than 10 years. According to Bates and Pilgrim (1983), a bankfull discharge would have an ARI of 2-5 years. A 10-year flood would be well over the bankfull level and can be regarded as having fully developed floodplain flow.
It has been shown that the new model, code named LSRM (linear storage routing model), can provide good reproduction of observed large floods without the need for parameter fitting or direct search optimization (Zhang et al, 2000a, 2000b). The application of the model does not require too much data. The need for such a simple model for practical flood estimation is quite obvious (Pilgrim and Cordery, 1993).
This paper briefly describes the structure of the LSRM and presents some application results. The aim of the paper is to compare the performance of the model to that of the more conventional modeling schemes such as RORB and WBNM.
LSRM is a storage routing model with a simple soil moisture component, in which the so-called initial loss and continuous loss method for estimation of effective rainfall is used. The aim of the model is to provide an adequate routing scheme that is appropriate for the estimation of large floods. The storage routing is based on a linear storage-discharge relationship as follows,
(1)
where S is storage (m3), Q is discharge (m3/s) and K is the linear storage delay time (hour) of a catchment. A simple method has been proposed for estimation of K using the recession curves of large floods without using any direct fitting optimization (Zhang and Cordery, 1999a). After the separation of base flow from the selected recession curves, backward integration can be performed to obtain (S, Q) pairs. A plot of these values can give a graphical presentation of the observed storage-discharge relationship. The global matching of the S-Q plot by Equation (1) gives the value of K.
Like RORB and WBNM, LSRM also has a distributed routing structure. It is assumed that for each subcatchment the S-Q relationship is still linear, but with different storage delay parameters (ki). These parameters are estimated from the catchment storage delay time, K, using a volume law that offers quantitative estimation of the ratios between the volume of a channel segment and the total volume of the entire channel system (Zhang et al, 2000a). The volume ratios are estimated using the following formulae:
(2)
(3)
(4)
(5)
where hi is the ratio between the volume of the ith channel segment and the total volume of the channel network. hi¢ is the remaining volume ratio above the ith channel segment. N is the number of channel segments. h0¢ is the total volume ratio. li and Li are the corresponding channel lengths. The calculation starts from the lowest channel segment and progresses upstream. Details can be found in Zhang and Cordery (1999b), and Zhang et al (2000a). After all ratios are calculated, the ki values are simply interpolated by the following equation:
(6)
Equation (6) shows that ki is specified in such a way that storage increases downstream. In contrary, RORB and WBNM specify storage decreasing downstream, which logically does not appear to match geomorphological reality.
LSRM provides a general routing solution expressed in a succinct mathematical form as follows (Zhang and Cordery 1999b, Zhang et al 2000a),
(7)
where N is the number of routing elements. m(i) is the number of rainfall ordinates for input number i. U(t) is a unit step function. DIi,j =Ii,j - Ii,j-1 and Ii,,j refer to the jth value of ith rainfall input. Equation (7) means that the outflow contributed from each subcatchment is separately routed through the storage network, and the sum of these outflows gives the total discharge of the catchment. Before carrying out the routing task, parallel cascades of storage elements should be arranged. Details can be found from Zhang and Cordery (1999b), and Zhang et al (2000a).
Let’s now examine the capability of the volume law in enhancing the estimation of the rising limbs of large floods. Suppose a catchment is divided into two equal parts. The overall value of K is 1 hour. Let’s consider four scenarios. (1): the storage parameter for the upper part k1, is equal to the storage parameter for the lower part, k2. Both are 0.5 hours. (2): (volume law) k1 and k2 are 0.125 hour and 0.875 hour respectively using Equations (2) to (5), or ki increases downstream. (3): the values of k1 and k2 are inversed, with k1 taking the value of 0.875 and k2 0.125, or ki decreases downstream (as in RORB and WBNM). (4): k1 and k2 are assumed equal but the input is lumped at the top of the catchment (as in the Nash model (Nash, 1957)). Assume the rainfall input is uniformly distributed and I1 = I2 =100 m3/s. The routing results of these scenarios are compared in Figure 1.
