## Abstract

The traditional instantaneous unit hydrograph (IUH) is very useful for theoretical analysis and practical forecasting of floods owing to its linear assumptions. Although various revised methods to overcome the unphysical assumptions have been proposed, it is still difficult to obtain efficiently a nonlinear IUH of diverse rainfall excess intensities in a watershed. In this study, we proposed practical and physical interpolation techniques to derive new IUHs from at least two existing IUHs corresponding to diverse rainfall excess intensities in a watershed. To interpolate the new IUHs, mass conservation law and power–law relationships between rainfall excess intensities and the peak flow and time to peak of IUHs were used. By employing convolution integration, surface rainfall–runoff hydrographs for timely varying rainfall events were derived. For verification, we applied the proposed technique to three real watersheds with different sizes ranging from 0.036 to 1,047 km^{2}. All flood prediction procedures were completed instantly, stably and the prediction results showed the accuracy of Nash–Sutcliffe efficiency (*NSE*) = 0.55–0.93 and coefficient of determination (*R*^{2}) = 0.72–0.94.

## HIGHLIGHTS

A new instantaneous, physical, and stable rainfall–runoff prediction technique based on IUH is proposed.

A power–law relationship between peak flow and time to peak of IUH and rainfall excess intensity is employed.

## NOTATIONS

watershed area

wetted cross-sectional area

constant [−]

curve number [−]

cumulative infiltration

infiltration rate

acceleration due to gravity

time-varying rainfall excess intensity in a watershed

representative rainfall excess intensity

arbitrary rainfall excess intensity

static moment of the wetted area

variation in the static moment along the distance

initial abstraction (initial loss)

hydraulic conductivity

Manning's roughness coefficient

Nash–Sutcliffe efficiency, [−]

accumulated rainfall

wetted cross-sectional perimeter

accumulated rainfall excess

direct surface runoff

discharge at

observed discharge

predicted discharge

correlation coefficient [−]

coefficient of determination [−]

hydraulic radius

effective saturation [−]

friction slope [−]

representative

*S*-hydrographbottom slope [−]

potential maximum retention

lateral discharge per unit distance

time

characteristic time

time to peak of

time to peak of

representative IUH

peak flow of

new IUH

peak flow of

distance along the channel

exponent of power function [−]

exponent of power function [−]

coefficient of power function

porosity [−]

exponent of power function [−]

initial abstraction ratio [−]

exponent of power function [−]

wetting front soil head

time variable of the integration

moisture content

effective porosity [−]

initial moisture content [−]

residual moisture content [−]

change in the soil moisture content [−]

coefficient of power function

## INTRODUCTION

Accurate and efficient flood forecasting techniques are indispensable to human society. Thus, various flood prediction models with three typical forms of conceptual, empirical, and physical groups have been developed. Empirical rainfall–runoff models are based on observation, usually without searching for the detailed physical processes during rainfall–runoff events. A common first step to develop an empirical model is to observe input and output data, that is, the rainfall on a watershed and discharge at the watershed outlet, respectively. Next, a relationship between the input and output data is established. Finally, the rainfall–runoff discharge for the input rainfall can be predicted. Since empirical models are based on observation, they are inherently effective under the same conditions in which they were developed. One of the most famous and widely used empirical models is the unit hydrograph proposed by Sherman (1932) and various modified versions such as synthetic unit hydrographs (UHs) (Snyder 1938; Clark 1945; SCS 1972) and numerically calculated UHs (Bellos & Tsakiris 2016) have been proposed.

Conceptual rainfall–runoff models (e.g., Sugawara & Funiyuki 1956; Nash 1957) are based on pertinent physics and use conceptualized mathematical expressions for rainfall–runoff events. The conceptual models are usually lumped types and their prediction accuracy is strongly dependent on observed data due to parameter calibration. Instantaneous unit hydrograph (IUH) models such as conceptual IUHs (Nash 1957; Dooge 1959), geomorphological IUHs (GIUHs) (Rodriguez-Iturbe & Valdés 1979; Gupta *et al.* 1980), width function-based IUHs (WFIUHs) (Naden 1992; Kumar *et al.* 2007; Grimaldi *et al.* 2012), and the kinematic wave based GIUH (KW-GIUH) (Lee & Yen 1997) have been developed and are used for rainfall–runoff prediction. These models are fully or partially conceptual and also dependent on refined rainfall–runoff observed data for calibration.

Physically based rainfall–runoff models are based on the physics of fluid flow, such as conservation of mass and momentum, to describe the rainfall–runoff processes. Various kinematic (Tsai 2003; Nguyena *et al.* 2016), diffusive (Jain *et al.* 2005; Park *et al.* 2019), and dynamic (Mignot *et al.* 2006; Kim & Seo 2013; Fernández-Pato *et al.* 2016) wave models have been developed and successful results have been reported. Theoretically, physically based models can be applied to almost any kind of rainfall–runoff process (Bellos & Tsakiris 2016; Yu & Duan 2017; Bellos *et al.* 2020; Costabile *et al.* 2021; Barbero *et al.* 2022; Zhu *et al.* 2022).

