Introduction

Located at the intersection of the Eurasian Plate and the Philippine Sea Plate, Taiwan frequently experiences earthquakes and other geological activities. Complex geological structures, such as widely and densely distributed faults, folds, and joints, fracture rocks into unstable piles, which in turn provide materials for debris flows. In addition, Taiwan’s small overall area (total 36,000 km2) and steep topography (highest elevation: 3952 m above sea level) together make streams in Taiwan short yet torrential. Moreover, the annual rainfall of Taiwan, an island simultaneously in the subtropics and in the western Pacific monsoon area, reaches 2500 mm. This rainfall and heavy rains brought by typhoons provide sufficient water for debris flows to occur, and when storms strike Taiwan, debris flows often take lives and damage property. It is thus very important to predict and prevent debris flows. One way to do so is to clarify certain relations between debris flow occurrence and certain physiographic factors. On the whole, the essential conditions for a debris flow to occur include sufficient water, deposits, and slope steepness. Of these factors, sufficient water is the major triggering factor in the occurrence of a debris flow, while the others represent the potential for a debris flow. The higher the potential is for debris flow, the less is the water required to trigger it. Therefore, physiographic factors such as geological conditions, slope steepness, and landslide area significantly affect the occurrence of debris flows. The main purposes of this research were:

  1. 1.

    To identify the most significant and correlated physiographic factors in debris flow occurrence.

  2. 2.

    To examine the relations among such factors (including physiographic factors and rainfall) and the potential for debris flow occurrence, and further to establish a model of such relations.

  3. 3.

    To demonstrate that the established model can be used to mitigate and prevent debris flows.

The establishment of a critical rainfall value for debris flows has been studied in many different ways with different parameters. Caine (1980), Cannon and Ellen (1985), Wieczorek (1987), and Keefer et al. (1987) used rainfall intensity and rainfall duration, without considering antecedent precipitation, to establish the critical rainfall value for debris flows. Aboshi (1972) applied the daily rainfall of the day on which a debris flow occurred and the antecedent precipitation of 14 days before that day. Wilson (1997) used the daily rainfall of the day on which a debris flow occurred and the annual average precipitation. Senoo and Yokobe (1978) developed a model that used effective rainfall intensity and effective rainfall to distinguish between rainfall events with or without debris flows. Using effective accumulated rainfall and effective rainfall intensity as indices, Shieh (1991) and Shieh et al. (1995) assessed the critical rainfall value for debris flows in Hualien County, Taiwan. Jan and Lee (2004) used rainfall intensity and accumulated rainfall to define the critical rainfall threshold of debris flows. Taking into account the effects of antecedent precipitation on the occurrence of debris flows, Fan et al. (1999) and Fan and Wu (2001) used a recession coefficient to address the effect of antecedent rainfall on effective accumulative rainfall, and further determined the relationship between the effective accumulated rainfall and the rainfall duration. Fan et al. (2003) used five physiographic and mechanical factors, along with the effective accumulated rainfall and rainfall duration, to evaluate the critical rainfall value for debris flows in Nantou County, Taiwan. Pradhan and Lee (2010) investigated the landslide susceptibility in Malaysia using GIS, remote sensing techniques, and several physiographic or topographic factors.

It therefore appears that the models for debris flow warnings developed by different researchers for different study sites do not have similar forms. The selection and definition of factors, such as the division method of rainfall events and the incorporation and selection of physiographic factors, will affect the prediction of debris flow occurrence. In addition, since the main causes of debris flows include both physiographic and hydrologic properties, considering physiographic factors is a more reasonable way to identify a mechanism. Accordingly, in this study, the model developed by Fan et al. (2003) was selected and was further integrated with the conversion of physiographic factors using membership functions.

The model proposed by Fan et al. (2003) considers five physiographic factors: the land use factor, the percentage of soil particles greater than Sieve No. 4, the main stream length, the effective watershed area, and the slope steepness of the streambed. However, the model does not consider some other factors significantly affecting debris flow occurrence, such as landslide area. Therefore, the main purposes of this study are to investigate mutually independent physiographic factors that are determinants of the occurrence of debris flows, and to build a model for evaluating the effects of these physiographic factors on the probability that debris flows will occur.

