As previously described, landslide susceptibility is commonly expressed as the spatial correlation between predisposing factors (terrain, climatic, tectonic, human activity) and the distribution of observed landslides in a region (Brabb 1984; Crozier & Glade 2005; Thiery et al. 2007). Since the parameters involved in landslide susceptibility assessment are fuzzy in nature and are commonly classified by using fuzzy descriptions such as low, moderate, high, steep, favourable, close to etc., and the slope failure mechanism is a complex phenomenon not completely understood, particularly
77 at the regional scale (Guzzetti et al. 1999), the application of fuzzy logic seems appropriate in order to deal with uncertainties and non-linear relationships between conditioning factors and landslide occurrence. Fuzzy logic has been applied for spatial data integration in mineral exploration (Bonham-Carter 1994; Luo & Dimitrakopoulos 2003; Porwal et al. 2006) and landslide susceptibility assessment as either “knowledge-driven” (Juang et al. 1992; Pistocchi et al. 2002; Saboya et al. 2006;
Miles & Keefer 2007; Champati et al 2007; Wang et al. 2009) or “data-driven” (Ercanoglu &
Gockeoglu 2002, 2004; Lee & Lee 2006; Song et al. 2006; Kanungo et al. 2006; Lee 2007b; Kanungo et al. 2008; Kanungo et al. 2009; Pradhan 2010, 2011a, 2011b; Ercanoglu & Temiz 2011; Bui et al.
2012), based on the assumption that spatial data (i.e. predisposing factors) are members of a set, either mineral favourability or landslide susceptibility.
The fundamental component of any fuzzy logic model lies in deriving the membership function. The membership function essentially associates the fuzzy linguistic terms (low, moderate, high, steep, favourable, close to etc.) to degrees of membership, quantifying the “degree of belonging” of a variable to a set. In most of the fuzzy models, membership functions are chosen arbitrarily by the modellers based on their experience and perspectives. Thus, the membership functions given by two modellers could be quite different. Alternatively, membership functions can be derived based on available datasets in order to minimize the subjectivity of knowledge-driven models. In the present study, the membership functions between conditioning factors and landslide occurrence have been derived by applying both knowledge- and data-driven techniques.
3.2.2.1 “Data-driven” fuzzy memberships
Frequency-ratio
Frequency-ratio (FR) is defined as the relative frequency of landslides in a factor category (e.g.
landslides within the 20o-25o slope category) to the relative frequency of all landslides in the area or alternatively, as the landslide density within each factor category normalized by the landslide density over the entire area.
( )⁄ ( )
( )⁄ ( )
(3.3)
where N(Li) is the number of landslide pixels in the category i, N(Ci) is the total number of pixels in the category i, N(L) is total number of landslide pixels in the study area and N(A) is the total number of pixels of the study area.
78 The concept of frequency-ratio has been extensively used in statistical approaches such as the weights of evidence method (Bonham-Carter 1994), in order to evaluate the importance of each factor in the occurrence of landslides and calculate the corresponding weights. Frequency-ratio values greater than 1 indicate higher densities of landslides in the category compared to the density of landslides in the entire study area (landslide inventory) and a higher correlation between the category and slope instability, whereas values < 1 indicate lower correlation. It also provides a simple means to derive “data-driven” fuzzy membership functions between factors and landslide occurrence. In order to derive the fuzzy membership function the FR values of the different categories are normalized by the higher FR value within the factor (Lee 2007b; Pradhan 2010, 2011a, 2011b). Therefore, the degrees of membership essentially reflect the landslide density in each factor category (Fig. 3.3).
Figure 3.3 Landslide frequency ratio plotted against slope angle (classified in 5o intervals). The shape of the distribution represents the fuzzy membership function (black line) of slope angle.
