Journal of Hydrology 535 (2016) 256–269

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Journal of Hydrology

journal homepage: www.elsevier .com/locate / jhydrol

A nonlinear spatio-temporal lumping of radar rainfall for modelingmulti-step-ahead inflow forecasts by data-driven techniques

http://dx.doi.org/10.1016/j.jhydrol.2016.01.0560022-1694/� 2016 Elsevier B.V. All rights reserved.

⇑ Corresponding author.E-mail address: changfj@ntu.edu.tw (F.-J. Chang).

Fi-John Chang ⇑, Meng-Jung TsaiDepartment of Bioenvironmental Systems Engineering, National Taiwan University, No. 1, Sec. 4, Roosevelt Rd., Taipei 10617, Taiwan, ROC

a r t i c l e i n f o

Article history:Received 19 November 2015Received in revised form 15 January 2016Accepted 21 January 2016Available online 1 February 2016This manuscript was handled by AndrasBardossy, Editor-in-Chief, with theassistance of, Ashish Sharma

Keywords:Data-driven modelsAdaptive Neuro-Fuzzy Inference System(ANFIS)Rainfall–runoff processesGamma testProbability information of forecasts

s u m m a r y

Accurate multi-step-ahead inflow forecasting during typhoon periods is extremely crucial for real-timereservoir flood control. We propose a spatio-temporal lumping of radar rainfall for modeling inflow fore-casts to mitigate time-lag problems and improve forecasting accuracy. Spatial aggregation of radar cells ismade based on the sub-catchment partitioning obtained from the Self-Organizing Map (SOM), and thenflood forecasting is made by the Adaptive Neuro Fuzzy Inference System (ANFIS) models coupled with a2-staged Gamma Test (2-GT) procedure that identifies the optimal non-trivial rainfall inputs. TheShihmen Reservoir in northern Taiwan is used as a case study. The results show that the proposed meth-ods can, in general, precisely make 1- to 4-hour-ahead forecasts and the lag time between predicted andobserved flood peaks could be mitigated. The constructed ANFIS models with only two fuzzy if-then rulescan effectively categorize inputs into two levels (i.e. high and low) and provide an insightful view (per-spective) of the rainfall–runoff process, which demonstrate their capability in modeling the complexrainfall–runoff process. In addition, the confidence level of forecasts with acceptable error can reach ashigh as 97% at horizon t+1 and 77% at horizon t+4, respectively, which evidently promotes model relia-bility and leads to better decisions on real-time reservoir operation during typhoon events.

� 2016 Elsevier B.V. All rights reserved.

1. Introduction

Flood forecasting is an extremely crucial non-structuralapproach for real-time reservoir operation in Taiwan due to itsunique topographical features and heterogeneous typhoon pat-terns. As a result of steep slope and short rivers in Taiwan, flashflood occurs typically within few hours (Chang et al., 2007;Chiang et al., 2007; Hsiao et al., 2013; Tsai et al., 2014) and reser-voirs could easily and quickly be filled up with mass inflow in atyphoon event. Such conditions make real-time reservoir operationvery challenging and reveal an urgent need for efficient and accu-rate multi-step-ahead inflow forecasting models. Rainfall–runoffrelationship is one of the most popular yet complex practices ofdata-driven models. The major challenge arises from the highdegree of spatio-temporal heterogeneity in storm rainfall and thehighly nonlinear nature of rainfall–runoff relationship, such asthe complex interaction among rainfall intensity, landcover, ter-rain and antecedent moisture (Pathiraja et al., 2012), and itrequires a substantial amount of spatial and temporal generaliza-tion on the spatio-temporal complexity of typhoon rainfall to build

an effective model. Data-driven techniques such as artificial neuralnetworks (ANNs) and fuzzy inference systems have been widelyapplied with success to modeling runoff based on rainfall data inoperational hydrology (Abrahart et al., 2012; Chang et al., 2003;Elshorbagy et al., 2010; Lohani et al., 2014; Nourani and Komasi,2013; Nourani et al., 2014; Talei et al., 2013). Data-driven modelsare particularly good at flood forecasting due to their ability indetermining the optimal relationships that relate inputs to outputs(Badrzadeh et al., 2015; Nayak et al., 2004, 2013; Nourani et al.,2014; Pappenberger et al., 2015; Talei et al., 2010; Valipour et al.,2013), and they are also flexible enough to accommodate the spa-tial and temporal heterogeneity of model inputs (Tsai et al., 2014).Spatially and temporally heterogeneous rainfall can be suitablyestimated subject to the availability of spatially continuous radarrainfall data while the non-linearity problem can be effectivelytackled by data-driven techniques. When using spatially continu-ous radar rainfall for modeling, a semi-distribution framework isoften adopted, in which an optimized spatio-temporal lumping isa key process to reduce input dimension (Tsai et al., 2014).Teschl and Randeu (2006) proposed a data-driven radar rainfallmodeling procedure and demonstrated that radar rainfall data pro-vided a better indication of areal precipitation in succession for therunoff volume than those of a single rain gauge, which revealed a

Fig. 1. Locations of the Shihmen Reservoir and the centers of radar cells within thestudy area.

F.-J. Chang, M.-J. Tsai / Journal of Hydrology 535 (2016) 256–269 257

physical and structural enrichment made by the constructed ANNmodel.

Input selection for constructing ANN models is commonly per-formed by a linear correlation analysis, in which the lagged rain-falls posting higher correlations with inflow are selected asinputs. This, however, may reduce the models’ ability in dealingwith the non-linearity of the input–output patterns. The problemof linear thinking in the modeling process is usually unnoticed,while we expect data-driven models could effectively extract thenon-linear relationship between the input and output patterns.The identification of rainfall time lags commonly depends onexperts’ knowledge with a consideration of physical meanings, towhich linear thinking is commonly applied (Maier et al., 2010).Even though the linear correlation is a crucial and effective factorfor modeling, the accompanied phase-shift problem (time shift)between observed and estimated flood hydrographs could not beeasily solved by the constructed models in many flow forecastingpractices (Chen et al., 2013; Khatibi et al., 2012; Nguyen andChua, 2012; Pan et al., 2013). Nevertheless, an accurate estimationon the timing of flooding, especially for extreme storm events, iscrucial to ensure a proper warning period available for reservoirauthorities to prepare for evacuation procedures in advance. Sev-eral studies have proposed to alleviate the timing error problem.For example, some studies attempted to consider the time shiftof peak flow forecasts through a correction process for model out-puts (Abrahart et al., 2007; Liu et al., 2011) while other studiestried to reduce the effect of timing errors on forecasts within neu-ral network frameworks (Chen et al., 2013).

