Mathematical Problems in Engineering

Volume 2016, Article ID 1612901, 12 pages

http://dx.doi.org/10.1155/2016/1612901

## Application of the Multitype Strauss Point Model for Characterizing the Spatial Distribution of Landslides

^{1}National Remote Sensing Centre, Department of Space, Government Of India, Hyderabad 500 037, India^{2}Department of Earth Observation Science, Faculty of Geo-Information Science and Earth Observation (ITC), University of Twente, P.O. Box 217, 7500 AE Enschede, Netherlands

Received 16 July 2015; Revised 30 March 2016; Accepted 20 April 2016

Academic Editor: Filippo Ubertini

Copyright © 2016 Iswar Das and Alfred Stein. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

Landslides are common but complex natural hazards. They occur on the Earth’s surface following a mass movement process. This study applies the multitype Strauss point process model to analyze the spatial distributions of small and large landslides along with geoenvironmental covariates. It addresses landslides as a set of irregularly distributed point-type locations within a spatial region. Their intensity and spatial interactions are analyzed by means of the distance correlation functions, model fitting, and simulation. We use as a dataset the landslide occurrences for 28 years from a landslide prone road corridor in the Indian Himalayas. The landslides are investigated for their spatial character, that is, whether they show inhibition or occur as a regular or a clustered point pattern, and for their interaction with landslides in the neighbourhood. Results show that the covariates lithology, land cover, road buffer, drainage density, and terrain units significantly improved model fitting. A comparison of the output made with logistic regression model output showed a superior prediction performance for the multitype Strauss model. We compared results of this model with the multitype/hard core Strauss point process model that further improved the modeling. Results from the study can be used to generate landslide susceptibility scenarios. The paper concludes that a multitype Strauss point process model enriches the set of statistical tools that can comprehensively analyze landslide data.

#### 1. Introduction

Landslides are defined as the movement of a mass of rock, debris, or soil along a downward slope, due to gravitational pull. The inherent properties of the Earth material, encompassing various geoenvironmental factors, can make a particular area susceptible to landslides. Landslides are among the most common natural hazards. They exhibit themselves in different mass movement processes and are considered as complex natural hazards occurring on the Earth’s surface [1]. Although individual landslides are not as spectacular or damaging as earthquakes, floods, and hurricanes, they are widespread and frequently occurring. Over the years they have caused great loss of life and property and their effects on the economy are growing at a rapid pace in many countries.

Spatial zonation of landslide occurrences, also known as landslide susceptibility mapping, aims to differentiate a land surface into homogeneous areas according to their degree of failure caused by mass movement at specific locations [2]. It relies on understanding complex mass movement processes and their controlling factors [3]. Approaches to the spatial modeling of landslides can broadly be divided into two groups [4]. The first approach consists of deterministic, dynamic modeling of the physical mechanisms that control slope failure, using mathematical methods. This approach is highly localized because of the detailed data requirements. The second approach uses the relation between the locations of previous landslides and geoenvironmental variables, to predict areas of different landslide susceptibility, using heuristic or statistical methods. The statistical methods used successfully in landslide susceptibility mapping to date include discriminant analysis [5, 6], multivariate statistics [7], likelihood ratio [8], information value method [9], and logistic regression methods [10]. These methods allow the analysis of geoenvironmental variables controlling landslide occurrence with respect to previous landslides without looking at the mutual interactions of landslides and their distribution patterns. Commonly applied generalized linear modeling uses a maximum likelihood estimation that results into point parameter estimates with standard errors [11]. As individual landslides cover only a small fraction of an unstable area, landsliding can be considered as a spatial point process that is controlled by a number of surface and subsurface spatial variables.

