1 Introduction

Every year, many lives are lost due to landslides and debris flows. Heavy rainfall is the major cause of these sediment-related disasters (SRDs), and it has been pointed out that climate change may increase risks of SRDs. In order to predict landslides and debris flows based on rainfall observations, various method to analyze rainfall anomalies have been proposed. One of the widespread methods is to select two parameters that represent rainfall characteristics (rainfall parameters), such as rainfall intensity, rainfall duration, and cumulative event rainfall, and set a rainfall threshold on a diagram showing their correlation (Fig. 1). In this method, if the plot on the diagram exceeds the threshold, the risk of SRDs is considered high.

Fig. 1
figure 1

Schematic diagram showing correlation between two rainfall parameters, and a rainfall threshold for SRD risks. Envelope of the plots is shown by blue line which represents the historical maximum

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Previous studies have discussed different rainfall thresholds, such as the rainfall intensity versus duration (I-D), the cumulative event rainfall versus duration (E-D), the cumulative event rainfall versus intensity (E-I), the normalized intensity versus duration (NI-D) thresholds (Maturidi et al. 2021b). Among them, perhaps the most frequently used one is the I-D threshold. For instance, Maturidi et al. (2021a) established the I-D threshold for landslide forecasting in Peninsular Malaysia. Zhang et al. (2022) proposed a screening method for defining the I-D threshold for shallow landslides in Guizhou Province, China. Other than these studies, the method is applied worldwide, including India (Mandal and Sarkar 2021; Abraham et al. 2021), Italy (Berti et al. 2012; Sala et al. 2021), Japan (Saito et al. 2010), and USA (Dennis et al. 2015). On the other hand, the E-D threshold was established for landslides in Chuncheon, Republic of Korea (Lee et al. 2021). Calvello et al. (2015) analyzed the correlation between cumulative event rainfall and duration for studying landsides in Rio de Janeiro, Brazil.

With regard to rainfall parameters used for SRD predictions, previous studies have considered various parameters other than rainfall intensity, duration, and cumulative event rainfall. For example, Kalubowila et al. (2021) used daily rainfall on 0 to 4 days before landslide occurrences for analyzing landslides in Matara district of Sri Lanka. In Japan, the soil water index (SWI) was proposed by the Japan Meteorological Agency as a rainfall parameter considering effects of current and antecedent precipitation (Okada et al. 2001). A three-layer tank model with fixed parameters is used to calculate SWI, where SWI represents the sum of the water depths in each layer. SWI is practically used under the current Japanese system for issuing alert information against SRDs. SWI is also used in the study for establishing a SRD warning system in Taiwan (Chen et al. 2022). More detailed analyses on rainfall parameters have conducted by Zhao et al. (2022), who examined 143 parameters related to rainfall observed in a catchment located in central China and concluded that the use of more rainfall parameters can improve accuracy and performance of debris flow prediction models. Moreover, based on comparisons between rainfall parameters and groundwater levels observed on mountain slopes, Kosugi (2015) showed that parameters for predicting SRDs should vary by location. Thus, results of these studies suggest that, in order to assess the risk of SRDs, it is essential to analyze rainfall features based on various criteria by using many rainfall parameters.

As for the determinations of threshold for SRD forecasting, the most common approach is to use the receiver operating characteristic (ROC) analysis which can identify the optimal threshold that minimize the false alarms while maximizing the correct alarms. Realistically, however, it is necessary to find an appropriate threshold that takes into account overall social costs of missed- and false-alarms. By developing cost-sensitive rainfall threshold curves, Sala et al. (2021) showed that the most appropriate threshold is derived by lowering the ROC-based threshold and reducing the cost of missed-alarms which is usually greater than the cost of false-alarms. In practices in Japan, thresholds are often set low so that landslides and debris flows will not be missed. However, this leads to many cases of false-alarms; it is reported that the ratio of false-alarm is 95.3% over the 10-year period from 2010 to 2019 (The Asahi Shimbun Company 2021). As a result, the reliability of alarm is dramatically reduced, preventing municipalities and residents to make appropriate evacuation actions.

In view of this current situation, Kosugi (2015) proposed to set the threshold to the envelope of plots on a diagram drawn by a combination of two rainfall parameters (Fig. 1). Based on this threshold, the announcement of a warning alarm implies that rainfall is on a scale never experienced in the past and, therefore, provides a clear rationale to residents for taking evacuation actions. On the other hand, this method arises a concern that the threshold becomes too high, increasing the number of missed-alarms. In order to cope with this problem, Kosugi (2015) proposed to obtain an envelope for each of various diagrams drawn by using different combinations of various rainfall parameters. If every envelope is used as the threshold, an alert will be issued when the current rainfall exceeds the historical maximum at least in one diagram. This method seems reasonable because, in order to assess the risks of SRDs, it is essential to analyze rainfall features based on various criteria, as described above. Furthermore, a recent study by Kosugi (2022) proposed a novel rainfall parameter, the unprecedented rainfall index, TP, which is defined as the time stamp representing the situation that “current rainfall is the largest since TP”. By using this index, it is possible to issue alarms based on the knowledge of at what point in time in the past rainfall was about the same as or greater than the present.

