H-fields and Exponentiation

We prove a result that gives positive evidence towards the universality of the field of surreal numbers as an exponential H-fiel

Suscettività di frana “Studio della capacità predittiva del metodo di analisi condizionale applicato agli orli delle scarpate principali delle frane (OSP)”

University of Pisa - Department of Earth Sciences Program of Earth Sciences, XXVI Cycle 2009 – 2011 Abstract Doctor of Philosophy dissertation “Predictive power analysis of the MSUE-Conditional-Method” Marco Capitani INTRODUCTION Landslides are a major source of damage in economic terms and in human lives and represent the world's second natural hazard after earthquakes (Yalcin et al., 2011). In fact, since the beginning of the 21st century landslides have involved worldwide about 1.5 million of people and caused a total economic loss of more than $ 875 million (EM-DAT, 2010). The economic and human losses continue to grow steadily for the continue population growth and the resulting urban sprawl in unstable areas (Pasuto & Soldati, 1996; Schuster, 1996; Guzzetti et al., 1999; NRC, 2004). Even in Italy, landslides are an extremely frequent natural event (AVI data, project-GNDCI CNR) and a major cause of risk to the country socio-economic fabric. Only in the last century, in fact, landslides have caused 5,939 deaths and 1,860 injuries with an average of 60 deaths / year. This condition puts our country in the fourth place in the world, behind the Andean countries, China and Japan (Guzzetti, 2000). In 2010 88 major landslide occurred and caused 17 dead, 44 injured as well the evacuation of 4431 people. Ventotene, Meran and Maierato are just some of the serious events that have marked Italy, while Liguria, Campania, Tuscany, Sicily, Calabria and Lombardy are the regions mostly affected (data Ispra, 2011). For Italy, the estimated damage to public finance amounts to around 1-2 billion per year (Department of Civil Protection, 1992; Luino, 2005), placing our country second only to Japan. For Italy, the estimated damage grows up to about 3-4 billion per year if we also take into account indirect costs, such as the lost of productivity and the reducing of real estate value (Reed & Fanti, 2005). To cope with this situation, the scientific community is increasingly interested in the development of methods for the landslides susceptibility zonation, able to constitute a valid basis for proper land management and adequate risk prevention. In particular, the need to properly assess the propensity to collapse of areas not yet affected by landslides has brought scientific research to develop increasingly complex systems of investigation. Landslide Susceptibility (LS) is the spatial probability of landslide occurrence and differs from Landslides Hazard (LH), as it no provides information on the timing and magnitude of predicted landslide event (Carrara et al., 1995; Guzzetti et al., 2005). LS presents the first step in assessing Landslide risk (Clerici et al., 2010). The methods currently used to LS zonation can be synthetically grouped into three main types (Carrara et al., 1992; Guzzetti et al., 1999): a) Heuristic methods that are qualitative o semi quantitative methods in which the quality of the results is strictly depends on the knowledge and the experience of the operators. b) Deterministic methods that base their prediction over the empirical geotechnical laws requiring a large amounts of data. For the latter reason they are generally used only for areas not very extended. c) Statistical methods that analyze the historical link between landslide-controlling factors and landslide distribution. To modeling the LS for large areas statistical methods are the common used technique (Carrara et al., 1995; Ayalew and Yamagishi, 2005). Many different methods of statistical analysis are applied to LS assessments. More information on the different methodologies and nomenclatures can be found in review papers (e.g., Soeters and van Westen, 1996; Aleotti and Chowdhury, 1999; Guzzetti et al., 1999; Dai et al., 2002; Chung and Fabbri, 2005; van Westen et al., 2006; and reference therein). The main assumption on which are based all the statistical methods is that “the past and the present landslide locations are the key to the future” (Carrara et al., 1995; Hutchinson, 1995; Zêzere, 2002). Landslides will occur in the areas where the boundary conditions are the same of the areas where landslides have occurred in the past. In other words, the probability of landslide occurrence over landslide free areas is carried out by the study of the similarity geo-environmental conditions between these areas and those in which landslides have been occurred. Therefore, for all statistical methods the conceptual work behind to the LS zonation consists in (Carrara et al., 1995; Vijith and Madhu, 2008): a) The knowledge of landslides distribution and their mapping. b) The mapping of a set of factor that are supposed to be directly or indirectly predisposing the landslide occurrence. c) The assessment of statistical relationships between predisposing factors and landslides. d) The classification of degree of LS on the basis of the observed statistical relationships. The spatial analysis of the relationships between predisposing factors and landslides needs of the definition of the mapping unit on which the statistical observations are made and matching the observed data. Various methods have been proposed to partition the landscape for LS assessment and mapping (Meijerink, 