Development of Flash Flood Forecasting System for Small and Medium‐Sized Rivers

Abstract

ABSTRACT

Owing to the increased frequency of short‐duration extreme rainfall events caused by climate change, peak flood flows are expected to increase substantially in small and medium‐sized rivers (SMRs) with a short time of concentration for a catchment (Tc). Accurate flood forecasts and corresponding evacuation are effective in reducing the number of casualties caused by flash floods in SMRs. Currently, flood forecasting using observed rainfall in SMRs has a short lead time, which often delays the issuance of evacuation orders by local governments. Moreover, the large number of SMRs necessitates a system that can be widely used by local governments for disaster response tasks, such as issuing evacuation orders. Therefore, we developed a system that can accurately predict when river water levels will reach the Flood Risk Level (FRL). This forecasting approach uses the rainfall–runoff–inundation (RRI) model and the H–Q equation. The parameters in the RRI model were optimized using the Shuffled Complex Evolution algorithm developed at the University of Arizona (SCE‐UA) to reduce the required time and effort. The system uses real‐time water level observation data to sequentially modify the basin state quantities in the RRI model using the particle filter method to improve the water level forecast accuracy. The system was implemented in 200 rivers in Japan with diverse rainfall and geological characteristics and was tested during the flood season. Accuracy verification was conducted when the forecasted water levels were operated within a range of ± 50 cm. The results showed that 75% of the flood events could be forecasted more than 2 h before reaching the FRLs. Furthermore, 89% of the flood events could be predicted with a lead time (LT; time that water levels reach the FRL—time of first forecast) of 2 h or more or a lead time equal to the Tc or more. These findings show that this system has the potential to enhance and strengthen flood warning and evacuation systems.

Introduction

Climate change will increase the frequency of short-duration extreme rainfall events (Fowler et al. 2021; Ministry of Land, Infrastructure, Transport and Tourism [MLIT] 2021). In small and medium-sized rivers (SMRs) with short Tc, large increases in peak flood flows are expected according to the rational equation principle that peak flood flows are determined by the maximum rainfall during Tc. In Japan, torrential rains in Hokkaido and Tohoku in 2016 (MLIT 2017) and northern Kyushu in 2017 (MLIT 2018a) caused extensive damage owing to flash floods in several SMRs. Floods in Germany and Belgium in 2021 (International Fire Service Information Center [IFSIC] 2022) and floods in northern Italy in 2023 (Ghiglione and Bettiza 2023) can also be considered flood damage caused by SMRs. To reduce the number of casualties caused by flooding in SMRs, it is important to make accurate flood forecasts and issue corresponding evacuation orders.

Numerical forecasting of localized heavy rainfall still faces several challenges, making it difficult to provide highly accurate information. Therefore, flood forecasting systems should use observed rainfall while considering the uncertainty of the forecasted rainfall. The issue with flood forecasting using observed rainfall is that SMRs have a short Tc, which requires that the forecast calculation time be kept within a few minutes to effectively utilize this limited lead time.

Moreover, if the number of SMRs to be covered is extremely large, the system needs to be developed for widespread use because local governments must use this for disaster response actions, such as evacuation orders. Although many flood forecasting models have been developed (e.g., Kim et al. 2007; Sayama et al. 2008; Tachikawa et al. 2011), none can address these issues altogether.

Therefore, this study aimed to assist local governments in issuing evacuation orders appropriately to ensure the time necessary for evacuation in the event of SMR flooding and live up to the trust of local residents. To achieve this goal, we developed a system that can provide accurate flood forecast information with a lead time of at least 2 h before the river water level reaches the Flood Risk Level (FRL), which is the standard water level for issuing evacuation orders.

There are many challenges in developing such a system, typically: (1) rapid assimilation of multi-source rainfall, (2) reasonable selection of models for site-specific cases, (3) practical and accurate terrain modeling, (4) rational resolution of parameterization for areas lacking rainfall or discharge records, and (5) uncertainty identification and processing.

Rapid Assimilation of Multi-Source Rainfall

The Japan Meteorological Agency (JMA) Radar/Raingauge-Analyzed Precipitation (R/A) was used as the rainfall product input for the model. R/A is a rainfall product generated by correcting weather radar observation data owned by the JMA and MLIT with ground rain gage data owned by the JMA, MLIT, and local governments, and is provided every 30 min with a spatial resolution of 1 km (JMA n.d.-a).

