Grey Forecast Rainfall with Flow Updating Algorithm for Real-time Flood Forecasting

The dynamic relationship between watershed characteristics and rainfall-runoff has been widely studied in recent decades. Since watershed rainfall-runoff is a non-stationary process, most deterministic flood forecasting approaches are ineffective without the assistance of adaptive algorithms. The purpose of this paper is to propose an effective flow forecasting system that integrates a rainfall forecasting model, watershed runoff model, and real-time updating algorithm. This study adopted a grey rainfall forecasting technique, based on existing hourly rainfall data. A geomorphology-based runoff model can be used for simulating impacts of the changing geo-climatic conditions on the hydrologic response of unsteady and non-linear watershed system, and flow updating algorithm were combined to estimate watershed runoff according to measured flow data. The proposed flood forecasting system was applied to three watersheds; one in the United States and two in Northern Taiwan. Four sets of rainfall-runoff simulations were performed to test the accuracy of the proposed flow forecasting technique. The results indicated that the forecast and observed hydrographs are in good agreement for all three watersheds. The proposed flow forecasting system could assist authorities in minimizing loss of life and property during flood events.


Introduction
Flood forecasting systems are nonstructural methods for reducing flood damage. An efficient forecasting system can assist with mitigating imminent disasters by providing information that can be disseminated rapidly to the flood-threatened areas. Standard practices in flood forecasting systems include hydrometeorological data transmission, database management, rainfall prediction, runoff estimation, and forecast information dissemination.
Brath et al. [1] indicated that quantitative rainfall forecasting plays a primary role in extending the lead time of river flow forecasting, which can improve the timeliness of flood control mechanisms. Because of improvements in the accuracy of weather radar systems, radar-based rainfall forecasting systems have superseded traditional rain gauges that provide measurements at only several locations for flood forecasting [2][3][4]. Toth et al. [5] indicated that radar detection is particularly difficult in mountainous regions because of the effect of ground occultation and altitude. Consequently, radar-based measurement techniques are limited under topographic conditions where radar reflectivity is poor [6,7]. Thus, hydrologists typically use probabilistic and stochastic methods for rainfall forecasting based on current and past rainfall measurements (e.g., [8][9][10][11][12][13]). Because these methods typically require large volumes of rainfall data for calibrating and training the model parameters, Deng [14] proposed a grey system theory-based model for future data prediction. Moreover, grey theory-based models require relatively few observations to predict outcomes [15][16][17]; thus, they are suitable for rainfall forecasting.
Rainfall forecasts are inputted into rainfall-runoff models to provide flood warning information for authorities. In recent decades, artificial neural networks (ANNs) have become a well-known tool for hydrologic forecasting [18][19][20][21][22][23][24][25][26][27][28][29]. However, ANNs require a large amount of hydrologic data to determine the adaptive weights, which are inadequate to be applied to data-sparse areas. Although fully distributed grid-based routing models can provide detailed information on flood wave transports, they may be unsuitable for real-time flood forecasting systems because the simulation process is typically time-consuming [30]. Hence, lumped and semi-distributed hydrological models are acceptable practical alternatives. For example, the Sacramento model was adopted by the National Weather Service River Forecast System in the United States [31,32], the tank model has been widely applied for runoff forecasting in Japan [33], the Hydrologiska Byråns Vattenbalansavdelning (HBV) model was adopted in Europe [34], and the Xinanjiang model was introduced in China [35]. To further minimize the requirement of observed flow data to develop semi-distributed models, hydrologists have adopted geomorphology-based runoff models [36][37][38][39][40].
Rodriguez-Iturbe and Valdes [35] proposed the geomorphologic instantaneous unit hydrograph (GIUH) model, which can be employed to derive the instantaneous unit hydrograph (IUH) of a watershed based on information from a topographic map or digital elevation data set. Subsequent studies have modified the GIUH model by incorporating kinematic-wave approximation, thereby providing reasonable estimations of flow velocity, which augments rainfall-runoff simulation used in both gauged and ungauged sites [41][42][43].
It is necessary to implement the effective data assimilation in the forecast process to bridge the immense gap between the theory and operational practice [44]. Comprehensive reviews of data assimilation approaches in operational hydrologic forecasting were presented by Liu et al. [45]. Since uncertainty is an inherent characteristic of watershed hydrodynamics, an ideal flood forecasting system should incorporate a real-time updating algorithm that revises the model state to improve the forecasting accuracy. Refsgaard [46] reported that techniques for updating real-time forecasting can be classified into the following four categories: (1) updating input variables; (2) updating state variables; (3) updating model parameters; and (4) updating output variables (i.e., error prediction). Previous studies have developed updating techniques based on time-series analysis [47,48], statistical methods [17,49], multiple regression analysis [50], dimensional variational algorithms [51,52], and the filter approaches [53][54][55][56][57][58]. Selection of an appropriate updating algorithm depends on the availability of real-time feedback data and the structure of the rainfall-runoff model employed for flow forecasting.
The purpose of this study is to develop an effective flood forecasting system for midsize rural watersheds. We adopted the grey rainfall forecasting technique based on existing hourly rainfall data to avoid poor radar reflectivity in mountainous watersheds. In performing the watershed runoff simulation, a geomorphology-based runoff model which can account different geomorphologic and hydrological characteristics of the watershed was used in this study. Furthermore, a flow updating algorithm was linked to the runoff model to estimate watershed runoff in the next three hours. The proposed flow forecasting system can operate with high efficiency to meet the requirements of real-time flow forecasting. The system was applied to three watersheds; one in the United States (Goodwin Creek) and two in Northern Taiwan (Heng-Chi and San-Hsia). The results of the flood forecasting were compared with official records to confirm the validity of the proposed system. In the following sections, Section 2 describes the analytical methods including the short-term rainfall forecasting and the geomorphology-based runoff model with an updating algorithm. The application of the proposed methods and the forecast results are presented in Section 3. Section 4 summarizes the conclusions of this study.

