To obtain critical rainfall (CR) estimates similar to the rainfall value that causes minor basin outlet flooding, and to reduce the flash flood warning missed/false alarm rate, the effect of unit hydrographs (UHs) and rainfall hyetographs on computed threshold rainfall (TR) values was investigated. The Tanjia River basin which is a headwater subbasin of the Greater Huai River basin in China was selected as study basin. Xin’anjiang Model, with subbasins as computation units, was constructed, and time-variant distributed unit hydrographs (TVUHs) were used to route the channel network concentration. Calibrated Xin’anjiang Model was employed to derive the TVUHs and to obtain the maximum critical rainfall duration (Dmax) of the study basin. Initial soil moisture condition was represented by the antecedent precipitation index (Pa). Rainfall hyetographs characterized by linearly increasing, linearly decreasing, and uniform hyetographs were used. Different combinations of the three hyetographs and UHs including TVUHs and time-invariant unit hydrographs (TIVUHs) were utilized as input to the calibrated Xin’anjiang Model to compute the relationships between TR and Pa (TR-Pa curves) by using trial and error methodology. The computed TR-Pa curves reveal that, for given Pa and UH, the TR corresponding to linearly increasing hyetograph is the minimum one. So, the linearly increasing hyetograph is the optimum hyetograph type for estimating CR. In the linearly increasing hyetograph context, a comparison was performed between TR-Pa curves computed from different UHs. The results show that TR values for different TIVUHs are significantly different and the TR-Pa curve gradient of TVUHs is lower than that of TIVUHs. It is observed that CR corresponds to the combination of linearly increasing hyetograph and TVUHs. The relationship between CR and Pa (CR-Pa curves) and that between CR and duration (D) (CR-D curves) were computed. Warnings for 12 historical flood events were performed. Warning results show that the success rate was 91.67% and that the critical success index (CSI) was 0.91. It is concluded that the combination of linearly increasing hyetograph and TVUHs can provide the CR estimate similar to the minimum rainfall value necessary to cause flash flooding.
Flooding is the worst weather-related hazard, causing loss of life and excessive property damage [1–3]. In general, flash floods are characterized by their rapid onset, leaving very limited effective response opportunities [3–5]. Flood damage mitigation is provided through a variety of structural and nonstructural methods. A significant nonstructural method is the operation of flood warning systems .
Currently, three criteria are used for an expected flooding determination: critical discharge, critical runoff, and critical rainfall (CR). Critical rainfall criterion is used by most flood warning systems [3, 6–9].
Given an initial soil moisture condition and a rainfall duration (D), different hyetographs show the diverse areal rainfall volumes over the study basin necessary to cause minor basin outlet flooding which is defined as threshold rainfall (TR), and the minimum of these TR values is referred to as CR. That is to say, TR is a function of initial soil moisture condition, rainfall duration, and the form of rainfall hyetograph, but CR is a function of only initial soil moisture condition and rainfall duration.
By comparing real-time observed or predicted rainfall volume of a given duration to the CR value, the CR-based flood warning systems decide whether to issue a warning. For early warning, the consequences of under- or overestimating the CR value are extremely different. Adopting a CR value higher than the rainfall volume that actually produces flood damage leads to missing such events and failure to issue an alarm. Underestimating the CR may instead determine the issue of false alarms . For flood warning systems development, it is important to obtain as accurate as possible CR estimates. The most significant way to reduce missed/false alarm rate is to have the CR estimates be comparable to the minimum rainfall value necessary to cause flooding.
The false warning costs are commonly not only much lower than the avoidable flood loss, but also cannot match up to indirect and/or intangible flood damages such as serious injury or loss of life [11, 12]. So considering that an error will always be present, it is better to underpredict rather than overpredict the CR estimate for safety reasons .
Currently, there are three CR value computation methods: inverse, positive, and empirical. The Flash Flood Guidance (FFG) system  is representative of an inverse method. The computational process of FFG is divided into three steps. First, the critical discharge value is estimated. In the second step, threshold runoff estimates for various rainfall durations are obtained based on critical discharge and unit hydrograph (UH) peak, which belongs to runoff concentration computation of hydrology. In the third step, the FFGs are obtained based on threshold runoff values where the rainfall vs. runoff curves as a function of soil moisture conditions are needed , which belongs to runoff generation computation of hydrology. A significant disadvantage of FFG is that uniform rainfall over rainfall duration is presumed .