Outside of Japan, data from the GSMaP (Kubota et al. 2020) can be utilized. These data are stored in a Data Integration and Analysis System (DIAS) in real time and can be bias-corrected in real time using local historical ground rain gage data or real-time observed data (Zhou et al. 2020). A method that combines these two approaches has also been developed. Bias-corrected data can be used as inputs for runoff models to perform flood forecasting calculations and provide calculation results (DIAS Office n.d.). This system operates in large river basins such as the Cal River Basin in Sri Lanka, the Niger River Basin in West Africa, and the Pampanga River Basin in the Philippines (Rasmy et al. 2023; Rasmy et al. 2024; Miyamoto et al. 2022). The rainfall input for the model is detailed in Section 3.1.

Reasonable Selection of Models for Site-Specific Cases

Typically, sufficient data on water levels and flow rates during past floods are unavailable for SMRs; therefore, it is necessary to use models that include processes with a physical background. When flood forecasting is performed using the observed rainfall, the forecast calculation time should be maintained within a few minutes so that the short Tc of the SMRs can be effectively utilized. The RRI model (described in Section 3.3) fulfills these requirements as it covers physical hydrological processes and the basic equations for river channel flow and can calculate inundation in the same format using a diffusion model (Sayama et al. 2012). Additionally, the model needs to be user-friendly and accessible because of the large number of SMRs and users it will cover. The International Centre for Water Hazard and Risk Management under the auspices of UNESCO (ICHARM) created a user interface for the RRI model and opened the source code, which helped to promote the spread of the RRI model. The selection and description of the runoff models are further detailed in Sections 3.2 and 3.3.

Practical and Accurate Terrain Modeling

The RRI model requires topographic data from the following: digital elevation model (DEM), DEM-based flow direction (DIR), and accumulation (ACC), which indicate the cumulative number of meshes upstream of the mesh created from the DIR. The Japan Flow Direction Map (Yamazaki et al. 2018; Yamazaki 2022) is free, publicly available hydrographic topographic data. DEMs, DIRs, and ACCs are created using elevation data from basic mapping information for all of Japan and have been used in many flood forecasting studies (e.g., Sayama et al. 2020; Yamada et al. 2022). The topographic data to be input into the model are described in Section 3.4.

High-precision elevation measurement data are usually unavailable in many areas of the world. In such areas, the global high-resolution “AW3D” and “MERIT DEM” (Yamazaki et al. 2017), which combine multiple satellite observation data such as “SRTM: Shuttle Radar Topography Mission (NASA)” and “ALOS (JAXA)” with original terrain analysis algorithms, can be used. For hydrological analysis, “HydroSHEDS v1” (Lehner et al. 2008) and “MERIT Hydro” (Yamazaki et al. 2019) datasets can be used because they provide hydrological terrain correction with a spatial resolution of 3″ for the whole globe.

Rational Resolution of Parameterization for Areas Lacking Rainfall or Discharge Records

To select reasonable parameters in areas lacking rainfall or discharge records, it is necessary to develop a method that extracts existing model rivers with basin characteristics similar to the modeled river using principal component analysis and uses their parameters to estimate the conditions of the target river (described in Section 3.7).

Uncertainty Identification and Process

Runoff analysis should be conducted by accurately determining the initial rainfall loss (IRL). However, the estimation of IRL is subject to uncertainty. The amount of IRL depends on the soil moisture content. To evaluate the IRL indirectly, this uncertainty must be reduced by applying a particle filter method for data assimilation using the observed river water levels to estimate the soil moisture content in the model (described in Section 3.8).

Observing water levels in SMRs has been challenging due to the limited number of water-level observation stations, mainly because of the high installation and maintenance fees of conventional water-level gages. Therefore, a 3 L Water Level Gage (3 L WLG) specialized in obtaining water level information during floods was recently developed in Japan to reduce costs (MLIT 2018b). The number of 3 L WLGs installed on SMRs in Japan is rapidly increasing, and it is expected that many other countries will use these gages in the future.

Basic Design of Flood Forecasting System for SMRs and Case Studies

Basic System Design

Rainfall Observation

Rainfall data from ground rain gages and radar rain gages were used. In countries and regions where such data are not available, rainfall data from satellite observations were used. Rainfall forecasting data are available for some countries, but the accuracy of rainfall forecasting in small areas such as small and medium-sized river (SMR) basins is insufficient. Therefore, it was necessary to perform calculations as quickly as possible using the observed rainfall.

River Water Level Observation

The water level observation data used the observation values of the water level gages at existing observation stations. Water-level observation stations have not been installed in some SMRs. In these locations, the installation of 3 L WLGs was considered. The recently developed 3 L WLG has the following features: low unit price, long-term maintenance-free operation (operating for more than 5 years without a power supply), and low communication costs (MLIT 2018b).