Analytical Methods
The framework of the proposed flood forecasting system incorporates a grey rainfall forecasting model [14], kinematic-wave-based GIUH (KW-GIUH) model [36], and flow updating algorithm that is linked to the KW-GIUH model to improve the flow forecasting accuracy.

Short-Term Rainfall Forecasting
The grey rainfall forecasting is adopted herein for two reasons: (1) a short-term rainstorm system is too complex to be simulated by using deterministic approaches; (2) the grey system provides an efficient way for rainfall prediction using only small amount of past observed rainfall data. Consequently, the grey model proposed by Deng [14] is appropriate for the present system for rainfall forecasting.
Although hydrological time-series data typically exhibit random forms, a systematic trend can be observed after a repeatedly accumulated generating operations (AGO). According to the AGO concept, Deng [14] developed an effective method for predicting future data based on a limited number of observations. A series of raw rainfall data can be expressed as follows: R t denotes the raw time-series rainfall data set, and ( ) ( ) 0 r k is the kth observed rainfall. The first-order AGO series can be defined as [14]: R t is the first-order AGO rainfall time series, and ( ) ( ) 1 r t is the first-order AGO observed rainfall data at time t. A first-order differential equation is employed to fit the cumulative rainfall data, which can be expressed as: where a and b are the grey system model parameters. Deng [14] indicated that the whitening of the grey derivatives of discrete data with unit time intervals can be expressed as: The whitening value of (1) where (1) ( ) Z k denotes the whitening value of (1) ( ) t k R t = . Next, Equations (4) and (5) are substituted into Equation (3) to obtain a grey discrete differential form as: The grey parameters a and b can be estimated using the least square method. Thus, the solution of Equation (6) is expressed as: is the forecast value of the first-order AGO series. Consequently, the rainfall depth at the subsequent time step can be obtained from the equation: is the forecast rainfall depth at time k t + Δ . The grey parameters in Equation (7) are updated when the new observed rainfall data are obtained. Yu et al. [17] indicated that the accuracy of rainfall forecasting decreases when the lead time is increased because the forecast error is cumulated from previous lead-time forecasting. Consequently, an algorithm called single-time-step forecasting proposed by Yu et al. [17] was used to overcome the shortcoming.
Several studies have successfully applied the first-order grey model for hydrological forecasting [14][15][16][17]. Because forecast errors are cumulative, grey forecasting models become less reliable as the forecast lead time increases. However, when a grey rainfall forecasting model is combined with a watershed rainfall-runoff model, the forecast lead time can be extended because of the time lag in the transporting of a flood wave from upstream to downstream.
This study adopted four criteria to evaluate the performance of the grey rainfall forecasting model. First, the error of total cumulative rainfall (ETCR) is defined as: where t r  is the forecast rainfall at time t; t r denotes the observed rainfall at time t; and n represents the number of time steps to be estimated. A more accurate forecast can be obtained when ETCR is approximately zero. Second, the relative root mean square error (RMSE) is defined as: where t R denotes the observed cumulative rainfall at time t;  t R is the forecast cumulative rainfall at time t. A more accurate forecast can be obtained when RMSE is approximately zero. Third, the coefficient of efficiency (CE) is defined as [56]: R represents the mean of the observed cumulative rainfall. A more accurate fit between the forecast rainfall and the observed rainfall can be obtained when CE is approximately one. Moreover, the coefficient of correlation (CC) is defined as: where  R is the mean of the forecast cumulative rainfall at time t. A more accurate forecast can be obtained when the value of CC is approximately one. The ETCR and RMSE represent a quantitative judgment of model performance. The CE is used to measure the similarity between the predicted and observed accumulated rainfall. The CC is used to measure the correlative relationship between the predicted and observed accumulative rainfall.