Figure 1 illustrates that, even if the rainfall is uniformly distributed over a catchment, the results are different because of the ki values. For example, the peak discharges corresponding to scenario three (ki decreasing downstream) and scenario two (ki increasing downstream) are 133 m3/s and 78 m3/s respectively, which indicates that the volume law could help to reduce the chance of overestimating flood peaks.
The Shihe River at Zushimiao is located in the Southwest of Henan Province, China. The gauging station, which monitors the flows entering the downstream reservoir, the Banqiao, is located at 113°34¢E, 33°02¢N. The catchment area of the Shihe River is 71.2 km2. Three pluviometers were used in the application. The aim is to reproduce a historical flood occurred in August 1975. According to Historical floods in China (1991), this flood was the biggest since 1593, or had an ARI of at least 400 years.
The catchment map, channel network and routing paths are shown in Figure 2. The whole catchment is divided into three subcatchments because there are three pluviometers. Three storage elements are adopted. Since the details of the channel lengths are not available (the map is only a rough sketch), an assumption is made here that the channel lengths the three subcatchments are identical, and so do the areas. Therefore, the volume ratios for subcatchments 3, 2 and 1 can be readily calculated. The catchment storage delay parameter (K) is estimated using the recession curve of an earlier flood. Figure 3 shows the S-Q values and the fitted relationship, which is approximately linear at high flows. The coefficient of the matched S-Q relationship is 8663s, so K=2.4 hours. This value is used in the model for the estimation of the historical flood.
The routing is performed using Equation (7) along the routing paths specified in Figure 2 (c) at a 1-hour time step. Figure 4 shows the observed and estimated hydrographs. Because none of the parameters is estimated from the current flood, the estimated hydrograph can be regarded as a verification result. The observed and estimated peak discharges are 2470 m3/s and 2345 m3/s respectively. The rising limb of the observed hydrograph is almost perfectly matched by LSRM.
This example provides evidence that LSRM is capable of estimating extreme floods under design conditions. The good fit to the observed hydrograph in this case can be attributed to three factors: the marked linearity of the catchment (Figure 3), the reasonably correct model structure, and the reliable data. The estimation of the storage delay time (K) using the recession curve analysis method is also proven to be successful. Without this method, the estimation of K would be somewhat difficult.
RORB and WBNM were not applied to Shihe River because of the model restrictions and the lack of detailed geomorphologic information.
LSRM has been applied to a number of other catchments in Australia and China (Zhang et al, 2000b). The results for South Creek, Hacking River and Cawleys Creek are discussed here.
The South Creek catchment (89.7 km2) located southwest of Sydney, Australia, has been used in the RORB manual as an example for a FIT run (Laurenson and Mein, 1995), where the parameters of RORB, kc and m, were found as 16 and 0.8 respectively. Using the recession curve analysis method based on other floods, Zhang and Cordery (1999a) identified the value of K as 6.3 hours.
Both RORB and LSRM were used to simulate the observed flood that occurred in April 1988 (a 100 year flood) and to estimate the PMF. The optimal values of kc (16) and m (0.8) in the RORB model, and the optimal value of K (6.3) in the LSRM were adopted for this purpose. The linear version of RORB, with kc=6.3, and m=1.0, was also used to estimate the PMF.
The input rainfall and loss model were identical for all three models. For the April 1988 event, the rainfall input is spatially non-uniform. A 6-hour duration and uniformly distributed PMP was adopted for estimation of the PMF. The total PMP rainfall was 480 mm and the temporal distribution was taken from the RORB Windows Interface contained in RORB version 4.
Figure 5 compares the observed and estimated hydrographs of the April 1988 flood. Both RORB and LSRM underestimated the observed flood peak. This appears to be due to input errors.