In the past decades, physically based models have not been practically suitable for flood forecasting in real world applications due to their numerical instability and high computational cost (Kim *et al.* 2012; Xia *et al.* 2017; Lu *et al.* 2018). However, several methods have been proposed recently to solve these issues. To increase the computational speed, various parallel computing techniques using MPI, OpenMP, and GPU were proposed, and approximately times speedup has been reported (Park *et al.* 2019; Xia *et al.* 2019; Ming *et al.* 2020; Buttinger-Kreuzhuber *et al.* 2022). In particular, GPU-accelerated hydrodynamic models were able to predict floods in real time or with 26–36 of lead time (Ming *et al.* 2020; Schubert *et al.* 2022). In addition, various techniques, like a fully implicit algorithm for stiff friction source terms (Zhao & Liang 2022), were developed to prevent numerical stability. Costabile & Costanzo (2021) proposed a heuristic procedure for non-uniform grid generation based on the river network to decrease the computational cost. Besides, García-Alén *et al.* (2022) suggested that the vertical accuracy of bathymetry may be more important than the vertical grid resolution.

Despite the significant progress made in resolving the issues associated with physically based models as noted above, conceptual models are still necessary for practical and instantaneous forecasting purposes. For example, only spatially averaged watershed data appropriate for lumped models are often available. Therefore, in some cases, lumped models like the IUH model may demonstrate better performance and reliability than physically based models in predicting rainfall–runoff (Sitterson *et al.* 2017; Vilaseca *et al.* 2021). Moreover, while the efficiency and stability of physically based models have improved recently, lumped models or conceptual models are still faster and more stable from a computational perspective. In addition, the physically based models require calibration for several parameters, as well as the empirical and conceptual models. That is, the physically based models also require measured flood and rainfall data. Due to the limitations we described, physically based models do not always produce satisfactory results and sometimes need to be supported by another rainfall–runoff model (Freire Diogo & Antunes do Carmo 2019).

Recently, machine learning techniques such as artificial neural network and deep learning have been extensively studied for rainfall–runoff modeling (Van *et al.* 2020; Ha *et al.* 2021; Frame *et al.* 2022) and their prediction accuracy is inherently dependent on the quality and quantity of measured data.

Each type of flood forecasting model mentioned above has its own strengths and weaknesses, but one common challenge is to secure sufficient, reliable observed flood data. Unfortunately, floods happen rarely and it is never easy to obtain sufficient, reliable rainfall–runoff processes data under heavy rainfall conditions. Therefore, it will be very useful if a technique to overcome the limited number of data sets is proposed.

*et al.*(1995), Ding (1974, 2011), and Paik & Kumar (2004) reported the nonlinear watershed response to rainfall excess intensity using measured data. In particular, Minshall (1960) first described that the peak flow and time to peak of hydrographs were power functions of rainfall excess intensity, as shown in Figure 1. Later, Ding (1974, 2011) and Paik & Kumar (2004) demonstrated that the IUH of a watershed is a function of the rainfall excess intensity. In turn, the peak flow and time to the peak are power functions of rainfall excess intensity, as shown in Figure 1. Additionally, although not based on measured data, the KW-GIUH in Lee & Yen (1997) and the IUHs derived using dynamic wave simulations (DIUH) in Jeong

*et al.*(2021) also resulted in the power–law relationships as shown in Figure 1.

The previous research results presented in Figure 1 clearly show that the peak and time to peak of IUH are power functions of the rainfall excess intensity. Notably, it was found that the power–law relationship can be derived from the dynamic wave modeling results (Jeong *et al.* 2021). Building upon these findings, the focus of this study extends beyond the peak point of IUH to consider the entire profile of IUH. The primary objective of this study is to propose an efficient technique that utilizes the nonlinear relationships depicted in Figure 1, to interpolate new IUHs for arbitrary rainfall excess intensities from existing IUHs. The rest of the sections are organized as follows: first, an interpolation technique to derive new IUHs from existing IUHs is presented. The generation of rainfall–runoff hydrographs is then described. Finally, the proposed technique is applied to real watersheds and the results are discussed.

## POWER–LAW-BASED IUH INTERPOLATION TECHNIQUE

### Interpolation of IUH

Although the above derivation procedures for the interpolation were expressed only for IUH, the interpolation Equations (1), (2), and (8) can be applied to UH interpolation by dividing the ordinate value of UH with rainfall depth or volume.

