1 Introduction

Many debris flows were triggered in the Chenyulan River Watershed in Taiwan in a rainstorm caused by the Typhoon Toraji. Most of these debris flows were gully type debris flows (Chen et al. 2005; Chen 2007). Gully type debris flows are dangerous and cause enormous risks. For example, the debris flows in Venezuela in 1999 with 15,000 deaths (Lopez et al. 2003) and the debris flows in China in 2010 causing 1744 casualties (Yu et al. 2010) are all gully type debris flows. Gully type debris flows differ from the so-called unconfined “hill slope debris flows” (VanDine 1985). Gully type debris flows take place in areas with significant gully topography (Liu et al. 2009). The gully type debris flows in the study area were triggered most of the time by flash floods (Chen et al. 2012) causing a so-called “fire hose” effect (Godt and Coe 2007). The fire hose effect is caused by the entrainment of material due to high shear stresses induced by a concentrated flow of water, as if the material had been washed away by a “fire hose” (Johnson and Rodine 1984; Coe et al. 1997; Griffiths et al. 2004).

To mitigate and prevent hazards induced by debris flows and related risks, one must understand the formation of these in order to make reliable forecasts. Many factors are related to the occurrence of debris flows like the basin gradient, the percentage of basin area with slopes greater than or equal to 30 %, basin ruggedness, additional measures of gradient, slope aspect, rainfall intensity, and soil properties, including the clay percentage, the percentage of organic matter, the soil granulometry and sorting, and the soil liquid limit. These were selected as the strongest determining factors for debris flow response in the forest fire burned basins of the Intermountain West (Cannon et al. 2010). The authors identified four groups of variables related to the occurrence of debris flows in burned basins: topography, degree of burning, soil properties, and rainfall variables.

Liu et al. (2009) among others stated that there are three groups of factors playing a major role in the formation of ordinary gully type debris flows. They are related to topography, geology, and hydrology. The topographical factors include watershed area, channel length, elevation difference, average slope, slope curvature, and a form factor (Lin et al. 2002; Lin 2009; Lan et al. 2004; Catani et al. 2005; Chang and Chao 2006; Chang 2007; Lu et al. 2007; Lee and Pradhan 2007; Chang and Chien 2007; Tiranti et al. 2008; Tunusluoglu et al. 2008; Ranjan et al. 2004; Akgun et al. 2008). The geological factors include among others lithology, soil type, fault length, distance to fault, and landslide area (Lin et al. 2002; Ohlmacher and Davis 2003; Lan et al. 2004; Catani et al. 2005; Lu et al. 2007; Lee and Pradhan 2007; Tiranti et al. 2008; Ranjan et al. 2004; Akgun et al. 2008). The hydrological factors include among others rainfall intensity, daily rainfall, cumulative rainfall, and antecedent rainfall (Chang and Chao 2006; Chang 2007; Lee and Pradhan 2007; Chang and Chien 2007; Tiranti et al. 2008). These factors were obtained from statistical correlation analyses of debris flow occurrences with topographic, geological, and hydrological aspects. Because of the uniqueness of these characteristics in each area, the correlation structure found in one area is not valid in other areas. This is why all these studies show so many different factors related to debris flow occurrence. It is therefore difficult to obtain a unique universal relationship between debris flows and topographic, geological, and hydrological characteristics. Past research revealed more than 47 factors related to debris flows (Lee 2006). The formation of debris flows will be much more predictable when topography, geology, and hydrology can be reduced to one single factor.

By focusing on the study of the process mechanisms, one can find significant and probably more general relationships between debris flow formation and the topographic, geological, and hydrological factors. In our case, they are the flash floods in large and steep channels which cause a so-called “fire hose” effect eroding especially the small size sediments (Godt and Coe 2007) leading to the formation of debris flows. So the major topographic, geological, and hydrological factors must refer to steep channels, small grain sizes of the sediments, and the development of large flash floods.