Geomorphology-Based Runoff Model with an Updating Algorithm
This study adopted an IUH model to provide an efficient method for estimating watershed runoff. The KW-GIUH model [36] was used because the IUH can be derived only by using watershed geomorphologic information obtained from a topographic map or digital elevation dataset. The hydrological response function of the watershed can be expressed analytically as follows [35]: where ( ) u t is the IUH of the watershed; W is the flow path space, which is expressed as , , ,..., Kinematic-wave approximation can be employed to express the runoff travel time for the ith-order surface flow region as follows [59]: where o n is the overland-flow roughness coefficient; where i B is the ith-order channel width; c n represents the channel roughness coefficient; i c L is the mean ith-order channel length; i c S denotes the mean ith-order channel slope; and i co h is the inflow depth of the ith-order channel caused by water transporting from upstream reaches. Hence, the runoff travel times for different orders of overland-flow paths and channels can be estimated, and the watershed IUH can then be derived by using Equation (13). Consequently, the watershed runoff simulated by using KW-GIUH model can be expressed as: where , sim t Q is the simulated direct runoff at time t; ( ) i τ is the rainfall intensity; and ( ) u t τ − is the unit impulse response function derived from the KW-GIUH model. Uncertainty is an inherent hydrodynamic characteristic of watershed; therefore, this study adopted a real-time updating algorithm to improve the accuracy of flow forecasting. The change in measured discharge between time t and t t + Δ can be expressed as: where , rec t Q and , rec t t Q +Δ denote the measured discharges at time t and t t + Δ , respectively; and rec Q Δ represents the change in measured discharge between time t and t t + Δ . Assuming that a reliable runoff model is used and short-term rainfall forecasting data (i.e., with a lead time of several hours) are available, the value of rec Q Δ in Equation (17) is assumed equal to the change in simulated discharge between time t and t t + Δ ; this is illustrated as follows: where , sim t Q and , sim t t Q +Δ denote the simulated discharges at time t and t t + Δ , respectively; and sim Q Δ represents the change in simulated discharge between time t and t t + Δ . Consequently, the forecast discharge at time t can be approximated by: where , fore t t Q +Δ is the forecast discharge at time t t + Δ and sim Q Δ is obtained from Equation (18). The schematic of the updating algorithm is shown in Figure 1. In the case that the real-time measured discharge , rec t Q cannot be transmitted successfully through the telemetric system during the rainstorm, is replaced by the model generated runoff discharge ( sim, t t Q +Δ ) without using the updating techniques. Although the proposed updating algorithm is simple, it is an efficient method for watershed runoff forecasting. To evaluate the suitability of the KW-GIUH model, two criteria were chosen to determine the goodness-of-fit between the observed and simulated flow hydrographs. The coefficient of efficiency Q CE is defined as follows [60]: where ( ) rec Q t and ( ) fore Q t denote the recorded and forecast discharges at time t, respectively; rec Q is the mean recorded discharge during a storm event, and n is the number of discharge records during the storm event. The Q CE is used to measure the similarity between the predicted and observed discharge hydrographs. A more accurate fit is obtained when the value of Q CE is approximately one. The peak discharge error is defined as follows: where ( ) P fore Q is the forecast peak discharge, and ( ) P rec Q denotes the recorded peak discharge. The EQp is used to measure the error of peak discharge directly. The error of time to peak discharge, P ET , is defined as: where ( ) P fore T and ( ) P rec T are the forecast and recorded time to peak discharge, respectively.