1988; Carrara et al., 1995). Nowadays, the most used mapping units are the Grid Cells (Chung and Fabbri 1993), the Slope Units (Guzzetti et al., 1999) and the Unique Condition Units (UCUs) (Bonham-Carter, 1994; Chung and Fabbri, 1995; Carrara et al., 1995). Over the last few decades, many researchers have produced landslide susceptibility maps using different methods of statistical analysis applied to the Unique Condition Units (UCUs) (Carrara et al., 1995; Chung et al., 1995; Guzzetti et al., 1999; Irigaray et al., 1999; Clerici et al., 2002; Falaschi et al., 2008). However, no one has emphasized how the choice of representing the dependent variable (which is defined into the landslides inventory) can lead to the construction of models with un-definable predictive power. Theoretically, since susceptibility assessment tries to identify under what conditions landslides are generated, the dependent variable should be represented in the landslides inventory as the area where landslides originated, i.e., the detachment zones (Nefeslioglu et al., 2008). Moreover, if the landslides are also represented with their accumulation zone, the environmental factors so acquired are erroneously considered to be prone to landsliding (Clerici et al., 2006, 2010; Magliuolo et al., 2008). Due to the fact that the detachment zones are only partially visible, their definition into an inventory map is highly subjective. Furthermore, without a geophysical prospecting, there is not the possibility to define how our representation differs from reality. This fact introduces unquantifiable errors in the dependent variable resolution that imply unquantifiable errors both in the definition of the UCU type involved in landslides and in the definition of the UCU instability extension (independent variables). Therefore, this way of variable dependent representation makes uncertain the results of the predictive power validation for the models so built. In fact, the dependent variable is used both in the construction of models and in their validation (Chung and Fabbri, 2003, 2008; Guzzetti et al., 2006; Guzzetti et al., 2009; Rossi et al., 2010). Basically, if the “undefined” detachment zone is considered as way to correlate the UCUs to the events of instability, the best predictive model, that is chosen from a validation process, could actually have a low predictive power for the future landslides occurrence. So, only a certain dependent variable should be used into landslides susceptibility analysis. Among the systems of representation of the landslide inventory that do not introduce large margins of subjectivity and therefore unquantifiable errors, the Main Scarp Upper Edge method (MSUE) (Clerici, 2002) is the only that could be better adapted to the purposes of the landslide susceptibility analysis. The applicability of statistical methods to the MSUE has been poorly studied in the past and only with a not rigorous way. In fact, the predictive ability of models was analyzed using a validation data set that was generated with a random split method of the occurred landslides, or regardless of the pre-landslide conditions in the relationship between predisposing factors and phenomena landslide. The random selection of a validation data set from the occurred landslides may lead us to the selection of the type and amount of the UCU involved in landslides not really representative of the landslide susceptibility image, that is the focus of the investigation. The statistical sample used for the analysis of the predictive ability of a forecasting model should be considered a part of all statistical units making up the population, chosen so as to give us a small but faithful image of the population characteristics (Bucciante et al., 2003). The forecasting model should be created using a dataset of landslides related to a period of time prior that to which belong the landslides used for its validation (Chung & Fabbri, 1999, 2008; Zêzere et al., 2005; Chung, 2006; Guzzetti et al., 2006b; Irigaray et al., 2007; Akgün et al., 2008; van Westen et al., 2008; Blahut et al., 2010; von Ruette et al., 2011 ). On the other hand, among the methods of statistical analysis used to create landslide susceptibility maps, the conditional method appears to be one of the easiest to understand and to read for non-specialists. Therefore, this study represents an attempt to assess the predictive capability of the MSUE Conditional Method and its advantages and limitations. STUDY AREAS The MSUEs Conditional Method was applied to the Milia and Roglio basins, situated in the southern-central Tuscany, Italy. The Milia basin has an extension of 101 Km2 and an elevation ranging from 39 m to 913 m above sea level, with an average value of 336 m (standard deviation = 167.5 m). The basin is stretched out to SE direction and shows a prevalent hilly character. Hillslopes are generally not very steep. The highest value of the slope gradient is present in the eastern part of the Milia basin where the carbonatic formations outcrop. The physiographic structure of the area is typical of landscapes in which the zones between the valley floor and slopes have a significant extension (Fig. 8). In fact, approximately 50% of the study area is located between the altitude of 325 m and the minimum value of the basin that characterizes the corresponding closing section. Only near the eastern side of the study area, where we observe the presence of a high morpho-structural, the altitude value tends to increase until the maximum of 913 m. Areas