A spatial point process underlies a pattern of spatial point data within a region. Spatial point patterns are characterized by the 1st- and 2nd-order effects of a point process [12], specifying the intensity and point interactions, respectively. Nearest neighbor functions between pairs of points commonly model the 2nd-order effects as a function of positions of points and their distances. Such functions usually consider the relative position of two points in a bounded region and are taken as a function of distance only [13]. Spatial point processes play a fundamental role in spatial statistics and exhibit an active area of research. Disciplinary applications occur in forestry addressing positions of trees in a forest and log-landing sites [13–15], ecology addressing locations of bird’s nests [13], seismology addressing earthquake epicentres [16, 17], astrophysics addressing locations of stars in a constellation [18], and environmental modeling for peak concentrations of a pollutant in a geographical region [19]. In this study we use a spatial point process model to analyze the landslide data for inferring properties of the spatial distribution pattern.

The objective of this study was to identify significant factors for landslide susceptibility by applying a multitype Strauss point model. In this way the landslide occurrence patterns could be better understood. The point process model used information at the level of detail offered by the landslide data. That information was combined with geoenvironmental covariates to explain the underlying process for determining landslide susceptibility within an area. The model output was compared with the output of logistic regression model. The study was applied to a landslide prone area in the northern Himalayas, India.

#### 2. Methods

A landslide distribution pattern can be considered as a collection of point data spread irregularly in an essentially planar region. A basic assumption for the analysis is that the data can be regarded as a realization of a stochastic point process [13]. This process is characterized by the intensity at location , , defined aswhere is the area of a small region , is expectation operator, and is number of points in the region *.*

Landslide occurrences, considered as a spatial point pattern, show variation in the relative frequency as a function of the distance between positionsAssuming stationarity, (2) depends only upon the relative position of the two landslides between positions and , that is, on their distance [13]. A landslide process is second-order stationary if its intensity is independent upon translation over , so that and the second-order intensitydepends only upon the distance vector between and and not on their locations.

##### 2.1. Conditional Intensity and the Gibbs Model

For a landslide process that exhibits inter-landslide interaction the pairwise interaction models define the intensity in the form of probability densities:where is a normalizing constant, , is the intensity or the first-order term, and , , is the pairwise interaction or second-order term in a bounded window . Pairwise interaction models are a special case of Markov point process, called Gibbs point process models. Analysis of a Gibbs point process model is based on its conditional intensity [20]. The probability of the occurrence of a landslide at location is determined by the conditional intensity defined by . For a landslide process in a bounded area the conditional intensity is related to the probability density byFor the general pairwise interaction process the conditional intensity is Landslides of a particular zone with a specified radius of influence, however, might have a homogeneity condition with respect to each other. A Strauss process emphasizes such homogeneity conditions for deriving the relationship among the events. For the Strauss process, a simple model of dependence between landslides has the conditional intensitywhere is the number of points of the landslides that lie within a distance of the location . The conditional intensity is useful in modelling because the two distinct components in the functional form represent the interaction of landslides that can be interpreted in a straightforward way, following Baddeley [20].

A major restriction of Gibbs models is that the parameters cannot be estimated using maximum likelihood estimation, and hence a maximum pseudolikelihood is returned for each model [20]. The unknown scaling factor is intractable; hence the calculated pseudolikelihood does not involve any unknown factor and it is easier to use when estimating the parameters.

##### 2.2. Nearest Neighbour -Function

Methods based upon the distances between landslides can be used for investigating inter-landslide interactions, for example, to identify second-order effects of the landslide pattern data. These second-order properties are specified by the pair correlation function that is assessed using the inter-landslide interaction methods like the -and -functions [20]. In this way, the nature of the departure from complete spatial randomness (CSR) can be identified. This, in turn, is useful in determining the kind of interaction and interaction distances between the landslides [21]. Alternatively, Ripley’s -function can be used for detecting deviations from spatial homogeneity. The shape of this function indicates the specific type of pattern displayed by the data, that is, indicating whether the landslides show inhibition or occur as a regular or a clustered pattern.

The -function quantifies the distance distribution of a landslide to the nearest other landslide. It is expressed aswhere is the radius of a disk centered at the location of the th landslide , is an edge correction weight such that is approximately unbiased, is the distance of each landslide location to its nearest neighbor, and is the indicator function equal to 1 if and it is 0 otherwise.