Yu et al. (2011) and Li (2012) proposed a topographic factor to describe the role of topography in the formation of debris flows triggered by the flash flood in channels. Yu et al. (2012) suggested a geological factor to represent the role of geology in the formation of this kind of debris flows. Shieh et al. (2009), Wu et al. (1990), and Tan and Han (1992) used the 1-h or 10-min rainfall intensity and effective cumulative precipitation to predict this kind of debris flows. These studies formed the base to analyze the formation of debris flows of 2001 in the Chenyulan River Watershed, which were triggered by flash floods in channels, using only 3 factors related to topography, geology, and hydrology. One critical parameter is derived from these three dimensionless factors in order to predict the occurrence of debris flows triggered by a “fire hose” mechanism.

2 The debris flow development in the Chenyulan River Watershed, Taiwan, during the Typhoon Toraji in 2001

Debris flows in the Chenyulan River Watershed were first reported during the Typhoon Wayne on August 22, 1986. During Typhoon Herb on July 29, 1996, debris flows occurred in more than 32 gullies (Lin et al. 2003; Jan and Chen 2005). The Chi–Chi earthquake (ML = 7.3), with a focal depth of 8.0 km, was triggered by reactivation of the Chelungpu fault in central Taiwan on September 21, 1999. This earthquake triggered many large landslides in central Taiwan (Kaima et al. 2000). Consequently, a great deal of loose sediments was produced, which in turn promoted heavy debris flows during subsequent typhoons and heavy rains. The most outstanding example was a large debris flow causing the deaths of more than 240 people during the Typhoon Toraji on July 30, 2001 (Lin et al. 2003). The rainfall threshold for debris flow initiation was significantly lower after the earthquake and recovered gradually in subsequent years (Shieh et al. 2009). The Typhoon Toraji, which occurred 2 years after the Chi–Chi earthquake, had a maximum rainfall intensity of 140 mm/h, a total rainfall of 700 mm, and a duration of 15 h in central and east Taiwan. It resulted in a severe debris flow development in the Chenyulan River Watershed (Lin et al. 2003).

Chen (2007) investigated after the Typhoon Toraji 117 gullies in the Chenyulan River Watershed with significant gully topography using SPOT images. She found debris flows in 65 out of a total of 117 surveyed gullies. Lin et al. (2003) used also SPOT images and carried out field investigations as well. He indentified debris flows in 55 out of the 117 gullies. The differences between the two studies are not only the number of debris flows, but also the uncertainty about the presence of debris flows in some gullies. In 43 cases, both studies agreed on the presence of debris flows in gullies, and in 40 cases, the two references agreed on the absence of debris flows, which means the presence of debris flows is uncertain in 34 gullies, because of the different judgment by both studies. Here, we define the occurrence of debris flows in the Chenyulan River Watershed during the Typhoon Toraji as “Yes” or “No” if the judgment of the two studies is consistent and as “uncertain” in case of a contrary decision of the two references (see Table 1; Fig. 1).

Table 1 Parametric values related to different debris flow forming factors and the presence of debris flows in gullies of the Chenyulan River Watershed
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Fig. 1
figure 1

The investigated gullies in the Chenyulan River Watershed

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The Chenyulan River is a main branch of the Chuo-Shui River, which is one of the three large rivers of Taiwan. The watershed lies in Central Taiwan and covers an area of 449 km2. The river with a total length of 42 km flows in a south–north direction (Fig. 1). The junction with the Chuo-Shui River lies at an altitude of 310 m. The highest peaks in the catchment have altitudes between 2,500 and 3,000 m. The channel gradients in the upstream part of the gullies are mostly more than 20°. These are suitable topographic conditions for debris flow outbreaks.

The four lithological units of the Chenyulan River Watershed are as follows: quartz sandstones alternated with hard shales, hard shales alternated with quartz sandstones, slates and siltstones interbedded with shales. The Chenyulan River follows closely the Chenyulanchi fault line. Some other faults can be found in the study area (Chen et al. 2005; Bai 2007; see Fig. 2; Table 1).