Most previous studies on debris flow occurrence and warning systems have treated the critical rainfall threshold as the end result. In this study, the results are presented in the form of the occurrence probability of debris flows. This feature allows the setting of an acceptable rainfall threshold based on risk tolerance. The study area was the watershed of the Chenyulan stream in Nantou County in central Taiwan. Statistical tests were conducted to evaluate the independency and significance of different factors, and logistic regression was used to build a model. The model can estimate the probability that a debris flow will occur in a specific torrent with known physiographic and hydrologic factors.

Materials and methods

Study area

The study site was the watershed of the Chenyulan stream, located in the southeastern part of Nantou County in Taiwan and having an area of 448 km2. The Chenyulan stream, with a length of 42 km, is one of the main tributaries of the Jhuoshuei stream. The Jhuoshuei stream, having a length of 186.6 km, is the longest in Taiwan. The southeastern part of the watershed is at a higher altitude than the northwestern part. The average slope steepness of the Chenyulan stream is 6.75 %, and the elevation ranges from 305 to 3926 m. The average annual precipitation in the watershed is 2097 mm (from 1923 to 2010). The location and the stream system of the watershed are shown in Fig. 1. Chenyulan stream is located in central Taiwan at the junction of the Shuehshan Range and Western Foothills. The stream closely follows the Chenyulan fault, which is a reverse fault. Due to the influence of the reverse fault, the vastly fractured strata are prone to landslides and debris flows (Lin et al. 2003a). The rock formation in the western part of the watershed is mainly Neogene sedimentary rock, while that in the eastern part is mainly Paleogene metamorphic rock. In addition, there are differences between the geological conditions of the western part and the eastern part of the watershed. A geological map and a geomorphological map of the watershed of Chenyulan stream are provided in Figs. 1 and 2.

Fig. 1
figure 1

The stream system and geological map of the watershed of Chenyulan stream, Nantou, Taiwan (modified 1:250,000 geological map of Taiwan, Central Geological Survey 2003)

Full size image
Fig. 2
figure 2

The geomorphological map of the watershed of Chenyulan stream, Nantou, Taiwan

Full size image

The flows that occur in this area are mostly debris flows consisting mainly of colluvial soils. As indicated by USGS (2004), the debris flow is a form of rapid mass movement in which a combination of loose soil, rock, organic matter, air, and water mobilize as a slurry that flows downslope. Debris flows include <50 % fines. Debris flows are commonly caused by intense surface-water flow, due to heavy precipitation or rapid snowmelt, which erodes and mobilizes loose soil or rock on steep slopes. Debris flows also commonly begin from other types of landslides that occur on steep slopes. They are nearly saturated and consist of a large proportion of silt- and sand-sized material. Debris-flow source areas are often associated with steep gullies, and debris-flow deposits are usually indicated by debris fans at the mouths of gullies. The watershed of Chenyulan stream is close to fault zones, where the materials consist mainly of fractured or weathered rocks and partly of soils (including gravels, sands, and fines). Fan et al. (2009) analyzed the particle size distribution of materials in this area. The results showed that the percentages of materials coarser than No. 4 sieve (4.75 mm), finer than No. 4 sieve and coarser than No. 200 sieve (0.075 mm), and finer than No. 200 sieve were approximately 69, 20, and 11 %, respectively. As indicated by the aforementioned information and as shown in Fig. 3, the flows in this study are referred to as debris flows.

Fig. 3
figure 3

a Aerial photo of the watershed of potential debris-flow torrent no. DF199 (The Soil and Water Conservation Bureau of Taiwan, 2009). b A picture taken at potential debris-flow torrent no. DF200 in Nantou after a debris flow occurred on August 8, 2009 (The Soil and Water Conservation Bureau of Taiwan, 2009). c A picture taken at potential debris-flow torrent No. DF198 in Nantou after a debris flow occurred on August 8, 2009 (The Soil and Water Conservation Bureau of Taiwan, 2009)

Full size image

Data source

Rainfall data

In this study, hourly rainfall data were used for the analyses. The rainfall data were obtained from seven weather observation stations of the Central Weather Bureau of Taiwan located in Shenmu Village, Xinxing Bridge, Fongciou, Sinyi, Shangan Bridge, Heshe, and Longshen Bridge. The rainfall data used were recorded from January 1996 to May 2011. The locations of the weather observation stations are shown in Fig. 4.