Description of Study Watersheds
Three watersheds were selected to investigate the applicability of the proposed model; one in the United States (Goodwin Creek) and two in Northern Taiwan (Heng-Chi and San-Hsia). Goodwin Creek is a tributary of Long Creek that flows into the Yocona River, which is one of the main rivers of the Yazoo River Basin. Figure 2a shows the watershed stream network and locations of hydrological gauging stations. The terrain elevation of the Goodwin Creek watershed ranges from 71 to 128 m above sea level (mean). The land area is composed of cultivated land (13.79%), forests (26.00%), pastures (59.80%), and water (0.41%). The climate of the Goodwin Creek watershed is humid with hot temperatures during summer and mild temperatures during winter. The mean annual temperature and rainfall are approximately 17 °C and 1399 mm, respectively. Most of the rainfall occurs during winter and spring. Hydrological data were obtained from the Agricultural Research Service of the United States Department of Agriculture. Among the 32 rain-gauging stations in the area, this study obtained rainfall records from nine stations. The Thiessen polygons method [61] was employed to calculate the hourly spatial-average rainfall intensities. Fourteen flow gauging stations were set up in the Goodwin Creek watershed area. The control areas of the flow gauging stations ranged from 0.06 to 21.39 km 2 . In this study, Flow-gauging Station No.1 (STA01), which has a drainage area of 21.39 km 2 , was selected as the test site to verify the model.
The Heng-Chi and San-Hsia watersheds are subwatersheds in Ta-Han Creek, which is one of the main rivers of the Tam-Sui River Basin in Northern Taiwan. Figure 2b shows the watershed stream networks and locations of the hydrological gauging stations. The elevation of the Heng-Chi (San-Hsia) watershed ranges from 20 to 970 m (30 to 1770 m), and the land is composed of 70% (75%) forest, 25% (20%) cultivated land, and 5% (5%) buildings/road. The mean annual precipitation in these areas is approximately 3000 mm. Most of the severe storm events are from typhoon activity between May and October, and intense rainfall (>50 mm/h) occurs every year. The geomorphologic factors were obtained from a digital elevation model [62] based on datasets of the Goodwin Creek watershed (30-m resolution) and the Heng-Chi and San-Hsia watersheds (40-m resolution). Table 1 shows the geomorphologic factors of the watersheds used in the KW-GIUH model.  Table 2 shows the details of storm events that occurred in the study watersheds; these details were used for parameter calibration and model verification. In performing the grey rainfall model, parameters a and b (Equation (7)) can be estimated by using a least square method only based on small amount of past observed rainfall data. The watershed geomorphological factors in performing the KW-GIUH model are shown in Table 1, which can be obtained by applying a digital elevation model. The calibrated model parameters of the KW-GIUH model for the Heng-Chi and San-Hsia watersheds are no = 0.6 and nc = 0.05, and no = 0.2 and nc = 0.02 for the Goodwin watershed. The values of model parameters were stable for the test storms in the watersheds. Sensitivity analysis for the model parameters of KW-GIUH can be found in Lee and Yen [42].  Table 3 and Figure 3 show the performance of the grey rainfall forecasting model for the three watersheds. The ETCR and RMSE from Equations (9) and (10) represent the quantitative evaluation of the model performance, and CE from Equation (11) indicates the performance of the model based on cumulative rainfall. The performance of the model was assessed qualitatively based on the value of CC (Equation (12)) relative to the correlation between the forecast and observed cumulative rainfall. The results showed that ETCR is less than 0.24, RMSE is less than 0.38, CE is greater than 0.85, and CC is greater than 0.90, indicating that the forecast and recorded hyetographs are in good agreement. The forecast and recorded hyetographs in Figures 4 and 5 show the performance of the grey rainfall forecasting model based on lead times ranging from 1 to 3 h. Although the accuracy of the forecast rainfall decreases as the lead time increased, the results indicate that the proposed grey model is suitable for rainfall forecasting.