The empirical methods are based on historical rainfall and streamflow data . Miao et al.  proposed an empirical method to determine TR value by using a linear binary classification based on long-term historical rainfall and flood data. Enough flood event data are necessary to derive the binary classification. So this method cannot be implemented in ungauged basins.
The positive methods based on a watershed hydrological model estimate the TR values from critical discharge by trial and error [16, 17]. The hydrological responses of different cumulative rainfall values, for fixed duration, initial soil moisture condition, and hyetograph type, are simulated by calibrated watershed hydrological model. The cumulative rainfall value generating the critical discharge is taken as the TR estimate. The computation process of the positive method has an explicit hydrology theoretical basis, so the disadvantages inherent in the inverse method may be overcome. In this study, the calibrated Xin’anjiang Model with its subbasins being used as computation units  is employed to compute TR and CR values.
Montesarchio et al.  estimated TR values for the Mignone River cross section using an entropy-based decision approach and a simulation approach based on radar data and rain gauge data. Results show that the TR values computed using various methods are obviously different and that, for the fixed watershed hydrological model, the type of rainfall data source used for model calibration significantly affects the TR estimates.
According to hydrological rainfall-runoff formation theory, for a fixed computing method, the TR estimate is generally a function of initial soil moisture condition, rainfall duration, and hyetograph. The effect of initial soil moisture conditions has been taken into account in the vast majority of current methods [6, 7, 16, 17, 19, 20]. In , the rainfall vs. runoff curves were taken as a function of initial soil moisture content to take into account the effect of initial soil moisture conditions on TR estimates. In [3, 7, 16], initial soil moisture conditions were classified into antecedent moisture classes AMC I, AMC II, and AMC III, representing dry soil, moderately saturated soil, and wet soil, according to the total amount of accumulative rain. In , the initial soil moisture conditions were taken into account by imposing an initial discharge value of the watershed hydrological model, and the effect of different initial conditions is analyzed by varying the initial discharge in the model simulations. In , a probability distributed moisture model was used to estimate the soil moisture content as the initial soil moisture condition of a rainfall-runoff model employed to compute TR values. In , both AMC and antecedent precipitation index (API) were utilized to estimate the initial soil conditions.
Given an initial soil moisture condition, for the same rainfall volume, the hydrographs and peak discharge rates may be significantly different when different rainfall hyetographs are adopted [16, 17, 21]. Consequently, for the same initial soil moisture conditions, various hyetograph types result in different TR values [16, 17]. In [17, 21], three synthetic hyetograph types characterized by linearly increasing intensity, decreasing intensity, and linearly increasing-decreasing intensity were employed to investigate the effect of rainfall hyetograph type on the hydrograph at given rainfall volume. In , four standards hyetographs including step hyetograph, triangular increasing rate hyetograph, triangular decreasing rate hyetograph, and isosceles triangular hyetograph were used to analyze the effect of rain type on TR estimates. If the hyetograph type corresponding to the minimum rainfall necessary to cause flooding is found, it can be used to directly compute CR value. So determining the hyetograph type corresponding to the minimum rainfall to cause flooding is one objective of this work.
No matter whether an inverse or positive method is used, TR values are always computed by routing surface runoff using the UH method [1, 6, 17, 19, 22]. So deriving the UHs representing the true basin concentration characteristics is a key to calculating the CR estimate matching the minor rainfall value necessary to cause flooding. For more than 75 years since the inception of UH theory was presented by Sherman, it is still one of the most widely used methods for flood prediction and warning system development in gauged basins with observed rainfall and runoff data, but this data-driven traditional approach limits the UH derivation only to gauged watersheds. Synthetic UHs may only be used in basins whose hydrographs have a single peak [23–25]. Geomorphologic UHs, regardless of time-invariant (TIVUH) [26–31] or time-variant (TVUH) [32, 33], do not take the dynamic factor (flow velocity) spatial distribution into account. Distributed UHs based on a spatially distributed velocity field can adequately take the nonuniformity of basin characteristics into account [34–36]. Formulas defined as a function of rainfall intensity are adopted to compute spatially distributed velocity fields so as to derive TVUHs [18, 37, 38] that can solve to a certain extent the nonlinear problem of runoff concentration.