An estimate of derived from a spatial landslide pattern dataset can be used in exploratory data analysis and formal inference about the pattern [22, 23]. The shape of this function provides information about the way the landslides are clustered in a particular area. If the landslides are clustered, increases rapidly at short distances, whereas for landslides that are evenly spaced, increases slowly up to the distance at which most events are spaced, and only then it increases rapidly [24]. For a homogeneous Poisson point process of intensity , the nearest neighbor distribution distance function for landslide distribution equalsIf then the landslide pattern is clustered, whereas if the landslide pattern is regular.

##### 2.3. Strauss Point Process for Marked Point Pattern Analysis of Landslides

A multitype pairwise interaction process is a Gibbs process which assumes symmetric interactions of landslides have the probability density of the formwhere is a normalizing constant, is a function determining the first-order trend for landslides of the th type of patterns, indicating whether the landslides show inhibition or occur as a regular or a clustered pattern, and are symmetric functions that describe the interaction between a pair of landslides and of given types and ; that is, and [20]. Thus, the conditional intensity of two categories of landslides* Small* and* Large* as defined in (11) for a multitype Strauss process is given byThe multitype Strauss process has pairwise interaction termswhere are interaction radii as above and are interaction parameters [20].

To fit a multitype Strauss process model to landslides, the matrix of interaction radii between individual landslides is specified based on field conditions and distribution patterns. Model fitting generates values for the interaction parameter and the model coefficients describing spatial inhomogeneity and inter-landslide interactions.

##### 2.4. The Multitype/Hard Core Strauss Point Process Model

The multitype/hard core Strauss point process is a hybrid of the multitype Strauss process and the hard core process, for example, the case that , , or of the Strauss process. A pair of points of types (*Small*) and (*Large*) must not lie closer than units apart; if the pair lies more than and less than units apart, it contributes a factor to the probability density similar to the pairs and the pairs. For landslides, this extension makes sense, as the large landslides are usually well separated and small distances do not occur; the same applies to the smaller landslides, although to a lesser degree as they can be closer. Moreover, a small landslide does not occur below a large one, whereas, if the location of a landslide is indicated as a point, the landslides cannot occur immediately close to each other.

##### 2.5. Goodness of Fit

Akaike’s Information Criterion (AIC) is a measure of the goodness of fit of an estimated statistical model. The AIC is defined as , where is the maximum likelihood value for the estimated model and is the number of parameters in the model. It can be interpreted as the trade-off between bias and variance in model construction indicating that of accuracy and complexity of the model [25].

The study system is based on the point process modeling of landslide data along with the geoenvironmental covariates influencing landslide such as lithology, topography, or geology. The models are selected based on the AIC. The study makes use of the spatial distribution of landslide to make the exploratory data analysis, their interactions using - and -functions, and the possible susceptibility intensity in the area using the Strauss point model. The AIC is a test between models and hence it may serve as a tool for model selection. Given a dataset, several competing models may be ranked according to their AIC, with the one having the lowest AIC being the best [25].

#### 3. Site Characteristics and Data Description

##### 3.1. Study Site

The study area lies between 30°47′29′′N and 30°54′45′′N latitude and 78°37′41′′E and 78°44′03′′E longitude in the northern Himalayas, India, in the catchment of the river Bhagirathi, a tributary of the river Ganges (Figure 1). This study area of a 12 km long road corridor with a total area of 8.88 km^{2} was selected judiciously with corroboration that any landslide that occurs in the area affects the road. In the Himalayan terrain rock strength and geological structures play a major role in the landslide activity. The dominant rock types in the area include low grade metamorphic rock such as chlorite schist, schistose quartzite, and quartz mica schist along with high grade migmatites and gneisses. Rock mass properties, such as intact rock strength (IRS) computed for the area, varies between 50 and 200 MPa and corresponding cohesion of rock mass varies between 9 and 29 KPa [26]. Detailed assessment showed that the IRS varies due to compositional changes, the spacing and orientation of the joints present in these rocks, and the degree of weathering in each rock type. Elevation in the area ranges between 1550 and 2100 m with a high relative relief; average elevation of the area is around 1900 m.