Apart from time shift problems, another issue of ANN modelsfor flood forecasting is that the probability information of quanti-tative forecasts made by deterministic ANN models is often over-looked. In a decision making procedure, nevertheless, probabilityinformation and/or risk level is usually needed. Probabilistic fore-casts take the form of a predictive probability distribution overfuture quantities of interest, which enables risk-based warning offloods for quantifying predictive uncertainty and enhancing fore-casting paradigm, and has gained increasing attention lately(Krzysztofowicz, 2001, 2002; Gneiting and Katzfuss, 2014). A num-ber of studies provided probability information either in the way ofcombining multiple deterministic forecasting models to yield anensemble output and the spread of possible results (Alfieri et al.,2014; Bowler et al., 2008; Hamill et al., 2008; Fan et al., 2014;Tiwari and Chatterjee, 2010; Yang et al., 2015) or in the way ofbeing deduced from probability distribution of errors (Chen andYu, 2007). The main reason to use the probability information offorecast errors obtained from a deterministic forecast model toassess model uncertainty is that it can be directly used to quantifythe uncertainty of forecasting and derive the confidence intervalsof forecasts accordingly. This approach has been adopted in manystudies (e.g. Chen and Yu, 2007; Montanari and Brath, 2004; Tameaet al., 2005) and can provide useful information for flood defense.

This study intends to explore the inclusion of spatial rainfalldistribution into data-driven rainfall–runoff models to enhancetheir predictive performance by well capturing the spatial andtemporal heterogeneity of typhoon rainfall. To model the complexrainfall–runoff processes and mitigate the time lag problem, wepropose to implement the Self-Organizing Map (SOM; Kohonen,1982) and Gamma test (GT) to determine the optimal spatio-temporal lumping of radar rainfall as the inputs for a data-drivenmodel. To achieve this goal, this study is conducted on three-folds. Firstly, the non-linearity between rainfall and runoff is ana-lyzed through evaluating the travel time and spatial aggregationof rainfall in a catchment when cell-based radar rainfall data areused. Secondly, the time lag problems in data-driven models areexplored and adjusted by implementing effective rainfall informa-tion through a 2-staged GT process. In this way, a simplified hybrid

model that achieves the optimal degree of generalization on thespatio-temporal variability of typhoon rainfall–runoff processeswhilst avoiding an overly complex model structure could beobtained. Finally, probability information about forecast accuracy(error) is provided to aid decision-making on real-time reservoiroperation. The reliability and usefulness of the constructed modelscan then be greatly improved.

2. Methods and materials

2.1. Study area and materials

The Shihmen Reservoir is an important reservoir in northernTaiwan and is designed for multiple purposes including flow con-trol, water supply for irrigation, industrial and domestic uses,and hydropower generation. Its upstream basin area occupies763.4 km2, ranging in elevation from 157 m to 3514 m (Fig. 1),with average slope angles of about 30 degrees. Due to the steepslopes and fast flowing rivers, decisions about the timing andamount of reservoir releases must be made within a matter ofhours by the controlling agency during a typhoon. In this study,the continuous radar rainfall data set was produced from Quantita-tive Precipitation Estimation and Segregation Using Multiple Sen-sors (hereafter termed QPESUMS) with a temporal resolution of10 min and a spatial resolution of 1.25 km. In total, there were434 radar cells covering the study area (Fig. 1). This radar dataset had already been calibrated based on ground observations bythe Central Weather Bureau, Taiwan. The 10-min precipitationmap was temporally accumulated into an hourly sequence, corre-sponding to the hourly reservoir inflow series.

The reservoir inflow series comprised 637 hourly observationsof nine typhoon events during 2007 and 2012, in which the seriesranged from a maximum peak inflow of 5385 m3 s�1 (SAOLA) to aminimum peak inflow of 203 m3 s�1 (KALMAEGI) (Table 1). Thestudy also utilized a 40-m resolution digital elevation model(DEM, see Fig. 1) provided by the National Land Surveying andMapping Centre, Taiwan for sub-catchment segmentation analysis.

2.2. Modeling approach – input selection

Developing reliable and accurate multi-step-ahead flood fore-cast models driven by the optimized spatio-temporal lumping of

Table 1Typhoon datasets.

Event Name Period Peak inflow (m3 s�1)

1 SEPAT 2007/08/16–08/19 18442 KROSA 2007/10/04–10/07 53003 KALMAEGI 2008/07/16–07/18 2034 SINLAKU 2008/09/11–09/16 33515 MORAKOT 2009/08/05–08/10 18386 WIPHA 2007/09/17–09/19 27887 FUNG-WONG 2008/07/26–07/29 20408 JANGMI 2008/09/26–09/29 32929 SAOLA 2012/07/31–08/03 5385

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radar data is the major goal of this study. There are two key stepsto accomplish this goal. One is the spatial lumping of radar data, inwhich the catchment is divided into several sub-catchments byclustering approaches. The SOM is employed here to determinethe partition of the catchment and to cluster the sub-catchments.The other key step is the temporal lumping of model inputs includ-ing radar rainfall and inflow data. The purpose of temporal lump-ing is to determine the radar rainfall with proper time lagreflecting the rainfall–runoff relation in the catchment. Throughthe proposed spatio-temporal lumping process, the optimal inputcombination accommodating spatial heterogeneity of radar rainfalland rainfall–runoff relation can then be obtained.