Figure 6 shows the estimated PMFs for South Creek where the linear version of RORB produced the smallest peak and the nonlinear version of RORB produced the highest peak. It seems that the nonlinear version of RORB tends to produce a quick rising limb. It is hard to decide which result is the most realistic one, but given the observed linearity in Shihe River at high flows, the result produced by LSRM may be acceptable.
The Hacking River catchment (40 km2) is located in the southern suburbs of Sydney, Austalia. Cawleys Creek (5.24 km2) is a tributary of Hacking River. The value of K for Hacking River was identified as 4.4 hours, which was used for estimation of six large floods. In order to test the ability of the volume law in enhancing the matching of the rising limbs of observed floods, two methods were used for estimation of ki: the volume law method and the RORB method. Except for the differences in ki, other aspects of the models are identical.
Figures 7 and 8 show the matching results for Hacking River and Cawleys Creek. In general, the estimated hydrographs using the volume law agree with the observed hydrographs better than the estimated hydrographs using the RORB scheme for both Hacking and Cawleys, especially at the peaks. The RORB scheme overestimates the flood peaks and rising limbs in most of the cases for Hacking River, and the accuracy of the matching of the rising limbs was improved by the use of the volume law. The results for Cawleys Creek are generally poorer than those for Hacking River. This is not surprising since the estimated floods for Cawleys Creek were intermediate results from models of the Hacking River.
WBNM was also applied to South Creek, Hacking River and Cawleys Creek, results for which are discussed in the following section.
In addition to Shihe River, South Creek, Hacking River and Cawleys Creek, LSRM, WBNM and RORB were applied to Eastern Creek (24 km2), Bobo River (80 km2) and Thomson River (518 km2) in Australia.
In order to portray a clear picture of model performances, the results produced by LSRM, WBNM and RORB were further analyzed and compared using different error and efficiency criteria.
Since the rainfall and runoff data contains errors of different levels for each catchment, it is necessary to use an index to represent the data error before comparing the model performances. The data error index adopted in this study is a subjective data quality rating method. The quality range is between 1 and 10, from the poorest quality to the highest quality. The factors used in the rating include the number of rain gauges (3 points), the quality of rainfall data (3 points), the quality of rating curves (3 points), and other error sources (1 point). It should be noted that this rating has been developed based on a visual inspection of the available data, and cannot be taken as definitive. However, as this rating is not used in any of the model or error calculations, it is found to be a useful indicator of the cause of some of the large errors.
The model performance indices include,
(1) The Nash efficiency criterion (NEC) (Nash and Sutcliffe, 1970),
(2) The root mean square error (RMSE),
(3) The absolute relative error in flood peak (AREP), and
(4) The absolute error in time to peak (ARET).
NEC and RMSE give more attention to large discharge values, which suites the purpose of this study. The normal range of NEC is 0 to 100%. A perfect match means 100% efficiency. On the other hand, if the values of RMSE, AREP and ARET are large, that implies that the model performs poorly. The AREP and ARET are important indicators since the accuracy of the estimated peak discharge is a key factor affecting many engineering applications of flood event modeling.
A comprehensive analysis of model performance can be found in Zhang et al (2000b). Some key statistics include:
· Average efficiency (NEC). For the Australian catchments, the average efficiency of LSRM is 69.5%, which is slightly better than the average efficiencies of both WBNM (66.2%) and RORB (67.2%). For the 24 floods in Australian catchments, LSRM performs better than WBNM in 15 cases, and better than RORB in 14 cases.
· Average root mean square error (RMSE). For the Australian catchments, the average RMSE of LSRM is 19.8 m3/s. In comparison, the RMSE of WBNM and RORB are 21.3 m3/s and 20.9 m3/s respectively, which indicates LSRM performs better than both WBNM and RORB by this criterion.