Fig. 2
figure 2

Geologic setting of Chenyulan River Watershed. Q6 Alluvium; MS Siltstones interbedded with shales; EH Slates; EO Quartz sandstones alternated with hard shales; OM Hard shales alternated with quartz sandstones

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The 32 debris flows which were triggered during the typhoon Herb in 1996 arose mainly by lateral and vertical erosion along the gully bed. Only a few landslides on steep and moderate slopes have been detected, which did not contribute to the debris flow development. Loose debris sediments were originally deposited in gully beds and these produced also during the Typhoon Toraji debris flows by severe and deep bed erosion. But also in areas without significant gully topography, the so-called slope debris flows (VanDine 1985; Lin et al. 2003) could develop in the original loose debris material. These slope debris flows are not considered in this study.

Long-duration rainfall such as produced by the Typhoon Herb in 1996 (more than 48 h), the Typhoon Midulle in 2004 (more than 48 h), and the Typhoon Morakot in 2009 (72 h) is the main cause for the development of large landslides in the study area (Chen et al. 2012). Short-duration high-intensity rainfall is the main inducing factor for the development of gully type debris flows with a “fire hose” mechanism like the Typhoon Toraji (duration 15 h; maximum intensity 140 mm/h) (Chen et al. 2012).

The Chenyulan River Watershed, located at some 12 km east of the epicenter, experienced 400–600 cm/s2 of ground acceleration during the Chi–Chi earthquake, which caused large-scale rock fracturing and landslides (Lin et al. 2003). Huang and Li (2009) pointed out that in the Wenchuan earthquake area (Sichuan Provence, China), the co-seismic landslides are mainly concentrated within a distance of 7 km from the triggering fault. A second zone at a distance between 7 and 11 km from the fault was distinguished with a medium landslide density and a third effected zone with the lowest landslide density at a distance of more than 11 km from the fault. Since the Chelungpu fault, which triggered the Chi–Chi earthquake, is about 15–45 km away from the study area, the distance from the fault must have only a slight differential effect on the landslide density distribution in the study area. Therefore, the effect of the Chi–Chi earthquake is assumed to be the same throughout the study area.

3 The formation factors of debris flows with a “fire hose” mechanism

3.1 The topographic factor

Generally, the catchment of a debris flow is subdivided into a formation or source section, a transport or passing section, and a deposition section. The topographical factors influence in different ways the processes in these three sections of the catchment. In this study, we will focus on the role of topographical factors in the formation (source) section of the catchment. Yu et al. (2011) and Li (2012) obtained a dimensionless topographic factor describing the role of topography in the formation of debris flows with a “fire hose” mechanism:

$$ T = FJ\left( {{\raise0.7ex\hbox{$A$} \!\mathord{\left/ {\vphantom {A {A_{0} }}}\right.\kern-0pt} \!\lower0.7ex\hbox{${A_{0} }$}}} \right)^{0.2} = J\left( {{\raise0.7ex\hbox{$A$} \!\mathord{\left/ {\vphantom {A {L^{2} }}}\right.\kern-0pt} \!\lower0.7ex\hbox{${L^{2} }$}}} \right)\left( {{\raise0.7ex\hbox{$A$} \!\mathord{\left/ {\vphantom {A {A_{0} }}}\right.\kern-0pt} \!\lower0.7ex\hbox{${A_{0} }$}}} \right)^{0.2} $$
(1)

in which T is the dimensionless topographic factor; F (=A/L 2) is the form factor of the formation section of the gully; J is the average slope of the channel in the formation section of the gully; A is the area of the formation section of the gully (km2); A 0 is the unit area of the gully (=1 km2), and L is the stream length in the formation section of the gully (km).

The stream length is determined by accumulating the length of a series of straight line segments along its flow direction. The form factor is calculated as the ratio between the area of the formation section and the square of the stream length in the formation section. The form factor is highly related to the distribution of the hydrograph: a larger form factor produces a larger discharge and velocity than a smaller form factor. Therefore, under the same conditions, a watershed area with a large form factor has a higher likelihood to generate debris flows (Chang 2007). The average slope of a stream is calculated as the elevation difference between the upslope end of the stream and the outlet of the section, divided by the stream length. These parameters also influence the surface flow discharge and the flow velocity and thus the resulting down slope movement of sediments. The topographic factors are calculated using Google Earth. The values of the topographic factor T for all the 117 gullies are listed in Table 1.