Inflow of the Shihmen Reservoir is modeled at forecast horizonsup to 4 hours ahead (Qtþ1;Qtþ2;Qtþ3;Qtþ4) by using the five model-ing approaches described in Table 2, and the model input selectionflowchart is shown in Fig. 2. Model 1, which is obtained from ourprevious study (Tsai et al., 2014), is used as a benchmark, and itsresults are compared with those of Model 2 for the purpose of eval-uating the two spatio-temporal lumping approaches. The compar-ison between Model 3 and Model 4 focuses on how the differentrainfall inputs affect the outputs of ANFIS models, while the atten-tion of the comparison between Model 2 and Model 5 is paid to thetime lags of forecasted peak inflow.

2.3. Spatio-temporal lumping of radar inputs

Although radar data provide spatially continuous rainfall infor-mation, using massive radar cells as inputs to the ANFIS may result

Table 2Model input description.

Model Inputs Desc

Model 1 (GIS + CC) Qt A lumradachantempandsame

DQR(I,t�5), R(II,t�6), R(III,t�7), R(IV,t�7)(lagged rainfall of 4 sub-catchments bythe GIS + CC approach)Note that the catchment partition isdifferent from that of the other models

Model 2 (SOM + GT) Qt Simithe sspaticonccond

DQR(I,t�5), R(II,t�6), R(III,t�6), R(IV,t�7)(lagged rainfall of 4 sub-catchments bythe SOM + GT approach)

Model 3 (SOM + GT) R(I,t�5), R(II,t�6), R(III,t�6), R(IV,t�7)(lagged rainfall of 4 sub-catchments bythe SOM + GT approach)

ThederivGamselec

Model 4 (SOM + 2-staged GT) Lagged rainfall of 4 sub-catchments by theSOM + 2-staged GT approach (see Table 4)

Thesub-inpu

Model 5 (SOM + 2-staged GT) Qt Simithe cDQ

Lagged rainfall of 4 sub-catchments by theSOM + 2-staged GT approach (see Table 4)

in an excessively complex structure. Thus, a spatial lumping ofradar data in a catchment is needed to reduce the structural com-plexity of the ANFIS. In this study, the SOM is applied to clusteringthe radar cells. The SOM is a powerful classifier and has been suc-cessfully implemented in mapping various pattern recognitiontasks (Chang et al., 2010; Chang et al., 2014a,b; Kang and Yusof,2012; Wallner et al., 2013). In order to tackle the nonlinear naturein the interaction between rainfall and terrain, the inputs of theSOM consist of the rainfall of nine events in each radar cell andthe geographical features presented by each radar cell, includinglatitude, longitude and elevation. Tsai et al. (2014) suggested thatthe Shihmen catchment could be partitioned into 4 sub-catchment polygons according to stream segmentation producedfrom ArcGIS based on DEM data. Therefore, we also partition thecatchment into 4 categories by using the SOM and the spatiallumping of radar cells simply takes the summation of cell-basedhourly rainfall in each category.

Travel time, also known as time of concentration, represents thetime for a raindrop traveling from any point of the catchment tothe outlet of the catchment. Travel time is a function of terrain fea-tures and is a highly non-linear parameter in a catchment withcomplex terrain (Zuazo et al., 2014). Some attempts utilized thecorrelation analysis between the lagged rainfall of each sub-catchment and reservoir inflow to determine the travel time atwhich the highest correlation coefficient (CC) was produced(Pianosi et al., 2014; Pramanik and Panda, 2009). However, a linearcorrelation analysis may fail to describe the non-linearity of therainfall–runoff relationship. Instead of using a linear correlationanalysis to determine the time of concentration, a non-linearapproach is proposed here. In this study, the GT is used to identifythe non-linearity of the rainfall–runoff relationship and determinethe optimal input combination for a data-driven model.

2.4. 2-staged Gamma Test (GT)

The GT is an input selection technique for assessing the extentto which a given set of M data points can be modeled by anunknown smooth nonlinear function (Jones et al., 2007; Koncar,1997). The Gamma statistic (C) is an estimate of the modeloutput’s variance that cannot be accounted for through a smooth

ription

ped rainfall–runoff model with inputs comprising lagged rainfall derived fromr data that spatially averaged across 4 sub-catchments, inflow at time t (Qt), thege in inflow between Qt and Qt�1 (hereafter termed D(). Notably, the spatio-oral lumping method combines DEM data for spatial aggregation using ArcGISthe correlation analysis for estimating the time of concentration. This model is theas the model D in Tsai et al. (2014) used here as a benchmark

lar to Model 1, the inputs comprising lagged rainfall and inflow (Qt and Da), butpatio-temporal lumping method is different. It combines the SOM classifier foral aggregation and the Gamma test (GT) for the optimal estimation on the time ofentration. For each sub-catchment, one input is selected. Gamma test isucted for each sub-catchment separately

SOM + GT approach with inputs comprising only lagged rainfall, without inflow,ed from radar data and spatially averaged radar rainfall across 4 sub-catchments.ma test is conducted for each sub-catchment separately, and only 1 input isted for each sub-catchment

inputs comprising only lagged rainfall. All lagged rainfall (t�1 up to t�8) for 4catchments (32 in total) are used in the 2-staged GT, and the number of the bestts in a combination is not limited to 4

lar to Model 4, using SOM + 2-staged GT but including inflow at time t (Qt) andhange in inflow between Qt and Qt�1 (D()

Fig. 2. Flowchart of model input selection in this study.

F.-J. Chang, M.-J. Tsai / Journal of Hydrology 535 (2016) 256–269 259

function. The GT is assessed based on the kth ð1 6 k 6 pÞ nearestneighbor XNði;kÞ for each vector Xi, and then the GT can be derivedfrom the Delta function of input vectors:

dMðkÞ ¼ 1M

XMi¼1

jXNði;kÞ � Xij2 ð1 6 k 6 pÞ ð1Þ

where j � � � j is the Euclidean distance, and the correspondingGamma function of the output value is given in Eq. (2). The numberof p depends on the density of sampling (Koncar, 1997).

cMðkÞ ¼12M

XMi¼1

jyNði;kÞ � yij2 ð1 6 k 6 pÞ ð2Þ

where yNði;kÞ is the corresponding y-value for the kth nearest neigh-bor of Xi, in Eq. (1). For computing C, a least squares regression lineis constructed for p points ðdMðkÞ; cMðkÞÞ as Eq. (3):

c ¼ Adþ C ð3Þwhere A is the gradient.

