A framework for event-based flood scaling analysis by hydrological modeling in data-scarce regions

Flood scaling theory is important for flood predictions in data-scarce regions but is often applied to quantile-based floods that have no physical mechanisms. In this study, we propose a framework for flood prediction in data-scarce regions by event-based flood scaling. After analyzing the factors controlling the flood scaling, flood events are first simulated by a hydrological model with different areally averaged rainfall events and curve number (CN) values as inputs, and the peak discharge of each subcatchment is obtained. Then, the flood scaling is analyzed according to the simulated peak discharge and subcatchment area. Accordingly, the relationship curves between the scaling exponent and the two explanatory factors (rainfall intensity and CN) can be drawn. Assuming that the flood and the corresponding rainfall event have the same frequency, the scaling exponent with a specific flood frequency can be interpolated from these curves.


GRAPHICAL ABSTRACT INTRODUCTION
Flood scaling, which is extensively applied to flood prediction in ungauged and poorly gauged river basins of different sizes, is usually expressed by an equation that relates peak discharge to catchment descriptors such as the drainage area, slope of the watershed, land use, and rainfall characteristics; among these descriptors, the drainage area is the most widely used explanatory variable to predict flood peak discharge ( where Q is the peak discharge for a given drainage area A, α is the scaling intercept, and θ is the scaling exponent (Gupta et al. ; Medhi & Tripathi ). If a reference river basin with drainage area A 1 has a flood peak discharge Q 1 and θ is determined, then the flood peak Q 2 in an ungauged basin with drainage area A 2 can be estimated by the following equation: Q 1 /Q 2 ¼ (A 1 /A 2 ) θ .
Most previous studies focused on the relation between the quantile-based peak discharge and drainage area. The peak discharge shows simple scaling or multiscaling for snowmelt-generated or rainfall-generated floods, which means that the scaling parameters (especially the scaling exponent) remain invariant or vary, respectively, with the flood return period (Gupta & Dawdy ). This hypothesis has been tested in many river basins of different sizes from small-scale to mesoscale basins (Ogden &  Although the quantile-based scaling exponent is widely used for future flood prediction, the physical mechanism cannot be interpreted because the peak discharges of floods with the same return period in different catchments may not be generated by the same rainfall events (Liu et al. ). Therefore, Gupta et al. () proposed performing flood scaling research by using event-based floods for statistical analysis. As such, observed floods that occur simultaneously in different catchments can be selected, and the spatial and temporal distributions of rainfall can be considered. Furey & Gupta () presented a new index to reflect the temporal variability of excess rainfall, and the results showed that the scaling parameters depended mainly on the duration and amount of excess rainfall.
Together with statistical analysis, hydrological modeling can provide intuitive results regarding how meteorological and land surface factors impact flood scaling parameters (Yang et al. ). For simulated flood events, other controlling factors of the scaling parameters, such as the channel slope, flow velocity, and rainfall distribution, can be identified by changing the hydrological model parameters (Mantilla ; Venkata ). Therefore, it is better to use physically based distributed hydrological models that contain all the important parameters reflecting flood generation processes. For example, Ayalew et al. () applied the CUENCAS rainfall-runoff model to three river basins in Iowa to investigate how the rainfall intensity, duration, hillslope overland flow velocity, and channel flow velocity influenced the scaling parameters; they found that the key factors affecting the scaling parameters were the rainfall duration and overland flow velocity.
The main aims of this paper are to (1) perform a simulated flood scaling analysis and evaluate the effects of the rainfall characteristics and land surface and (2) propose a framework for estimating the scaling exponent by hydrological modeling. Compared with previous studies, the novelty of this study is that a framework for determining the flood

HEC-HMS model calibration and validation
The Hydrologic Engineering Center-Hydrologic Modeling

System (HEC-HMS) model developed by the US Army
Corps of Engineers is widely used for watershed hydrological simulations. This model has been verified to be applicable in semi-arid regions throughout China (Wang & Sun ). In addition, the Soil Conservation Service curve number (SCS-CN) method is often used to calculate the amount of generated runoff, and the SCS unit hydrograph method is used for flow routing. Therefore, we selected these two modules for the model. The CN value is deter-  (1) and (2), respectively, are used to evaluate the performance of the model.
where q i andq i are the observed and simulated discharges, respectively, and q i is the mean observed discharge of the flood event. If the value of the NSE coefficient is in the range of 0.6-1, the hydrological model is considered to perform well in simulating the flood event.

Flood scaling analysis
This study aims to analyze the flood scaling exponent in a data-scarce region. Therefore, the Zijingguan catchment with a drainage area of 1,760 km 2 (Figure 1), which has only one hydrological station (Zijingguan station), is selected. Scaling analysis is conducted by using the simulated flood peak discharges from 11 subcatchments. The HEC-HMS model, which is applicable in semi-humid and semi-arid regions, is employed to simulate flood events due to its data availability and structure (Alfy ). The simulated peak discharges and corresponding subcatchment areas are fitted with power functions, and then the scaling exponent is obtained for each flood event.
Framework for determining the scaling exponent based on simulated flood events Based on the simulated flood events and scaling analysis, a framework that links event-based scaling exponents to quantile-based scaling exponents can be proposed. There are two main assumptions: (1) the input rainfall at each rain gauge is spatially uniform and (2) the rainfall events have the same frequency as the generated flood events. The idea and procedures are as follows: 1. A typical rainfall event is selected from the observed data.
A typical rainfall event should have a large rainfall amount and high rainfall intensity. Generally, the largest rainfall event is selected for its rare occurrence as the low-frequency design flood. Then, using the relationship between the scaling exponent and rainfall intensity (total amount), the CN value is obtained, and the existence of simple scaling or multiscaling is concluded.
5. If the flood shows simple scaling, then the scaling exponent can be used in quantile-based flood scaling.
6. If the flood shows multiscaling, then the rainfall and the corresponding flood events are assumed to have the same frequency. The rainfall with a specific frequency is obtained from rainfall frequency analysis, and the corresponding scaling exponent is calculated according to the relationship established in step 4.
The framework is illustrated in Figure 2.