3.2 The geological factor

Accumulation of solid material into channels forms the source for the debris flows, which will be triggered by the erosion of flash floods (Wang and Fan 2006; Tan et al. 1994): The harder the rock, the greater the particle size of the solid material, and the longer the accumulation time and thus the more difficult to trigger debris flows. On the other hand, soft rocks have a larger potential to supply solid source material delivered by landslides. The smaller the particle size of solid materials, the faster the deposition of source material, and the easier debris flows are activated (Yu et al. 2012). By using this idea, Yu et al. (2012) obtained a dimensionless geologic factor to represent the role of geology in the formation of debris flows triggered by flash floods in channels:

$$ G = F_{0} C_{1} C_{2} C_{3} $$
(2)

in which G is the dimensionless geologic factor; F 0 is the average firmness coefficient of the lithology in the formation section of the gully; C 1 is a correction coefficient for seismic intensity in the formation section of the gully; C 2 is a correction coefficient for tectonics (faults); C 3 is the correction coefficient for weathering.

Based on field investigations, the average firmness coefficient for lithology F 0 was revised from the Protodrakonov Coefficient (Protodyakonov 1962) for rock strength (Yu et al. 2012, see Table 2). The values of this coefficient for the 117 gullies are given in Table 2, Figs. 1 and 2. As was mentioned before: four lithological units can be distinguished in the study area: (1) quartz sandstones with alternating hard shales, (2) hard shales with alternating quartz sandstones, (3) slates, and (4) siltstones interbedded with shales. In group (1), two-third consists of quartz sandstones (F 0 = 9) and one-third of hard shales (F 0 = 5) which gives a weighted average firmness coefficient of F 0 = 7.67 for group (1). Two-third in group (2) consists of hard shales (F 0 = 5) and one-third of quartz sandstones (F 0 = 9). Thus, the weighted average firmness coefficient of group (2) F 0 = 6.33. F 0 = 9 is the average firmness coefficient for group (3) (slates). The siltstones (F 0 = 6) and shales (F 0 = 4) are equally distributed in group (4) which delivers a weighted average firmness coefficient F 0 = 5. When the lithology in the formation section of a gully belongs to more than one group, the average firmness coefficient of the major group is taken as the average firmness coefficient of the gully. The mean value of the average firmness coefficients of groups is taken when the groups have nearly the same importance. The lithologies for the 117 gullies are listed in Table 1.

Table 2 The classification of rock types by firmness
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The correction coefficients C 1, C 2, and C 3 (see Eq. 2) of, respectively, seismic intensity, tectonics (faults), and weathering for the formation area of the gullies are listed in Table 3 (Yu et al. 2012).

Table 3 Correction factors for seismic intensity, faults (tectonics), and degree of weathering
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The seismic intensity in the study area is VIII, which gives a correction coefficient C 1 of 0.93 (see Table 3). The correction factor with respect to the tectonics of all the 117 gullies is determined with the Figs. 1 and 2 by counting the faults crossing the formation section of the gullies. The correction factor for the weathering in Table 3 is based on physical weathering. Physical weathering plays an important role in the formation of debris flows: the mechanically damaged and frigid-weathered clasts of blocks form the ideal solid source material. Important for physical weathering are the differences in day and night temperatures around 0 °C, which causes frost splitting of rock material. Rock damages and enlarging cracks are caused by frequently freezing and melting of water in fissures (Fookes et al. 1971). The intensity of weathering is mainly affected by the average annual rainfall and temperature of an area. The intensity of physical weathering is determined not only by the average annual rainfall and temperature but also by lithology. So C 3 and F 0 in Eq. 2 include the temperature and rainfall difference, and lithology for the physical weathering. The average annual rainfall lies in a range between 1,858 and 3,912 mm, and the average annual temperature varies between 15 and 24 °C in the study area. Thus, the coefficient C 3 in the research area is 1 according to Fookes et al. (1971). The resulting final geologic values for G for all the 117 gullies are listed in Table 1.