The GT can provide the noise estimate (C value) for each subsetof input variables. When the subset for which its associated Cvalue is closest to zero, it can be considered as ‘‘the best combina-tion” of input variables. It is effective to implement the GT for iden-tifying non-trivial input variables and thus reduces the inputdimensions as well as produces precise outputs of ANNs (Changet al., 2014a,b; Noori et al., 2010).

There are three steps for identifying non-trivial inputs by theGT:

Step 1: Derive C values corresponding to all possible input com-binations (2M � 1) through the GT.Step 2: Sort the derived C values in an ascending order, and cat-egorize the C values smaller than the 10th percentile (C10) asthe best group (FC6C10 ) whereas the C values bigger than 90thpercentile (C90) as the worst group (FCPC90 ).Step 3: Derive the ratio of the occurrence frequency of each vari-able in the best group to that of the worst group. The input vari-ables associated with higher ratios represent non-trivialvariables.

260 F.-J. Chang, M.-J. Tsai / Journal of Hydrology 535 (2016) 256–269

Although the GT is a fast algorithm, however, identifying non-trivial inputs for a case with many inputs can still consume hugecomputational time. For example, if there are 32 variables, the totalnumber of all possible combinations is 0.4295 billion (232 � 1),which will take huge computational time to derive all C values.To overcome this problem, a 2-staged GT procedure is proposed,shown as follows.

First stage:

� Decide the number of variables (N) selected from M variablesand then perform the GT. The number of possible combinationsis CM

1 þ CM2 þ CM

3 þ . . .þ CMN . In this study, M = 32 and N = 4.

� Repeat Steps 2 and 3 mentioned above.

In this stage, allM variables compete with each other for a smal-ler number of possible combinations (CM

1 þ CM2 þ CM

3 þ . . .þ CMN out

of 2M � 1). The setting of N is not necessary to be large in order tosave computational time. After a series of trials, N is set as 4 in thisstudy.

Second stage:

As mentioned earlier, the ratio of the occurrence frequency canbe used to determine non-trivial variables. However, there is nostandard rule to determine a proper threshold for the ratio orhow many non-trivial variables should be selected. Practically(based on our previous experiences), we could take the descendingsorted ratios and visually identify the elbow point. Once the elbowpoint in the sorted ratios is identified, the rank of the elbow point isconsidered as the potential number of variables (K) to form thebest input combination. If K 6 N, the optimal combination of Kvariables is the one with the lowest C value that has been evalu-ated in the first stage. In the case of K > N, the optimal combinationof K variables is not yet evaluated by the GT process in the firststage. Thus, a second stage of the GT is needed.

� Identify the number of K input variables out of M variables byusing the elbow point method. The total number of ways toselect K input variables from M variables is CM

K . For this study(M = 32 and K = 7), the total number of combinations is3,365,856 (C32

7 ). As compared with the original 0.4295 billionpossible combinations, less than 1% (i.e. 1/127) of all possiblecombinations would be executed. Apparently, the computa-tional time is largely reduced.

� The combination with the lowest C value is identified as thebest one.

The proposed 2-staged GT is a fast process that can save hugecomputational time and efficiently identify the non-trivial vari-ables among a large number of candidates.

2.5. Adaptive Neuro Fuzzy Inference System (ANFIS)

A major drawback of ANNs is the inherent black box approachesthat provide adequate solutions without delivering heuristic inter-pretation of solutions. Fuzzy logic can easily provide heuristic rea-soning while has difficulty in providing exact solutions. The ANFISmerges the neural network and fuzzy logic techniques to provideadequate solutions while delivering qualitative, heuristic knowl-edge about the solutions. Its fuzzy if-then rules may provideinsights to explore the non-linear rainfall–runoff relationship(Chang et al., 2005). The optimal structure for each ANFIS modelused in this study is determined by heuristic search, where the

best-performing configuration is identified as the configurationwith the minimum root mean square error (RMSE). The nine events(typhoons) for modeling are divided into three sub-sets: Events1–5 form the training dataset; Events 6–7 form the validationdataset; and Events 8–9 form the testing dataset (Table 1). Earlystopping (Coulibaly et al., 2000) is used to avoid over-fitting.

2.6. Evaluation metrics

The performance of each model is evaluated and comparedusing six different metrics including CC, RMSE, mean absolute error(MAE), coefficient of efficiency (CE), skill score (SS) and time lag offorecasted peak inflow (TLag).

CC ¼PN

i¼1ðXf ðiÞ � Xf ÞðXoðiÞ � XoÞffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiPNi¼1ðXf ðiÞ � Xf Þ2

q ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiPNi¼1ðXoðiÞ � XoÞ2

q ð4Þ

RMSE ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiPN

i¼1ðXf ðiÞ � XoðiÞÞ2N

sð5Þ

MAE ¼PN

i¼1jXf ðiÞ � XoðiÞjN

ð6Þ

CE ¼ 1�PN

i¼1ðXoðiÞ � Xf ðiÞÞ2PNi¼1ðXoðiÞ � XoÞ2

ð7Þ

SS ¼ RMSEref � RMSEi

RMSEi� 100% ð8Þ

TLag ¼ Tp;forecast � Tp;obs ð9Þwhere N is the number of observations; Xf ðiÞ is the forecasted

inflow at time i; XoðiÞ is the observed inflow at time i; and Xf and

Xo are the mean values of forecasted and observed inflow, respec-tively. RMSEref is the RMSE of the benchmark model (Model 1 in thisstudy), while RMSEi is the RMSE of another comparative model.These indices serve as performance criteria adopted to identifythe optimal spatio-temporal aggregation for modeling multi-step-ahead inflow. The delay time of forecasted peak inflow (TLag) isthe time difference between forecasted (Tp;forecast) and observed(Tp;obs) peaks.