Fig. 4
figure 4

The distribution of rainfall observation stations and potential debris-flow torrents in the Chenyulan stream watershed, and the locations of debris-flow events

Full size image

Potential debris-flow torrents and debris-flow events

In this study, the potential debris-flow torrents in the watershed of the Chenyulan stream were investigated. As announced in 2011 by the Soil and Water Conservation Bureau of Taiwan, the Chenyulan Stream watershed has 47 potential debris-flow torrents, including 31 torrents of high potential, 15 torrents of medium potential, and 1 torrent of low potential. The spatial distribution of the torrents is shown in Fig. 4.

In order to determine the relationship between landslides and debris flow occurrence, annual reports by the Soil and Water Conservation Bureau of Taiwan for debris flows, site investigations, and treatment plans were collected. The collected information showed that 100 debris-flow events occurred in the watershed from January 1985 to May 2011. The information collected in this study included the location and occurrence time of debris flows, and rainfall events. However, only 40 of the 100 debris-flow events had clearly recorded occurrence times, so only those 40 events were used as the occurrence samples for building the model. The occurrence times of debris flows, the potential debris-flow torrent numbers, and the rainfall observation stations of the 40 debris-flow events are shown in Table 1. The locations of the debris-flow events are also shown in Fig. 4. From Table 1 and Fig. 4, it can be seen that the 40 debris-flow events were distributed evenly throughout the study area. They were not related to any specific region or rainfall, nor were they specially chosen. Hence, they stand well as research samples.

Table 1 Occurrence time of debris flow, potential debris-flow torrent no. and rainfall observation stations
Full size table

Methods

Division methods of rainfall events and calculation of effective rainfall

For this research, methods to calculate the effective cumulative rainfall from several studies were compared, and finally those from the Soil and Water Conservation Bureau of Taiwan (SWCB 2003) and Fan et al. (2003) were selected. The main difference between the two is that SWCB’s method takes antecedent rains into consideration (with a decay coefficient of 0.8) and neglects the decay of the present rain, while the method from Fan et al. (2003) does the opposite. Given that streams in mountain areas in Taiwan are torrential and have steep streambeds (for example, the streams considered in this research have average streambed steepness of up to 17.2°), the runoff of such streams flows downstream rapidly. In these conditions, neglecting the decay of the present rain would lead to overestimation of the stream runoff. Since previous studies have indicated that the method from Fan et al. (2003) increases the accuracy of debris-flow prediction models, that method was used in this study.

According to the definition by Fan et al. (2003), and for the purposes of this study, an effective rainfall event is defined as beginning when the preceding accumulated 24-h rainfall is no less than 10 mm, and ends when the following accumulated 24-h rainfall is less than 10 mm. A schematic diagram of the division method is shown in Fig. 5. The effective accumulated rainfall (ER) is calculated as shown in Eq. (1).

$${\text{ER}} = \sum\limits_{t = 0}^{i} {\alpha_{h}^{t} d_{t} }$$
(1)

where i indicates the ith hour after the starting time, α h indicates the hourly recession coefficient, and d t indicates the hourly rainfall at a given time. According to the study by Fedora and Beschta (1989), the recession coefficient for a time period of 2 h is calculated as shown in Eq. (2).

$$K = 0.881 + 0.00793 \times \ln \left( A \right)$$
(2)

where A indicates the area of the watershed (in hectares), and α h is equal to \(\sqrt K\). Using Eqs. (1) and (2), the effective accumulated rainfall at the ith hour after the starting time for a certain watershed can be calculated. Because rainwater will infiltrate into the soil during the time between the beginning of rainfall and the occurrence of debris flows, it is reasonable to consider rainfall decay. The method of obtaining the decay coefficient proposed by Fedora and Beschta (1989) was used in this study.