The purpose of this study is to evaluate effectiveness of the unprecedented rainfall index in achieving appropriate evacuation actions in the face of landslide and debris flow risks. In the following, we first present the definition and calculation method of TP. Next, TP values are computed for every storm event that caused landslides and debris flows which resulted in fatalities in Japan in 2021. Based on the results, we discuss usefulness of the unprecedented rainfall index and how it can be used to reduce losses of human lives due to SRDs.

2 Materials and methods

2.1 Definition and behavior of API

The antecedent precipitation index (API), which is a rainfall parameter to evaluate the residual effects of previous precipitation, has been used frequently to analyze soil moisture dynamics (Sittner 1969), groundwater behaviors (Kosugi et al. 2008), storm runoff generations (Fedora and Beschta 1989) as well as landslide occurrences (Suzuki and Kobashi 1981). The value of API, X(M, t) [mm], with the half-life M [h] at time t [h] is calculated as

$$X\left( {M,\;t} \right) = X\left( {M,\;t{-}1} \right) \, 0.5^{1/M} + R\left( t \right)$$
(1)

where X(M, t–1) is API 1-h ago, and R(t) [mm] is the total precipitation from t–1 to t (equivalent to the 1-h rainfall intensity, I [mm/h]). The value of M indicates that precipitation decays to one-half of its initial value M h after its occurrence (Suzuki and Kobashi 1981; Descroix et al. 2002). Figure 2 exhibits the behavior of X(M, t) for various M values. When M is short, API strongly reflects the intensity of the current rainfall. As M is lengthened, the effect of past rainfall becomes more persistent, and API is an index that more strongly reflects the cumulative rainfall.

Fig. 2
figure 2

a One hour rainfall intensity, I, and the antecedent precipitation index (API) with the half-life M of b 1.5, c 72, and d 720 h. Modified from Fig. 1 of Kosugi (2015)

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2.2 The API diagram

In this study, the API diagram is defined as the correlation diagram with two types of API (X(M1,t) and X(M2,t)) with different half-lives M1 and M2 (M1 ≥ M2) on the horizontal and vertical axes, respectively (Fig. 3). Values of X(M1,t) on the horizontal axis evaluate effects of cumulative rainfall, and X(M2,t) on the vertical axis evaluates effects of rainfall intensity. In Japan, the API diagram had long been used to announce warning alarms for SRDs. That is, when a plot on the API diagram exceeds a threshold, the risk of disaster is considered high. The employed M1 and M2 values had been 72 h and 1.5 h, respectively (Fig. 3a).

Fig. 3
figure 3

API diagrams drawn by combinations of a APIs with half-lives of 72 and 1.5 h, and b APIs with half-lives of 1061 and 77 h. Color of each plot represents the range of groundwater level H. Modified from Fig. 2 of Kosugi (2015)

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Since groundwater levels on slopes dominantly control landslide occurrences, Kosugi et al. (2013) analyzed correspondences between API diagrams and groundwater levels observed on slopes. Figure 3 shows an example: with values of M1 and M2 of 72 h and 1.5 h, respectively, the API diagram could not reproduce groundwater levels observed on a steep mountain slope with high accuracy (Fig. 3a). Then, Kosugi et al. (2013) optimized M1 and M2 to obtain values of 1061 h and 77 h, respectively, which reproduced the groundwater levels with higher accuracy (Fig. 3b). Furthermore, Kosugi (2015) compared API diagrams and groundwater hydrographs observed at 33 locations on mountain slopes and estimated the optimal M1 and M2 ranges to be 2.7–1398.2 h and 2.7–1138.7 h, respectively. Based on these results, Kosugi (2015) concluded that rainfall parameters for predicting SRDs should vary by location. In other words, in order to assess SRD risks, it is essential to analyze rainfall features based on various evaluation criteria by using various API diagrams with different values of M1 and M2.

2.3 Definition and calculation of the unprecedented rainfall index

The unprecedented rainfall index, TP, is defined as the time stamp representing the situation that “current rainfall is the largest since TP”. Kosugi (2022) proposed the following procedure for computing TP.