2.7. Probability information derived from error analysis

Probabilistic forecasting gains increasing attention nowadays.Many studies pointed out the advantages of adopting probabilisticforecasts in contrast to deterministic forecasts in hydrology andsuggested that probabilistic flood forecasts would lead to betterdecisions related to operational hydrological concerns (Boucheret al., 2011, 2012; Dale et al., 2012; Dietrich et al., 2009;McCollor and Stull, 2008; Ramos et al., 2013; Schellekens et al.,2011; Verkade and Werner, 2011; Younis et al., 2008). Decisionmaking for reservoir operation during flood periods is a complexprocess, which becomes an urgent issue in Taiwan. Under such cir-cumstance, quick and reliable flood information is crucial for reser-voir operation. In this study, we intend to provide probabilisticinformation of flood forecasting based on a deterministic ANNmodel and to provide confidence level of forecast by applyingpre-determined error thresholds. For simplicity, the normal distri-bution is used to describe the errors between forecasts and obser-vations, which can be very useful to reservoir operationalauthorities. The major attention here focuses on how to determinea proper error threshold that really fits the operational demand ofthe reservoir management agency.

F.-J. Chang, M.-J. Tsai / Journal of Hydrology 535 (2016) 256–269 261

A top-down (from modeller to user) method is often adopted toforecast the near future as accurately as possible. However, thismethod may not meet the real needs of decision makers. Therefore,a bottom-up (from the need of user) method is desired, in whichthe key point is the concept of risk or probability information aboutforecasts. To make forecasts useful to decision makers, it is crucialto provide a confidence level of model outputs. Assume that theerrors between forecasts and observations follow normal distribu-tion with zero mean and certain variance (Nð0;rÞ). If a pre-determined error threshold (Te) is given, the probability of fore-casting error within the threshold can be determined as:

PðXo � Te < Xe < Xo þ TeÞ ¼ FðXo þ TeÞ � FðXo � TeÞ ð10Þ

where F represents the cumulative density function (CDF), which isof normal distribution.

The method mentioned above is fundamental and easy toimplement; however, a critical issue is how to determine a propererror threshold for decision makers. To this end, we utilize reser-voir operational rules (Fig. 3) as an operational guideline becauseit is the most important reference for real-time reservoir operation.The Shihmen reservoir is a pivotal multi-objective reservoir innorthern Taiwan and is operated by the Northern Region WaterResources Office (NRWRO) of Water Resource Agency in Taiwan.Its operation is based mainly on the operational rules, as shownin Fig. 3. During typhoon events, dam safety is the major concernsuch that flood control gains the highest priority in reservoir oper-ation, while in non-typhoon periods we move to consider the otherobjectives such as water supply and power generation as priorityobjectives for better managing reservoir. Because flood control(dam safety) and water supply are two competing operationalobjectives, error thresholds for these two objectives should be con-sidered in different ways. In this study, we separate these twoobjectives, dam safety and water supply, by using the upper limitof operational rules as a boundary. When the water level is higherthan the upper limit of operational rules, which is a critical condi-tion, the major operational objective focuses on dam safety (floodcontrol). In contrast, when water level is lower than the upperlimit, the operational objectives would include flood control, watersupply and power generation simultaneously. Based on historicalwater level records, most of the water levels of the reservoir duringJuly and October, i.e. typhoon seasons in Taiwan, fell within therange of 225–245 m. Therefore, we determine two error thresholdsaccording to the two conditions, i.e. the water level is higher orlower than the upper limit of operational rules. Under each condi-tion, the standard deviation of historical water levels is calculatedand selected as the error threshold to reflect the natural variationof water levels. The confidence level of forecasts is presented bythe percentage of forecast errors that fall within the designed errorthreshold.

Fig. 3. Operational rules of the Shihmen Reservoir.

3. Results and discussion

As mentioned earlier, the spatial and temporal heterogeneitycan be alleviated with the spatially continuous radar rainfall datawhile the non-linearity problem can be tackled by data-driventechniques. A semi-distribution modeling framework is adoptedto reduce input dimension, where the watershed is divided intofour sub-catchments as suggested in a previous study. The SOMis applied to clustering the radar cells. Using geographical features(including longitude, latitude and elevation) and the average rain-fall of each radar cell as model inputs, all variables are normalizedwith values ranging between 0 and 1, and an SOM with 2 � 2 neu-rons is constructed. Its topology map is shown in Fig. 4, and theclassified four sub-catchments are shown in Fig. 5. We find that:(1) elevation is the most important factor for this clustering, wherecluster 4 has the highest elevation while cluster 1 has the lowestelevation; (2) longitude and latitude play important geophysicalguidance in this clustering; and (3) the average rainfall valuesassociated with clusters 2, 3 and 4 are similar, but the average rain-fall value of cluster 1 is relatively low (Fig. 4). This partitioning ismeaningful and suitably reflects the terrain and river pattern ofthe catchment.

The spatial lumping simply calculates the spatial average ofhourly radar rainfall over each sub-catchment. For temporal anal-ysis, we select a proper time lag of rainfall for each sub-catchment as an input to the ANFIS, and the GT is used to selectthe inputs for each sub-catchment. In each GT procedure, eight dif-ferent rainfalls, i.e. R(t�1), R(t�2),. . ., and R(t�8), are evaluated fortheir suitability as model inputs. A total of 255 ð28 � 1Þ C valuescorresponding to all possible combinations are derived throughthe GT for each sub-catchment. The non-trivial variables can thenbe identified as the variables associated with higher occurrence inthe best group and lower occurrence in the worst group. Beingguided by the GT ratios of each sub-catchment, one can determinewhich variables should be selected for each sub-catchment.

The exploration here focuses on the spatio-temporal lumpingapproaches for GIS + CC (Model 1- a benchmark model) and SOM+ GT (Model 2). To make Model 2 comparable to Model 1, we alsotake the physical meaning of time lag into account and finally fourlagged rainfalls, R(I,t�5), R(II,t�6), R(III,t�6), R(IV,t�7), are selectedsince they also have high GT ratios. The comparison between theforecasts and observations of the testing events shows that theconstructed models could well fit the observation (i.e. very highCC and CE values in all the cases), while the time lag phenomenonof forecasting is obvious (Fig. 6). The performance between Model2 and Model 1 reveals that the hybrid model (SOM + GT) haspotential to improve the long-term forecasting at t+4 (Fig. 7).

