Fabbri, Andrea G.

This contribution proposes iterative cross-validation as an approach to assess the quality of spatial predictions of hazardous events. Given the complexity of mathematical procedures and the diversity of geomorphologic applications made to date, STM, the Spatial Target Mapping, is a piece of software, ancillary to a geographical information system and a spreadsheet, that constrains such complexity into a clearly structured framework optimized for modelling. Spatial relationships are established between the distribution of hazardous occurrences and their physical settings to represent in part the slope failure process. They are used in the modelling to anticipate the location of future occurrences. Procedural aspects and computational options are discussed by means of an application to a database developed for landslide susceptibility prediction in northern Italy. Two mathematical models of spatial relationships, fuzzy set function and logistic discriminant function, are applied to generate prediction patterns, prediction-rate tables, and subsequently compute target and uncertainty patterns. The two processing strategies used are sequential elimination and random selection of occurrences for iterative crossvalidations.

This contribution analyses the spatial support of sampling points used to express the presence or absence of NO3 ˉ pollution in the water table. A spatial database constructed for the assessment of ground water vulnerability is re-analysed with a different predictive strategy. In practice, a case study area surrounding the city of Milan in northern Italy becomes an opportunity to point at a very general prediction modelling problem in which the basic direct evidence of a process is obtained only by sampling with point like measurements of nitrate concentration, as the ones from drill holes or water wells. The main questions are: “What is the functional spatial support for the modelling?” and “What happens if different spatial supports are assumed?” The answers to these questions are counterintuitive. Over the area of study of about 2,000 km2 , the distribution of 305 water wells delimits a training area in which 133 wells are considered as impacted by nitrate pollution, i.e., direct supporting patterns of the modelling. The remaining 172 wells are considered as non-impacted. In the training area, nine natural and anthropogenic map data are assumed, as indirect supporting patterns of the modelling, to reflect both the potential source of nitrates and the relative ease in which nitrates may migrate in ground water. They cover the entire area of study. A mathematical model is used that computes spatial relationships between the direct and indirect supporting patterns based on empirical likelihood ratios. The relationships are integrated into prediction patterns and, by iterative cross-validations, into target and uncertainty patterns. These are then extended from the training area over the remaining much larger study areas for analysis and visualization. Square neighbourhoods of dimensions 20 × 20 m, 60 × 60 m, 180 × 180 m and 1,020 × 1,020 m around the 305 wells are used to delimit four training areas of different sizes. Surprisingly, the smaller spatial support appears as the most reliable.

his contribution exposes the relative uncertainties associated with prediction patterns of landslide susceptibility. The patterns are based on relationships between direct and indirect spatial evidence of landslide occurrences. In a spatial database constructed for the modeling, direct evidence is the presence of landslide trigger areas, while indirect evidence is the presence of corresponding multivariate context in the form of digital maps. Five mathematical modeling functions are applied to capture and integrate evidence, indirect and direct, for separating landslide-presence areas from the areas of landslide assumed absence. Empirical likelihood ratios are used first to represent the spatial relationships. These are then combined by the models into prediction scores, ordered, equal-area ranked, displayed, and synthesized as prediction-rate curves. A critical task is assessing how uncertainty levels vary across the different prediction patterns, i.e., the modeling results visualized as fixed, colored groups of ranks. This is obtained by a strategy of iterative cross validation that uses only part of the direct evidence to model the pattern and the rest to validate it as a predictor. The conducted experiments in a mountainous area in northern Italy point at a research challenge that can now be confronted with relative rank-based statistics and iterative cross-validation processes. The uncertainty properties of prediction patterns are mostly unknown nevertheless they are critical for interpreting and justifying prediction results.

This contribution proposes a cautious way of constructing the susceptibility classes obtained from favourability modeling of landslide occurrences. It is based on the ranks of the numerical values obtained by the modelling. Such ranks can be displayed in the form of histograms, cumulative curves, and prediction patterns resembling maps. A number of models have been proposed and in this contribution the following will be compared in terms of their respective rankings for equal area classes: fuzzy set function, empirical likelihood ratio, linear and logistic regression, and Bayesian prediction function. The analyses performed and contrasted exemplify a generalized methodology for comparing predictions that should allow evaluating prediction patterns from any model. Unfortunately, many applications in the scientific literature use methods of characterizing prediction quality that make comparison hard or impossible. A database from a study area in the Mountain Community of Tirano in Valtellina, Lombardy Region, northern Italy, is used to illustrate how the results of the different models and strategies of analysis show the relevance of the properties of the database over those of the models.