Step 1: Calculate APIs, X(M1, t) and X(M2, t), for two different half-lives of M1 and M2 (M1 ≥ M2), respectively, using rainfall data for the entire period from the start of observation to the present time. Then, draw a API diagram as shown in Fig. 4, by plotting data during the current and past storm events with X(M1, t) on the horizontal axis and X(M2, t) on the vertical axis. In this study, a storm event is considered to be in progress when the rainfall in the past 24 h is greater than zero, and a storm event is delimited by a rainfall-free period of 24 h or longer.

Fig. 4
figure 4

Schematic diagram showing the calculation method of a TP candidate. Modified from Fig. 2 of Kosugi (2022)

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Step 2: On the API diagram shown in Fig. 4, define the upper right region (indicated by the red square) of the plot at the current time. Then, extract individual plots during the past storm events that are located in this region (including the boundary). As you can see from the figure, the plot at the current time is located in the same or smaller region in comparison with each of the extracted plots.

Step 3: Among the plots extracted in the Step 2, chose the plot which occurred closest to the current time. The occurrence time of the chosen plot is a candidate for TP. In the case of the example shown in Fig. 4, the candidate TP is “3:00 a.m., Jul. 4, 2004”.

Step 4: Set the combination of M1 and M2 to a different combination from that in the Step 1. Then, by performing the Steps 1 through 3 above, determine a new candidate TP. This process is repeated for various combinations of M1 and M2, and a TP candidates is determined for each of the various API diagrams.

Step 5: The earliest TP among the candidates obtained from the above processes is designated as TP.

The use of various API diagrams in the Step 4 corresponds to the aforementioned analysis of rainfall features using various evaluation criteria effective for predicting SRDs (Kosugi 2015). Adopting the time stamp that is farthest back in time among the various TP candidates (i.e., the Step 5) results in that, for at least one criterion, the rainfall at the current time is of a magnitude never experienced in the period from TP through the present (TP is not included). In other words, if we go back to time before TP (TP is included), the rainfall at the current time is of a magnitude that has been experienced in the past (i.e., the situation is within the historical range) in every evaluation criterion used in the analysis. In this sense, we can say that the rainfall at the current time is the largest since TP.

Regarding the evaluation criteria effective for predicting SRDs, 1326 different API diagrams were used to analyze rainfall features in this study. That is, the half-lives M1 and M2 were set in the range of 0.1 to 1500 h, in a total of 51 ways so that the change in the logarithmic value is constant. Every combination of M1 and M2 (M1 ≥ M2) was employed, resulting in the 1326 different API diagrams. The minimum value of M1 and M2 (i.e., 0.1 h) was determined so that API is almost the same as the 1-h rainfall intensity, while the maximum value was determined to be 1500 h with reference to the maximum value (1398.2 h) estimated by (Kosugi 2015).

We used the following procedure to obtain an initial X value in Eq. (1) for calculating the API time series for each half-life value. First, the API time series was calculated by assuming that the initial value is equal to zero. We found that, regardless of the half-life value, effects of this initial value on API were almost eliminated after three years from the start of the calculation. Therefore, from the API time series for the period after 3 years from the start of calculation, we extracted APIs with the same month, day, and hour as the start of the time series. Then, the average value of the extracted APIs was used as the initial value for calculating the API time series that was subsequently used for the TP calculations.

2.4 Studied disasters

In Japan, the number of SRDs in 2021 was reported to be 972, which resulted in death of 32 people and one person missing as reported by Ministry of Land, Infrastructure, Transport and Tourism (MLIT) (2022). These victims were brought by 3 disasters shown in Fig. 5. This study computed TP values for these 3 disasters for the purpose of discussing usefulness of the unprecedented rainfall index for reducing death and missing of people. Precipitation data from reliable stations close to each disaster site were used to calculate TP values. As a result of collecting data as far back in time as possible, we were able to obtain data for 66.5, 47.4, and 43.4 years for the Atami, Unzen, and Okaya disasters, respectively.

Fig. 5
figure 5

Locations of the three disasters studied. Background map was provided by Geospatial Information Authority of Japan

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2.4.1 The Atami disaster

From Jun. 28 through Jul. 6, 2021, a stationary front brought a heavy storm event with a total precipitation of 528 mm and the maximal 1-h intensity of 24 mm/h at Atami City, Shizuoka Prefecture, Japan. In the morning of Jul. 3, a large debris flow occurred at the Aizome River in Izusan district in Atami City. According to Imaizumi et al. (2022), the debris flow occurred owing to a failure of an embankment slope in the upper reaches of the basin (Fig. 6a) which was constructed in or around 2010. In the downstream region from the valley outlet, the debris flow engulfed houses on both sides of the channel, and although some concrete structures remained, many houses in the center part of the flow area were washed away (Fig. 6b). Thus, the debris flow caused extensive damages, resulting in the death of 26 people, leaving one person missing.