3.1. 2-staged GT procedure

The variables selected by the GT for the four sub-catchmentsare not necessary to form the optimal input combination for themodel applied to the whole catchment because the GT is con-ducted separately for each sub-catchment and the selected vari-ables from the four sub-catchments do not compete with eachother. Therefore, a 2-staged GT procedure is utilized to obtain theoptimal input combination for the whole catchment.

To better address the time lag issue in modeling rainfall–runoffprocesses, Model 3 and Model 4 are designed with the inputs com-prising only lagged rainfall identified by two different GTapproaches and the flow information is not considered as an input.For Model 3, only the lagged rainfalls at R(I,t�5), R(II,t�6), R(III,t�6), R(IV,t�7) (same as Model 2-obtained from previous1-staged GT procedure) are used. For Model 4, the 2-staged GTprocedure is applied to reducing computational time for input

1: Longitude 2: Latitude 3: Elevation 4: Average Rainfall

Fig. 4. Topology maps of the SOM (2 � 2). Y-axis represents the weight of each variable.

Fig. 5. Catchment partition made by the SOM.

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selection among 32 variables (four sub-catchments with 8 timelags). In the first stage, the GT with 1 to 4 variables selected fromthe 32 variables would be performed through a total of 41,448(C32

1 þ C322 þ C32

3 þ C324 ) times. Among these corresponding 41,448

C values, the ratio of the occurrence frequency of each variablein the best group to that in the worst group could then be derived.By sorting the ratios, we find the change in ratio between the 7thvariable and the 8th variable becomes small, which forms an elbowpoint (Table 3). Consequently, the number of variables in the bestcombination is seven, i.e. k = 7.

Note that the first 7 variables in Table 3 are not necessary toform the best combination because the combinations of these 7

variables (C327 ) is not yet evaluated by the GT algorithm. Therefore,

in the second stage the best combination of the 7 variables is iden-tified through performing the GT 3,365,856 (C32

7 ) times, in whichthe combination associated with the lowest C value (0.0226) isconsidered as the best combination (Table 4). Comparing the vari-ables in Tables 3 and 4, it can be found that only R(IV,t�6), rank 4in Table 3, remains in the best combination. This implies that thefirst 7 variables in Table 3 are not necessary to form the best com-bination as they may contain similar information (lagged rainfall att�6 up to t�8 at the upper basin).

For different horizons of multi-step-ahead forecasting, theaforementioned GT procedures could be duplicated to select theirbest input combinations. Nevertheless, for the sake of simplicity,the observations unavailable at current time t are simply removed.For the horizon of t+2, for example, the observations of R(II,t�1)and R(IV,t�1) do not exist at time t and thus are removed fromTable 4. We also notice that some selected variables (Table 4) seemunreasonable (or, have no physical sense), such as the rainfall ofthe upper stream sub-catchment with 1-hour lag (R(IV,t�1)), butmay play a certain role in tackling noises while those variables thatsound reasonable (or, have a physical sense, e.g. R(I,t�4) and R(IV,t�6)) may essentially contribute to the prediction of the inflowtrend.

According to the comparison between Model 3 and Model 4 forthe two testing events, it is obvious that the use of the optimal 7lagged rainfalls as model inputs makes a great improvement onthe ANFIS forecasting (Figs. 7 and 8). The improvements can beshown not only in the error-based performance metrics (Fig. 7)but also in the timing of peak flow (Fig. 8), where the delay of peakflow reduces by 1–2 hours for Model 4. This reveals that the timedelay problem arisen in peak inflow prediction for the ANFIS modelcould be mitigated by incorporating the optimal lagged rainfallvariables identified by the 2-staged GT as model inputs.

Because the flow information is crucial and commonly used asthe key input for rainfall–runoff modeling, the incorporation of

t+1

t+2

t+3

t+4

Fig. 6. Comparison of observed and predicted inflow of Models 1, 2 and 5 for 1- up to 4-step-ahead forecasting based on the testing dataset.

F.-J. Chang, M.-J. Tsai / Journal of Hydrology 535 (2016) 256–269 263

the optimal combination of radar rainfall variables with inflowvariables as the inputs to the ANFIS (Model 5) might significantlyimprove inflow forecasting accuracy. In Model 5, we add the cur-rent inflow (Qt) and the change in inflow between Qt and Qt�1

(DQ) as another inputs to Model 4. The phase-shift (or timingerror) problem is mitigated in Model 5, as compared with Models1 & 2, (Fig. 6). Besides, the overall comparison shown in Fig. 7 indi-cates that Model 5 largely outperforms the other four models,which can be attributed to the input combination of Model 5, i.e.

the optimal lagged rainfall variables identified by the 2-stagedGT and inflow variables (Qt and DQ). Notably, for 3- and 4-step-ahead forecasting, CC values are still higher than 0.94 while CE val-ues are higher than 0.88 for the testing dataset. The improvementis quite clear when comparing the SS values of Model 2 and Model5 (Fig. 9). The improvement can also be shown in the multi-step-ahead forecasts of peak inflow (Table 5), where the timing errorof peak inflow is typically less than 2 hour for 4-step-ahead fore-casting. The achievement is made by using the 2-staged GT to

Fig. 7. Performace of 20 ANFIS models based on the testing data set in terms of four metrices.

Table 3GT ratios for which the number of variables in all combinations ranges from 1 to 4 out of the 32 lagged rainfalls.

Rank 1 2 3 4 5 6 7 8

Variable R(IV,t�8)a R(IV,t�7) R(III,t�8) R(IV,t�6) R(II,t�8) R(III,t�7) R(I,t�8) R(III,t�6)Ratio Inf.b Inf. 428.7 57.6 32.2 26.8 7.8 7.6

a R(IV,t�8) represents the lagged rainfall at t�8 of the 4th sub-catchment.b Infinity.

Table 4Top 7 lagged rainfalls used in Model 4 and Model 5.

Horizon Rank1 2 3 4 5 6 7

T+1 R(I,t�4) R(II,t�1) R(II,t�3) R(III,t�2) R(IV,t�1) R(IV,t�4) R(IV,t�6)T+2 R(I,t�4) R(II,t�3) R(III,t�2) R(IV,t�4) R(IV,t�6)T+3 R(I,t�4) R(II,t�3) R(IV,t�4) R(IV,t�6)T+4 R(I,t�4) R(IV,t�4) R(IV,t�6)

264 F.-J. Chang, M.-J. Tsai / Journal of Hydrology 535 (2016) 256–269

identify the best input combination of lagged rainfall based on thespatial aggregation of radar rainfall data obtained from the SOM.