Achieving Both Diffusion and Accuracy Through a Diffusion-Type Runoff Model and Improved Initial Values

For runoff analysis, we used the RRI model, which analyzes rainfall-runoff and flood inundation in an integrated manner based on a physical model. The water level was calculated from the calculated discharge using the water level– discharge conversion equation (H–Q equation).

To ensure forecasting accuracy, it is necessary to estimate the initial value of the soil moisture content accurately during runoff forecasting. Therefore, we adopted a method to sequentially revise the initial value of soil moisture content in the water-level forecasting model using real-time water-level observation data to improve forecasting accuracy.

The flood forecasting system must be easily constructed, with minimal labor, to become widely applicable in many SMRs. Therefore, the SCE-UA method was adopted to reduce the labor required to estimate the parameters of the runoff models.

Water level observation data for SMRs during past floods are often unavailable. Therefore, to enable parameter estimation when water level data from past floods are unavailable, we developed a method to estimate runoff model parameters based on the characteristics of the target river basin through principal component analysis. This model uses runoff model parameters from rivers where the models have been calibrated.

Case Study

In this case study, 200 SMR basins in Japan were selected as the target area. The Japanese archipelago hosts a variety of climate zones, from subarctic in the north to subtropical in the south. MLIT (2015) classified Japan into 15 climate categories based on a cluster analysis of heavy rain characteristics and organized their causes and scale. The causes of the largest to third largest rainfall events in terms of rainfall duration and basin area vary from typhoons, low pressure, atmospheric instability, and rainy fronts in the south and north, Pacific side, and Sea of Japan side of the Japanese archipelago, respectively (see Figure S1 of Supporting Information). A difference of more than twofold between the catchment area and 3-h rainfall intensity in each climate category was observed (see Figure S2). Due to the diverse and complex geological structure of river basins, their discharge characteristics vary.

We ran the developed models in real time across 200 rivers with diverse climatic and runoff characteristics, verified their performance, and refined the methodology. These changes will help to realize a general-purpose model that can be applied to rivers with various climates and basin characteristics both locally and overseas.

Figure 1 presents a histogram of the catchment areas of the 200 rivers. SMRs refer to rivers with a basin area equivalent to those of prefecturally managed rivers in Japan, generally with a basin area of several 100 square kilometers or less. The knowledge obtained during the model-building process for 200 rivers was integrated and compiled as a manual for building flood forecasting models for SMRs (Study Group for Development of Flood Forecasting Models for Small and Medium-sized Rivers 2023). The details of this system are described in Chapter 3.

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Development of Flood Forecasting System

Rainfall and Water Level Data for Runoff Analysis

The system developed here uses five precipitation products provided by the JMA depending on the catchment area and Tc. Radar/Raingauge-Analyzed Precipitation (R/A), Immediate R/A, and High-resolution Precipitation Nowcasts (HRPNs) were used for observed precipitation, and Very-Short-Range Forecasts of Precipitation (VSRFs) (JMA n.d.-b), Immediate VSRFs, and HRPNs (JMA n.d.-c) were used to forecast precipitation.

R/A is a highly accurate precipitation product obtained by correcting weather radar observation data owned by the JMA and MLIT with data from ground rain gages owned by the JMA, MLIT, and local governments. It is generated and provided every 30 min, with a spatial resolution of 1 km.

VSRFs are rainfall products that predict precipitation amounts up to 6 h in advance by considering the development and weakening of precipitation and the speed of movement of the rain area. They are generated and provided every 30 min with a spatial resolution of 1 km.

This water level prediction system uses R/A and VSRFs. The delay time of rainfall data delivery varies depending on the network environment and rainfall conditions; however, in this experimental environment, the delay time for R/A was approximately 14 min, and approximately 19 min for VSRFs. The water level forecasting system begins the forecasting and assimilation processes after collecting the data.

For SMRs in general, the combination of R/A and VSRFs data can provide accurate flood forecasts. However, the distribution interval and delay of rainfall data for rivers with very small catchment areas and very short Tc must be shortened to improve forecast accuracy. In such cases, immediate R/A and VSRFs, which have a delivery interval of 10 min and a short delivery delay, or HRPNs, which have a shorter delay (5 min), are used. Thus, in this study, the forecasting accuracy was improved using an appropriate combination of each rainfall product according to the characteristics of the watershed. In addition, the observed values from existing water level gages were used for the locally observed water levels in this study.