Prediction patterns are generated using different data sets from a database for landslides hazard in northern Italy. A direct supporting pattern of the distribution of 28 complex landslides was previously used to obtain their spatial relationships with five categorical indirect supporting patterns representing the spatial context of the landslides: geology, land use, and permeability in addition to internal relief and slope, the latter two categorized into five classes. The five indirect supporting patterns were selected to minimize the effects of conditional dependence on prediction patterns by a Weight-of-Evidence model. The same set of patterns is reanalysed applying the Empirical Likelihood Ratio model using also uncategorized continuous supporting patterns: aspect, curvature, and digital elevation, in addition to internal relief and slope. The resulting prediction patterns are compared in terms of prediction rates and target-uncertainty patterns.

This contribution stems from the exposure of two different approaches to the representation of natural hazard: regression analysis on one side and favourability models on the other. As a consequence a spatial database for landslide hazard prediction in central Slovenia was shared to experiment on spatial prediction via cross-validation techniques. Due to the peculiarities of the database three types of analyses were selected: (i) predictions using an Empirical Likelihood Ratio model and four types of landslides in a training area, and extended to a surrounding study area; (ii) iterative cross-validations to obtain target, uncertainty and their combination patterns; and (iii) separation of one type of landslides into two groups of well predicted and poorly predicted occurrences by a cross-validation with the target pattern. The importance is underlined of sharing databases to encourage broader views of methodologies and strategies in spatial modelling.

This study applies mathematical models for assessing landslide susceptibility around Lake Albano, a volcanic crater and resort area near the city of Rome, Italy. The hazards are mass movements of many different types, recorded for more than 2,100 years that continue occurring to date encroaching with expanding urbanization and socioeconomic activities. The study area surrounding the lake occupies 30 km2, in the form of a digital raster of 1002 pixels × 1202 lines at 5 m resolution: 975,093 above the water level and 229,311 below it. Of those, 8,867 pixels indicate the location of 150 sub-aerial landslides and 34,028 pixels that of 65 sub-aqueous landslides, respectively, that is, high densities of mass movements. A database collected the most available information on the landslides: distributions, types, linear and polygonal forms, and sub-aerial or sub-aqueous locations. Digitized categorical maps of land use classes and lithology units, in addition to a continuous field of high-resolution topographic elevation data, represented their physical settings. From a dense grid of elevation points, continuous value maps at 5 m resolution were the following: aspect, digital elevation model, slope, curvature, planform, and profile. The results of prediction modelling by a fuzzy set membership function and a logistic discriminant function were digital images ranking the study area into relative levels of susceptibility. The spatial support of the settings varied with landslide types and physiographic conditions. The levels integrated empirical likelihood values representing the contrast in settings for all the pixels in the presence of the landslides with the pixel in their absence for each landslide type within the study area. Such ranks tend to overlap in predictions from the two models and for different types of landslides. Predicting landslide susceptibility for the area is feasible and with low uncertainty; however, the volcanic and socioeconomic context is a main challenge to measures of hazard and risk avoidance. Keywords: landslide susceptibility, volcanic crater, fuzzy sets, logistic discriminant functions, spatial support, prediction modelling.

This contribution proposes a cautious way of constructing the susceptibility classes obtained from favourability modeling of landslide occurrences. It is based on the ranks of the numerical values obtained by the modelling. Such ranks can be displayed in the form of histograms, cumulative curves, and prediction patterns resembling maps. A number of models have been proposed and in this contribution the following will be compared in terms of their respective rankings for equal area classes: fuzzy set function, empirical likelihood ratio, linear and logistic regression, and Bayesian prediction function. The analyses performed and contrasted exemplify a generalized methodology for comparing predictions that should allow evaluating prediction patterns from any model. Unfortunately, many applications in the scientific literature use methods of characterizing prediction quality that make comparison hard or impossible. A database from a study area in the Mountain Community of Tirano in Valtellina, Lombardy Region, northern Italy, is used to illustrate how the results of the different models and strategies of analysis show the relevance of the properties of the database over those of the models.