This study proposes a framework for developing a probabilistic lag time (PLT) equation by taking into account uncertainty factors, including the rainfall factor (i.e., the maximum rainfall intensity), the hydraulic factor (i.e., the roughness coefficient in the river), and the geometrical factors (the catchment area, the length, and the basin slope). The proposed PLT equation is established based on the lag time equation by means of the uncertainty and risk analysis, i.e., the advanced first-order and second-moment approach. Hourly rainfall data in the Bazhang River watershed are used in the model development and application. The results indicate that compared with observed lag times extracted from historical events, the observed ones are mostly located within the 95% confidence interval of the simulated lag times. In addition, the resulting underestimated risk from the PLT equation can reasonably represent the degree of the difference between the estimated lag time and observed lag time. Consequently, the proposed PLT equation not only estimates the lag time at specific locations along the river, but also provides a corresponding reliability.

## INTRODUCTION

In hydrological analysis and relevant applications (e.g., in flood early warning operation), lag time plays an important role in the hydrological process, such as the rainfall–runoff (RR) modeling (e.g., Snyder 1938; Gioia et al. 2008; Sudharsanan et al. 2010; Talei & Chua 2012; Munoz-Villers & McDonnell 2013; Alfieri et al. 2014). A number of definitions of lag time (e.g., Rao & Delleur 1974; Hall 1984; Yu et al. 2000; Talei & Chua 2012) are proposed in the literature. Hall (1984) summarized seven definitions of the lag time based on the time difference between the rainfall and the runoff of events. In general, lag time is defined as the time difference between centroids of the effective periods from a specific rainfall derived from in a hyetograph and time when peak discharge occurs. For example, as summarized in Table 1, the second lag time (tlag,2) is defined as the distance in time between the centroid of the hyetograph and runoff hydrograph, while the third lag time (tlag,3) is a time difference between the centroid of the hyetograph and the peak discharge. Among the seven definitions of lag time listed in Table 1, tlag,2 and tlag,3 are the most widely adopted (Talei & Chua 2012).

Table 1

Summary of definitions of lag time (Hall 1984)

Lag timeDefinition
tlag,1 Difference in time from the initial rainfall to the centroid of runoff hydrograph
tlag,2 Difference in time from the centroid of effective rainfall to the peak discharge
tlag,3 Difference in time from the centroid of the effective rainfall to the centroid of the runoff hydrograph
tlag,4 Difference in time from centroid of the total rainfall to the centroid of the runoff hydrograph
tlag,5 Difference in time from the initial rainfall to the peak discharge
tlag,6 Difference in time from the ending rainfall to the inverse point of runoff hydrograph
tlag,7 Difference in time from the centroid of total rainfall to the peak discharge
Lag timeDefinition
tlag,1 Difference in time from the initial rainfall to the centroid of runoff hydrograph
tlag,2 Difference in time from the centroid of effective rainfall to the peak discharge
tlag,3 Difference in time from the centroid of the effective rainfall to the centroid of the runoff hydrograph
tlag,4 Difference in time from centroid of the total rainfall to the centroid of the runoff hydrograph
tlag,5 Difference in time from the initial rainfall to the peak discharge
tlag,6 Difference in time from the ending rainfall to the inverse point of runoff hydrograph
tlag,7 Difference in time from the centroid of total rainfall to the peak discharge

In general, lag time reflects the runoff velocity and storage effect of a river length and slope from the upstream to the downstream points (Watt & Chow 1985; Leopold 1991; Thompson et al. 2004; Roussel et al. 2005; Loukas & Quick 2009). Using the lag time with the appropriate RR model, the resulting runoff characteristics (i.e., runoff volume, peak discharge, and time to the peak) are essential information for designing hydraulic structures, such as dikes and spillways. Talei & Chua (2012) evaluated the effect of the lag time on the performance of the data-driven RR modeling developed using adaptive network-based fuzzy inference system, and found that their proposed model which takes the lag time into account simulates the peak discharge better than models which do not consider the lag time. Therefore, the lag time should be an essential feature for estimating runoff from a catchment. In addition, the lag time can be applied in flood warning to qualify the flash flood response time (e.g., Marchi et al. 2010; Borga et al. 2014).

Numerous equations for estimating the lag time have been published (e.g., Rao & Delleur 1974; Hall 1984; Leopold 1991; Yu et al. 2000; Honarbakhsh et al. 2012; Talei & Chua 2012; Gericke & Smithers 2014). Most equations are derived using geographical features, such as the basin area, characteristic lengths (e.g., river length and width), and the basin slope. These lag time equations are mostly empirical. For example, Honarbakhsh et al. (2012) derived a power function of the lag time with the equivalent diameter of the watershed and river length. Loukas & Quick (2009) proposed a method for establishing the lag time equation by using the kinematic wave equation, and the resulting equation provides approximations of the basin lag calculated from the observed data than when estimated using empirical equations in the forested mountainous watershed.

Since the lag time equations widely used utilize watershed characteristics to estimate the lag time, it is a constant for a given watershed, and hence the equations can be treated as deterministic models. However, many investigations have indicated that the lag time should vary with the flow rate, rainfall factors, vegetation, land use, and associated factors (e.g., Diskin 1964; Askew 1970; Pilgrim 1976; Yu et al. 2000). For example, Diskin (1964) concluded that there are different estimated lag times in a watershed due to various rainfall events. Askew (1970) presented a non-linear relationship of the lag time with the average total discharge. Yu et al. (2000) indicated that the variation in the roughness coefficient in the river due to the surface treatment impacts on the lag time estimate. According to Loukas & Quick (2009), an error in the lag time would lead to errors in the estimation of peak discharge. Therefore, the variation and error in the lag time due to various geographical features, rainfall and the surface cover in the watershed should be taken into account when estimating lag time. In summary, the lag time should be a function of the geographical features of the watershed, the roughness coefficient on the river bed, and rainstorm events.

Therefore, the purpose of this study is to develop a relationship between the lag time and the watershed features (e.g., area, river length, and basin slope) and the rainfall events. Since the uncertainties exist in the flow rate due to the rainstorm events and roughness coefficient (e.g., Loukas & Quick 2009; Wu et al. 2011), the reliability of the results from the lag time equation derived should be affected, so that the estimated lag time also involves uncertainty. In order to evaluate the effect of the aforementioned uncertainty factors on the estimation of the lag time, the proposed lag time equation would incorporate with the uncertainty and risk analysis methods to derive a probabilistic equation (the probabilistic lag time (PLT) equation). The proposed PLT equation is not only expected to estimate the lag times at specific locations in a watershed, but also to provide the exceedance probability which is defined as the probability of the lag time greater than the estimate (i.e., underestimated risk).

## METHODOLOGY

### Model concept

The definitions listed in Table 1 are for a single-peak hydrograph, but observed hydrographs are commonly multi-peak particularly on larger watersheds. Thus, the centroids of the hyetograph and runoff hydrograph are only determined with great difficulty. To solve the above problem, this study adopts the definition of the lag time proposed by Yu et al. (2000), which regards the lag time as the time period between the maximum rainfall intensity and the peak discharge. However, in reality, observed discharges are calculated from the observed water levels through the stage–discharge rating curve. In this study, lag time is defined as the time difference between the maximum rainfall intensity and the peak water level without considering the hysteresis effect which leads to uncertainty in the estimated discharge (e.g., Di Baldassarre et al. 2012; Sikorska et al. 2013).