Fig. 6
figure 6

a Embankment slope failure and b debris flow flooding in the Atami disaster. Photographs were provided by Asia Air Survey Co Ltd (2021a)

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For calculating TP values, we used the 1-h precipitation data from Mar. 1, 1985 through Jul. 3, 2021 observed at the Atami observatory operated by Shizuoka Prefecture, which is located approximately 2.9 km away from the collapsed slope. Because the data at the Atami observatory can only be traced back to the last 36 years, we supplementary used the 1-h precipitation data observed at the Taga observatory (from Jan. 1, 1982 through Mar. 1, 1985), Tanna observatory (from Jan. 1, 1976 through Dec. 31, 1981), and Ajiro observatory (from Jan. 1, 1955 through Dec. 31, 1975). The Taga, Tanna, and Ajiro observatories are located approximately 5.2, 5.6, and 6.1 km, respectively, away from the Atami observatory, and operated by Shizuoka Prefecture, MLIT, and the Japan Meteorological Agency, respectively. Missing data were either supplemented with data from the nearest station or interpolated based on daily precipitation.

2.4.2 The Unzen disaster

From Aug. 7 through 20, 2021, a stationary front brought a heavy storm event with a total precipitation of 1484.5 mm and the maximal 1-h rainfall intensity of 81 mm/h at Unzen City, Nagasaki Prefecture, Japan. In the early morning of Aug. 13, a landslide occurred at Kojigoku district in Unzen City (Fig. 7a). Debris from the collapsed slope flowed down a small valley, and 2 houses located at the exit of the valley were completely destroyed, resulting in 3 deaths (Fig. 7b) (Jitousono et al. 2021).

Fig. 7
figure 7

a Landslide and b debris deposit in the Unzen disaster. Photographs were provided by Asia Air Survey Co Ltd (2021b)

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For the calculation of TP, we used the 1-h precipitation data from Apr. 1, 1974 through Aug. 20, 2021 observed at the Unzendake observatory operated by the Japan Meteorological Agency, which is located approximately 0.6 km away from the collapsed slope. Missing data were either supplemented with data from the nearest station or interpolated based on daily precipitation.

2.4.3 The Okaya disaster

From Aug. 12 through 20, 2021, a stationary front brought a heavy storm event with a total precipitation of 423.5 mm and the maximal 1-h rainfall intensity of 44 mm/h at Okaya City, Nagano Prefecture, Japan. In the early morning of Aug. 15, debris flows occurred at Kawagishihigashi district in Okaya City. The debris flows were observed at the Honzawa Valley, Nakaohkubo Valley, and Ohkubo Valley as shown in Fig. 8a. Among them the debris flow at the Nakaohkubo Valley swept into the dwellings, resulting in a tragedy that killed 3 people (Fig. 8b) (Hiramatsu et al. 2022).

Fig. 8
figure 8

a Valleys where debris flows occurred (modified from Hiramatsu et al. (2022); the original figure was provided by Nagano Pref.) and b a house hit by debris flow (modified from Ministry of Land, Infrastructure, Transport and Tourism (2021)) in the Okaya disaster

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For calculating TP, we used the 1-h precipitation data observed at Kawagishi observatory operated by Okaya City, which is located approximately 0.7 km away from the collapsed slope. Because the data from the Kawagishi observatory are available only from Mar. 28, 2021, we used data from the Shimohama observatory for the period from Mar. 18, 1978 through Mar. 27, 2021. The Shimohama observatory is about 2.8 km away from the Kawagishi observatory, and is operated by MLIT. Missing data were supplemented with data from the nearest station.

3 Results

3.1 The Atami disaster

Figure 9a shows the hyetograph for the storm event which caused the Atami disaster (the Atami 2021 storm event). A report of the first debris flow was brought by local people at 10:28 a.m. on Jul. 3, that was about 110 h after the start of the storm event. The cumulative rainfall at 10:00 on Jul. 3 reached 488 mm, and the rainfall intensity from 9:00 to 10:00 a.m. was as high as 24 mm/h. About 10 debris flows, in total, were observed by around noon on Jul. 3 (Imaizumi et al. 2022).

Fig. 9
figure 9

a Observed 1-h rainfall intensity and b computed TP values for the Atami disaster

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Figure 9b shows the time series of computed TP values. Between 12:00 a.m. and 5:00 a.m. on Jul. 1, TP reached Jun. or May of 2021. This indicated that the magnitude of the Atami 2021 storm event was still small compared to the magnitude of storm events that occurred within about 1–2 months prior to it. Then, between 6:00 a.m. on Jul. 1 and 5:00 a.m. on Jul. 2, TP went back about a year, indicating that the Atami 2021 storm event was the largest in approximately one year. Thus, as time progresses, TP generally went back in time. From 3:00 a.m. through 6:00 a.m. on Jul. 3, TP reached Aug. of 1983. That is, the Atami 2021 storm event was the largest since the Aug. 1983 storm event. And, after 7:00 a.m. on Jul. 3, TP reached Jan. 1 in 1955, which corresponds to the beginning of the rainfall record. This indicates that, during the time period of the debris flow occurrences (i.e., around 10:30 a.m. to noon on Jul. 3), the Atami 2021 storm event was on a scale never experienced during the period for which rainfall records exist.