Cosine amplitude
The cosine amplitude method (Ross 1995) is a widely-applied similarity method (Ercanoglu &
Gockeoglu 2004; Song et al. 2006; Kanungo et al. 2006, 2009; Ercanoglu & Temiz 2011) used to establish relationships among elements of two or more data sets. Assuming that n is the number of data samples (categories of a factor used in the analysis) represented as an array X:
X = {x1, x2 … xn}, (3.4)
and that each of its elements, xi, is a vector of length m (i.e. the size of the raster image) and can be expressed as:
X = {xi1, xi2 … xim} (3.5)
79 then each element of a relation rij, results from a pair-wise comparison of a factor category xi with a category of the landslide distribution layer xj (landslide or non-landslide). According to the cosine amplitude method the strength (degrees of membership) of the relationship or similarity rij between categories of thematic data layers and the categories of landslide distribution layer are calculated by the following equation with values ranging from 0 to 1 (0 ≤ rij ≤1):
|∑ |
√(∑ )(∑ )
(3.6)
Based on the cosine amplitude concept Kanungo et al. (2006) defined the rij value for any given factor category as the ratio of the total number of landslide pixels in the category to the square root of the product of the total number of pixels in that category and the total number of landslide pixels in the area. Values of rij close to 1 indicate similarity whereas values close to 0 indicate dissimilarity between the two datasets.
3.2.2.2 “User-defined or knowledge-driven” fuzzy memberships
In knowledge-driven fuzzy models the membership values can be selected based on subjective judgment (Bonham-Carter 1994) using if-then rules (Miles & Keefer 2007), or they can be derived by various functions representing the relationships between factors and the phenomena being studied, such as “J-shaped”, “S-shaped”, ”triangular”, “trapezoidal” and “linear” functions.
Recently ESRI incorporated the Fuzzy Membership tool in the Spatial Analyst extension of ArcGIS 10 as an alternative way to transform the input data into membership values ranging from 0 to 1 by selecting a user-specified fuzzy membership function. These functions include:
Fuzzy large
Fuzzy Large is a sigmoid-shaped function used when the larger input values are more likely to be a member of the set, in this case landslide occurrence;
( )
( )
(3.7)
80 where μ(x) is the membership value of category x, f2 is the midpoint and f1 the spread of the function.
The defined midpoint identifies the crossover point (assigned a membership of 0.5) with values greater than the midpoint having a higher possibility of being a member of the set and values below the midpoint having a decreasing membership. The spread parameter defines the shape and character of the transition zone (Fig. 3.4). The spread and midpoint parameters are subjectively determined and reflect the expert opinion. The fuzzy large function is suitable for modelling parameters where increasing the parameter value often results in higher susceptibility (e.g. rainfall, soil drainage).
Figure 3.4 Fuzzy Large membership function.
Fuzzy small
Contrary to the Fuzzy Large, the Fuzzy Small function is used when the smaller input values are more likely to be members of the landslide susceptibility set (e.g. proximity to faults and streams).
( )
( )
(3.8)
where μ(x) is the membership value of category x, f2 is the midpoint and f1 the spread of the function (Fig. 3.5).
81 Figure 3.5 Fuzzy Small membership function.
Fuzzy Gaussian
The Fuzzy Gaussian function transforms the original values into a normal distribution. The midpoint here defines the strongest membership with the remaining input values decreasing in membership as they move away from the midpoint in both the positive and negative directions (Fig. 3.6).
( ) ( ) (3.9)
where μ(x) is the membership value of category x, f2 is the midpoint and f1 the spread of the function.
Figure 3.6 Fuzzy Gaussian membership function.
In the present study area the number of rainfall-triggered shallow landslides and debris-flows increases with slope angle up to approximately 35o-40o and then decreases close to the very steep
82 mountain ridges. Therefore, the slope factor can be approximately represented by a Gaussian fuzzy membership with the midpoint assigned to the class 35o-40o and an appropriate spread.
Fuzzy near
The Fuzzy Near function returns a curved peak of membership over an intermediate value, similar to the fuzzy Gaussian but decreasing at a faster rate, with a narrower spread.
( )
( )
(3.10)
It is suitable when a particular intermediate class or classes have significantly more influence compared to the others (Fig. 3.7).
Figure 3.7 Fuzzy Near membership function.
Fuzzy linear
Finally, the Fuzzy Linear function assumes a linear relationship between user-specified minimum and maximum values. Any value below the minimum will be assigned 0 (definitely not a member) and any value above the maximum 1 (definitely a member).