Empirical lag time equations are derived using observations, and hence they are influenced by the available record length and quality of the observed data. Moreover, the flow velocity in the river, which has a significant impact on the lag time, not only results from the basin slope and rainfall intensity, but also from the roughness coefficient of the channel. Thus, the rainfall intensity and roughness coefficient could influence the lag time. In order to quantify the above uncertainties in estimating the lag time in a watershed attributed to the above factors, this study develops a lag time equation by taking into account the geographical factors (i.e., watershed area, river length, and basin slope), rainfall factors and hydraulic factor (the roughness coefficient). Furthermore, the rainstorm can be characterized into three properties, including rainfall duration, depth, and storm pattern (temporal distribution of rainfall) (Wu et al. 2006). In summary, the proposed PLT equation integrates the rainfall factors, i.e., the rainfall characteristics (rainfall duration, depth, and storm pattern), the roughness coefficient of the channel and geographical factors, including the area, slope, and river length of the watershed. Thereby, in this study, the lag time equation would be a function of lag time with the rainfall, hydraulic as well as geographical factors, and it can be derived by means of the multivariate regression analysis.

Since the purpose of this study is to provide a stochastic method to estimate lag time and its exceedance probability, the uncertainty and risk analysis method is incorporated to quantify the reliability of the estimated lag time. When carrying out the uncertainty and risk analysis, the rainfall characteristics would be generated by using the Monte Carlo simulation method to produce the hyetograph. Then, the lag times estimated at various locations are obtained by the river routing model. Eventually, the PLT equation can be established using the estimated lag times at the specific locations associated with various hydraulic and geographical factors under simulated rainstorm events.

In summary, the proposed PLT equation involves two parts: one is the relationship of the lag time with the uncertainty factors, which primarily estimates the lag time using the known geographical, hydraulic, and rainfall factors. The other equation is the computation equation of the underestimated risk which quantifies the underestimated risk of the estimated lag time. The detailed concepts of the relevant methods used in the model development are detailed below.

### Simulation of rainfall characteristics

The hyetograph of a rainstorm event can be characterized by three components: rainfall depth, duration, and storm pattern, as shown in Figure 1 (Wu et al. 2006). Of the three rainfall characteristics, the rainfall depth and duration can be calculated directly from observations recorded by the rain gauges in a watershed. The storm pattern is the distribution of rainfall during a certain period of time; it varies with the storm duration and the rainfall amount from one event to another, as well as the locations of the rain gauges. Wu et al. (2006) proposed a dimensionless storm pattern which could be obtained by adjusting the rainfall depth and duration of a rainfall mass curve as (see Figure 2):in which is the dimensionless time; d is the storm duration; is the dimensionless cumulative rainfall ordinate; is the incremental dimensionless rainfall amount; Dt is the cumulative rainfall depth at time t; and Dd is the total rainfall depth. Accordingly, the dimensionless rainfall mass curve of a storm pattern shows the cumulative fraction of the storm depth ∈ [0, 1], over the dimensionless time ∈ [0, 1], and the corresponding dimensionless rainfall hyetograph represents the rainfall percentage increment at each time interval. Two special features of dimensionless rainstorm patterns must be observed: (1) summation of dimensionless rainfall (Pτ) must be equal to one: P1/12+P2/12+…+P12/12 = 1 as 0 ≤ F1/12F2/12F12/12 = 1; and (2) dimensionless rainfall (Fτ) must be between zero and one: 0 ≤ Pτ ≤ 1 for τ = 1/12, 2/12, …, 12/12. Thereby, the dimensionless storm pattern can remove the effects of the rainfall duration and amount and only consider the temporal variation of the rainfall.

The rainfall characteristics, i.e., the rainfall depth, rainfall duration, and storm pattern, are inherently correlated and their probabilistic distributions are likely to be non-normal; namely, the rainfall characteristics are multivariate non-normal random variates (Wu et al. 2006). The generation of rainfall characteristics should preserve the respective marginal statistical properties and correlation relations. However, as it is difficult to establish their joint distribution, this study applies a stochastic model for generating rainfall characteristics developed by Wu et al. (2006). This stochastic model mainly applies the multivariate Monte Carlo simulation for the non-normal random variables developed by Chang et al. (1997).

Note that the rainfall duration and depth can be directly generated by the above procedure. However, the storm pattern is the time distribution of rainfall, i.e., it is composed of dimensionless rainfall in which the time-axis is divided into 12 increments. Fang & Tung (1996) found that the Johnson distribution system is more flexible in fitting the rainstorm pattern than are other distributions. The Johnson distribution is a system of frequency curve consisting of four parameters (Johnson 1949). Eventually, these simulated rainfall characteristics can be combined as the rainstorm hyetographs, as shown in Figure 3.

### SOBEK river routing model

Using the estimated runoff hydrograph and tide level, the water level hydrographs at various cross sections of interest along the river can be calculated through the hydraulic routing. Many numerical models can be applied for the hydraulic routing, such as HEC-RAS (2006), SOBEK (WL Delft Hydraulic 2005), or MIKE 11 (Havno et al. 1995). The SOBEK model is a sophisticated one-dimensional open-channel dynamic flow and two-dimensional overland flow modeling system (named SOBEK model). It can be used to simulate problems in river management, flood protection, design of canals, irrigation systems, water quality, navigation, and dredging. This study uses the SOBEK model for the hydraulic simulation of the water level in the river network. It is well known that the hydraulic routing needs the geometric data of the channel section and the roughness coefficient in a watershed of interest.

In the SOBEK model, the Sacramento Soil Moisture Accounting (SAC-SMA) model (Burnash et al. 1973) is used to estimate runoff as boundary and lateral conditions. SAC-SMA is a deterministic, continuous, non-linear semi-distributed hydrologic model, containing two soil layers: an upper and a lower zone. Each layer includes the tension and free water storage, which interact to generate soil moisture states and five runoff components (Koren et al. 2000). Figure 4 shows the schematic representation of RR routing in the SAC-SMA model (Ajami et al. 2004). Since the SAC-SMA has 19 parameters, as presented in Table 2, it is difficult to calibrate the parameters optimally. Wu et al. (2012) modified the genetic algorithm to develop a parameter-calibration method by considering the sensitivity of the SAC-SMA parameters. This parameter-calibration is named the GA_SA method. In the GA_SA method, an objective function (Fobj) which computes the weighted mean square error, is used in the parameter calibration (Madsen 2000):where nf is the runoff period; qobs(t) and qest(t) are the observed and estimated discharge at time t; and is the mean of observed runoff volume. Consequently, this study utilizes the GA_SA method associated with the weighted mean square error as the fitness function to calibrate the ten sensitive parameters of the SAC-SMA model for the study area.
Table 2

Description of parameters of SAC-SMA model (Ajami et al. 2004)

ParametersDescription
UZTWM Upper zone tension-water capacity (mm)
UZFWM Upper zone free-water capacity (mm)
UZK Upper zone recession coefficient
PCTIM Percent of impervious area
SARVA Fraction of segment covered by streams, lakes, and riparian vegetation
ZPERC Minimum percolation rate coefficient
REXP Percolation equation exponent
LZTWM Lower zone tension water capacity (mm)
LZFSM Lower zone supplementary free-water capacity (mm)
LZFPM Lower zone primary free-water capacity (mm)
LZSK Lower zone supplementary recession coefficient (mm)
LZPK Lower zone primary recession coefficient (mm)
PFREE Percentage percolating directly to lower zone free water
SIDE Ratio of deep recharge water going to channel baseflow
RESERV Percentage of lower zone free water not transferable to lower zone tension water
SSOUT Fixed rate of discharge lost from the total channel flow (mm/Δt)
DF_L Period of runoff distribution function
DF_P Maximum ratio of runoff distribution function
ParametersDescription
UZTWM Upper zone tension-water capacity (mm)
UZFWM Upper zone free-water capacity (mm)
UZK Upper zone recession coefficient
PCTIM Percent of impervious area
SARVA Fraction of segment covered by streams, lakes, and riparian vegetation
ZPERC Minimum percolation rate coefficient
REXP Percolation equation exponent
LZTWM Lower zone tension water capacity (mm)
LZFSM Lower zone supplementary free-water capacity (mm)
LZFPM Lower zone primary free-water capacity (mm)
LZSK Lower zone supplementary recession coefficient (mm)
LZPK Lower zone primary recession coefficient (mm)
PFREE Percentage percolating directly to lower zone free water
SIDE Ratio of deep recharge water going to channel baseflow
RESERV Percentage of lower zone free water not transferable to lower zone tension water
SSOUT Fixed rate of discharge lost from the total channel flow (mm/Δt)
DF_L Period of runoff distribution function
DF_P Maximum ratio of runoff distribution function