3.3 The hydrological factor

The hydraulic triggering conditions are narrowly related to the rainfall characteristics. Therefore, in this paper, we use the rainfall characteristics instead of the hydraulic conditions, which are difficult to assess. Short-duration high-intensity rainfall is the main triggering factor for the gully type debris flows in the Chenyulan River Watershed during Typhoon Toraji. Shieh et al. (2009), Wu et al. (1990), and Tan and Han (1992) used the 1-h or 10-min rainfall intensity and effective cumulative precipitation to predict the debris flows with a “fire hose” mechanism. The critical rainfall can be expressed as:

$$ S = B + KI $$
(3)

in which S is the critical rainfall (mm); B is effective cumulative precipitation, the rainfall until the start of the debris flow (mm); K (h or min.) is the coefficient of rainfall intensity; I is the rainfall intensity per 1 h or 10 min (mm/h, or mm/10 min).

Wu et al. (1990) indicated that the 10-min rainfall intensity is strongly correlated with the triggering of debris flows with a “fire hose” mechanism. Also in some cases, 1-h rainfall intensities show a relationship with the formation of debris flows like the Zhouqu event in China (Yu et al. 2010). However, 10-min intensities show better correlations with debris flow initiation. Unfortunately, only 1-h rain intensities are available in the study area. So, for the hydrological factor, we have to use the 1-h intensity.

Shieh et al. (1995) analyzed the critical rainfall with 1-h intensities triggering debris flows and obtained a K-value (Eq. 3) for the coefficient of rainfall intensity K = 38.5 in eastern Taiwan before 1995, but a K-value of 10.5 in the central area of Taiwan after the Chi–Chi earthquake in 2000–2001 (Shieh 2001). Jiang and Lin (1991) obtained K values between 179 and 16.7 in different area of Taiwan during the 1980s. Fan and Yao (1997) found a K-value of 10.2 in eastern Taiwan before 1996, and Tan (1991) pointed out that the K-coefficient was 4.1 in the mainland of China during the period 1975–1985. Jan et al. (2002) obtained a K-value of 10 for the central area of Taiwan in the period 2000–2002 after the Chi–Chi earthquake. So these empirical analyses revealed a large range for the coefficient of rainfall intensity K varying between 4.1 and 179. For the Chi–Chi earthquake affected area, the K-coefficient lies within a smaller range between 10 and 13.7 (I in Eq. 3 is the rainfall intensity per hour in this case) (Shieh 2001; Shieh et al. 2009; Jan et al. 2002). After 7 years of observations at two sites, Shieh et al. (2009) obtained coefficient values of 11.3 and 13.7, respectively. In this study, the coefficient of rainfall intensity K is taken as the average of these two values: 12.5. The cumulative rainfall and 1-h maximum rainfall for each of the 117 gullies are obtained by interpolation of the spatial distribution over the area of the cumulative rainfall and 1-h maximum rainfall given by Chen (2007). Chou et al. (2002) pointed out that in Taiwan between 1981 and 2000, 77 % of the debris flows occurred about half way the duration of the rainfall. Since we do not know the effective cumulative precipitation (B in Eq. 3), we consider half of the cumulative rainfall as the effective cumulative precipitation. The critical rainfall S of the 117 gullies could then be calculated with Eq. 3.

Aleotti (2004) used the annual precipitation to normalize the critical rainfall. This kind of normalization is very important because rainfall values vary widely between the different areas. The difference may be reduced by introducing a normalization with the annual precipitation. Because of the scarcity of rainfall data in the research area during the Typhoon Toraji, the annual precipitation was the only option to normalize the critical rainfall. The normalized critical rainfall (Aleotti 2004) is used for the dimensionless hydraulic (rainfall) factor (Eq. 4):

$$ R = {\raise0.7ex\hbox{$S$} \!\mathord{\left/ {\vphantom {S {R_{0} }}}\right.\kern-0pt} \!\lower0.7ex\hbox{${R_{0} }$}} = {{\left( {B + KI} \right)} \mathord{\left/ {\vphantom {{\left( {B + KI} \right)} {R_{0} }}} \right. \kern-0pt} {R_{0} }} $$
(4)

in which R is the dimensionless hydraulic (rainfall) factor; R 0 is the annual precipitation of the site (mm). The annual precipitation R 0 for each gully is obtained from the spatial distribution of annual rainfall in the study area. The values of the hydraulic factor R for all the 117 gullies are listed in Table 1.