Fig. 5
figure 5

The division method for rainfall events proposed by Fan et al. (2003)

Full size image

Selection and analysis of physiographic factors

To build an appropriate set of factors for this study, the physiographic factors related to the occurrence of debris flows were collected from a literature review, and the significance of each factor to the occurrence of debris flows was tested. The effects of physiographic factors on the occurrence of debris flows have been studied (e.g., Fan et al. 2003; Glade 2005; Pradhan and Lee 2010) with different methods, such as field surveys, statistical analyses, empirical or a semi-empirical approaches, satellite image recognition, and the geographical information system (GIS). To find the most frequently used factors, 20 studies were collected, each having different combinations of factors, and a total of 46 factors were listed. The top eight most frequently used factors were selected to further undergo tests to determine their correlations with debris-flow events. Through this process, the most significant factors could be culled and used in this study. Based on the experience of previous researchers, factors that are easily obtained, easily quantified, and physically significant are generally preferred. The authors selected 8 of the 46 frequently used factors. In descending order of frequency of use, they were as follows: average steepness of the streambed, effective watershed area, watershed area, rock classification, form factor, landslide ratio, length of the stream channel, and average steepness of the watershed. These eight physiographic factors are defined and explained in Table 2. The selected factors were obtained via analysis of the digital terrain model and the software ArcGIS, except for landslide area, which was provided by the Soil and Water Conservation Bureau of Taiwan. These factors were surveyed and analyzed at different times, as shown in Fig. 6, together with the distributions of landslides in the watershed of the Chenyulan stream at different times. Certain physiographic factors, such as watershed area, form factor, and streambed slope, are unlikely to undergo significant changes with time. However, the landslide ratio does change with time significantly, especially after storm events. Hence, satellite images (provided by the Soil and Water Conservation Bureau) taken at nine different times were used in this study to determine the landslide ratios at the times debris flows occurred. Satellite images were used to find exposed regions, after which structures like roads and buildings were excluded. The remaining area was assumed to be the area of the landslide, on which the landslide ratio in this study is based. Thus, here the term “landslide” refers to all kinds of landslides, including falls, topples, slides, lateral spreads, and flows, as defined by Varnes (1978).

Table 2 The definition and explanation of the eight physiographic factors
Full size table
Fig. 6
figure 6

The distribution of landslides in the watershed of the Chenyulan stream at various times

Full size image

The eight selected factors were further tested for their associations with debris flows and independency. The Spearman rank correlation test and the Mann–Whitney U test are both nonparametric tests suited for analysis of non-normal-distribution populations or small sample sizes. Generally speaking, when statistical tests are applied, the significance levels are commonly set at 0.1, 0.05, or 0.01. A p value lower than the significance level indicates significant correlation between two sets of samples. In this study, the significance levels for both the Spearman rank test and the Mann–Whitney U test were set as 0.1 (90 % confidence interval).

The Spearman rank correlation method is used to test the association between two ranked variables, or between one ranked variable and one measurement variable. This method has been widely and successfully applied in biological and medical research (e.g., de Jong et al. 2006; Melfi and Poyser 2007). In a Spearman rank test, the correlation coefficient, indicated as r, ranges from −1 to 1. The closer |r| is to 1.0, the more the two variables are linearly correlated, whereas the closer |r| is to zero, the less the two variables are linearly correlated. Generally speaking, |r| < 0.4 indicates low linear correlation, 0.4 ≤ |r| < 0.7 indicates moderate linear correlation, and 0.7 ≤ |r| < 1.0 indicates high linear correlation. Furthermore, if test results show that the p value of two certain factors is less than 0.1, the significance level, then the correlation coefficient of those factors is significant. Therefore, if two factors share a high correlation coefficient and a low p value, they can be simply represented by one of the two. As shown in Table 3, watershed area and effective watershed area share a high correlation coefficient and a low p value, indicating high and significant linear correlation, so in the analysis thereafter, watershed area was excluded. Likewise, length of stream channel was also excluded.

Table 3 Analytical results for the independence of the physiographic factors related to the occurrence of debris flow
Full size table

The Mann–Whitney U test was used to test the association of the remaining factors with debris flow occurrence. Rainfall event samples were collected from 47 potential debris-flow torrents, consisting of 31 torrents with records of debris-flow occurrence and 16 torrents without. The collected samples were further divided into two groups: rainfall events that triggered debris flows and rainfall events that did not. The Mann–Whitney U test revealed whether a specific physiographic factor was significantly associated with debris-flow occurrence, based on two inputs: whether debris flows occurred, and the corresponding values of that physiographic factor. The Mann–Whitney U test resembles the parametric t test; both can be used to test if the two populations from which two sets of samples are drawn have the same mean (Haring et al. 1989; Karjalainen 2007; Michael et al. 2005). In this study, if the results of a Mann–Whitney U test showed that the p value of a certain physiographic factor was less than 0.1, the significance level, then the tested factor was significantly related to debris flow occurrence. This study compared the results of the Mann–Whitney U tests of specific physiographic factors to discover their significance in debris flow occurrence. As shown in Table 4, due to high p values, average steepness of the watershed and rock classification were excluded from further analysis.