characterized by altitudes above 550 m are mainly concentrated in this area of the basin and represent approximately the 9% of the total basin extension. The upward trend of the altitude from the eastern to the western sectors of the basin indicates that the basin physiographic evolution is strongly conditioned by the geological-structural situation. Most of the streams of higher order has a general anti-apenninic management type and shows a strong vertical erosion tendency at the far north-eastern areas of the basin while in western the river action evolves into a prevalent lateral erosion whose effects are manifested in a non-negligible way even along the T. Milia just before its merger into the Cornia River, one of the leading collectors of the central-southern Tuscany. The catchment area of the T. Roglio is one of the major sub-basins of the Era valley. The area covers about 160 km2 and an elevation ranging from about 20 m to about 500 m above the sea level, with an average value of 130.9 m. The basin is elongated in the N-NW direction and shows a predominantly hilly morphology with not very steep slopes. The highest values of the steepness are found mainly in areas of the basin where Pliocene formations with clayey and sandy-clayey facies outcrop. About 80% of the basin surface shows a steepness value less than 20°, the 20% denotes a slope value less than 6° and only 5% of the basin surface is characterized by a steepness more than 29°. The physiographic structure of the basin is typical of landscapes in which areas of the valley floor and the connection between these and the slopes are a very dominant feature. In fact, about 87% of the surface of the study area is located between 175 m and the minimum value which characterizes the basin with the corresponding closing section. The hypsometric curve indicates a situation more pronounced than that observed in the basin of the T. Milia, in which about 90% of the territory is confined within an interval of about 150 m elevation. Only near the eastern area of the basin, where the high morphological and structural of Iano occours, the value tends to increase until it reaches the maximum of 500 m. Areas characterized by altitudes above 375 m are mainly concentrated in this area of the basin and represent about 5% of the total. The concentration of the higher altitudes along the eastern sectors of this basin indicates that the physiographic evolution is strongly conditioned by the geological-structural situation. With the exception of the T. Roglio, which assumes a prevailing apenninic direction, most of the higher order streams generally presents an anti-apenninic direction and a strong tendency to vertical erosion. The erosive effects associated with the lateral rivers action occur in a non-negligible even along the T. Roglio, in the section between the central areas of the basin up to the confluence with the Era river, that is one of the main tributaries of the Arno river. MAIN GEOLOGICAL AND GEOMORPHOLOGICAL FEATURES In the T. Milia basin the compressional events occurring before and during the collisional apenninic episode originated the complex sheet stack where three allochthonous units are emplaced above the Tuscan Unit. The two units at the top of this complex are derived from the Ligurian Domain and are, from top to bottom, the Palombini Shale Ophiolitic Unit and the Monteverdi-Lanciaia Ophiolitic Unit, respectively. Between these units and the Tuscany Unit there is the Argille and Calcari Unit that belongs to the Sub-Ligurian Domain (Costantini et al., 1991). All the allochthonous units are characteristic of distal turbiditic and hemipelagic environments and are composed by altering siltitic, argillitic, and fine arenitic formations and by argillitic with inter-bedded limestone formations. The ophiolitic units also contain remains of the basalt, gabbros and serpentinites complex disseminate as blocks in their sediments. The Tuscan Unit is represented prevalently by the Mesozoic carbonatic succession, associated to very few outcrops of the middle and upper turbiditic and hemipelagic sequence. Ligurian units show a complex, pre-upper Oligocene deformation history related to subduction, accretion and later exhumation events. The deformation history includes almost two deformation phases of veining, folding and thrusting. (Marroni et al., 2004). These deformative structures are successively superimposed by the deformative structures due to collisional and post collisional events. In the collisional event the Ligurian and Sub-Ligurian Units overthrusted the Tuscan Domain and have been deformed, with the units belonging to this later domain, by a kilometric folds that in the Milia basin have a WNW-ESE axial general direction. Post collisional deformations are strictly related to the extensional tectonic, which began at the end of the Early Miocene and caused the collapse of a large part of the Apennines chain. This extensional event started with low and high angle normal faults as result of thinning of the upper crust. Then differential uplift, lowering and tilting phenomena have occurred since the middle Pleistocene (Bossio et al., 1993). The deposition of the Neogene-Quaternary successions has been largely controlled by vertical crustal movements. These successions are representative of coastal-marine and continental environments and are generally characterized by sandy clays and sandy conglomerates deposits. In the Roglio basin the Apennine compressional phase has originated a complex of thrust