4 The formation model for debris flows

The formation conditions for debris flows in the Chenyulan River Watershed during the Typhoon Toraji can now be expressed by the topographic factor T, the geological factor F, and the rainfall factor R. In order to arrive at one single factor for the formation of debris flows, the relation field between these three factors for gullies with debris flows and no debris flows needs to be analyzed. First, the areas with the same lithology are chosen to study the relation field between factor T and factor R for gullies with debris flows and no debris flows. Figure 3 shows the scatter plot of the gullies with debris flows and no debris flows (including the uncertain observations) for the four lithological units. Apart from the uncertain cases, gullies with debris flows and no debris flows can be found in three out of four lithological units: group (1) (quartz sandstones alternated with hard shales), group (2) (hard shales alternated with quartz sandstones), and group (4) (siltstones interbedded with shales). In gullies belonging to group (3) (slates), no certain debris flows were found. For the groups (1), (2), and (4), the function RT 0.2 = C (C is a constant) made an optimal distinction between debris flow and no debris flows. In Fig. 3a, b, the critical lines described with RT 0.2 = 0.47 and 0.44, respectively, can divide all “debris” from most of the “no debris” cases. In Fig. 3d, the critical line described with RT0.2 = 0.41 divides a large part of “debris” from all cases of “no debris”. So a first exponential graph with T and R (RT0.2) could be obtained, separating more or less debris flows from no debris flows for three lithological units.

Fig. 3
figure 3

Relationship of the topographic factor T and the hydraulic factor R for areas with different lithology. a Quartz sandstones alternated with hard shales. b Hard shales alternated with quartz sandstones. c Slates. d Siltstones interbedded with shales

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The average firmness coefficients F 0 of the three lithological groups (4), (2), and (1) are 5, 6.33, and 7.67, respectively (see section “Geological factor”). Now one can set up a relationship between the critical values RT 0.2 = 0.41, 0.44, 0.47, and the average firmness coefficients for the three lithological units. The graph can be described with the function RT 0.2/F 0.50  = 0.18 by regression analysis, (R 2 = 0.86), which makes a reasonable fit with the 3 critical values (see Fig. 4). The formation factor P can be obtained from this relationship between T, G, and R (Eq. 5):

$$ P = {\raise0.7ex\hbox{${RT^{0.2} }$} \!\mathord{\left/ {\vphantom {{RT^{0.2} } {G^{0.5} }}}\right.\kern-0pt} \!\lower0.7ex\hbox{${G^{0.5} }$}} \ge C_{r} $$
(5)

in which P is the formation factor; C r is a critical value for the formation of debris flows.

Fig. 4
figure 4

Relationship of RT 0.2 and 1/F 0

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The formation factor P for all the 117 gullies is listed in Table 1. Figure 5 shows a scatter plot of R against T 0.2/G0.5 (from Eq. 5) for all the 117 gullies and three graphs for three different values of P: marking three critical probability values for debris flow formation: C r1 = 0.13, C r2 = 0.15, and C r3 = 0.19. These critical values deliver a division into four classes of the probability of debris flow occurrence. Debris flows are hardly formed in the area with P < C r1. This area can be considered as a very low probability or safe area. Few debris flows are formed in the area between 0.13 ≤ P < 0.15, which is a low probability area where one has to watch out. Some debris flows are formed in the area between 0.15 ≤ P < 0.19, which makes this area a medium probability or an alarm area. When P ≥ 0.19, debris flows are triggered in most gullies, which makes it a high probability area. In this area, people have to be evacuated to safer places. Table 4 shows the number and percentages of debris flows and no debris flows for the different classes distinguished by Eq. 5. There is 1 debris flow (2.3 % of all debris flows) and there are 12 no debris flows (30 % of all no debris flows) in the safe area class. There are 27 debris flows (62.8 % of all debris flows) and 6 no debris flows (15 % of all no debris flows) in the high probability class.