### Risk quantification of estimated lag time

In accordance with the section ‘Model concept’, the proposed PLT equation can provide the probability of the lag time exceeding the estimated value (underestimated risk) by using the advance first-order second-moment method (AFOSM) (Mays & Tung 1980). In calculating the exceedance probability, the relationship between the lag time and the associated uncertainty factors can be established using the multivariate regression method as follows:where denotes the lag time; denotes the geographical factors (i.e., the river length and the slope of watershed); denotes the hydraulic factor (i.e., Manning roughness coefficient); and represents the rainfall factors (i.e., maximum rainfall intensity and total depth).
This study uses the AFOSM to analyze the underestimated risk of the lag time resulting from the uncertainties in the geographical, rainfall and hydraulic factors. Since the underestimated risk of the lag time is defined as the probability of the lag time exceeding the estimation in this study, the risk can be quantified by the following equation:in which Z denotes the performance function; and denote the mean and standard deviation of Z; represents the standard normal distribution; and is the reliability index. are calculated by the following equations:where stands for the failure points of the ith uncertainty factor when the performance function is z = 0; and are the mean and standard deviation of the ith uncertainty factor; denotes the lag time using the uncertainty factors’ failure points which lead to the equal to zero; is the sensitivity coefficient of the ith uncertainty factor. In the case of the mean variance of and uncertainty factors being given, the probability of the lag time exceeding s specific value, , can be calculated. Figure 5 is a graphical illustration of the derived exceedance probability (i.e., underestimated risk) for the lag time.

## RESULTS AND DISCUSSION

### Study area and data

To demonstrate the applicability of the PLT equation, this study selects a watershed as the study area, Bazhang River located in Southern Taiwan, as shown in Figure 7. The Bazhang River is an important river in Southern Taiwan and its length is about 80 km long with a drainage basin of 475 km2. In addition, the average slope of the river bed reaches 1/40. Within the Bazhang River watershed, there are two main branches: Chilan Creek and Touqian Creek. The average annual rainfall is approximately 2,277.4 mm, and the peak discharge in the rainy and dry seasons reaches 1,800 cm and 0.4 cm, respectively.

There are nine automatically recording rain gauges and six water level gauges in the Bazhang River watershed, as shown in Figure 7, and their detailed information can be referred to in Table 3.

Table 3

Summary of rain gauges used in Bazhang River watershed

No.GaugeLocation
Record period
X_TWD97_TM2 (m)Y_TWD97_TM2 (m)
RG1 Xiao-Gong-Tian 207,154 2,584,414 1967–present
RG2 Zhong-Bu 200,981 2,591,267 1988–present
RG3 Nan-Jing 187,180 2,590,024 1988–present
RG4 Tou-Dong 209,468 2,589,336 1988–present
RG5 An-Nei 173,530 2,581,217 1988–present
RG6 Jia-Yi 191,213 2,599,557 1969–present
RG7 Nei-Bu 205,488 2,598,140 1993–present
RG8 Dong-Hou-Liao 173,163 2,585,418 1988–present
RG9 Guan-Zi-Ling 198,753 2,581,219 1959–2008
No.GaugeLocation
Record period
X_TWD97_TM2 (m)Y_TWD97_TM2 (m)
RG1 Xiao-Gong-Tian 207,154 2,584,414 1967–present
RG2 Zhong-Bu 200,981 2,591,267 1988–present
RG3 Nan-Jing 187,180 2,590,024 1988–present
RG4 Tou-Dong 209,468 2,589,336 1988–present
RG5 An-Nei 173,530 2,581,217 1988–present
RG6 Jia-Yi 191,213 2,599,557 1969–present
RG7 Nei-Bu 205,488 2,598,140 1993–present
RG8 Dong-Hou-Liao 173,163 2,585,418 1988–present
RG9 Guan-Zi-Ling 198,753 2,581,219 1959–2008

In this study, the upstream area at each water level gauge is regarded as the gauged sub-basin, so that the average areal rainfall used in the estimation of runoff from the branches is calculated by means of the Thiessen polygon method with the observed data at rain gauges within the sub-basins. Table 4 lists the geographical characteristics of gauged sub-basins and associated rain gauges as well as their areal Thiessan's weights.

Table 4

Summary of gauged sub-basins (water level gauges) and associated rain gauges, including Thiessen's areal weights in Bazhang River watershed

Gauged sub-basinLocation
Geomorphic characteristics
Rain gaugesGeomorphic characteristics
X_TWD97_TM2 (m)Y_TWD97_TM2 (m)Area (km2)L (km)Lca (km)S
Chu-Kou 209,787 2,592,898 80.47 17.49 8.86 0.068 Tou-Dong 1.00
Jun-Hui 194,654 2,595,164 119.7 36.04 22.1 0.039 Zhong-Bu 0.14
Nei-Bu 0.10
Jia-Yi 0.04
Tou-Dong 0.72
Tou-Qian-Qi 175,633 2,583,667 26.59 17.32 9.73 0.029 Guan-Zi-Ling 1.00
Chang-Pan 192,965.2 2,592,095 98.39 22.59 16.34 0.031 Tou-Dong 0.20
Xiao-Gong-Tian 0.27
Zhong-Bu 0.53
Ba-Zhang-Qi 193,806 2,591,886 313.42 49.14 25.13 0.029 Xiao-Gong-Tian 0.08
Nei-Bu 0.04
Guan-Zi-Ling 0.06
Zhong-Bu 0.30
Jia-Yi 0.04
Nan-Jing 0.13
Tou-Dong 0.35
Qing-Shui-Gang 176,501 2,583,447 421.73 64.98 36.19 0.022 Xiao-Gong-Tian 0.06
Nei-Bu 0.03
Zhong-Bu 0.22
Guan-Zi-Ling 0.04
Jia-Yi 0.05
Nan-Jing 0.25
Dong-Hou-Liao 0.03
An-Nei 0.06
Tou-Dong 0.26
Gauged sub-basinLocation
Geomorphic characteristics
Rain gaugesGeomorphic characteristics
X_TWD97_TM2 (m)Y_TWD97_TM2 (m)Area (km2)L (km)Lca (km)S
Chu-Kou 209,787 2,592,898 80.47 17.49 8.86 0.068 Tou-Dong 1.00
Jun-Hui 194,654 2,595,164 119.7 36.04 22.1 0.039 Zhong-Bu 0.14
Nei-Bu 0.10
Jia-Yi 0.04
Tou-Dong 0.72
Tou-Qian-Qi 175,633 2,583,667 26.59 17.32 9.73 0.029 Guan-Zi-Ling 1.00
Chang-Pan 192,965.2 2,592,095 98.39 22.59 16.34 0.031 Tou-Dong 0.20
Xiao-Gong-Tian 0.27
Zhong-Bu 0.53
Ba-Zhang-Qi 193,806 2,591,886 313.42 49.14 25.13 0.029 Xiao-Gong-Tian 0.08
Nei-Bu 0.04
Guan-Zi-Ling 0.06
Zhong-Bu 0.30
Jia-Yi 0.04
Nan-Jing 0.13
Tou-Dong 0.35
Qing-Shui-Gang 176,501 2,583,447 421.73 64.98 36.19 0.022 Xiao-Gong-Tian 0.06
Nei-Bu 0.03
Zhong-Bu 0.22
Guan-Zi-Ling 0.04
Jia-Yi 0.05
Nan-Jing 0.25
Dong-Hou-Liao 0.03
An-Nei 0.06
Tou-Dong 0.26