Table 4 Analytical results for the correlation among the physiographic factors related to the occurrence of debris flow
Full size table

Quantification of physiographic factors

Two methods of quantification of factors were used to determine the influence of each physiographic factor on the occurrence of debris flows. The first method used the genuine value of each factor, whereas the second method used the degree of membership in fuzzy theory to represent each factor, as proposed by Wang et al. (2011). The membership function for each physiographic factor was established using information on the potential debris-flow torrents in the watershed of the Chenyulan stream. In addition, in this study, the models, built according to two quantitative methods, were compared for the assessment of the occurrence probability of a debris flow (i.e., the genuine value method, PF1, and the degree of membership method, PF2. The methods of PF1 and PF2 will be compared and discussed in more detail in “The model for occurrence probability of debris flows”.

The model for the occurrence probability of debris flows

In this study, to analyze the occurrence of debris flows, a logistic regression method from multivariate statistical analysis was used. Since the method of logistic regression is a statistical approach, it lacks a sound mechanical base. Nevertheless, using this method, a large amount of hydrological and physiographic factors can be integrated, and the binary data (presence or absence of a debris flow) can be well processed. Logistic regression has been used in some studies evaluating the occurrence of landslides in the watersheds of the Chenyulan stream and the Baishi stream, such as Chang et al. (2007) and Chang and Chiang (2009). The results demonstrated that the accuracy in identifying landslides for the two watersheds was 78 and 84 %, respectively.

A logistic regression model, in which the variables are binary, is a special form of log-linear model (Agresti 2002). The model can be expressed as shown in Eq. (3). In this equation, p i indicates the occurrence probability for the ith sample for a given series of independent variables, and α and β k are constants. In this study, x ki indicates the factor vector of the ith sample. If a debris flow occurs within the sample, then p i  = 1, whereas if a debris flow does not occur within the sample, then p i  = 0. Using the training data and regression analysis, α and β k can be obtained. The variables in this model consist of physiographic factors, which are already known, and rainfall factors, which can be obtained with the method proposed by Fan et al. (2003).

$$\begin{gathered} Z_{i} = \ln \left( {\frac{{p_{i} }}{{1 - p_{i} }}} \right) = \alpha + \sum\limits_{k = 1}^{k} {\beta_{k} x_{ki} } \hfill \\ p_{i} = \frac{{\exp \left( {Z_{i} } \right)}}{{1 + \exp \left( {Z_{i} } \right)}} \hfill \\ \end{gathered}$$
(3)

For the training data for this model, the results of the study by Can et al. (2005) were used. For logistic regression models applied to landslides, one can obtain the best results if the training data consist of samples of occurrence and of non-occurrence in equal quantities. Accordingly, in this study, equal numbers of two kinds of binary data samples were selected to comprise the training data set, whereas the remaining data were then used for verification. A classification error matrix was used to evaluate the performance of the model, as shown in Table 5. That table also presents the three indices for the evaluation of model performance: sensitivity, specificity, and overall accuracy.

Table 5 The evaluation of model performance using a classification error matrix
Full size table

Results and discussion

Analytical results of representative physiographic factors

In this study, eight physiographic factors were selected and tested: the average steepness of the streambed, the effective watershed area, the watershed area, the rock classification, the form factor, the landslide ratio, the length of the stream channel, and the average steepness of the watershed. Statistical analyses were conducted on the representative physiographic factors so that their influences on the occurrence could be evaluated more appropriately. These analyses are described as follows.

Test of independence

The results of the Spearman rank correlation test are shown in Table 3. As shown in that table, the coefficients of association for the effective watershed area versus the watershed area, the effective watershed area versus the length of the stream channel, and the watershed area versus the length of the stream channel were 0.99, 0.80 and 0.81, respectively. It can also be seen that their coefficients of significance were less than 0.1, indicating that the three physiographic factors were highly correlated.