It can be found that the models with inputs of non-sequentialrainfall time series and the antecedent discharge (Models 1, 2and 5) perform particularly well at shorter lead times (up to twotime steps ahead). Model 4 with the optimal rainfall combinationselected by the 2-staged GT shows a smaller time shift error, ascompared to Model 3 with input combination selected by CC. Inter-estingly, Model 5 combining the positive features of antecedentdischarge and the selected optimal rainfall combination couldimprove the goodness-of-fit for discharge at shorter lead timesand reduce time shift errors at longer lead times simultaneously.

3.2. Fuzzy rules of ANFIS models

The fuzzy if-then rules used in the ANFIS models are of theSugeno’s type (genfis3 function in the Matlab 7.0). Take Model 5as an example, the inputs consist of Qt and DQ , and three laggedrainfalls. Model 5 has two clusters (rules), and its Gaussian mem-bership functions of the inputs (Qt , DQ and R(IV,t�6)) at t+1 upto t+4 are shown in Fig. 10. Only the membership function of thelagged rainfall R(IV,t�6) is shown because the remaining rainfall

membership functions are similar to each other. For the member-ship functions of Qt , it is clear that cluster 1 represents low flowwhile cluster 2 represents high flow. For the membership functionsof DQ at the horizons of t+1 up to t+3, cluster 1 represents negativeDQ values (i.e. the recession limb of the inflow hydrograph) whilecluster 2 represents positive DQ values (i.e. the ascending limb ofthe inflow hydrograph). The membership functions for lagged rain-falls are also associated with the two clusters that represent higherrainfall amount and lower rainfall amount (Fig. 10).

As mentioned earlier, the ANFIS can provide qualitative andheuristic knowledge about the solutions. We would like to explorethe non-linear rainfall–runoff relationship based on its built fuzzyif-ten rules. The output weights of ANFIS models are shown inTable 6. By comparing the weights for each cluster at various hori-zons, the contribution of the inputs to forecasts mainly comes fromcurrent inflow (Qt), with which associated the highest weights,except cluster 1 at t+4; while these weights decrease from 0.903(t+1) to 0.519 (t+4) for cluster 1 and from 0.858 (t+1) to 0.661(t+4) for cluster 2, respectively. It indicates that the contributionof inflow at time t decreases when the horizon increases, whichis consistent with the concept of auto-correction of inflow. Fromanother point of view, the importance of lagged rainfall arises for

t+1

t+2

t+3

t+4

Fig. 8. Comparison of observed and predicted inflows of Models 3 and 4 for 1- to 4-step-ahead forecasting based on the testing dataset.

Fig. 9. Skill scores of Models 2 and 5 over Model 1 (benchmark model) based on thetesting data set, respectively.

F.-J. Chang, M.-J. Tsai / Journal of Hydrology 535 (2016) 256–269 265

longer horizon forecasting. This could be found in Table 6, in whichthe weights of R(I,t�4) at horizon t+1 increase from �0.003 (t+1) to0.281 (t+4) for cluster 1 and from �0.019 (t+1) to 0.033 (t+4) for

cluster 2, respectively. Similar changes could be found for theweights of R(IV,t�4) and R(IV,t�6). Notably, the output weightsof DQ have different signs in two clusters, which means they canreduce forecast values and thus mitigate the problem of overesti-mation. For example, the weights of DQ at horizon t+4 for clusters1 and 2 is 0.742 and �0.263 accordingly, which would decrease theforecast values since the membership function of DQ movestoward negative values for cluster 1 but has a tendency towardpositive values for cluster 2 (Fig. 10).

3.3. Probability information derived from error analysis

Decisions on reservoir operation during flood periods must bemade within few hours in Taiwan, and consequently the probabil-ity information of inflow forecasts is crucial to decision makers.

Table 5Comparison of observed and predicted peak inflows for the testing events.

Difference of peak inflow (cm s) Time-step error of peak inflow (h)

Model 1 Model 2 Model 3 Model 4 Model 5 Model 1 Model 2 Model 3 Model 4 Model 5

JANGMIt+1 53.7 105.6 392.1 �173.9 94.2 1 1 2 �2 0t+2 265.5 177.0 393.4 429.2 301.6 2 1 2 �2 1t+3 583.9 360.0 393.5 447.5 490.8 2 2 2 �2 1t+4 561.3 321.5 392.5 �115.7 551.4 3 3 2 �2 2

SAOLAt+1 �76.7 97.4 1941.1 1025.9 �55.9 1 1 3 1 1t+2 61.4 430.2 1939.3 1032.1 �152.1 2 2 3 1 2t+3 126.9 559.7 1938.4 1037.5 �173.3 3 3 3 1 2t+4 730.8 742.3 1937.9 1046.1 379.0 3 3 3 1 1

t+1t+2

t+3t+4

Fig. 10. Membership functions of the inputs (Qt, DQ, R(IV,t�6)) in Model 5 at horizons t+1 up to t+4.

266 F.-J. Chang, M.-J. Tsai / Journal of Hydrology 535 (2016) 256–269

The most important issue in the decision-making process is to pro-vide a quantitative probability value presenting the confidence offorecasts that meets the requirement of decision makers. Althoughthe proposed data-driven model is a deterministic model, the prob-ability information in terms of confidence level for model output isalso evaluated by the error analysis. The error analysis is based on abottom-up process, in which the acceptable error for multi-step-ahead forecasting is deduced from the water level of the reservoir.By evaluating the historical water level records of the nine typhoonevents, the standard deviation of the records below the upper-limit

of operational rules is 330 cm s while that above the upper limit is275 cm s, respectively. These two standard deviations are used aserror thresholds to represent the acceptable errors of inflow fore-casting. The smaller acceptable error (275 cm s) for the water levelabove the upper-limit of operational rules reveals that decisionmakers would bear a smaller risk under this situation.