## VERIFICATION

*i*= 3, 50, 50, 4, and 50 mm/h , respectively, as shown in Figure 3. Finally, we transform the representative hydrographs using Equation (8) to the IUHs and UH as shown in Figure 3. To test the performance of the proposed interpolation technique, we compared the original and interpolated IUHs quantitatively as follows:where the correlation value between and ranges from and the and are strongly correlated as the

_{k}*r*approaches one. For the tested cases in Figure 3, very close agreements were achieved with . More interpolated IUHs for are provided in Supplementary Material, Figure S1, where all .

In addition to the quantitative evaluations, we could observe the nonlinearity in the watersheds of which areas are as shown in Figures 1 and 3, Robinson *et al.* (1995) demonstrated that this nonlinearity might be independent of watershed size. Therefore, we can expect that the proposed method will produce almost the same IUH with the existing one, regardless of the watershed areas.

## FLOOD PREDICTION

### Rainfall–runoff prediction

*i*is the time-varying rainfall excess intensity in a watershed.

*et al.*(2015). The performance was considered to be very good for or , good for or and generally satisfactory for or . The range of 1.0–0 was regarded as acceptable.

### Application to the Lucky Hills 103 watershed

*n*, of the entire watershed is (Kim

*et al.*2013). For the flood prediction test, we selected two observed rainfall–runoff events. The rainfall durations were approximately 100 and 200 , respectively, and the maximum rainfall excess intensities were approximately 120 and 100 , respectively.

*et al.*(2021), where

*S*-hydrographs were first generated by simulating a two-dimensional dynamic wave model, then the

*S*-hydrographs were differentiated by time to derive IUHs. As shown in Figure 5, during a rainfall event, the rainfall excess intensity continuously varies. That is, we need many IUHs for the various rainfall excess intensities. Consequently, it requires a huge computational cost if we adopt the DIUH method to generate all IUHs for the diverse rainfall excess intensities observed during the rainfall events. On the other hand, the present interpolation technique can generate a number of IUHs instantly only if there are at least two existing IUHs.

*K*is the hydraulic conductivity. Referring to Kim

*et al.*(2013) and Jeong

*et al.*(2021), we used , , , , and for the entire watershed.

Figure 5 shows the predicted flood discharges. For the case in Figure 5(a), the and . For the case in Figure 5(b), the and , respectively. Considering the evaluation criteria of Moriasi *et al.* (2015), the proposed technique showed generally satisfactory and good performance, respectively.

### Application to the Keelung River watershed

Figure 7 shows the comparison between the discharges by measurements and the proposed technique. For the case in Figure 7(a), the and are 0.94 and 0.93, respectively. For the case in Figure 7(b), the and are 0.89 and 0.83, respectively. Thus, the proposed technique showed very good performance.

### Application to the Naerin River watershed

#### Dividing watersheds considering non-uniform rainfall distribution

^{2}and the Naerin River flows through the watershed as shown in Figure 8. The average topography elevation is 724.6 and average watershed slope is 0.265, which is relatively steep. The riverbed slope is relatively gentle, approximately 1/285. Agricultural land (9% of the total watershed area) is located along the riverside and most of the remaining area is forest. The average annual rainfall varies from 907 to 1,294 mm. Approximately 70–80% of the annual rainfall precipitates during the rainy season (June–September) in the watershed.

Naturally, due to the large area, it is not reasonable to assume that the spatial rainfall distribution is uniform. Thus, we divided the Naerin River watershed into 14 sub-basins, as shown in Figure 8. The average rainfall excess intensities of each sub-basin were calculated using the Thiessen polygon method (Thiessen 1911).

#### IUH derivation

*et al.*2007). The computational domains of each basin were composed of 848–28,127 nodes and 1,668–56,199 cells, and the computational time step was 0.1 s. Thus, it took a very long time for the generation of each ‘existing’ IUH using the DIUH method. To estimate

*n*of the sub-basins (Supplementary Material, Figure S2(a)), a parameter of the dynamic wave model, we used a land cover map of 5 m resolution (Supplementary Material, Figure S2(b)) supplied by the Ministry of Environment of the Republic of Korea (2013) and the land cover type in Supplementary Material, Table S2 (Vieux 2004). Figure 9 shows the existing and interpolated IUHs derived by the DIUH method and proposed technique. More detailed results are presented in Supplementary Material, Figures S3–S6.

#### Runoff prediction

*et al.*(2015) for various Korean watersheds, we adopted in this case. is the potential maximum retention and is related with the curve number as follows:

Since the value of a sub-basin can vary temporally depending on the antecedent rainfall (USDA-NRCS 2004) and the discharge was observed only at the entire watershed outlet, we calibrated the values by considering the total rainfall–runoff volume observed at the watershed outlet. The calibrated and values are listed in Supplementary Material, Table S3.