According to the studies by Takahashi (1978) and Shieh (1991), for all of the streams in which debris flows had ever occurred, the steepness of the streambed exceeded 15°. Therefore, the effective watershed area represents the three physiographic factors. The coefficients of association for the factor of effective watershed area and the other five factors are relatively lower. Accordingly, in this study, six physiographic factors—the average steepness of the streambed, the effective watershed area, the form factor, the average steepness of the watershed, the rock classification and the landslide ratio—were used for the test of association in order to evaluate the correlation between the physiographic factors and the occurrence of debris flows.

Test of association

The Mann–Whitney U test was used to examine the correlation between the six physiographic factors and the occurrence of debris flows. The results are shown in Table 4. As described in “Selection and analysis of physiographic factors”, a significance level of less than 0.1 indicates that the association between that factor and the occurrence of debris flows is significant. Among the six physiographic factors, the landslide ratio was found to have the lowest value, 0. Therefore, the landslide ratio was the most highly correlated with the occurrence of debris flows. For the other five physiographic factors, the average steepness of the watershed and the rock classification were comparatively less correlative with the occurrence of debris flows. There may be two reasons why the correlation of the rock classification and the occurrence of debris flows is comparatively lower. First, rock classifications throughout the study area have low variation; second, the materials of the debris flows in the study area are mainly colluvial soils. For the average steepness of the streambed, the effective watershed area, and the form factor, the results did not reach significance, but they were correlated with the occurrence of debris flows to a certain degree. Based on the test results of association, we selected four physiographic factors—the landslide ratio, the average steepness of the streambed, the effective watershed, and the form factor—for our physiographic factor set (Table 5).

Quantification of the factors

In this study, the landslide ratio, the average steepness of the streambed, the effective watershed area, and the form factor represent the physiographic factors. In order to determine the influence of the factors on the occurrence of debris flows, the degree of membership from fuzzy theory was used. Each factor was quantified as follows:

  1. 1.

    Landslide ratio

    The landslide ratio is defined as the ratio of the landslide area in the watershed of a potential debris-flow torrent to the watershed area. The membership degree distribution and the membership function of the landslide ratio are shown in Fig. 7a and Eq. (4), where D is the landslide ratio (%) and D N is the membership degree of the landslide area.

    $$\begin{array}{*{20}c} {D\, < 1.77\; \% ,\quad D_{N} \; = \;0} \\ {D_{N} = 0.334\ln \left( D \right) - 0.19} \\ {D\; > \;35.3\; \% ,\quad D_{N} \; = \;1} \\ \end{array}.$$
    (4)
    Fig. 7
    figure 7

    The membership degree distribution and the membership function for the physiographic factors

    Full size image
  2. 2.

    The average steepness of the streambed

    The average steepness of the streambed is defined as the arc tangent of the ratio of the difference in the elevation between upstream and downstream of the stream and the horizontal length of the stream. In order to quantitatively evaluate the influence of the average steepness of the streambed on the occurrence of debris flows, potential debris-flow torrents in which debris flows have occurred were used for analysis and for establishing the membership function. The results are shown in Fig. 7b and Eq. (5), where S is the average steepness of the streambed (in degrees) and S N is the membership degree of the average steepness of the streambed.

    $$\left\{ {\begin{array}{*{20}c} {S < 4.8^{ \circ } ,\quad S_{N} = 0} \\ {S_{N} = 0.04S - 0.193} \\ {S > 29.8^{ \circ } ,\quad S_{N} = 1} \\ \end{array} } \right.$$
    (5)
  3. 3.

    The effective watershed area

    The area of the watershed upstream of a certain point is called the effective watershed area if the average steepness of the streambed upstream of that point is greater than 15°, as defined previously. The membership degree distribution and the membership function of the effective watershed area are shown in Fig. 7c and Eq. (6), where A is the effective watershed area (in ha) and A N is the membership degree of the effective watershed area.

    $$\begin{aligned} \left\{ {\begin{array}{ll} {A < 16.6\,{\text{ha}}, \quad A_{N} = 0} \\ {A_{N} = 0.204\ln \left( A \right) - 0.573} \\ {A > 2232.3\,{\text{ha}},\quad A_{N} = 1} \\ \end{array} } \right.\end{aligned}$$
    (6)
  4. (4)

    The form factor

    The form factor is defined as the ratio of the watershed area to the square of the length of the main stream, which is a dimensionless parameter. The membership degree distribution and the membership function are shown in Fig. 7d and Eq. (7), where F is the form factor and F N is the membership degree of the form factor.