In the Bazhange River watershed, the corresponding hourly rainfall data for the five rainstorm events are used as the study data (see Figure 8) and their information is summarized in Table 5. In view of Table 5, the typhoon events selected from 2009 to 2013 show a large range of rainfall depths (on average from 186.2 mm to 1,181.4 mm) associated with the rainfall maximum intensities (on average, from 19.3 mm to 55.8 mm), respectively. Specifically, the rainfall durations range from 27 hours to 118 hours. Hence, using the statistical properties of rainfall characteristic obtained from the five rainfall events, it is possible to simulate the rainstorm events associated with various magnitudes. Given this variation, it is assumed that this study takes into account the effect of temporal and spatial variations in the rainfall events on estimating the lag time.
Table 5

Summary of rainstorm events used in the simulation of rainfall characteristics in Bazhang River watershed

No. of eventRainstorm eventOccurrence periodDuration (hours)Areal average rainfall depth (mm)Areal average maximum rainfall intensity (mm/hr)
Morakot 06-08-2009 14:00–11-08-2009 11:00 118 1,131.4 48.8
Saola 01-08-2012 14:00–03-08-2012 12:00 47 248 19.3
Soulik 12-07-2013 14:00–13-07-2013 17:00 27 186.2 20.1
Trami 21-08-2013 10:00–22-08-2013 21:00 36 388.9 55.8
Kong-Rey 27-08-2013 23:00–01-09-2013 12:00 110 732.6 47.7
No. of eventRainstorm eventOccurrence periodDuration (hours)Areal average rainfall depth (mm)Areal average maximum rainfall intensity (mm/hr)
Morakot 06-08-2009 14:00–11-08-2009 11:00 118 1,131.4 48.8
Saola 01-08-2012 14:00–03-08-2012 12:00 47 248 19.3
Soulik 12-07-2013 14:00–13-07-2013 17:00 27 186.2 20.1
Trami 21-08-2013 10:00–22-08-2013 21:00 36 388.9 55.8
Kong-Rey 27-08-2013 23:00–01-09-2013 12:00 110 732.6 47.7

### Simulation of rainstorm events

Before developing the proposed PLT equation, the extract rainfall characteristics (rainfall duration, depth, and storm pattern) are extracted from the hyetographs of the five rainstorm events shown in Table 5, and the corresponding statistical properties are then calculated. For example, Table 6 contains the mean, standard deviation, skewness, kurtosis, and correlation coefficients of the rainfall characteristics at the Zhong-Bu gauge (RG2), Tou-Dong gauge (RG4), An-Nei gauge (RG5), and Nei-Bu gauge (RG7). From Table 6, it is evident that, on average, the rainfall duration of a rainstorm event is approximately 68 hours (about 3 days) with a rainfall depth of 385 mm. This implies that 3-day rainstorm events frequently take place in the Bazhang River watershed. In addition, the standard deviation of the rainfall duration is approximately 38 hours with its coefficient of variation (CV) being about 0.6. In regard to the rainfall depth, its CV is approximately 0.3; this indicates that there is more significant variation in the rainfall duration than there is in the depth. Table 6 also presents the statistical properties among the rainfall characteristics. It can be seen that the correlation coefficient between the rainfall duration and depth approximately exceeds 0.9, and the correlation coefficients among dimensionless rainfalls range from −0.95 to 0.95. This definitely indicates that within a storm pattern, the dimensionless rainfall at a dimensionless time step is highly correlated with those at other dimension time steps.

Table 6

Summary for resulting statistical properties of rainfall characteristics from eight rainstorm events