We match the prediction errors of training, validation and test-ing datasets, with sample sizes of 293, 75 and 133 accordingly, in anormal distribution and applied the thresholds to determining theprobability of the prediction error falling within the acceptable

Table 6Output weights of Model 5.

Input t+1 t+2 t+3 t+4

Cluster 1 Cluster 2 Cluster 1 Cluster 2 Cluster 1 Cluster 2 Cluster 1 Cluster 2

R(I,t�4) �0.003 �0.019 0.016 0.028 0.004 0.035 0.281 0.033R(II,t�1) 0.033 �0.003 – – – – – –R(II,t�3) 0.034 0.039 0.069 0.015 0.127 0.049 – –R(III,t�2) �0.001 0.010 0.022 0.041 – – – –R(IV,t�1) �0.003 0.009 – – – – – –R(IV,t�4) 0.020 �0.035 0.044 �0.033 0.046 0.012 0.002 0.012R(IV,t�6) 0.011 0.030 0.034 0.027 0.069 0.012 0.075 0.019Qt 0.903 0.858 0.811 0.775 0.712 0.725 0.519 0.661D 0.306 �0.384 0.312 �0.426 0.655 �0.430 0.742 �0.263Constant �147.30 15.44 �264.30 22.48 �294.70 22.93 313.50 86.49

Table 7Probability (%) for the prediction error of inflow within the acceptable error.

Te = 275 Te = 330

Model 1 Model 2 Model 5 Model 1 Model 2 Model 5

Training (N = 293)t+1 91.3 91.5 93.9 96.0 96.1 97.5t+2 75.5 76.9 82.4 83.8 84.8 89.5t+3 67.8 66.9 74.7 76.5 75.7 83.1t+4 66.0 65.6 75.0 74.9 74.4 83.4

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error (Fig. 11 and Table 7). For the training data set, the confidencelevel of forecasts for Model 5 can reach as high as 97.5% and 83.4%at forecast horizons of 1- and 4-hour-ahead, respectively. Thisimplies that the risks of the predicted inflow at 1- and 4-hour-ahead horizons failing to fall within the pre-defined threshold(i.e. Te = 330) are less than 3% and 17%, respectively. We notice thatthe confidence level of 1-hour-ahead forecast remains the samewhile that of 4-hour-ahead forecast, however, decreases to 65%for validation and testing datasets. It is worth noting that Model

Model 1

Model 2

Model 5

Fig. 11. Normal distributions of forecasting errors for Models 1, 2, and 5 based onthe testing events.

Validation (N = 75)t+1 91.8 92.1 95.5 96.2 96.4 98.4t+2 75.2 72.4 81.6 83.3 81.0 88.7t+3 60.8 70.0 72.6 69.5 78.7 81.0t+4 50.7 52.1 58.0 58.8 60.4 66.8

Testing (N = 133)t+1 87.7 85.5 96.1 93.5 92.0 98.6t+2 63.6 64.3 76.3 72.5 72.7 84.4t+3 49.4 50.3 68.3 57.7 58.4 77.2t+4 47.0 49.8 56.5 54.8 57.8 65.2

Alla (N = 501)t+1 91.0 90.9 94.4 95.9 95.7 97.8t+2 74.1 74.6 81.4 82.6 83.0 88.8t+3 63.7 63.5 73.2 72.4 71.9 81.9t+4 60.2 59.0 68.0 69.0 67.7 76.9

a All means using all typhoon events to derive probability.

5 has the highest confidence levels for all horizons. Notably, beingcompared with those of Model 2, the prediction confidence levelsof Model 5 improve up to 36% at horizons t+3 and t+4 for all data-sets. This is a remarkable improvement for long-term inflow fore-casting using data-driven models.

4. Conclusions

A novel spatio-temporal lumping approach combining the SOMfor spatial data aggregation and the 2-staged GT for the optimalinput combination selection of a data-driven model (i.e. ANFIS)for multi-step-ahead inflow forecasting is presented in this study.The results reveal that the proposed approach is capable to allevi-ate the timing error problem and improve the accuracy and relia-bility of flood forecasting. Probabilistic forecasting can lead tobetter decisions in response to operational hydrological concerns.We provide probabilistic information (confidence level) of fore-casts based on a deterministic ANFIS model through pre-determined error thresholds. The major findings are summarizedas follows.

(1) The SOM and the proposed 2-staged GT serve as the optimalapproach to integrating continuous radar rainfall data andproviding the optimal input combination for the ANFISmodel.

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(2) A 2-staged GT procedure is developed to reduce the compu-tational time for selecting the key inputs from a large num-ber of potential input variables, 32 variables in this case. Thenon-trivial variables selected by this procedure prove to bevery efficient in identifying the best input combination formodeling.

(3) The constructed ANFIS model with the optimal input combi-nation can well establish the complex rainfall–runoff rela-tionship in a large watershed through only 2 fuzzy if-thenrules, which can effectively categorize inputs into high andlow conditions. Evaluating the membership functions offuzzy rules provides useful insights to improve our knowl-edge on how the ANFIS algorithm simulates the rainfall–run-off relationship.

(4) The time lag of predicted peak inflow is greatly reduced bythe proposed approach (Model 5), and the occurrence ofpeak inflow is predicted precisely for 1-hour-ahead forecast-ing whereas the delay of predicted peak inflow is typicallyless than 1 hour for 2- and 3-hour-ahead forecasting. Thisis a significant improvement made by data-driven modelsfor inflow forecasting. The solution simply uses the GT toidentify non-trivial variables and the best input combinationfor data-driven models.

(5) From decision makers’ point of view, the confidence level aswell as reliability of the model output is more useful than asingle deterministic forecast value. The results reveal thatthe confidence levels of forecasts obtained in this study gen-erally reach 77% at horizon t+4 for all events and the pro-posed approach (Model 5) has the highest confidencelevels for all horizons.

Acknowledgements

This study was partially funded by the Ministry of Science andTechnology, Taiwan, ROC. (Grant numbers: 103-2313-B-002-016-MY3). We are grateful for the radar dataset provided by CentralWeather Bureau, Taiwan, ROC, and inflow data provided by WaterResource Agency, Taiwan, ROC.

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