Flow Forecasting
Four sets of tests were performed to evaluate the applicability of the proposed system for real-time flood prediction. The simulation results are detailed shown as follows.
(1) Flow forecasting by using measured rainfall and without flow updating This set of tests was conducted to evaluate the performance of the KW-GIUH model for simulating rainfall-runoff. Observed rainfall data were inputted into the KW-GIUH model and the flow updating algorithm was not used in the simulation. Figure 6 shows the results of runoff simulations for the Goodwin Creek and San-Hsia watersheds. As shown in Table 4, the simulated and observed hydrographs are in relatively good agreement in the study watersheds. The Q CE values of the simulated hydrographs for all storm events are greater than 0.82, and most of the P EQ and P ET are lesser than 10% and 2 h, respectively. The results indicate that the KW-GIUH model is reliable for rainfall-runoff simulation in these two watersheds. Figure 6a shows that the temporal distributions of the observed rainfall hyetograph and flow hydrograph were inconsistent; specifically, the rainfall peak occurred at 45 h, whereas the flow    (2) Flow forecasting by using forecast rainfall and without flow updating For the second set of tests, flow forecasting was performed by inputting the forecast rainfall (obtained from the grey model) into the KW-GIUH model. Table 5 and Figure 7 show that the flow forecasting accuracy decreased as the lead time increased from 1 to 3 h. For the t + 1 forecast, the forecast flow is in good agreement with the observed flow. For the t + 2 and t + 3 forecasts, the temporal variation of the flow hydrograph is adequately represented in the simulation although the simulated flow peak is higher than the observed flow peak because the forecast peak rainfall was overestimated in the hyetograph. Regarding the storm event at the Goodwin Creek watershed on 17 January 2001, the results shown in Figure 7a indicate that the KW-GIUH model forecast the first flow peak accurately. However, the second flow peak is underestimated because of the inconsistency between the rainfall hyetograph and flow hydrograph as mentioned. Table 5. Results of flow forecasting using forecast rainfall and without flow updating technique.  (3) Flow forecasting by using measured rainfall and flow updating technique

3-h Ahead
The third set of tests was conducted to evaluate the performance of the KW-GIUH model when the flow updating algorithm was used in the rainfall-runoff simulation, as shown in Equation (18). The measured rainfall at t + 1, t + 2, and t + 3 was inputted into the KW-GIUH model. Table 6 and Figure 8 show the simulation results, which were evaluated based on the coefficient of efficiency  Figures 7a and 9a show that the second peak was accurately forecasted when the flow updating algorithm is used, despite the recorded flow peak appearing to be unreasonable. The results show that using a purely deterministic approach to simulate watershed rainfall runoff is difficult without the assistance of a real-time adaptive algorithm.  The final set tests was conducted to confirm the performance of the proposed flood forecasting system. The forecast rainfall is generated by using the grey model, and the flow updating algorithm is included in the runoff simulation by using the KW-GIUH model to improve the forecasting accuracy. Table 7 and Figure 10 show that the mean Q CE ( P EQ ) values of the t + 1, t + 2, and t + 3 forecasts are 0.92 (4.50%), 0.80 (9.12%), and 0.72 (13.57%). The mean P ET values of the t + 1, t + 2, and t + 3 forecasts are 0.70 h, 1.47 h, and 2.13 h, respectively. The results of the storm event simulations in Figure 11 shows that the recorded and simulated hydrographs are in good agreement for all the three watersheds under various geoclimate conditions, even as the lead time increases from 1 to 3 h.

Conclusions
This study developed an integrated framework for flood forecasting by using a rainfall forecasting model, watershed rainfall-runoff model, and real-time flow updating algorithm. Considering that current numerical meteorological models used in Taiwan cannot provide a 3-h prediction of the temporal Recorded distribution of rainfall, this study adopted a grey rainfall forecasting model. Using the KW-GIUH model for runoff simulation is advantageous because it can be developed based on only geomorphologic factors of the watershed. Moreover, a real-time flow updating algorithm was incorporated into the KW-GIUH structure to account for the uncertainty of watershed runoff processes. The proposed flood forecasting system was tested based on hydrological records from three watersheds under different geomorphological and hydrological conditions. For the 1-h, 2-h, and 3-h ahead forecast cases, the simulated mean coefficient of efficiency (error of peak discharge) is 0.92 (4.5%), 0.80 (9.12%), and 0.72 (13.57%). The mean ETP values of the t + 1, t + 2, and t + 3 forecast cases are 0.70 h, 1.47 h, and 2.13 h, respectively. These results indicate that the proposed flood forecasting system can provide credible warning information for authorities. Furthermore, the proposed flow forecasting system can operate with high efficiency to meet the requirements of real-time flow forecasting. Nevertheless, in considering that the forecast rainfall is assumed to have the same tendency as previous rainfall; hence, only short-term prediction is applicable for this rainfall forecasting system. Since spatially-uniform rainfall is used in the KW-GIUH model, the proposed flow forecasting may be not used in a large watershed. Further validations to account for watersheds with various hydrological and geomorphologic characteristics are still required in future research.