3.2 The Unzen disaster

Figure 10a shows the hyetograph for the storm event which caused the the Unzen disaster (the Unzen 2021 storm event). The landslide at Kojigoku district reportedly occurred at around 3:55 a.m. on Aug. 13 (Nagasakishimbunsha 2021), that was about 128 h after the start of the storm event. The cumulative rainfall at 4:00 a.m. on Aug. 13 reached 863 mm. While the maximal 1-h rainfall intensity of 81 mm/h was observed between 11:00 p.m. to midnight on Aug. 12, the rainfall intensity around the occurrence of landslide (i.e., 3:00–4:00 a.m. on Aug. 13) was 28.5 mm/h.

Fig. 10
figure 10

a Observed 1-h rainfall intensity and b computed TP values for the Unzen disaster

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Figure 10b shows the time series of computed TP. The value of TP went back to Jul. of 1982 during the period of 7:00–11:00 p.m. on Aug. 12. Then, after 12:00 a.m. on Aug. 13, TP reached Apr. 1 in 1974, which corresponds to the beginning of the rainfall record, indicating that the Unzen 2021 storm event was on a scale never experienced before at the time of landslide occurrence.

3.3 The Okaya disaster

Figure 11a shows the hyetograph for the storm event which caused the the Okaya disaster (the Okaya 2021 storm event). The Shinano Mainichi Shimbun (2021) reported that the debris flow at the Nakaohkubo Valley occurred at around 5:15 a.m. on Aug. 15, that was about 54 h after the start of the storm event. The cumulative rainfall at 5:00 a.m. on Aug. 15 reached 375.5 mm. While the maximal 1-h rainfall intensity of 44.0 mm/h was observed between 3:00 to 4:00 a.m. on Aug. 15, the rainfall intensity just before the occurrence of debris flow (i.e., 4:00–5:00 a.m.) was not so large (i.e., 6.5 mm/h).

Fig. 11
figure 11

a Observed 1-h rainfall intensity and b computed TP values for the Okaya disaster

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The time series of the unprecedented rainfall index, shown in Fig. 11b, indicates that, at noon on Aug. 14, TP went back to Sep. of 1980. During the period of 1:00–8:00 p.m., TP went back to Jul. of 2006 or Sep. of 1999. Then, after 9:00 p.m. on Aug. 14, TP went back to Mar. 18 in 1978 (i.e., the beginning of the rainfall record). Thus, this figure clearly indicates that, at the occurrence of the debris flow, the Okaya 2021 storm event was on a scale never experienced before.

4 Discussion

4.1 Use of T P for appropriate evacuations

4.1.1 The Atami disaster

Under the current Japanese system, each prefectural government, in cooperation with each local meteorological observatory, issues alert information against SRDs, so-called the landslide alert information (LAI), based on rainfall analyses using a diagram that shows correlation between SWI and 60-min cumulative rainfall. In response to the issue of LAI, municipalities are recommended to issue the evacuation alert which require all residents to evacuate from possible affected areas (Japan Meteorological Agency 2022).

In case of the Atami disaster, although LAI was issued at 12:30 p.m. on Jul. 2, the evacuation alert was not issued before the disaster occurrence (Fig. 9b) (Imaizumi et al. 2022). The municipality issued the emergency alert, which is positioned as the next step after the evacuation alert, at 11:05 a.m. on Jul. 3, that was about 30 min after the first report of the debris flow occurrence (NHK news web 2021). Thus, in the case of the Atami disaster, the LAI announcement did not lead to effective evacuation actions by the municipality and residents.

The voluntary disaster prevention association, which is organized in the district directly hit by the debris flow, had received an award from the prefecture government in 2012 for its efforts to raise awareness of disaster risks and strengthen measures to prevent human casualties. The association has been aware that there are SRD warning zones in the area damaged by the Atami 2021 storm event, and the residents in this district have conducted evacuation drills against SRDs in 2016 and 2019 (Imaizumi et al. 2022). Thus, it has been believed that the residents have a good understanding of the risk of SRDs, whereas the 2021 disaster caused human casualties.