Selection of Runoff Analysis Models

Generally, flood forecasting methods can be classified into two categories (Tsubaki et al. 2013): a direct water-level assessment model that forecasts the water level at the assessment point based on the correlation between upstream and downstream water levels, or a runoff analysis method that evaluates runoff flow based on rainfall. Forecasting methods based on direct water-level assessment models are difficult to apply to many SMRs for which past flood data is scarce.

In contrast, many distributed models using a runoff analysis method have been developed since around 2000. Distributed models can be broadly divided into conceptual and physical types. Conceptual models are unsuitable for SMRs because previous flood flow data is often difficult to obtain. Physical models include land surface process models that rigorously simulate water and energy balance (e.g., WEB-DHM (Wang, Koike, Yang, Jackson et al. 2009; Wang, Koike, Yang, and Yeh 2009; Wang, Koike, Yang, and Yang 2009); WEB-RRI (Rasmy et al. 2019)) and terrestrial water cycle models that provide detailed simulations of water exchange in the subsurface layer (e.g., GET-FLOWS (Tosaka et al. 2000)). While these sophisticated models can reproduce water levels accurately across various conditions owing to the large number of physical processes that can be represented, the large computational burden in real-time forecasting makes them unsuitable for real-time flood forecasting for many SMRs with short Tc. Other physical models include 1K-DHM (e.g., Ichikawa et al. 2001; Tanaka and Tachikawa 2015), RRI model (Sayama et al. 2012), and LISFLOOD (e.g., Van der Knijff et al. 2008; Burek et al. 2013).

To forecast flash floods in SMRs, it is desirable to select a distributed hydrological model that includes the physical mechanisms of the rainfall –runoff process and can be computed quickly. The RRI model, while a widely used model, represents the hydrologic physical characteristics of floods (evapotranspiration/infiltration capacity (initial rainfall loss), sub-surface runoff, channel runoff, and inundation flow). The parameters of the RRI model can be calibrated and validated using historical flood data. In addition, using the observed rainfall until the start of the forecast calculation and inputting the forecast rainfall during the forecast period enable the accurate forecasting of water levels within Tc. Therefore, we selected the RRI model for this study.

RRI Model

The RRI model is a distributed runoff model that can analyze infiltration capacity, surface runoff, channel runoff, and inundation flow at the watershed scale using rainfall as an input (Sayama et al. 2012). The RRI model can handle three types of runoff processes for each mesh. For flat fields and rice paddies, vertical infiltration was considered using the Green-Ampt equation, and the excess that could not infiltrate into the soil was treated as surface runoff. In mountainous terrain with a slope, the method solves for lateral flow in the soil by using the flow-product relationship that considers unsaturated and saturated subsurface flows and surface flows in the continuous equation. In urban areas, where water does not infiltrate, only surface flow is used.

The RRI model is widely used in Japan and abroad and has high reproducibility (e.g., Yamamoto et al. 2017; Bhagabati and Kawasaki 2017; Sayama et al. 2020; Kakinuma et al. 2022). It is an open-source model, developed by the International Centre for Water Hazard and Risk Management under the auspices of UNESCO (ICHARM) and can be downloaded for free from the ICHARM website (ICHARM, 2022). The RRI model is included in the iRIC software (Shimizu et al. 2020) and has been used by many researchers and practitioners.

Base Model Construction and Initial Analysis With Default Parameters

The spatial resolution of the data input into the RRI model was set to 5″. The spatial resolution of the river channel/flood plain/ground/subsurface was set to 5″. The Japan Flow Direction Map (Yamazaki 2022) was used for topographic data, and digital national land information and land-use subdivision mesh data (MLIT 2014) were used for land-use data classified into mountainous areas, paddy fields, cropland, urban areas, and water bodies. This developed system makes this data readily available in the GUI of the RRI model.

As the spatial resolutions of the rainfall and ground meshes are different, the rainfall data given for each ground mesh in the calculation is the value of the rainfall mesh that includes the latitude and longitude of the center of the ground mesh. Before conducting the parameter optimization, the accuracy of the base model was checked using the default parameters listed in Table 1. The default parameters were set to general values for the SMRs based on the general values listed in the RRI model manual (Sayama 2022) and the values of the model parameters for rivers that were modeled on a trial basis. For the base model constructed in this manner, the consistency of the normal water level during no rainfall, the response of the hydrograph to rainfall, and the reliability of the H –Q equation were checked.

TABLE 1 Default parameters of base model.