Statistics Rainfall durationRainfall depthDimensionless rainfall amount
P1/12P2/12P3/12P4/12P5/12P6/12P7/12P8/12P9/12P10/12P11/12P12/12
(a) RG2 (Zhong-Bu gauge)
Mean  67.6 533.6 0.029 0.028 0.052 0.167 0.103 0.166 0.158 0.097 0.074 0.056 0.028 0.043
Standard deviation  38.495 371.512 0.03 0.019 0.041 0.133 0.065 0.085 0.083 0.066 0.073 0.041 0.031 0.041
Skewness coefficient  0.34 0.615 0.808 1.068 0.175 0.222 0.293 −0.453 0.838 0.406 0.426 0.172 0.499 0.391
Kurtosis coefficient  1.237 1.714 2.304 2.58 1.412 1.339 2.439 1.676 2.304 1.748 1.222 1.754 1.372 1.183
Correlation coefficient Duration
Depth 0.972
P1/12
P2/12   −0.177
P3/12   −0.481 0.681
P4/12   −0.258 0.31 0.292
P5/12   −0.161 −0.114 0.605 0.11
P6/12   0.326 −0.083 0.049 −0.903 0.197
P7/12   0.723 −0.151 −0.047 −0.633 0.318 0.831
P8/12   −0.256 −0.679 −0.521 −0.638 −0.214 0.302 −0.064
P9/12   −0.458 0.462 −0.109 −0.717 −0.103 −0.53 0.248
P10/12   0.006 −0.169 −0.606 0.494 −0.678 −0.788 −0.667 0.051 0.291
P11/12   0.076 −0.564 −0.838 0.183 −0.598 −0.555 −0.48 0.411 0.137 0.899
P12/12   −0.038 −0.522 −0.823 0.058 −0.684 −0.472 −0.523 0.544 0.32 0.861 0.975
RG4 (Toug-Dong gauge)
Mean  67.6 835.8 0.02 0.05 0.052 0.165 0.083 0.148 0.14 0.123 0.07 0.065 0.047 0.051
Standard deviation  38.495 653.412 0.016 0.041 0.04 0.13 0.04 0.082 0.057 0.073 0.032 0.055 0.051 0.043
Skewness coefficient  0.34 1.118 1.068 0.715 0.201 0.354 0.396 −0.485 −0.318 0.396 0.296 0.258 0.502 0.418
Kurtosis coefficient  1.237 2.669 2.277 1.973 1.637 1.349 2.326 1.851 1.203 1.85 2.007 1.672 1.371 1.286
Correlation coefficient Duration
Depth 0.904
P1/12
P2/12   0.944
P3/12   0.443 0.698
P4/12   0.037 −0.064 0.046
P5/12   0.935 0.964 0.651 0.086
P6/12   0.316 0.476 0.359 −0.853 0.388
P7/12   0.237 0.292 −0.009 −0.929 0.203 0.925
P8/12   −0.745 −0.615 −0.256 −0.51 −0.59 0.333 0.37
P9/12   −0.833 −0.689 −0.208 −0.453 −0.82 0.071 0.11 0.708
P10/12   −0.444 −0.634 −0.588 0.598 −0.612 −0.926 −0.746 −0.22 0.161
P11/12   −0.535 −0.717 −0.802 −0.008 −0.797 −0.508 −0.208 0.078 0.488 0.794
P12/12   −0.513 −0.751 −0.899 0.192 −0.748 −0.648 −0.327 0.05 0.296 0.865 0.949
(b) RG5 (An-Nei gauge)
Mean  67.6 355.7 0.019 0.025 0.041 0.188 0.107 0.146 0.233 0.133 0.04 0.121 0.018 0.017
Standard deviation  38.495 206.736 0.014 0.03 0.033 0.103 0.067 0.07 0.125 0.07 0.048 0.049 0.016 0.015
Skewness coefficient  0.34 −0.112 1.143 0.182 0.081 −0.46 0.212 0.36 0.483 1.327 1.133 0.895 0.563
Kurtosis coefficient  1.237 1.677 2.325 1.397 1.113 1.39 1.35 1.675 1.866 3.019 2.317 2.489 1.821
Correlation coefficient Duration
Depth 0.929
P1/12
P2/12   −0.033
P3/12   −0.182 0.438
P4/12   −0.221 0.497 0.043
P5/12   0.375 −0.354 0.264 0.16
P6/12   0.463 −0.004 0.703 −0.336 0.628
P7/12   0.303 −0.426 −0.356 −0.923 −0.383 0.045
P8/12   −0.65 −0.502 −0.129 −0.553 −0.446 −0.309 0.49
P9/12   −0.44 −0.302 −0.53 −0.454 −0.781 −0.659 0.619 0.821
P10/12   0.002 −0.05 −0.738 0.639 −0.034 −0.738 −0.361 −0.276 0.073
P11/12   −0.224 0.858 0.016 0.439 −0.728 −0.482 −0.222 −0.198 0.175 0.221
P12/12   −0.296 0.905 0.721 0.51 −0.178 0.15 −0.591 −0.339 −0.388 −0.244 0.671
RG7 (Nei-Bu gauge)
Mean  67.6 634.1 0.024 0.035 0.042 0.175 0.105 0.146 0.169 0.114 0.069 0.061 0.031 0.03
Standard deviation  38.495 375.854 0.038 0.028 0.035 0.144 0.062 0.088 0.08 0.099 0.056 0.051 0.035 0.029
Skewness coefficient  0.34 0.415 1.472 0.833 −0.062 0.257 −0.127 −0.166 1.078 1.006 0.408 0.23 0.503 0.393
Kurtosis coefficient  1.237 1.903 3.213 2.598 1.367 1.309 2.358 1.549 2.851 2.536 1.393 1.294 1.38 1.175
Correlation coefficient Duration
Depth 0.917
P1/12
P2/12   0.016
P3/12   0.123 0.617
P4/12   −0.336 0.545 −0.026
P5/12   0.097 0.21 0.658 0.263
P6/12   0.56 −0.112 0.578 −0.779 0.291
P7/12   0.92 0.072 0.25 −0.567 −0.072 0.728
P8/12   −0.41 −0.756 −0.455 −0.636 −0.512 0.087 −0.222
P9/12   −0.612 0.071 −0.232 −0.117 −0.745 −0.304 −0.322 0.552
P10/12   −0.434 −0.47 −0.938 0.252 −0.603 −0.786 −0.551 0.435 0.383
P11/12   −0.36 −0.647 −0.933 0.183 −0.422 −0.678 −0.537 0.46 0.161 0.958
P12/12   −0.467 −0.684 −0.913 0.071 −0.503 −0.629 −0.572 0.615 0.32 0.955 0.979
Statistics Rainfall durationRainfall depthDimensionless rainfall amount
P1/12P2/12P3/12P4/12P5/12P6/12P7/12P8/12P9/12P10/12P11/12P12/12
(a) RG2 (Zhong-Bu gauge)
Mean  67.6 533.6 0.029 0.028 0.052 0.167 0.103 0.166 0.158 0.097 0.074 0.056 0.028 0.043
Standard deviation  38.495 371.512 0.03 0.019 0.041 0.133 0.065 0.085 0.083 0.066 0.073 0.041 0.031 0.041
Skewness coefficient  0.34 0.615 0.808 1.068 0.175 0.222 0.293 −0.453 0.838 0.406 0.426 0.172 0.499 0.391
Kurtosis coefficient  1.237 1.714 2.304 2.58 1.412 1.339 2.439 1.676 2.304 1.748 1.222 1.754 1.372 1.183
Correlation coefficient Duration
Depth 0.972
P1/12
P2/12   −0.177
P3/12   −0.481 0.681
P4/12   −0.258 0.31 0.292
P5/12   −0.161 −0.114 0.605 0.11
P6/12   0.326 −0.083 0.049 −0.903 0.197
P7/12   0.723 −0.151 −0.047 −0.633 0.318 0.831
P8/12   −0.256 −0.679 −0.521 −0.638 −0.214 0.302 −0.064
P9/12   −0.458 0.462 −0.109 −0.717 −0.103 −0.53 0.248
P10/12   0.006 −0.169 −0.606 0.494 −0.678 −0.788 −0.667 0.051 0.291
P11/12   0.076 −0.564 −0.838 0.183 −0.598 −0.555 −0.48 0.411 0.137 0.899
P12/12   −0.038 −0.522 −0.823 0.058 −0.684 −0.472 −0.523 0.544 0.32 0.861 0.975
RG4 (Toug-Dong gauge)
Mean  67.6 835.8 0.02 0.05 0.052 0.165 0.083 0.148 0.14 0.123 0.07 0.065 0.047 0.051
Standard deviation  38.495 653.412 0.016 0.041 0.04 0.13 0.04 0.082 0.057 0.073 0.032 0.055 0.051 0.043
Skewness coefficient  0.34 1.118 1.068 0.715 0.201 0.354 0.396 −0.485 −0.318 0.396 0.296 0.258 0.502 0.418
Kurtosis coefficient  1.237 2.669 2.277 1.973 1.637 1.349 2.326 1.851 1.203 1.85 2.007 1.672 1.371 1.286
Correlation coefficient Duration
Depth 0.904
P1/12
P2/12   0.944
P3/12   0.443 0.698
P4/12   0.037 −0.064 0.046
P5/12   0.935 0.964 0.651 0.086
P6/12   0.316 0.476 0.359 −0.853 0.388
P7/12   0.237 0.292 −0.009 −0.929 0.203 0.925
P8/12   −0.745 −0.615 −0.256 −0.51 −0.59 0.333 0.37
P9/12   −0.833 −0.689 −0.208 −0.453 −0.82 0.071 0.11 0.708
P10/12   −0.444 −0.634 −0.588 0.598 −0.612 −0.926 −0.746 −0.22 0.161
P11/12   −0.535 −0.717 −0.802 −0.008 −0.797 −0.508 −0.208 0.078 0.488 0.794
P12/12   −0.513 −0.751 −0.899 0.192 −0.748 −0.648 −0.327 0.05 0.296 0.865 0.949
(b) RG5 (An-Nei gauge)
Mean  67.6 355.7 0.019 0.025 0.041 0.188 0.107 0.146 0.233 0.133 0.04 0.121 0.018 0.017
Standard deviation  38.495 206.736 0.014 0.03 0.033 0.103 0.067 0.07 0.125 0.07 0.048 0.049 0.016 0.015
Skewness coefficient  0.34 −0.112 1.143 0.182 0.081 −0.46 0.212 0.36 0.483 1.327 1.133 0.895 0.563
Kurtosis coefficient  1.237 1.677 2.325 1.397 1.113 1.39 1.35 1.675 1.866 3.019 2.317 2.489 1.821
Correlation coefficient Duration
Depth 0.929
P1/12
P2/12   −0.033
P3/12   −0.182 0.438
P4/12   −0.221 0.497 0.043
P5/12   0.375 −0.354 0.264 0.16
P6/12   0.463 −0.004 0.703 −0.336 0.628
P7/12   0.303 −0.426 −0.356 −0.923 −0.383 0.045
P8/12   −0.65 −0.502 −0.129 −0.553 −0.446 −0.309 0.49
P9/12   −0.44 −0.302 −0.53 −0.454 −0.781 −0.659 0.619 0.821
P10/12   0.002 −0.05 −0.738 0.639 −0.034 −0.738 −0.361 −0.276 0.073
P11/12   −0.224 0.858 0.016 0.439 −0.728 −0.482 −0.222 −0.198 0.175 0.221
P12/12   −0.296 0.905 0.721 0.51 −0.178 0.15 −0.591 −0.339 −0.388 −0.244 0.671
RG7 (Nei-Bu gauge)
Mean  67.6 634.1 0.024 0.035 0.042 0.175 0.105 0.146 0.169 0.114 0.069 0.061 0.031 0.03
Standard deviation  38.495 375.854 0.038 0.028 0.035 0.144 0.062 0.088 0.08 0.099 0.056 0.051 0.035 0.029
Skewness coefficient  0.34 0.415 1.472 0.833 −0.062 0.257 −0.127 −0.166 1.078 1.006 0.408 0.23 0.503 0.393
Kurtosis coefficient  1.237 1.903 3.213 2.598 1.367 1.309 2.358 1.549 2.851 2.536 1.393 1.294 1.38 1.175
Correlation coefficient Duration
Depth 0.917
P1/12
P2/12   0.016
P3/12   0.123 0.617
P4/12   −0.336 0.545 −0.026
P5/12   0.097 0.21 0.658 0.263
P6/12   0.56 −0.112 0.578 −0.779 0.291
P7/12   0.92 0.072 0.25 −0.567 −0.072 0.728
P8/12   −0.41 −0.756 −0.455 −0.636 −0.512 0.087 −0.222
P9/12   −0.612 0.071 −0.232 −0.117 −0.745 −0.304 −0.322 0.552
P10/12   −0.434 −0.47 −0.938 0.252 −0.603 −0.786 −0.551 0.435 0.383
P11/12   −0.36 −0.647 −0.933 0.183 −0.422 −0.678 −0.537 0.46 0.161 0.958
P12/12   −0.467 −0.684 −0.913 0.071 −0.503 −0.629 −0.572 0.615 0.32 0.955 0.979