Facing threats of disasters, people tend to minimize the probability of potential threats or their dangerous implications (Omer and Alon 1994). Individuals and organizations tend to believe in the less alarming options whenever they are presented with conflicting or ambiguous information about danger (Perry et al. 1982). This tendency is called the normalcy bias. To counter the normalcy bias and allow for suitable preparation, it is necessary to deliver timely, repeated, and unambiguous warnings and instructions (Omer and Alon 1994).

As suggested by Kosugi (2022), who analyzed TP values for a storm event occurred in Kumamoto Prefecture, Japan, in 2020, causing 11 death and one missing of people, the information brought by TP time series shown in Fig. 9b has a large potential to reduce the normalcy bias, by unambiguously showing the abnormality of current rainfall in comparison with the previous storm event which caused historical catastrophes. For instance, Typhoon Ida, which hit this area in Sep. 1958, caused a catastrophe that left more than 1000 people dead or missing in the Izu Peninsula area (Fig. 5). By this typhoon, 4 people were killed and one was missing in Atami City (Shizuoka Prefecture 2016). As shown in Fig. 9b, the TP time series clearly indicates that, more than 3 h before the debris flow occurrences, the Atami 2021 storm event had exceeded the heavy rainfall brought by Typhoon Ida and reached a magnitude never experienced before. By delivering this information to local authorities and residents could have led to more appropriate evacuation actions.

4.1.2 The Unzen disaster

In case of the Unzen disaster, LAI was issued at 1:17 p.m. on Aug. 12 (Fig. 10b) (Kitamoto 2022), and in response to LAI, the municipality issued the evacuation alert at 4:00 p.m. (Nagasakishimbunsha 2021). However, this did not necessarily lead to appropriate evacuation actions by residents; 3 people were killed by the landslide occurred at around 3:55 a.m. on Aug. 13.

In this region, historical disasters occurred in Aug. 1993; torrential rains caused by a stagnant front triggered landslides and debris flows, resulting in two fatalities (Nagasaki Local Meteorological Observatory 2022). As shown in Fig. 10b, the TP time series clearly indicates that the Unzen 2021 storm event had exceeded the storm event in Aug. 1993 about 9 h before the landslide. Moreover, this figure indicates that, about 4 h before the landslide, the Unzen 2021 storm event had reached a magnitude never experienced before. Actually, an elderly woman who lives nearby made a comment on the Unzen 2021 storm event as “Since the evening on Aug. 12, it has been raining and thundering more heavily than I have ever experienced (The Yomiuri Shimbun 2021)”. By using the TP time series shown in Fig. 10b, the situation of such unprecedented heavy rainfall can be clearly informed to all residents based on the objective and quantitative analyses of rainfall data. Thus, the unprecedented rainfall index is expected to be effectively used to facilitate proper evacuations of residents.

4.1.3 The Okaya disaster

In case of the Okaya disaster, LAI was issued at 1:25 p.m. on Aug. 14 (Fig. 11b) (Hiramatsu et al. 2022). However, the municipality did not issue the evacuation alert until 6:00 a.m. on Aug. 15, about 45 min after the debris flow occurred at Nakaohkubo Valley which caused 3 victims (The Mainichi Newspapers 2021).

In this region, torrential rains from Jul. 15 to 24, 2006 initiated landslides and debris flows, and 13 people were killed or missing. At that time, debris flows reportedly occurred in the Honzawa Valley and Matoba Valley adjacent to the Nakaohkubo Valley. Another historical disaster in this region was caused by heavy rainfall on Sep. 28–29, 1983 resulting from Typhoon Forrest. Floods, landslides, and debris flows occurred, which resulted in 9 fatalities (Geospatial Information Authority of Japan 2009).

From the TP time series shown in Fig. 11b, it is clear that the Okaya 2021 storm event had exceeded both of the storm event in Jul. 2006 and Typhoon Forrest in 1983, and reached a magnitude never experienced before at 9:00 p.m. on Aug. 14, 2021, more than 8 h before the occurrence of the debris flow. Most likely, such information is effective to reduce the normalcy bias letting local authorities and residents to take more appropriate evacuation actions.

4.2 Storm evaluation criteria

In this study, 1326 different API diagrams were used for computing TP values. That is, the half-lives M1 and M2, which were used to calculate API on the horizontal and vertical axes of each API diagram, respectively, were set in the range of 0.1 to 1500 h, in a total of 51 ways so that the change in the logarithmic value is constant. The use of various API diagrams corresponds to analyzing rainfall features using various evaluation criteria effective for predicting SRDs. As shown in Figs. 9 through 11, for every disaster analyzed in this study, rainfall reached a magnitude never experienced in the past at the time of the SRD occurrence. Here, the question is in which evaluation criteria the rainfall reached a magnitude never experienced in the past.