	Parameter	Parameter in model	Unit	Mountainous area	Paddy field	Clop land	Urban area	Water body
Slope cell	Common parameter	n (Slope)	ns_slope	m−1/3・s	0.6	0.4	0.4	0.2	0.2
Soil depth	soildepth	m	0.5	1	1	—	—
Porosity	gamma_a	—	0.3	0.475	0.463	—	—
Green-Ampt model	Vertical hydraulic conductivity	ksv	m/s	—	1.67d-7	3.67d-6	—	—
Suction	faif		—	3.16d-1	8.89d-2	—	—
Subsurface flow	Lateral saturated hydraulic conductivity	ka	m/s	0.01	—	—	—	—
Unsaturated porosity	gamma_m	—	0.12	—	—	—	—
Hydraulic conductivity ratio	beta	—	4	4	4	—	—
	Parameter	Parameter in model	Unit
River cell		n (River)	ns_river	m−1/3・s	0.03

Setting Up the Water Level-Discharge (H–Q) Equation

The discharge calculated using the runoff analysis model must be converted into water level. The H–Q equation was used for rivers where flood discharge observations were conducted, and the H–Q equation was prepared. In many cases, SMRs managed by prefectural governments do not have past flood discharge or river channel geometry data. Therefore, this system enables water-level forecasting, even in such cases.

If the H –Q equation is not available, or if the H–Q equation has changed and river channel geometry data are available, the H–Q equation is created by uniform or non-uniform flow calculations. If the uniform flow approximation is valid at the station, a uniform flow calculation is used, and vice versa.

The channel roughness coefficient used in this process was estimated based on characteristics such as the riverbed gradient, representative grain size of the riverbed material, and revetment structure, by referring to existing literature (e.g., Japan Institute of Country-ology and Engineering [JICE] 2002; MLIT 2018c). In addition, the riverbed slope was estimated using survey data of the river channel shape when available; if not, a longitudinal section near the water level observation station was created from maps provided by the Geospatial Information Authority of Japan (GSI) (n.d.), and the average slope of the riverbed was calculated (Table 2).

TABLE 2 Table of the abbreviations and acronyms.

Abbreviations and acronyms	Terminology
3 L WLG	3 L Water Level Gage
CPU	Central Processing Units
DIAS	Data Integration and Analysis System
FRL	Flood Risk Level
GPGPU	General-Purpose computing on Graphics Processing Units
GSI	Geospatial Information Authority of Japan
GUI	Graphical User Interface
HRPNs	High-Resolution Precipitation Nowcasts
ICHARM	International Centre for Water Hazard and Risk Management under the auspices of UNESCO
IFSIC	International Fire Service Information Center
IRL	Initial Rainfall Loss
JICE	Japan Institute of Country-ology and Engineering
JMA	Japan Meteorological Agency
LT	Lead Time
MLIT	Ministry of Land, Infrastructure, Transport and Tourism
PCA	Principal Component Analysis
PF	Particle Filter
R/A	Radar/Raingauge-Analyzed Precipitation
RRI model	Rainfall-Runoff-Inundation model
SCE-UA	Shuffled Complex Evolution algorithm developed in University of Arizona
SMR	Small and Medium-sized River
Tc	Time of concentration for a catchment
VSRFs	Very-Short-Range Forecasts of Precipitation

If neither the H–Q equation nor channel geometry data are available, it is possible to calculate the water depth from the channel cross section based on the empirical equation used in the RRI model (Sayama 2022; Study Group for Development of Flood Forecasting Models for Small and Medium-sized Rivers 2023).

Parameter Optimization Using the SCE-UA Method

Traditionally, estimating parameters that significantly affect a model's reproducibility requires a lot of time and effort, which hinders the widespread dissemination of flood forecasting systems. Therefore, this system we developed employs the SCE-UA method (Duan et al. 1992), which is a parameter optimization technique. The SCE-UA method was adopted because it is easy to implement in hydrological models such as RRI and efficiently converges to optimal parameters with a relatively small number of iterations.

The SCE-UA method is an optimization technique that has been widely applied to many nonlinear hydrological models. It is characterized by its ability to converge more quickly from local to global optima through a search method that combines the concepts of competitive evolution and population mixing (Eckhardt and Arnold 2001; Zhang et al. 2016; Naeini et al. 2019). The parameters to be optimized in the RRI model were the channel roughness coefficient, slope roughness coefficient, hydraulic conductivity, and soil porosity. The search range of the parameter values was based on previous literature (e.g., Sayama 2022) and set within a range that did not impair the physicality of the parameters.