    $$\left\{ {\begin{array}{*{20}c} {F < 0.08, \quad F_{N} = 0} \\ {F_{N} = 0.361\ln \left( {F \times 100} \right) - 0.737} \\ {F > 1.23, \quad F_{N} = 1} \\ \end{array} } \right.$$
    (7)

The model for occurrence probability of debris flows

The models in this study were built based on two methods, one using genuine values of physiographic factors and the other using the degrees of membership of the factors. Both models were trained and verified 20 times with randomly selected sets of training data, and were evaluated with three items: specificity, sensitivity, and overall accuracy. The means and standard deviations of the three items for both models are shown in Table 6.

Table 6 A comparison of the accuracy of the combinations of the model for the critical rainfall value for the occurrence of debris flow
Full size table

Considering the immense damage a disaster may cause, it is reasonable for researchers to be conservative when building models. Therefore, a model with higher sensitivity will have greater potential to reduce disaster damage. Table 6 shows that in the training stage PF1 was better, with all three items greater than 90 %. However, in the validation stage, the values of specificity and overall accuracy plunged to 35.9 and 36.5 % respectively. For PF2, the values of the three items in the training stage were lower, but they were also more stable in the validation stage; despite the slightly lower sensitivity, PF2 had remarkably higher specificity and overall accuracy. Thus, due to a fair value of sensitivity and remarkably higher values of specificity and overall accuracy, it appears that PF2 is the better of the two. The reasons are stated as follows.

For PF2, when the values of the landslide ratio, the effective watershed area, and the form factor increase, their effects on the occurrence of debris flows initially increase significantly and then slowly level off at a maximum value. When the average steepness of the streambed increases, the effect on the occurrence of debris flows increases linearly. However, when it exceeds a certain value, its effect on the occurrence of debris flows remains constant. When the values of all four physiographic factors are below certain values, their effects on the occurrence of debris flows are zero. For PF1, the effects of the four physiographic factors on the occurrence of debris flows increase and decrease with the values of each of the four factors, without any limit. The results demonstrate that PF2 better reflects the effects of the factors on the occurrence of debris flows.

Consequently, PF2 was used to build the model for the critical rainfall value for the occurrence of debris flows, as shown in Eq. (8)

$$\begin{gathered} Z = 2.85\left( {S_{N} } \right) + 1.289\left( {A_{N} } \right) + 2.047\left( {F_{N} } \right) + 7.655\left( {D_{N} } \right) + 0.012\left( {\text{ER}} \right) - 8.027 \hfill \\ p = \frac{\exp \left( Z \right)}{1 + \exp \left( Z \right)} \hfill \\ \end{gathered}$$
(8)

In Eq. (8), p is the probability of the occurrence of debris flows.

In terms of the mechanism of the occurrence of debris flows, Z increases with the landslide ratio (D N ), the average steepness of the streambed (S N ), the effective watershed area (A N ), and the form factor (F N ), and a larger Z leads to a higher probability of a debris flow occurring (p). The minimum possible value of Z is −8.027, which leads to a very low occurrence probability, on the order of 0.0001. Since this value is comparatively insignificant in practice, the minimum value of p can be viewed as zero. Likewise, if the value of Z increases, it will lead to such a high a value of p that in practice, it can be viewed as one. Thus, the range of occurrence probability is fairly reasonable.

The effect of landslide ratio on the occurrence probability of a debris flows

In reality, the average steepness of the streambed, effective watershed area, and form factor are relatively less likely to undergo remarkable changes in value, leaving effective rainfall (ER) and landslide ratio (D N ) as the main variables. Thus, given the form factor, effective watershed area, and average steepness of the streambed, the relation between D N , ER, and occurrence probability of a debris flow (p) can be calculated. Since both ER and D N are positively correlated to occurrence probability, for a fixed occurrence probability, ER and D N must be negatively correlated. A potential debris-flow torrent (DF185) was selected for use as an example, as shown in Fig. 8. The physiographic conditions of the selected torrent are listed in Table 7. Comparing Table 7 and Fig. 8 reveals that the torrents can vary significantly in terms of physiographic properties, but will still share similarly varying patterns of occurrence probability.