In Fig. 12, each circle shows the combination of M1 and M2 with which the API diagram exceeded the historical maximum just before or at around the time of the disaster occurrence. In this figure, the red circles represent API diagrams in which both of the API value on the horizontal axis (the horizontal API) and the API value on the vertical axis (the vertical API) exceeded their historical maximums (i.e., API diagrams in which the plot was located in the red region shown in Fig. 13). Especially, the red circles on the 1 to 1 line represents that M1 is equal to M2, indicating that rainfall exceeded the historical maximum for a single API rather than a combination of two APIs. In addition, the green circles represent API diagrams in which only the horizontal API exceeded its historical maximum, and the yellow circles represent API diagrams in which only the vertical API exceeded its historical maximum, corresponding to the green and yellow regions in Fig. 13, respectively. Moreover, the gray circles represent API diagrams in which, although neither of the horizontal and vertical APIs exceeded their historical maximums, the combination of the two APIs indicated that the rainfall reached a magnitude never experienced in the past (i.e., the gray region shown in Fig. 13).

Fig. 12
figure 12

Combinations of M1 and M2 with which the API diagram exceeded the historical maximum just before or at around the time of disaster occurrences for the a Atami, b Unzen, and c Okaya disasters. Colors of plots correspond to colors of regions shown in Fig. 13

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Fig. 13
figure 13

Schematic API diagram showing 4 regions in the range above the historical maximum

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In case of the Atami disaster, APIs with the half-lives of 4.7 × 101 through 2.7 × 102 h exceeded their historical maximums just before the occurrence of debris flow (the red circles on the 1 to 1 line in Fig. 12a). The API diagrams which have one of these APIs on both of the horizontal and vertical axes (the red circles in Fig. 12a) marked a magnitude never experienced in the past. Moreover, API diagrams which has one of these APIs on either of the horizontal or vertical axis (the green or yellow circles in Fig. 12a, respectively) exceeded the historical maximum. Furthermore, in some of the API diagrams with the combinations of horizontal APIs with half-lives of 3.2 × 102 through 1.5 × 103 h and vertical APIs with half-lives of 3.2 × 100 through 3.9 × 101 h (the gray circles in Fig. 12a), rainfall reached a magnitude never experienced in the past. These results indicate that the Atami 2021 storm event is characterized by large values of APIs with middle to long half-lives. At the same time, excesses of historical maximums were marked in the API diagrams with the combinations of a horizontal API with the half-life of 3.9 × 101 h and vertical APIs with half-lives of 1.0 × 10–1 through 8.3 × 10–1 h, as well as in the API diagrams with the combinations of a horizontal API with the half-life of 3.2 × 102 h and vertical APIs with half-lives of 1.0 × 10–1 through 1.2 × 100 h (the gray circles in Fig. 12a). This was attributable to the largest 1-h rainfall intensity during the Atami 2021 storm event (i.e., 24 mm/h) from 9:00–10:00 a.m. on Jul. 3, just before the occurrence of the debris flow (Fig. 9a).

In case of the Unzen disaster, APIs with the half-lives of 5.7 × 100 through 5.7 × 101 h exceeded their historical maximums at around the time of landslide occurrence (the red circles on the 1 to 1 line in Fig. 12b). Moreover, in some of the API diagrams with the combinations of horizontal APIs with half-lives of 6.9 × 101 through 1.0 × 103 h and vertical APIs with half-lives of 8.3 × 10–1 through 4.7 × 100 h (i.e., the gray circles in Fig. 12b), rainfall reached a magnitude never experienced in the past. Thus, the Unzen 2021 storm event is characterized by large values of APIs with relatively short half-lives in comparison with the storm event that caused the Atami disaster.

In case of the Okaya disaster, APIs with the half-lives of 1.0 × 101 through 1.5 × 102 h and 5.7 × 102 through 1.5 × 103 h exceeded their historical maximums just before the debris flow occurrence (the red circles on the 1 to 1 line in Fig. 12c). Moreover, in many of the API diagrams with the combinations of horizontal APIs with half-lives of 1.8 × 102 through 4.7 × 102 h and vertical APIs with half-lives of 1.0 × 10–1 through 8.3 × 100 h, and with the combinations of horizontal APIs with half-lives of 3.2 × 102 through 4.7 × 102 h and vertical APIs with half-lives of 1.8 × 102 through 2.2 × 102 h (the gray circles in Fig. 12c), rainfall reached a magnitude never experienced in the past. Comparing Fig. 12c with Figs. 12a, b, it is clear that the Okaya 2021 storm event recorded historical maximums in more evaluation criteria than the Atami 2021 and Unzen 2021 storm events.

Thus, the results shown in Fig. 12 depicted that the combinations of M1 and M2, which indicate the exceedances of historical maximum, are different from each other in the three disasters studied. This suggests that, for evaluating anomaly in rainfall and detecting the excesses over historical maximum, it is important to analyze rainfall features based on various evaluation criteria by using various API diagrams with different combinations of M1 and M2 values.