To obtain a model with good prediction accuracy for floods of various scales, the parameters were optimized for two flood events: a major flood and a medium flood. The Nash-Sutcliffe coefficient was used to evaluate the goodness of fit of the prediction results of the model. The optimized parameter set was used to verify its applicability to flood events that were not used in the optimization. The model parameters were optimized using the SCE-UA method for offline calibration.

A case study of applying the SCE-UA method to an RRI model with a default parameter set is shown in Figure 2. In this case, the application of the SCE-UA method improved the Nash –Sutcliffe coefficient from 0.89 to 0.99.

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Parameterization in the Absence of Rainfall or Flood Water Level Data

In some rivers, rainfall or water level data from historical floods are not available, making the optimization of the RRI model parameters based on this data impossible. To set appropriate model parameters for such rivers, we developed a method using principal component analysis (PCA) to select other rivers with catchment characteristics similar to those of the target river and used the parameters of those rivers to represent the target river. The parameter estimation procedure is as follows:

• Data such as the characteristics of the watershed for each river for which models were constructed (area of the watershed above the observation points, dominant area by land use (mountainous area, paddy field, cropland, urban area, and water body)) and parameters after optimization.
• Conduct PCA based on river characteristics and calculate the principal component scores for each river and the eigenvectors for each characteristic.
• The principal component score of the river was calculated using the inner product of the features and the eigenvectors of the target river.
• Rivers with principal component scores similar to those of the target river were extracted.
• Among the extracted rivers, we set the parameters of the river with high Nash –Sutcliffe coefficients using the optimization parameters as the parameters of the target river.

To validate the parameter estimation method using PCA, PCA was conducted using data from 125 rivers and validated against historical floods in 72 rivers that were not included in the PCA. The results showed that the Nash –Sutcliffe coefficient was greater than 0.7 in 42% of the historical floods, indicating that this method can be used to some extent to forecast floods, even when water level data from historical floods are not available (Endo et al. 2023). This developed system can utilize PCA results from the data of 200 rivers (Study Group for Development of Flood Forecasting Models for Small and Medium-sized Rivers 2023).

Data Assimilation Using the Particle Filter Method

To improve the accuracy of the water level forecasting model, a particle filter (PF) was incorporated into the RRI model, and real-time water level observation data were used to sequentially modify the basin state quantities in the flood forecasting model (Nakamura et al. 2018).

The particle filter is a method for the sequential assimilation of numerical simulation models and observed data for state-space models with nonlinearity and non-Gaussianity. Compared with that of the Kalman filter and ensemble Kalman filter, the degree of freedom of the state-space model that can be handled is higher; however, the number of particles required to properly approximate the probability distribution increases significantly as the number of dimensions of the state space increases (Snyder et al. 2008). With the rapid development of Central Processing Units (CPUs) and general-purpose computing on Graphics Processing Units (GPGPU), an environment that allows parallel computation of system models is now in place, and implementation in practice is becoming possible.

The data assimilation technique using the particle filter method is illustrated in Figure 3. The particles in the figure refer to the state space of the watershed in the RRI model. The assimilation process proceeds to the right of the figure based on the initial distribution of the particles. In the figure, x is the state quantity, t is the current time, t−1 is the time one step before the current time, y is the observed value, w is the weight, i is the particle number, and u is the white noise. For example, $$$ {x}_{t\mid t-1} $$$ means the state quantity x at time t based on data up to time t-1. First, in the “Prediction” step, the known particle group (column $$$ {x}_{t-1\mid t-1} $$$; Figure 3) was used as the initial condition and rainfall observed as the boundary condition. The estimated river water level (column $$$ {x}_{t\mid t-1} $$$; Figure 3) was obtained using a system model that combines the RRI model and the H–Q equation. The observed river water level ($$$ {y}_t $$$; Figure 3) was obtained. In the next step, “Likelihood”, the likelihood of the estimated value for each particle was calculated from this observed value, and each weight ($$$ {w}_t $$$; Figure 3) from the likelihoods was calculated via normalization (second column $$$ {x}_{t\mid t-1} $$$; Figure 3). In the “Resampling” step, depending on the magnitude of the weight ($$$ {w}_t $$$), the particles were divided into those that replicated (i = 1, 2, 3, 4, and 5; Figure 3) and those that decay (i = 6, 7, and 8; Figure 3). In the “System noise” step, a slight noise ($$$ {u}_t $$$) was introduced to prevent the degeneration of the replicated particles. Finally, the current time was updated from $$$ t $$$ to $$$ t+1 $$$ in “Update time” and the process was repeated. Repeating this series of processes makes sequentially updating the plausible states of the RRI model in the system (H–Q equation is fixed) based on Bayesian estimation possible.