The storm pattern can be classified into four types based on the time to peak dimensionless rainfall, i.e., the advanced type (the time to the maximum dimensionless rainfall is earlier than the median of a duration), central type (the time to the maximum dimensionless ratio approaches the median of a duration), uniform type associated with identical dimensionless rainfall, and delayed type (the time to the maximum dimensionless ratio is at a later time step than the median of a duration) (Wu et al. 2006). Figure 9 shows storm patterns estimated from five rainstorm events at the nine rain gauges. According to Figure 9, all four types of storm patterns probably occur in the Bazhang River watershed. For example, at the Xiao-Gong-Tian (RG1) gauge, the storm patterns of Event 1 and Event 2 can be regarded as an advanced type, and the pattern of Event 4 can be identified as the uniform type. In addition, the patterns of Event 3 and Event 5 can be treated as the delayed type and central type, respectively. Similar results can also be seen at the remaining rainfall gauges. This reveals that uncertainty exists in the storm pattern (distribution of rainfall in time) in the Bazhang River watershed.

In order to generate the rainfall characteristics, this study employs the multivariate Monte Carlo simulation method (Wu et al. 2006) to simultaneously produce the 200 sets of rainfall durations, depths, and dimensionless rainfalls of a storm pattern at nine rain gauges based on their correlation structure, as shown in Table 6. Note that since the multivariate Monte Carlo simulation method (Wu et al. 2006) can be used to simultaneously generate various correlated variables, this study treats rainfall characteristics at the nine rain gauges as the 18 variables for the rainfall duration and depth and 108 variables for the 12 dimensionless rainfalls.

Accordingly, the hyetographs at the nine gauges can also be produced simultaneously. Thus, the 200 generated rainfall characteristics at nine rain gauges are then synthesized into 200 rainstorm hyetographs. Eventually, the area average hyetograph could be calculated using the Thessian's areal weights as shown in Table 4 for six gauged sub-basins of interest. For example, Figure 10 shows the 1st, 50th, 100th 150th, and 200th simulated areal average hyetographs in Qing-Shui-Gang sub-basin. From Figure 10, it can be seen that the simulation of rainstorm events with various temporal resolutions can be carried out in this study, implying that the proposed PLT equation should take into account the uncertainty in the rainfall characteristics of interest.

### Development of SOBEK river routing model

In this study, the determination of the lag time is based on the definition of the time period from the maximum intensity to the peak water level, so a fluvial routing model is needed to estimate the hydrographs using the simulated hyetographs. This study uses the SOBEK river routing model to carry out the rainfall–rainfall process and channel routing in the Bazhang River watershed (see Figure 11). Figure 11 indicates the river channels and computation nodes in the SOBEK model are set up in accordance with the geographical and hydrological data in the Bazhang River watershed. The roughness coefficients for the various channels and overland are listed in Table 7 (WRAP 2009).
Table 7

Summary for roughness coefficients in SOBEK model for Bazhang River watershed (Braanc1: Chilan Creek and Branch 2: Touqian Creek)

River reachRoughness coefficient
Upstream bound of mean channel–Chu-Kou gauge 0.035
Chu-Kou gauge–Jun-Hui gauge 0.035
Upstream bound of Branch 1–Chang-Pan gauge 0.044
Upstream bound of Branch 2–Tou-Qian-Qi gauge 0.038
Jun-Hui gauge–Ba-Zhang-Qi gauge 0.039
Ba-Zhang-Qi gauge–Qing-Shui-Gang gauge 0.039
River reachRoughness coefficient
Upstream bound of mean channel–Chu-Kou gauge 0.035
Chu-Kou gauge–Jun-Hui gauge 0.035
Upstream bound of Branch 1–Chang-Pan gauge 0.044
Upstream bound of Branch 2–Tou-Qian-Qi gauge 0.038
Jun-Hui gauge–Ba-Zhang-Qi gauge 0.039
Ba-Zhang-Qi gauge–Qing-Shui-Gang gauge 0.039

In addition to the geometric data on the channel section and hydraulic parameters (e.g., roughness coefficient), the parameters of the RR model (SAC-SMA model) are required. In this study, the calibration of the SAC-SMA parameters is carried out by using the genetic algorithm based on the sensitivity of the model parameter (GA_SA). The six rainstorm events (i.e., typhoons) recorded from 2007 and 2008 as listed in Table 8 are adopted. The observed peak discharges of the six rainstorm events are between 224 and 1,280 m3/s.

Table 8

Rainstorm events used in the parameter calibration of SAC-SMA model

No. of eventRainstorm eventOccurrence periodRainfall depth (mm)Peak discharge (cm)
Sepat 20070818–20070821 644.0 643.6
Krosa 20071006–20070920 286.8 224.5
Haitang 20071006–20071008 993.2 1276
Kaimaegi 20080717–20080720 670.9 830.84
10 Fung-Wong 20080728–20080730 386.5 410.6
11 Jangmi 20080928–20080916 1,049.6 523.83
No. of eventRainstorm eventOccurrence periodRainfall depth (mm)Peak discharge (cm)
Sepat 20070818–20070821 644.0 643.6
Krosa 20071006–20070920 286.8 224.5
Haitang 20071006–20071008 993.2 1276
Kaimaegi 20080717–20080720 670.9 830.84
10 Fung-Wong 20080728–20080730 386.5 410.6
11 Jangmi 20080928–20080916 1,049.6 523.83

Table 9 represents the optimal parameters of the SAC-SMA model with the corresponding objective function value (Fobj = 83.2 m3/s). Moreover, Figure 12 shows the comparison of the estimated runoff for the six rainstorm events in the calibration of the SAC-SMA model. It can be observed that the computed runoff, accordingly, has a good agreement with the observed values, except for Event 1 (Typhoon Sepat) and Event 3 (Typhoon Haitang). On average, the error of the peak discharge (%) approximates 5.98%, and the root of the mean square error of runoff is about 6.72 m3/s. In addition, this study calculates the coefficient of Nash–Sutcliffe efficiency (CE) (Krause et al. 2005) is calculated to evaluate the accuracy of the calibrated optimal SAC-SMA parameters as:in which Qobs,i and Qest,i are estimated and observed discharge at the time step i; and serves as the average observed discharge. A CE value approaching one implies a perfect match of modeled discharge to the observed data. Given that the average CE is 0.88, it is concluded that the change in the estimated runoff resembles the observed values. In summary, the SAC-SMA model calibrated in this study can well simulate the RR characteristics in the Bazhang River watershed.
Table 9

Parameters of SAC-SMA RR modeling used in SOBEK model for Bazhang River watershed

Value 51.9 (mm) 102.7 (mm) 0.162 0.19 0.135 0.01 20.6 0.009 471.8 (mm)
ParameterLZFSMLZFPMLZSKLZPKPFREESIDERESERVSSOUTOBJ_Value
Value 23 (mm) 40 (mm) 0.043 (mm) 0.009 (mm) 0.063 0.0001 0.3 0.001 (mm/hr) 83.2 (m3/s)
Value 51.9 (mm) 102.7 (mm) 0.162 0.19 0.135 0.01 20.6 0.009 471.8 (mm)
ParameterLZFSMLZFPMLZSKLZPKPFREESIDERESERVSSOUTOBJ_Value
Value 23 (mm) 40 (mm) 0.043 (mm) 0.009 (mm) 0.063 0.0001 0.3 0.001 (mm/hr) 83.2 (m3/s)

### Derivation of PLT equation

According to the proposed framework outlined above, the simulation of the hydrographs along the Bazhang River can be carried out by using the SOBEK river routing model with the 200 generated hyetographs. The lag times at specific locations (i.e., water level gauges), where the rainfall, hydraulic, and geographical uncertainty factors are available, are extracted based on the definition of the lag time used in this study. Results from the above can be used to derive the lag time equation with uncertainty factors. Thus, the quantile relationship (i.e., probability function) of the lag time can be derived by the AFOSM method with the uncertainties in the rainfall, hydraulic, and geographical factors to be applied in the risk analysis for the lag time.