4.3 Thresholds of T P

In the current system operated in Japan, LAIs are announced when the magnitude of rainfall exceeds the threshold for SRD risks according to the rainfall forecast for the next 2 h. In case of the three disasters studied in this study, the disasters occurred when rainfall continued more than about 15 h after the LAIs were announced (Figs. 9 through 11). This suggests that the disasters occurred after the magnitude of the rainfall exceeded the thresholds by a large margin. As a general trend, the thresholds are often set low so that landslides and debris flows will not be missed. As a result, there are many cases of false-alarm where no intensive landslide or debris flow occurs even though LAI is issued; it is reported that, when LAI was issued, the percentage of occurrence of intensive landslides or debris flow remained at 4.7% over the 10-year period from 2010 to 2019 (The Asahi Shimbun Company 2021). This can be one of the reasons why, as in the three disaster cases examined in this study, the evacuation alert was not issued by municipalities or residents were not evacuated even after LAI was issued.

In case of the Unzen disaster, Fig. 10 indicates that, about 6 h after the LAI issue, the Unzen 2021 storm event exceeded the historic storm event in Aug. 1993. This means that, in the land where landslides and debris flows occurred during the storm event in Aug. 1993, this is the first time that rainfall of this magnitude has been experienced since the land was devastated by the 1993 storm event. Similarly, in case of the Okaya disaster, Fig. 11 indicates that TP went back to the historic storm event in Jul. 2006 at the time of LAI announcement. Hence, the land where debris flow occurred during the storm event in Jul. 2006, this is the first rainfall of this magnitude since the land was devastated by the 2006 storm event. In addition to the LAI announcement, communications of such information brought by the unprecedented rainfall index may help to better clarify the reasons for evacuation and the areas targeted for evacuation. Consequently, using the historical disaster occurrence time as one of the thresholds and increasing the level of alertness as TP exceeds the threshold may be effective in guiding appropriate evacuation behaviors of municipalities and residents.

In case of the Atami disaster, it is reported that the debris flow was triggered by the failure of the embankment slope which was constructed in or around 2010 (Imaizumi et al. 2022). Figure 9 shows that TP went back further than 2010 at 1:00 p.m. of Jul. 2, 2021, that was 30 min after the issue of LAI. This means that immediately after the LAI was announced, the embankment slope was experiencing torrential rains never experienced before. This information brought by TP is expected to strongly encourage municipalities and residents to take evacuation actions. Thus, the time of the embankment slope construction can be one of the thresholds of TP.

As described above, a warning system based on TP can issue warning and evacuation information that takes into account the past history of individual locations. That is, the time of past land alteration at an individual location might be set as the TP threshold as suggested by Kosugi (2022). This study pointed out that the occurrence time of historical disaster and the construction time of embankment slope as examples of such threshold, while Kosugi (2022) noted that the time when trees on a slope fell due to strong winds or the time when a slope subjected to strong shaking by an earthquake could also be a threshold of TP.

5 Conclusions

This study evaluated effectiveness of the unprecedented rainfall index, TP, proposed by Kosugi (2022) in achieving appropriate evacuation actions by municipalities and residents for reducing victims caused by SRDs. Based on the analyses for all disasters that caused death and missing of people in Japan in 2021, i.e., the Atami, Unzen, and Okaya disasters, we derived the following conclusions.

  1. 1.

    The information brought by TP time series has a large potential to reduce the normalcy bias in all 3 disasters studied, by unambiguously showing the abnormality of current rainfall in comparison with previous storm events which caused historical catastrophes.

  2. 2.

    For evaluating anomaly in rainfall and detecting the excesses over historical maximum, it is important to analyze rainfall features based on various evaluation criteria by using different combinations of various rainfall parameters.

  3. 3.

    The time of past land alteration at an individual location, such as an occurrence time of historical storm event and a construction time of embankment slope, can be set as a TP threshold, which might be effective for clarifying reasons for evacuation and areas targeted for evacuation.

To obtain TP values, this study used combinations of two parameters for evaluating rainfall features. In order to calculate more accurate TP values, combinations of three or more parameters should be considered in future studies, although more computational resources are required. In regard to the rainfall parameter for calculating TP values, it is important to consider a variety of parameters other than the API used in this study. Here, suitable parameters and suitable combinations of parameters likely vary from location to location due to differences in geology, topography, land use, etc. Detecting the most appropriate parameters and most appropriate combinations for each location is expected to contribute to improving the accuracy of SRD predictions and considered to be another important future research topic. For practical application, the effectiveness of TP and related ideas should be verified by increasing the number of SRD analyses in future studies.