*g*is the acceleration due to gravity. is the static moment of the wetted area. is the variation in the static moment along the distance. is the bottom slope. is the friction slope. To solve the governing equations, we used the finite difference scheme of MacCormack (1969). We used the channel geometry data provided by the Ministry of Land Infrastructure and Transport of the Republic of Korea (2019) and assumed for the entire channel. The water surface level at the watershed outlet was used for the downstream boundary condition and the hydrographs of sub-basins were used for the lateral inflow into the main river channel shown in Figure 8.

Event . | 1 . | 2 . | 3 . | 4 . | 5 . | 6 . | 7 . | 8 . | 9 . |
---|---|---|---|---|---|---|---|---|---|

0.59 | 0.76 | 0.79 | 0.77 | 0.69 | 0.82 | 0.64 | 0.67 | 0.55 | |

0.72 | 0.84 | 0.90 | 0.79 | 0.85 | 0.88 | 0.83 | 0.81 | 0.83 |

Event . | 1 . | 2 . | 3 . | 4 . | 5 . | 6 . | 7 . | 8 . | 9 . |
---|---|---|---|---|---|---|---|---|---|

0.59 | 0.76 | 0.79 | 0.77 | 0.69 | 0.82 | 0.64 | 0.67 | 0.55 | |

0.72 | 0.84 | 0.90 | 0.79 | 0.85 | 0.88 | 0.83 | 0.81 | 0.83 |

## DISCUSSION

### Non-identical shapes of existing IUHs

This limitation can be partially solved by choosing multiple existing IUHs with different rainfall excess intensities. For example, if five of the and are known and we need IUHs for , the existing IUH of can be used to generate new IUHs for , and the IUH of can be used to generate new IUHs for , respectively. Then, the interpolated IUHs will be more similar to the original shape of existing IUHs than if only an existing IUH is used.

### Power–law relationship between IUH peaks and bottom roughness

The bottom roughness of a natural watershed spatiotemporally varies depending on the vegetation, land use, size of the raindrops, and microtopography (Li & Shen 1973; Liu & Singh 2004). These variations can change the speed and depth of the overland flow, which in turn can change the shape of the IUH of a watershed.

*n*using the DIUH method. As shown in Figure 13, the peak flow and time to peak of the IUHs are power functions of both and

*n*. In Figure 13(c) and 13(d), the exponents of the power functions are −0.6 and 0.6 because we adopted Manning's formula (Manning 1891). Thus, the peak flow and time to peak times of the IUHs can be expressed as power functions of both and

*n*as follows:where and are arbitrary constants. Therefore, the proposed technique can be applied to watersheds where the bottom roughness varies with time during the rainfall event using Equation (20).

## CONCLUDING REMARKS

In this study, we developed an instantaneous and physical rainfall–runoff prediction technique. IUHs corresponding to diverse rainfall excess intensities in a watershed could be derived instantaneously and physically using the mass conservation law and the power–law relationship between the rainfall excess intensity and peak flow and time to peak of IUH. The new interpolated IUHs were very similar to existing IUHs, with for all test cases. Surface rainfall–runoff hydrographs could be instantly and stably derived using a convolutional integration. The proposed technique was applied to three real watersheds with an area of 1,047 km^{2}. It was verified that all flood prediction procedures could be instantly and stably completed. Reasonable flood prediction accuracy was achieved with = 0.55–0.93 and . Therefore, the power–law relationship, which has been reported over the past decades, can be used for IUH interpolation and runoff prediction.

Although the efficiency and accuracy of the proposed technique were acceptable at least for the test cases, inherent limitations originating from the interpolation-based technique are apparent. All shapes of the newly derived IUHs by the proposed technique are identical if we use only an existing IUH. This problem can be partially solved by using multiple existing IUHs for various rainfall excess intensities. In addition, all physics missed in the existing IUH must be ignored in the IUH and flood predicted by the proposed technique.

## ACKNOWLEDGEMENT

This work was supported by the National Research Foundation of Korea (NRF) grant funded by the Korea government (MSIT) (RS-2022-00165287).

## DATA AVAILABILITY STATEMENT

All relevant data are included in the paper or its Supplementary Information.

## CONFLICT OF INTEREST

The authors declare there is no conflict.

## REFERENCES

*A Basic Study of the Linearity of the Rainfall-Runoff Process in Watersheds*

*Republic of Korea 37°39'50.39"N*,

*127°58'42.45"E, eye altitude 1868.77 km.*Google Maps Data layer. Available from: http://www.google.com/earth/index.html (accessed 2 July 2022)

*Taiwan 23°41'52.12"N*,

*120°57'37.85"E, eye altitude 1096.91 km.*Google Maps Data layer. Available from: http://www.google.com/earth/index.html (accessed 2 July 2022)

*.*

Arizona, USA 34°02'56.14"N,111°05'37.43"W, eye altitude 990.64 km.Google Maps Data layer. Available from: http://www.google.com/earth/index.html (accessed 2 July 2022)