#### Estimation of lag time from simulation cases

Six water level gauges are located from the upstream and the downstream along the Bazhang River. Figure 13 shows the resulting 200 simulations of the lag time at the six water level gauges obtained by the SOBEK model with the 200 simulated rainstorm events. Note that some simulation cases where the maximum water levels might occur at the last time step during the rainstorm events should be incomplete hydrographs. Thus, these water levels might not be the true maximum values. Therefore, these simulation cases are excluded in the derivation of the lag time equation in this study. Moreover, in referring to Figure 13, the lag time for the Chu-Kou gauge ranges between 1 hour and 2 hours; whereas, the lag time for the Qing-Shui-Gang gauge mostly exceeds 4 hours. This implies that the lag time, on average, increases from the upstream to the downstream.

Table 10 contains the statistical properties of the lag time at six water level gauges calculated from the 200 simulations cases. The average lag time increases with the distance from the upstream. For example, the average lag time at the upstream gauge (Chu-Kou gauge) is 1.05 hours which is significantly less than the average at the downstream Qing-Shui-Gang gauge (5.02 hours). Similarly, the standard deviation of the lag time exhibits an increase with the distance from the upstream boundary. Specifically, the deviations of the lag time at the Qing-Shui-Gang gauge are significantly larger than the values at other gauges as a result of the impact of the tide on the water level.

Table 10

Mean and standard of lag times at specific water level gauges in Bazhang River watershed (WRAP 2009)

Sub-basinMean (hour)Standard deviation (hour)
Chu-Kou gauge 1.049 0.117
Jun-Hui gauge 1.677 0.423
Chang-Pan gauge 1.738 0.602
Tou-Qian-Qi gauge 1.707 0.790
Ba-Zhang-Qi gauge 3.069 0.489
Qing-Shui-Gan gauge 5.018 1.036
Sub-basinMean (hour)Standard deviation (hour)
Chu-Kou gauge 1.049 0.117
Jun-Hui gauge 1.677 0.423
Chang-Pan gauge 1.738 0.602
Tou-Qian-Qi gauge 1.707 0.790
Ba-Zhang-Qi gauge 3.069 0.489
Qing-Shui-Gan gauge 5.018 1.036

#### Derivation of deterministic lag time equation

A number of investigations (e.g., Askew 1970; Watt & Chow 1985; Yu et al. 2000; Creutin et al. 2013) have shown that the lag time equation is supposed to be a non-linear function. Therefore, in accordance with the proposed framework for deriving the PLT equation in this study, the relationship between the lag time tlag (hour) and uncertainty factors considered in this study are treated as a non-linear function as follows:

In Equation (7), the uncertainty factors of interest involve the rainfall factors, i.e., maximum rainfall intensity ip (mm/hour) and average rainfall intensity ia (mm), hydraulic factor (i.e., roughness coefficient n), and geographical factors, including the catchment area A (km2), river length from the upstream boundary L (km), and the centroid and Lca (km) to the estuary and the basin slope .

Using the results from the simulations of the lag time at six water level gauges under the generation of uncertainty factors, the lag time equation for the Bazhang River watershed is derived using the multivariate regression analysis as follows:The coefficient of determination for the above regression equation is 0.83. In addition, Figure 14 shows the average lag time estimated using Equation (8) has a good agreement with the result from the SOBEK model. This reveals that Equation (8) can reasonably describe the change in the lag time with the uncertainty factors of interest. However, although Equation (8) can provide reasonable results, this might lead to overfitting due to the seven uncertainty factors considered. To solve this problem, this study performs sensitivity analysis to identify the appropriate uncertainty factors.
In referring to Equation (8), it can be known that the exponent of the uncertainty factor is related to the sensitivity coefficient for an exponential equation (Wu et al. 2010). Since the derived lag time equation belongs to the exponential function, the coefficients can be used in the evaluation of the sensitivity of the dependent variable (i.e., lag time) to the independent variables (i.e., the uncertainty factors). This suggests that the larger coefficient of the uncertainty factor implies that this factor makes a significantly greater contribution to the estimation of the lag time. In addition, the positive coefficient means that the lag time should be positively related to a specific uncertainty factor; whereas the negative coefficient indicates that the lag time varies inversely with a specific factor. Regarding the rainfall factors maximum rainfall intensity (Ip) and the average rainfall intensity (Ia), the absolute value of the Ip coefficient (0.191) is slightly greater than Ia value (0.122). Therefore, the Ip is selected as the rainfall uncertainty factor. Moreover, only one hydraulic factor is selected, i.e., the roughness coefficient (n) as its coefficient is 4.5. Hence, the roughness coefficient should be an essential factor for the estimation of the lag time. With respect to the geographical factors , the and L coefficients (1.46 and 2.09) obviously exceed the A and Lca coefficient (0.959 and 0.0029). This reveals that only the river length L and slope of the watershed are re-treated as the geographical factors. As a result, for the Bazhang River, the lag time equation with the rainfall factor (ip), hydraulic factor (n), and geographical factors (L and ) is determined by using the multivariate regression analysis as follows:
In Equation (9), the coefficient of determination R2 is approximately 0.78 and Figure 15 shows that the average lag time estimated by the SOBEK approach the result from Equation (9).

From Equation (9), it can be seen that the coefficients of the river length (L) and the roughness coefficient (n) are positive, and there are negative coefficients of the basin slope and the maximum rainfall intensity. This indicates that the lag time increases positively with L and n, whereas reduced lag time is obtained with a steeper basin slope and large maximum rainfall intensity ip. This conclusion is consistent with the aforementioned discussion on the effect of the rainfall, the roughness coefficient of the river bed, the river length, and the basin slope on the flow speed, which strongly impacts the lag time. Although the lag time estimated using Equation (9) matches with the observed value worse than results from Equation (9), Equation (9) also can provide a reliable estimation of the lag time with given L, n, , and ip.

#### Derivation of PLT equation

Using the AFOSM with the relationship of the lag time with the uncertainty factors in Equation (9), the quantile relationship of the lag time (i.e., cumulative probability distribution) can be established under the consideration of uncertainties in the rainfall, hydraulic, and geographical factors. Figures 16 and 17 show the cumulative probability distribution of the lag time at the Chu-Kou gauge (river length L = 14.79 km and slope = 0.068) in association with various assumptions of the mean and variance of the maximum rainfall intensity ip and the roughness coefficient n. It can be seen that the quantile of the lag time at the specific cumulative probability reduces with increasing the ip mean, but it reduces with rising n means. However, the quantile relationship of the lag time is not sensitive to the CV. For example, as the ip mean rises from 10 to 100 mm, the quantile of the lag time significantly decreases from 1.6 hours to 1.0 hour for the cumulative probability being 0.9; however, as the n means increases from 0.01 to 0.06, the quantile of the lag time slightly rises from 1.1 hour to 1.3 hour. In general, a higher CV could lead to the wider confidence interval at a particular significant level (such as 95%). Nevertheless, the width of the 95% confidence interval approximately stays at a constant (about 0.5 hour) with varying CV of the maximum rainfall intensity and roughness coefficient.
Figure 16

Cumulative probability distribution of lag time at the Chu-Kou gauge under varying mean and CV of maximum rainfall intensity (mean of the roughness coefficient = 0.03 and its CV = 0.05). (a) Varying mean of the maximum rainfall intensity from 10 to 100 mm under consideration of its CV being 0.5. (b) Varying CV of maximum rainfall intensity from 0.1 to 0.9 under consideration of its mean being 50 mm.

