Modeling probabilistic lag time equation in a watershed based on uncertainties in rainfall, hydraulic and geographical factors

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 (t_lag,2) is defined as the distance in time between the centroid of the hyetograph and runoff hydrograph, while the third lag time (t_lag,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, t_lag,2 and t_lag,3 are the most widely adopted (Talei & Chua 2012).

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).

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.

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).

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.

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.

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.

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.

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.

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.

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.

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.

In Equation (7), the uncertainty factors of interest involve the rainfall factors, i.e., maximum rainfall intensity i_p (mm/hour) and average rainfall intensity i_a (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 L_ca (km) to the estuary and the basin slope ⁠.

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 i_p. 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 i_p.

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.

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).

In summary, a positive difference (t_lag,obs-t_lag,est) between the observed lag time (t_lag,obs) and estimated one (t_lag,est) is in association with the high overestimated risk calculated by using the proposed PLT equation. On the contrary, the negative difference (t_lag,obs-t_lag,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.

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.