Fig. 5
figure 5

Formation model showing probability fields for the occurrence of debris flows induced by the Typhoon Toraji in the Chenyulan River Watershed

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Table 4 The probability areas of debris flow occurrence distinguished by Eq. 5
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Equation 5 shows that the hydraulic factor R has the largest influence on the formation factor P compared to the other two factors. The exponential values of G and T are 0.5 and 0.2, respectively. Therefore, the geological factor G is more important than the topographic factor T. This is why in many studies only the hydraulic factors are used empirically to analyze the occurrence of debris flows.

5 Discussion

In 2001, the Typhoon Toraji provided an unprecedented amount of data for studying the formation of debris flows with a so-called “fire hose” mechanism in the Chenyulan River Watershed. With the lack of a detailed field survey, some mistakes are made by two studies using only SPOT images to determine the occurrence of debris flows in this area. The category “uncertain” for the occurrence of debris flows is added in this study to account for the difference in judgment between the two studies.

Neglecting the impact of the uncertain cases in the results, the distinction between occurrence and no occurrence of debris flows by means of the formation factor P is acceptable despite the fact that one debris flow occurred in the area with very low probability and 6 gullies with no debris flows are found in the area with very high probability. However, the uncertainty in the detection of debris flows may affect the result of the formation model, and the hypothesis of a “fire hose” mechanism for all the gullies is also another source of errors.

The study area experienced 400–600 cm/s2 of ground acceleration during the Chi–Chi earthquake, which significantly decreased the triggering threshold for debris flows. In this study, the effect of the Chi–Chi earthquake is assumed to be the same all over the area. This may also form a source of errors for the judgment of the formation of debris flows with the formation factor P. The rainfall threshold for debris flows recovered gradually in subsequent years after the earthquake but is still lower than the original threshold before the earthquake. So the critical value C r in Eq. 5 will increase in the subsequent years after the Typhoon Toraji. Future investigations of the influence of earthquakes on the change of the critical value with the years are needed.

The evaluation of F 0, C 1, C 2, and C 3 was supported by some field investigations and verified by some references (Yu et al. 2012). But a multiplication of these four factors as presented in Eq. 2 may be not the best solution although it works quite well in some areas. Future investigations on analysis of the relationships among these factors are needed.

Landslides, channel bed erosion, and destruction of natural dams are three common causes that trigger debris flows (Takahashi 2000). In this study area, only the “fire hose” mechanism (channel bed erosion) is considered as the trigger mechanism of debris flows in the study area. The formation factor P is not suitable for the other mechanisms of debris flow formation like landslides and natural dam breaks which occurred during the Typhoon Herb, the Typhoon Midulle, and the Typhoon Morakot.

In this study, only 5 lithologies were involved in the determination of the geological factor. The applicability of the geological factor in other areas is an important subject for the future research. The relationship between RT 0.2 and F, and the function P = RT 0.2/G 0.5 is based on only 3 points (see Fig. 4). Future work with a larger number of lithological classes is needed to test the relationship between R, T, and G.

The critical value C r3 is 46 % higher than the critical value C r1, which shows a moderate performance of the formation model. For a more accurate prediction of the occurrence of debris flows, more research is needed to reduce the difference between C r1 and C r3.

6 Conclusions

The Typhoon Toraji, which occurred 2 years after Chi–Chi earthquake caused a severe debris flow activity in the Chenyulan River Watershed. Short-duration high-intensity rainfall was the main inducing factor for these gully type debris flows with a “fire hose” mechanism. This research proposed a formation model for debris flows based on a topographic, geological, and hydraulic factor. The type of these dimensionless factors is not deduced from the statistic analyses of a given area but based on the mechanism for the formation of debris flows. The formation model has a general nature and can be used in other areas, despite the fact that the formation factor P is based in this study on a statistical analyzes of the Chenyulan River basin.

With the combination of the topographic, geological, and hydraulic factor, a formation factor was obtained which delivered a good judgment of the occurrence of debris flows in the study area. In our view, the formation model offers a new and exciting way to forecast the probability of occurrence of debris flows with a “fire hose” mechanism. However, to improve the understanding of the occurrence and triggering mechanisms of debris flows, and for a better understanding of the long-term impact of earthquakes on the spatial and temporal frequency of debris flows, future research is needed to determine the set of errors in mapping debris flows with SPOT images, the change in critical probability values after successive years, and to reduce the difference between critical probability values.