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Generally, in time series analysis data assimilation, it is particularly important to decide which state quantity to estimate. In river-level forecasting, estimating state quantities that do not change over time or have a small effect on the rainfall-runoff response does not provide data assimilation benefits commensurate with the increase in computational cost. In addition, the sequential modification of directly measurable physical quantities, such as hydraulic conductivity, negates the original design concept of the hydrologic model. Here, we focused on the slope runoff process of the RRI model and used a model parameter that expresses soil moisture, hs, for each mesh as the state quantity sequentially modified by the particle filter. However, as hs exists as a spatial distribution for each mesh, the particle filter is a parameter vector of correction coefficient α that corrects hs. Direct observation of the spatial distribution of hs is not realistic, and a rigorous calculation of hs requires the calculation of the water and energy balance between the atmosphere and the land surface. The error in the initial value of hs increases over time, especially during long-term real-time calculations. Therefore, hs was selected as the state quantity to be estimated using data assimilation in this system.

As most SMRs managed by prefectural governments have only installed one water level gage, the water level gage installation point was used as the evaluation point, and the correction factor was uniformly applied to the entire catchment area. For likelihood evaluation, the RMSE calculated from each calculated and observed value for 3 h in the past was used as the evaluation index (Nakamura et al. 2018).

As there are as many forecasted water levels as there are particles, there are multiple ways to present them; for example, a single forecast water level with a maximum likelihood value, an ensemble forecast water level for all particles, or a weighted average forecast water level for all particles (Nakamura et al. 2018). In this study, the weighted average water level of all particles, which represents a relatively stable forecasted water level, is displayed as the forecasted water level at the relevant time.

For resampling, we adopted the D'Hondt method, which has a low computational load, based on the results of Tachikawa et al. (2011). The computation period of each cycle was set from 3 h in the past to 6 h in the future at the time of assimilation.

In general, the larger the number of particles in the particle filter, the smaller the forecasting error; however, this increases the calculation time. For flood forecasting of SMRs, the calculation time needs to be kept within a few minutes. In this study, forecasting results were evaluated using multiple particle cases (2, 4, 8, 16, 32, and 64 particles) for a model river with a basin area of 500 km². Based on the results, 64 particles were determined to be the optimal number to satisfy both the required accuracy and calculation time for flood forecasting. The number of particles, 64, was therefore adopted, and it was verified whether the existing computational resources could provide information with an acceptable level of accuracy that would contribute to the flood forecasting and warning for the target 200 rivers.

Real-Time Water Level Forecasting System Using DIAS

An automatic calculation and distribution system was built on DIAS to perform real-time water level forecasting using the developed model. DIAS stores nationwide rainfall data and water-level gage data in real time (DIAS Office n.d.; Kawasaki et al. 2018) Based on this data, real-time calculations were performed using the DIAS, and the calculation results were drawn and distributed; the delivery interval was set at 30 min, and an alert function was placed in response to forecasted water levels (Kakinuma et al. 2021). Currently, a flood forecasting system for 200 SMRs is being constructed, and real-time automatic distribution is being implemented. An example of a distribution screen is shown in Figure 4.

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To avoid delays in delivery time under such circumstances, we optimized the allocation of computing resources when the computing load was concentrated and shortened the computation time (Nemoto et al. 2020). Computation time was reduced by dynamically allocating free computer resources to water-level forecasting calculations in rivers, which take a long time to execute.

Evaluation of the Accuracy of Water Level Forecast Results

Experimental Results When JMA Rainfall Forecasts Were Accurate

We evaluated the experimental results when rainfall is accurately forecasted (using the observed rainfall (Radar/Raingauge-Analyzed Precipitation [R/A])) to verify whether the RRI model is appropriately modified by the application of the particle filter. The target river was the Kagetsu River in Kyushu (catchment area: 130.2 km²), and the target flood was the flood caused by the July 2017 torrential rainfall in northern Kyushu.

Figure 5 shows the forecasted water levels up to 6 h ahead with (red line) and without (blue line) the particle filter. Compared to the observed water level (▲), the results without the particle filter show deviations around 15:00 on July 5 and after 0:00 on July 6 in the declining phase. Contrarily, when the particle filter is applied, the results are in better agreement with the observed water level than when the filter is not applied, indicating that the application of the particle filter improves the accuracy of the water level forecasting.