Flood event simulations
The Zijingguan catchment was divided into 11 subcatchments, and 17 outlets were selected with drainage areas ranging from 3 to 1,760 km 2 . Fourteen observed flood events were selected for HEC-HMS calibration and validation, including large floods (>5-year return period with a peak discharge greater than 500 m 3 /s) and small floods (<5-year return period with a peak discharge smaller than 500 m 3 /s). The model parameters were adjusted artificially to fit the observed flood hydrographs, and comparisons between the simulated and observed flood events are listed in Table 1 to estimate the model performance. Some simulated and observed flood events are shown in Figure 3.  Note: Fp is the flood peak, Fpo is the observed flood peak, and Fps is the simulated flood peak. Fd is the flood depth, Fdo is the observed flood depth, and Fds is the simulated flood depth.

Effects of rainfall and CN parameters on the scaling exponent
Due to the large drainage area of the Zijingguan catchment, rainfall is not spatially uniform. Figure 6 shows the spatial distributions of the two rainfall events. Each rainfall event was distributed nonuniformly over the whole catchment, especially the small rainfall events. This may be the reason for the small values of the fitted R 2 in the above section.

Flood scaling with areally averaged rainfall as the model input
To research the effects of the spatial rainfall distribution on the flood scaling properties, the areally averaged rainfall calculated by the Thiessen polygon method was input at each of the seven rain gauges. Then, the simulated flood peak discharge at each subcatchment was subjected to flood scaling analysis, as shown in Figure 7 and Table 3. and thus, the initial soil moisture content may be higher than that in the other subcatchments, leading to a higher runoff generation capacity and higher peak discharge. However, for large flood events, the peak discharge of W150 shows obvious scaling properties. This may be because the subcatchments are unsaturated during small flood events but become saturated for large flood events.

Relationships between the rainfall characteristics and the scaling exponent
On the basis of the above 14 scaling exponents θ simulated by the areally averaged rainfall, the relationships between the maximum 1 h rainfall (P 1 ) and θ and between the total rainfall (P) and θ were analyzed (Figure 9). The scaling exponent has no obvious relation with P 1 or P, which is uncommon. This may be because the rainfall intensity and total rainfall amount from different rainfall events have different temporal distributions. Therefore, the effects of the rainfall intensity on flood scaling need to be studied with the same spatial and temporal distributions.
Effects of rainfall with the same temporal distribution and CN value on the scaling exponent The rainfall event that occurred on 6 August 1963 was Linking the framework to the quantile-based scaling exponent Based on the above flood scaling analysis, flood predictions can be conducted by implementing a hydrological

DISCUSSION
Flood scalings in ungauged basins should be analyzed by analyzing homogeneous regions with sufficient flood data.
In watersheds possessing only one or a few hydrological stations, flood scaling analysis cannot be conducted by fitting the observed flood peak and watershed attributes.
Therefore, hydrological modeling is an approach for performing scaling research by simulating the flood processes in each subwatershed.
During the flood scaling analysis, the spatial rainfall distribution was found to be a controlling factor for the scaling exponent. Spatially uniform rainfall resulted in better fitting of the simulated peak discharge and watershed area and resulted in a different scaling exponent. Hence, the difference between the scaling exponent obtained from fitting the observed rainfall and that obtained from fitting the areally averaged rainfall cannot be neglected. In practice, we use the scaling exponent obtained from fitting the observed peak discharge. However, this may lead to a significant error.
Similarly, land surface conditions significantly influenced the scaling exponent. However, for different rainfall events, the relationship between θ and the maximum 1 h or total rainfall is not good. Since all 14 flood events were simulated by using the same CN value, the temporal rainfall distribution must have a significant influence on θ. Furey & Gupta () presented a variable to represent the effective rainfall duration and analyzed the relationship with the scaling exponent. Therefore, the influence of the temporal rainfall distribution needs further research.  In addition, the simulated peak discharge of each subcatchment was used for flood scaling analysis. Thus, the result must be affected by the peak discharge simulation error. In this catchment, the peak discharge simulation errors ranged from À1.98 to 19.82%, with the largest simulation error obtained for the event that occurred on 6 August 1963. However, this flood event was selected to illustrate the multiscaling analysis framework. If the result is applied to a data-scarce region, the simulation accuracy should be improved further.
Gupta & Dawdy () noted that floods caused by rainstorms exhibited multiscaling, and the results of our study further verify their statements. However, our results were obtained in semi-humid and semi-arid regions with multiscaling, whereas in humid regions, the initial CN value is large, and simple scaling may be inferred from the findings in this study.

CONCLUSIONS
In this study, we proposed a method to conduct flood scaling analysis by hydrological modeling in data-scarce regions. The scaling exponents were very different between using the observed rainfall and the areally averaged rainfall as model inputs, and the coefficient of determination was larger for spatially uniform rainfall. Therefore, the spatial rainfall distribution has a significant impact on flood scaling.
Based on how the spatial rainfall distribution and CN values influence the scaling exponent, a framework was proposed to estimate the scaling exponent in a data-scarce region. We recommend that the areally averaged rainfall of an event be used as the input to the hydrological model; accordingly, the correlation curves relating the scaling exponent, rainfall intensity, and CN values can be drawn. As long as the flood frequency and CN value are determined, the scaling exponent can be obtained from these correlation curves.