between the formations belonging to the Succession Liguride and Tuscany (Costantini et al., 2002; Costantini et al., In press). In particular, in the eastern areas formations belonging to the Ofiolitifera Montaione Unit outcrop (Flysch of Montaione, a complex of M. Carulli) above those of the no-metamorphic Tuscan Succession, which in turn are over-thrusting above the Tuscan Metamorphic Unit. The units belonging to the Ligurian Domain are representative both of distal turbiditic and pelagic environments and of the oceanic crust that formed the ancient Ligurian-Piedmont Basin. These units are composed mainly by formations characterized by alternating shales, limestone and shales interspersed with basalts, gabbros, and serpentinites. The Tuscan Nappe is mainly composed by non-metamorphic mesozoic carbonate formations and with limited outcrops of paleocenic-miocenic turbidite formations. All pre-Neogene formations emerge only on the east of the basin, at the high morphological of Iano, while over the 80% of the area studied is formed by outcrops of Plio-Pleistocene deposits, mainly characterized by marine facies. The extensional phase conditioned the hydrographic evolution of each basin. In particular, from the Pleistocene the tectonic evolution was followed by a rapid sinking of the hydrographic network, with the development of considerable level difference. The lowering of the network base level is suggested by numerous erosive terraces that are located at different altitudes along the basins. The lateral erosion action of the Milia and the Roglio river is an important still-active morphogenetic process. The morphology of the study areas is also strongly conditioned by the numerous mass movements related to a prevalent rotational slide, translational slide and flow types (Cruden and Varnes, 1996). Moreover, in the Milia basin many phenomena of Deep-seated Gravitational Slope Deformation (DGSDs) are present and their evolution appears strictly related to the Pleistocenic tectonic evolution and the base level fluvial lowering. From a classification point of view, the type of movement of these DGSDs could be considered similar to Block-slide and Sackung (Zischinsky, 1969; Sorriso-Valvo, 1988; Dramis and Sorriso-Valvo, 1994; Cruden and Varnes, 1996). Most of these DGDSs are involved in landsliding processes. MATERIALS AND METHODS MSUE-Conditional Method The Conditional Method is based on Bayes Theorem (Morgan, 1968) where the probability of the future occurrence of an event with determinate boundary conditions is determinated by the same type of events that occurred in the past with the same boundary conditions (conditional probability). In particular, for the landslide susceptibility (LS) assessment the conditional probability of landslide occurrence at specific UCU (boundary conditions) is assumed equivalent to the currently landslide density in that UCU (Carrara et al., 1995). Considering the problems for the use of the detachment zones as dependent variable representation, in this study the landslides have been identified by their MSUEs (Clerici, 2002; Clerici et al., 2006, 2010). Furthermore, an upstream buffer of 10 m is used for each MSUEs, in order to consider the UCUs involved in the landslide process as potential representative of the boundary conditions existing before the landslides development (Clerici, 2002; Süzen and Doyuran, 2004a,b; Clerici et al., 2006; Nefeslioglu et al., 2008; Clerici et al., 2010). Therefore, in the method applied to the LS zonation of the Milia and Roglio basins the conditional probability of landslide occurrence for a given UCU is assumed as the ratio of the sum of the areas of each UCU that fall within the MSUEs buffer and the total area for each specific UCU. Considering the different orders of magnitude between the areas of the UCU inside the MSUE buffer and the total of the specific UCU, the landslide density has been expressed in m2/km2. Landslides mapping The landslide map is the result of a two-year detailed geological and geomorphological field survey. The landslides were mapped on a 1:10,000 Tuscany Region topographic map. This process has been carried out also with the aid of the aerial photographs taken in 1975 at the scale 1:13,000 (flight EIRA75). The use of these aerial photographs was necessary to split landslides into two temporal groups. The landslides occurred before the 1975 have been used to create the models, while the landslides occurred after the 1975 have been used to validate their predictive capability. Following the division proposed by Keefer (1984), only 2039 deep-seated (≥ 3m) landslides were considered for the Milia basin. They occupy a surface of about 22.66 Km2, representing the 22.43% of the whole study area. In accord to Guzzetti (1999), LS analysis should be carried out for different landslide types. For this reason, the landslides have been mapped into separate datasets on the base of their prevalent movement typology. Because in this study the MSUE is considered as the dependent variable, the landslides belonging to the complex typology have been classified on the base of their initial prevalent movement. So, in the Milia basin 2,039 landslides have been divided into three typologies: translational slide (1,577), flow type (155), and rotational type (307). Among these, 128 translational landslides, 31 flow landslides and 46 rotational landslides have been occurred after the 1975. In the Roglio basin, only the 4.137 deep-sea

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