Fig. 8
figure 8

a The relationship between landslide ratio, effective rainfall, and occurrence probability of debris flow of DF185; b the contour map of occurrence probability of debris flow of DF185

Full size image
Table 7 Physiographic conditions of the selected torrents
Full size table

As shown in Fig. 8, ER and D N are positively correlated to occurrence probability in a similar way. First, as ER or D N increases, p increases mildly initially, and then rises steeply. Second, as ER or D N exceeds a certain value and keeps increasing, the drastic rise is slightly eased as p finally reaches 1.0. This result indicates that to mitigate the threat of debris flows, an effective measure would be to reduce the landslide area. Note that when D N is sufficiently high, even a limited amount of rainfall can cause debris flows. Thus, a key aspect of debris flow countermeasures is monitoring and managing the landslide area, which mostly consists of relatively loose topsoil.

For example, the current landslide ratio for DF185 is 4.26 %, and the 24-h rainfall of the Chenyulan stream watershed, caused by a 25-year return period storm event, is approximately 400 mm. Under these conditions, the value of p for DF185 is about 0.84. However, given the same amount of rainfall, if the landslide ratio is reduced to 2 %, whether by engineering or vegetation, the value of p is reduced to 0.39, which indicates a significant drop in debris flow risk. Thus, if the physiographic factors change in the future, the potential for debris flow occurrence can be rapidly updated using the results of this study. Moreover, given a hydrological condition and an acceptable debris flow risk, the targeted landslide ratio for disaster prevention can also be evaluated.

Table 8 lists some of the randomly selected rainfall events used for establishing the accuracy of the model built in this study. The probabilities of debris flow occurrence were calculated by Eq. (8) and compared with the actual situations. Although the model did not yield 100 % accurate predictions, the overall accuracy of prediction is fairly acceptable. From Table 8, as a whole, if the calculated probability is higher than 0.5, it is reliable enough for related authorities to announce a debris flow warning; such warnings will result in only 3 out of 10 being false alarms, and only 2 out of 10 debris-flow events will occur without warning.

Table 8 Comparison of the calculated probability with actual situation for selected rainfall events
Full size table

Conclusions

In this study, in order to investigate the effects of the physiographic factors, such as landslide area, on the occurrence probability of debris flows in central Taiwan, statistical tests were conducted to select the physiographic factors that are significant determiners of the occurrence of debris flows. In addition, two quantifying methods of physiographic factors were also applied and compared. A logistic regression analysis was then used to establish a model for the evaluation of the occurrence probability of debris flows. The findings of this study are as follows:

  1. 1.

    After review of previous studies and tests for independence and association, four physiographic factors—the landslide ratio, the average steepness of the streambed, the effective watershed area, and the form factor—were selected as the representative factors. Of these four factors, the landslide ratio was found to have the highest correlation with the occurrence of debris flows.

  2. 2.

    Two quantifying methods for physiographic factors, namely genuine values and membership functions, were applied and compared. The results showed that if the overall performance of three items—specificity, sensitivity and overall accuracy—are considered, method PF2 may be the better and more stable of the two. Since it is reasonable to develop rather conservative models for disaster prevention and damage mitigation, PF2 was used to establish the model for the evaluation of occurrence probability of debris flows, as shown in Eq. (8).

  3. 3.

    To mitigate the risk of debris flows, managing the landslide ratio is a key factor. Removing landslide areas and preventing accumulation, whether by engineering or vegetation, can significantly reduce the occurrence probability of debris flows.

  4. 4.

    The model for evaluating the probability of debris flow occurrence established in this study was validated with 20 randomly selected historical rainfall events, including 10 events with the occurrence of debris flows and 10 events without. The validation results indicated that if p = 0.5 is set as the threshold probability to announce warnings, the accuracy of the model ranges from 70 to 80 %, with only three false alarms and two unpredicted debris flows.

  5. 5.

    The model established in this study is consistent with the mechanism of the occurrence of debris flows. That is to say, the probability of a debris flow occurring increases as the landslide ratio (D N ), the average steepness of the streambed (S N ), the effective watershed area (A N ), and the form factor (F N ) increase.

  6. 6.

    The model can be used to adjust the rainfall threshold by updating physiographic factors after significant land reform due to earthquakes, typhoons, human exploitation, or other reasons. Aside from this, given a hydrological condition and an acceptable debris flow risk, the targeted landslide ratio for disaster prevention can also be evaluated.