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Experiments Using Actual Rainfall Forecasts

We verified the accuracy of this method by conducting forecast simulations up to 6 h ahead using actual forecast rainfall (Very-Short-Range Forecasts of Precipitation [VSRFs]), which were also distributed by the JMA in real time. The target river and target rainfall were the same as those in Section 4.1, but the observed rainfall (R/A) and forecast rainfall (VSRFs) available at each time were used as input values.

The forecast time of the Flood Risk Level (FRL) exceedance using actual rainfall forecasts at 16:00 was 20:00 when the particle filter was not applied. In contrast, when the PF was applied, the forecasted time was 17:50, which was closer to the actual time (18:00) when the observed water level exceeded the FRL (Figure 6).

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Figure 7 shows the errors in the forecasted water levels with and without the PF by the preceding time. The figure shows the error distributions of the mean, maximum, and minimum water levels. Comparisons of the mean values of the forecasting errors show predictions for 1 and 2 h ahead were −0.32 and −0.26 m without PF, and −0.09 and −0.09 m with PF, respectively. For the minimum values, the forecasting errors for 1 and 2 h ahead were −1.83 and −1.09 m without PF, and −0.73 and −0.71 m with PF, respectively. Therefore, applying PF improved the accuracy of forecasts 1 and 2 h ahead. However, this effect had a negligible impact on the accuracy of forecasts more than 3 h ahead.

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Verification of Test Operation Results for 200 Rivers

Using the above method, a flood-forecasting system for 200 SMRs in Japan was constructed (Figure 8) and is currently under test operation with real-time inputs of observed and forecasted rainfall. Of the 200 rivers, 195 had FRLs, which are standard water levels for issuing evacuation orders. To date, 291 flood events have been reported from 195 rivers. For these 291 flood events, the lead time (LT; time that water levels reach the FRL—time of first forecast) was verified.

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In the actual operation of the flood forecasting system, the forecasted water level is planned to be maintained within a range of ± 50 cm with respect to the calculated value, considering errors in local water level observations due to wind waves and other factors. Based on these considerations, we verified the accuracy of this forecasting system when operated within a ±50 cm range.

Figure 9 shows the results of the accuracy verification of the forecasted water level, and Figure 10 shows a histogram of the LT.

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For 219 of the 291 events (75%), we were able to forecast that the river level would reach the FRL more than 2 h in advance. In other words, LT was ≧ 2 h. The remaining 72 events included rivers with small catchment areas and a Tc of less than 2 h. In these rivers, the LT was checked, and in 39 events, the LT was longer than Tc. Ultimately, 258 flood events (89%) could be predicted with an LT of 2 h or more or an LT of Tc or more. For 13 of the remaining 33 events, system improvements allowed the LT to be longer than Tc for that river. We are still considering methods to improve the system further to forecast the remaining 20 events accurately.

Conclusions

In this study, we developed a system that can accurately predict the time at which the water level will reach the standard level for issuing evacuation orders by local governments for floods in SMRs. This system can be widely disseminated both domestically and internationally.

Using the RRI model as the base model, we optimized the parameters using the SCE-UA method, applied the particle filter method to sequentially modify the soil moisture content in the model using observed river water levels, and developed a GUI to create a forecasting system that meets development requirements.

In many cases, H–Q equations and observed flood flow data were insufficient for SMRs. Therefore, we present a method to easily obtain the H–Q equation and parameters of the RRI model, even for data-deficient rivers.

The flood forecasting system developed by this study shows reasonable performance in responding to the request from society. The system is currently being tested and distributed for real-time operation on 200 rivers. There are approximately 1500 rivers managed by prefectural authorities in Japan where water levels are not forecasted despite the significant risk to lives due to flooding. By applying the system developed in this study, we aim to establish a flood forecasting system for these rivers and enhance and strengthen the flood warning and evacuation systems.

Acknowledgments

In conducting this research and development, we received funding from the Ministry of Land, Infrastructure, Transport and Tourism, as well as data and other support from related prefectural governments and the National Institute for Land and Infrastructure Management.

Data Availability Statement

Part of the data that support the findings of this study are available from the references in this article.

References

Bhagabati, S. S., and A. Kawasaki. 2017. “Consideration of the Rainfall‐Runoff‐Inundation (RRI) Model for Flood Mapping in a Deltaic Area of Myanmar.” Hydrological Research Letters 11, no. 3: 155–160. [DOI: https://dx.doi.org/10.3178/hrl.11.155].