( ) {
(3.11)
83 where μ(x) is the membership value of category x and α, b is the minimum and maximum values respectively (Fig. 3.8).
Figure 3.8 Linear fuzzy membership function.
Although its linearized sigmoid shape provides a simplified model, it can be particularly useful when there is no adequate information on the factor - landslide susceptibility relationship and only the effect of the minimum and maximum values is known. For example, provided that no other information exists, the proximity to faults factor can be represented by the linear membership function based only on the observation that within 100 m distance from a fault, landslides have the highest density and significantly decrease up to 3 km distance.
3.2.2.3 Aggregation
To derive the landslide susceptibility index (LSI) map, the “fuzzified” factors represented by information layers in raster format with values ranging from 0 to 1 need to be combined.
Aggregation operations on fuzzy sets are used to combine them to a single set (Dubois & Prade 1985; Zimmermann 1991). Bonham-Carter (1994) discusses five operators, the fuzzy AND, fuzzy OR, fuzzy algebraic Product, fuzzy algebraic Sum and fuzzy Gamma operator.
Fuzzy AND
This is equivalent to a Boolean AND (logical intersection) operation on classical set values of (1, 0). It is defined as:
( ) ( … ) (3.12)
where ( ) is the combined membership value, is the membership value for thematic layer A at a particular location, is the membership value for thematic layer B, and so on. The effect of this
84 operator is to cause the output map to be controlled by the lowest fuzzy membership value occurring at each location (e.g. if a location has a membership value of 0.9 according to map A and 0.5 according to map B, then the membership for the combination using fuzzy AND is 0.5).
Fuzzy OR
The fuzzy OR operator is similar to the Boolean OR (logical union) in that the output membership values are controlled by the maximum values of any of the input thematic layers, for any particular location. The fuzzy OR is defined as:
( ) ( … ) (3.13)
Using this operator, the combined membership value ( ) at a location is influenced only by the most suitable (or susceptible to landsliding) of the thematic layers.
Fuzzy Algebraic Product This function is defined as:
( ) ∏
(3.14)
where μi is the fuzzy membership function for the ith map, and i = 1, 2, . . . n are the number of thematic layers to be combined. The combined fuzzy membership value using this operator tends to be very small, due to the effect of multiplying several numbers less than 1. The output is always smaller than, or equal to, the smallest contributing membership value.
Fuzzy Algebraic Sum
This operator is complementary to the fuzzy algebraic product, being defined as:
( ) ∏( )
(3.15)
85 The result is always larger than (or equal to) the largest contributing fuzzy membership value.
Therefore if two thematic layers both favouring a hypothesis strengthen one another, and their combined result is more supportive than either thematic layer taken individually.
Fuzzy Gamma Operation (γ)
This is defined in terms of the fuzzy algebraic product and the fuzzy algebraic sum by the following formula: algebraic sum; and when γ is 0 the combination equals the fuzzy algebraic product.
In this study, the fuzzy gamma operator was applied as it establishes the relationships between the multiple input criteria and does not simply return the value of a single membership set as the Fuzzy Or and Fuzzy And do. It also provides a compromise between the increasing tendencies of the fuzzy algebraic sum and the decreasing effect of the fuzzy algebraic product (ESRI 2011). The γ value is a user-defined parameter introducing a degree of subjectivity, even to the data-driven models. In the context of landslide susceptibility the γ value determines how much the favourable and non-favourable factors will affect the overall landslide susceptibility. Similar to the negative and positive factor weights in statistical methods, in fuzzy logic, values close to 0 and 1 tend to decrease or increase respectively the overall landslide susceptibility of a site. The difference however is that the user decides the “importance” of the favourable and non-favourable conditions by adjusting the γ value. Therefore, the selection of the appropriate γ is a critical step in the modelling procedure in order to derive a realistic output.
3.3 Data
The fundamental data layers required in any landslide susceptibility or hazard and risk assessment can be subdivided into four main groups: landslide inventory data, environmental (controlling) factors, triggering factors and elements at risk (Van Westen et al. 2006).
86