· Average absolute relative error in peak (AREP). For the Australian catchments, the average AREP of LSRM is 20.9%, better than WBNM (24.0%) and RORB (26.7%).
· Average absolute relative error in time to peak (ARET). For the Australian catchments, the average ARET of LSRM is 6.4%, better than WBNM (8.1%) and RORB (8.3%).
The analysis also illustrates that, for the majority of the catchments, the quality of the input data cannot be guaranteed. The average quality rating for the Australian rainfall runoff data is only 7.5. Consequently, for floods with poor quality data, the models produce poor results, which in turn affect the inter-comparison of model performances. The capability of the volume law in improving the matching of the rising limbs of the observed hydrographs is masked because of this problem.
LSRM is capable of reproducing large floods. The physical realism of the linear storage-discharge relation and the volume law give the model the potential to provide better estimates of large floods than other models. The examination of that possibility by applying LSRM and other models to a range of large floods on a number of catchments has shown that this potential is realized in practice.
LSRM also has other practical advantages. It appears likely that the nonlinear models overestimate extreme floods. Parameter estimation for this linear model is simple, objective, and can be representative of large floods, but for the nonlinear models it involves subjective fitting using observed data from small floods which are not representative of the floods the models are usually used to estimate. The structure of the nonlinear models is not fully logical in that internally they have the rate of change of runoff production being greater than the rate of change of the input rainfall.
Given the simple model structure and the method of parameter estimation, LSRM could be useful for the estimation of large floods.
References
Anon, Historical floods in China, Hydro-power Press, Beijing, 1991. (in Chinese)
Bates B C and D H Pilgrim, Investigation of storage-discharge relations for river reaches and runoff Routing models. Civil Eng. Trans. IEAust, CE25 (3), 153-161, 1983.
Boyd M J, B C Bates, D H Pilgrim and I Cordery, WBNM a general runoff routing model computer programs and user guide. Water Research Lab. Report No.170. The University of New South Wales, 1987.
Institution of Engineers, Australia, Australian Rainfall and Runoff ¾ A Guide to Flood Estimation. 3rd edition. 1987.
Laurenson E M and R G Mein, RORB-version 4 runoff routing program user manual, Dept. Civil Eng., Monash Univ. in conjunction with Montech Pty Ltd, 1995.
Nash J E, The form of the instantaneous unit hydrograph, Int. Assoc. Hydrol. Sci., Proc. of the Toronto General Assembly, publication, 45 (3-4), 114-121, 1957.
Pilgrim D H and I Cordery, Flood routing, In: Handbook of Hydrology, edited by D. R. Maidment, Chapter 9, McGraw-Hill, 1993.
Zhang S Y and I Cordery, The catchment storage-discharge relationship: non-linear or linear? Australian Journal of Water Resources, 3(1): 155-165, 1999a.
Zhang S Y and I Cordery, Travel time and storage-discharge relations for flood estimation. Proceedings of the Water 99 Joint Conference, Brisbane, 483-488, July 1999b.
Zhang S Y, I Cordery and A Sharma, A volume law for specification of linear channel storage for estimation of large floods, Water Resources Research, 36 (6): 1535-1543, 2000a.
Zhang
S Y, I Cordery and A Sharma, Results of use of an improved linear runoff routing
model, presented at the 26th national and 3rd
international hydrology and water resources symposium (Hydro2000), Perth, 20-23
November, 2000b.
Fig.1 The effect of the volume law on the routing result with uniform rainfall input
Fig.3 The S-Q relationship for Shihe River, China
Fig.4 The match of Aug 75 flood hydrograph, Shihe
River
Fig.5
The reproduction of April 88 event, South Creek 
Fig.6
The estimated PMF hydrographs, South Creek
Fig. 7 LSRM
results using the volume law and RORB scheme for ki, Hacking River
Fig. 8 LSRM results using the volume law and RORB scheme for ki Cawleys Creek using parameters derived for Hacking River