For a fixed runoff generation computing method, different rainfall-runoff transformation methods may lead to different TR estimates. By doing this work, investigating the effect of UHs on TR estimates and suggesting a reasonable UH used in computing CR value is another study objective.
In this study, three rainfall hyetograph types (linearly increasing, linearly decreasing, and uniform) and two UH types (TIVUHs and TVUHs) are used to investigate the effects of hyetographs and UHs on TR/CR estimates and warning results. The objectives are (1) to explain, for fixed duration and initial conditions, that rainfall hyetographs and UHs significantly affect the TR estimates; (2) to suggest that the rainfall hyetograph type leading to minimum TR estimate and the UH resulting in optimal simulation results should be adopted to compute CR estimates; and (3) to propose a method for CR computation.
Determining TR and CR value is a hydrological problem. The uncertainties of TR and CR estimates related principally to the method (including runoff generation and runoff concentration), parameters, data sources (including rainfall and discharge), and adopted rainfall hyetograph types [3, 16, 17, 21]. In this work, the method based on watershed hydrological model was used. For the fixed study watershed and data sources (observed data), in order to obtain the CR estimate approximate as far as possible with the minimum rainfall value necessary to cause flooding, the appropriate model (Xin’anjiang Model and UHs) and opportunely calibrated parameters were employed. For nonlinear types, no matter increasing or decreasing, the forms of rainfall hyetograph are innumerable. In this study, the linearly change hyetograph types were used. In fact, the rainfall hyetograph corresponding to the minimum rainfall value causing flooding is nonlinear.
This paper consists of six sections: Section 2 introduces the methods used in the study including the watershed hydrological model structure, TR and CR computation methods, and warning system assessment method. Section 3 describes the study basin and model calibration. Section 4 introduces the computation process and results of CR. In Section 5, the application of CR values in warnings historical flood event is performed. Finally, some conclusions are presented in Section 6.
2.1. Watershed Hydrological Model
2.1.1. Structure of Model
The Xin’anjiang Model is employed in this study and the basin is divided into a series of subbasins as computation units. Runoff generation and flow concentration computations are performed within the subbasins and the runoff from each subbasin is routed to the main basin outlet. The total hydrograph at the main basin outlet is equal to the sum of all subbasin hydrographs.
2.1.2. Rainfall Computation Component
The Kriging interpolation method is used to derive rainfall depth of all cells within subbasins from that of rainfall gauges. Mean areal rainfall values of each subbasin are computed by arithmetic mean method using rainfall depth for all cells within the subbasin.
2.1.3. Runoff Generation Computation Component
A tension water storage capacity curve, also named a parabolic curve, is used to represent the nonuniform distribution of tension water storage capacity and to compute the runoff value. The computational equations are as follows:where PE is the rainfall value, mm; R is the runoff value, mm; W is the mean areal tension water storage, mm; WM is the mean areal tension water storage capacity, mm; WMM is the maximum tension water storage capacity of the watershed, mm; and b is the tension water capacity distribution curve exponent (parabolic curve).
A free water storage capacity curve is used to represent the nonuniform distribution of free water storage capacity over runoff-producing areas and to separate runoff R into surface flow, interflow, and groundwater. Computational equations are as follows:where PE is the rainfall value (equal to excess rainfall R in the runoff-producing area), mm; RS, RI, and RG are surface flow, interflow, and groundwater, respectively, mm; S is the equivalent free water storage over the runoff-producing area, mm; SM is the area mean free water storage capacity (maximum possible deficit of free water storage), mm; EX is the free water storage capacity curve exponent (parabolic curve); SMM is the maximum watershed free water storage capacity, mm; FR is runoff-producing area; KI is the outflow coefficient of free water storage to interflow; and KG is the outflow coefficient of free water storage to groundwater.
Estimating initial soil moisture conditions is necessary for runoff generation computation. In this study, the antecedent precipitation index (Pa), used in the past as a moisture content indicator, is used and calculated at the beginning of every storm event as follows:where Pa,i is the antecedent precipitation index, mm; is daily rainfall, mm; i is day of estimate; and k is depletion constant (in this study, k = 0.83 ).