Figure 16

Cumulative probability distribution of lag time at the Chu-Kou gauge under varying mean and CV of maximum rainfall intensity (mean of the roughness coefficient = 0.03 and its CV = 0.05). (a) Varying mean of the maximum rainfall intensity from 10 to 100 mm under consideration of its CV being 0.5. (b) Varying CV of maximum rainfall intensity from 0.1 to 0.9 under consideration of its mean being 50 mm.

Figure 17

Cumulative probability distribution of lag time at the Chu-Kou gauge under varying mean and CV of roughness coefficient (mean of the maximum rainfall intensity = 0.03 and its CV = 0.05). (a) Varying mean of the maximum rainfall intensity from 10 to 100 mm under consideration of its CV being 0.5. (b) Varying CV of maximum rainfall intensity from 0.1 to 0.9 under consideration of its mean being 50 mm.

Figure 17

Cumulative probability distribution of lag time at the Chu-Kou gauge under varying mean and CV of roughness coefficient (mean of the maximum rainfall intensity = 0.03 and its CV = 0.05). (a) Varying mean of the maximum rainfall intensity from 10 to 100 mm under consideration of its CV being 0.5. (b) Varying CV of maximum rainfall intensity from 0.1 to 0.9 under consideration of its mean being 50 mm.

Figure 18 shows the cumulative probability distribution for the lag time estimate at six water level gauges as the associated mean and CV of the maximum intensity are hypothesized as 40 mm and 0.65 mm, respectively, while the mean and CV for the roughness coefficient are assumed as 0.04 s/m1/3 and 0.15 s/m1/3, respectively. Accordingly, the 95% of confidence interval, i.e., the difference of lag times at the cumulative probability of 2.5% and 97.5%, increases with the distance from the upstream bound (i.e., river length). This implies that the variation in the lag time rises with the river length and it may be attributed to the more complicated influx characteristic and geographical property at downstream locations.

As a result, according to the above results from evaluating the effect of statistical properties of uncertainty factors on the lag time, especially for the mean value, the maximum rainfall intensity contributes more variation in the lag time than the roughness coefficient does. It is evident that the proposed PLT equation can be applied in quantifying the impact of the considered uncertainty factors to the lag time.

### Model application

In this section, the proposed PLT equation is used in the risk quantification of the lag time for Typhoon Matmo (2014/07/22 08:00–2014/07/24 08:00) as a case of model application. The lag times at six water level gauges are extracted based on the definition of the lag time considered in this study as being 1 hour (Chu-Kou gauge), 2.2 hours (Jun-Hui gauge), 1.2 hours (Tou-Qian-Qi gauge), 1.2 hours (Chang-Pan gauge), 4.3 hours (Ba-Zhang-Qi gauge), and 8.6 hours (Qing-Shui-Gang gauge).

As compared with the 95% confidence interval of the lag time computed by the proposed PLT equation as shown in Figure 19, the estimates of the lag time are located within the 95% confidence interval, except for the Ba-Zhang Qi and Qing-Shui-Gang gauges. This reveals that the proposed PLT can capture the observed lag time with high likelihood. However, the Ba-Zhang Qi and Qing-Shui-Gang gauges are located near the river outlet (i.e., the tidal watercourse). Thus, the associated water level might be influenced by the tide. In other words, the water levels at the Ba-Zhang-Qi and Qing-Shui-Gang gauges are deemed to result from the rainfall and tide. This may lead to the variation in the lag time corresponding to the temporal difference between the maximum rainfall intensity and the peak water level.
Through the proposed framework for the PLT equation, the estimated lag times at six water level gauges and their corresponding exceedance probabilities (i.e., underestimated risk) are shown in Figure 20. Figure 20 indicates that the estimated lag time with exceedance probability (i.e., underestimated risk), in which t* stands for the estimated lag time according to Equation (9), could reasonably reflect the difference between the estimated and observed lag times, except for the Ba-Zhang Qi and Qing-Shui-Gang gauges. For example, the estimated lag time (0.95 hours) at Chu Kou gauge in association with a high underestimated risk is less than the observed value (1 hour). In contrast, at the Chang-Pang gauge, the underestimated risk is approximately 0.47. Hence, there is a low probability that the corresponding estimated lag (1.8 hours) is greater than the observed one (1.2 hours). However, the difference between estimation and observation of the lag time at the Ba-Zhang-Qi and Qing-Shui-Gang gauges exceeds 3 hours, but their corresponding only reaches a value of 0.4. As mentioned earlier, the Qian Ba-Zhang-Qi and Qing-Shui-Gang gauges are located within the tidal watercourse; thus, the lag time might be impacted not only by the uncertainty factor considered in this study, but also by the tide.
Figure 20

Exceedance probability corresponding to the estimated lag time associated with the observed ones at six water level gauges for Typhoon Matmo.

Figure 20

Exceedance probability corresponding to the estimated lag time associated with the observed ones at six water level gauges for Typhoon Matmo.

In summary, a positive difference (tlag,obs-tlag,est) between the observed lag time (tlag,obs) and estimated one (tlag,est) is in association with the high overestimated risk calculated by using the proposed PLT equation. On the contrary, the negative difference (tlag,obs-tlag,est) corresponds to low underestimated risk . Therefore, using the proposed PLT equation, the lag time estimate can be provided and its underestimated risk can also be quantified. In addition, according to the unreasonable underestimated risk, i.e., the large difference of lag time in association with the exceedance probability , the uncertainty in the lag time can be identified.

## CONCLUSION

The purpose of this study is to develop a PLT equation to provide the lag time at specific locations along the river and its reliability by taking into account uncertainty factors, which include the rainfall, hydraulic, and geometrical factors. After sensitivity analysis, the rainfall factors including the average and maximum rainfall intensity and the roughness coefficient (i.e., Manning n coefficient) in the river serve as the hydraulic factor. Using the quantile relationship of the lag time estimated by the proposed PLT equation, the observed ones mostly locate within the 95% confidence interval when compared with observed lag times extracted from historical events in the study area (i.e., Bazhang River watershed). In addition, the underestimated risk of the estimated lag time by the proposed PLT equation can reasonably reflect the degree of difference between the observed lag time and estimated value. Therefore, the proposed PLT equation can provide useful information on quantifying the underestimated risk of the estimated lag time as compared with the observed value.

To enhance the generality of the proposed PLT equation, the PLT equation for other watersheds would be derived based on the proposed framework in the case of the geographical data and rainfall data being given. Moreover, hourly rainfall data are used in this study, but finer time resolution data would be required to estimate lag times in smaller sub-basins. In deriving the PLT equation, the only hydraulic factor used is the roughness coefficient of the river bed. However, other factors, such as the downstream boundary (i.e., the tide) and the initial soil moisture (Singh 1988) could be included as predictor variables. The parameters of the RR model might influence the accuracy and reliability of the estimated runoff and water level (e.g., Montanari & Brath 2004; Wu et al. 2010, 2011), and hence the RR model parameters should be considered as uncertainty factors. In addition to more uncertainty factors considered, numerous investigations (e.g., McCuen et al. 1984; Ward & Elliot 1995; Sadatinejad et al. 2012; Grimaldi et al. 2012) indicated that the lag time should be related to the time of concentration, which plays an important role in the control and management of flood. Thus, it is recommended that future research is undertaken to establish a probabilistic equation for the concentration time based on the proposed lag time equation. Eventually, by incorporating the PLT equation with the rainfall forecasts, the stochastic lag time forecast could be estimated for use in flooding warning operation.

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