Because the tension water storage capacity curve is a parabolic type, the relationship between Pa and W is as follows:
2.1.4. Concentration Computation Components
(1). Hillside concentration. The surface flow passing directly into the channel systems is treated as TRS, and the interflow RI and groundwater RG are routed through linear reservoirs into channel systems as TRI and TRG. The computational equations are as follows:where TRS, TRI, and TRG are inflow into channel systems of surface flow, interflow, and groundwater, respectively, m³/s; TTR is the total inflow, m³/s; U is the transformation coefficient transforming runoff depth to discharge rate, , where F is the drainage area, km², and is computational time interval, h; CI is the recession constant of interflow storage; CG is the recession constant of groundwater storage.
(2). Channel network concentration. The channel network routing within a subbasin is represented by convolution of TTR(t) with a dimensionless UH as follows:where Q(t) is the subbasin outlet discharge rate, m³/s; UH is the ordinate of dimensionless unit hydrograph; and N is the number of dimensionless UH time intervals. The method presented by Kong et al.  is used to derive the TVUHs of each subbasin.
Given a velocity within a cell of DEM, the travel time through the cell is computed as follows:where τk is the travel time within cell k, s; L is cell size, m; and Vk is the velocity within cell k, m/s. The travel time of a cell to subbasin outlet is computed as follows:where Tj is the travel time of cell j to the subbasin outlet, s; m is the number of cells along the drainage network to subbasin outlet; τk is the travel time within cell k, s.
Based on the travel time of all cells to the subbasin outlet, the S-hydrograph of dimensionless UH can be obtained as follows:where St is the ordinate value of S-hydrograph at time t; F is the subbasin area, km²; Aj is the area of cell j, km²; and n is the number of cells whose T being less than or equal to t. The ordinate of dimensionless UH is computed as follows:where Δt is computational time interval, h.
It can be seen that the key to obtain UH is deriving the velocity of each cell. The fundamental equation form used to compute velocity is as follows:where k is a coefficient based on the flow type; Sorrell et al.  provide values of k for several flow types; and S is the flow path slope.
To take the effect of excess rainfall intensity on velocity into account, equation (17) is modified as follows:where i is the excess rainfall intensity, mm/h; i0 is the excess rainfall intensity corresponding to the coefficient k, determined by calibration; and c is a parameter, on the basis of literature [37, 38], with 0.4 being adopted.
Equation (18) can not only take into account the effect of excess rainfall intensity on velocity, but also can employ the values of k provided by Sorrell et al. .
By using equation (18), the time-variant spatially distributed velocity fields and TVUHs of different rainfall duration in a rainfall event are derived.
2.1.5. Channel Routing Method
The dynamic Muskingum method presented by Tewolde et al.  is used to route channel flow from subbasin outlets to the basin outlet.
2.2. CR Computation Method
2.2.1. TR Computation
TR estimate (mean areal rainfall value) depends on not only D and Pa, but also rainfall hyetograph and computation methods (runoff concentration). For a given Pa, the TR value of D is not unique. A primary goal of this study is to investigate the influence of rainfall hyetographs and runoff concentration computation methods on the TR estimates. Using the watershed hydrological model above, TR values are computed by trial and error.
For a given D (taken as an integer multiple of Δt), Pa, and a combination of hyetograph type and UH, computation steps of TR value are as follows: (1) D is divided into n time intervals; (2) according to the hyetograph, the proportion of each time interval rainfall to total rainfall of D (P) is determined; (3) trial and error: for a given P, running hydrological model to derive discharge hydrograph and comparing peak rate and critical discharge value. The P making the computed peak rate equal or close to critical discharge value is taken as the TR value.
2.2.2. CR Computation
CR is related only to D and Pa. For a given Pa and D combination, the minimum of all TR values is taken as the CR value. The relationship between CR and Pa of different D and the relationship between CR and D of different Pa are obtained, being the basis of flash flood warning.
2.3. Representation of Warning Results
The contingency table (Table 1) shows four possible results for a single flash flood warning, X denotes an event occurred and warning was issued (hits), Y denotes an event occurred but warning was not issued (missed events), Z denotes an event did not occur but warning was issued (false alarms), and W denotes an event did not occur and warning was not issued.