Michele Santangelo I Ivan Marchesini I Mauro Cardinali I Federica Fiorucci I Mauro Rossi IFrancesco Bucci I Fausto Guzzetti
A method for the assessment of the influence of beddingon landslide abundance and types
Abstract Bedding planes are a known factor that controls thetype, abundance and pattern of landslides. Where layered rockscrop out, the geometrical relationships between the attitude of thebedding and the geometry of the terrain is crucial to understandlandslide phenomena. Obtaining information on bedding attitudefor large areas through field surveys is time-consuming, and re-source intensive, hampering the possibility of quantitative inves-tigations on the control of bedding planes on landslides. Wepropose a GIS-based method to extract information on beddingplanes from the analysis of information captured through thevisual interpretation of stereoscopic aerial photographs and adigital representation of the terrain. We tested the method in theCollazone study area, Umbria, Central Italy, where we used spa-tially distributed information on beddings and terrain informationobtained from a 10×10-m DEM to determine morpho-structuraldomains. We exploited the morpho-structural terrain zonation, incombination with landslide information for the same area, toinvestigate the role of beddings in controlling the distributionand abundance of landslides in the study area. We found thatbeddings condition the location and abundance of relict anddeep-seated landslides, most abundant in cataclinal slopes, anddo not condition significantly the shallow landslides. We expectthe method to facilitate the production of maps of morpho-struc-tural domains in layered geological environments. This will con-tribute to a better understanding of landslide phenomena and tofoster the preparation of advanced landslide susceptibility andhazard models.
Keywords Bedding plane . Bedding domain . Landslide . Model
IntroductionInvestigators have shown the importance of lithology and struc-ture to control the type, abundance and pattern of landslides (e.g.Fookes and Wilson 1966; Zaruba and Mencl 1969; Varnes 1978;Crozier 1986; Koukis and Ziourkas 1991; Guzzetti et al. 1996, 2008;Günther 2003; Grelle et al. 2011). The presence and abundance ofdiscontinuities in the rock mass, including bedding planes andfoliation (Guzzetti et al. 1996) and faults, joints, cleavages andother fracture systems (Günther 2003; Goudie 2004), are knowngeological factors controlling the stability of slopes (Guzzetti et al.1996; Günther 2003; Grelle et al. 2011).
Where hard rocks crop out, beddings, joints and fracture sys-tems condition the formation and volume of rock falls, topples,rock slides and rock avalanches (Goodman and Bray 1976; Cruden2003; Günther 2003). Where layered rocks crop out, the geometri-cal relationships between the slopes and the discontinuities in theslopes favour (e.g. along cataclinal slopes) or limit (e.g. alonganaclinal slopes) the formation of deep-seated slides (Guzzetti etal. 1996) and determine the area and volume of the landslides(Katz and Aharonov 2006). At the regional scale, the presence ofstructurally complex rocks characterized by abundant and
intricate fabrics is a known condition that controls the location,type and abundance of landslides (Esu 1977; Grelle et al. 2011;Santangelo et al. 2013). At the local scale, the superposition ofrocks characterized by different permeability, a typical setting inlayered sedimentary and volcanic rocks, can generate hydrogeo-logical conditions that favour or limit slope instability (Carrara etal. 1992; Guzzetti et al. 1996).
In landscapes dominated by layered rocks, defining the geo-metrical relationships between the attitude (strike and dip) ofbedding planes and the attitude (gradient and aspect) of theterrain is crucial to understand landslide phenomena, to explaintheir types, patterns and abundance (Guzzetti et al. 1996) and todetermine landslide susceptibility at different geographical scalesusing heuristic (Ruff and Czurda 2008), statistical (Carrara et al.1992; Thiery et al. 2007; Rossi et al. 2010), physically based(Günther 2003) or combined (Frattini et al. 2008) methods.Geometrical information on the attitude of bedding planes is mostcommonly obtained through geological field mapping (Clegg et al.2006; De Donatis and Bruciatelli 2006; Bodien and Tipper 2013).The method is time-consuming, particularly where large areas(hundreds to thousands of square kilometres) have to be coveredby the survey. This hampers the possibility of using bedding andstructural data for regional landslide modelling. Geological map-ping is also conditioned by practical and operational constrains(e.g. inaccessible areas, poor visibility, limited number of out-crops) that can hamper, locally severely, the reliability and repre-sentativeness of the geological information (Grelle et al. 2011). Inaddition, bedding measurements are typically point measure-ments and, as such, they are conditioned by local anomalies (e.g.minor folds, minor faults, mass movements), and the single meas-urements may not represent the general geological trend thatcontrols the presence (or the absence) of landslides in an area(Guzzetti et al. 1996; Grelle et al. 2011; Marchesini et al. 2013).
Photo-geological techniques can be used to obtain informationon bedding attitude, in combination or as an alternative to geo-logical field mapping. This is achieved through the visual inter-pretation of stereoscopic aerial photographs (Rib and Liang 1978;Antonini et al. 2002; Brardinoni et al. 2003; Weirich and Blesius2007) or satellite images of sufficient ground resolution. The use ofremote sensing imagery (aerial photographs and satellite images)emphasizes spatial continuity or repetition of geological and mor-phological features, facilitating their recognition and mapping(Rib and Liang 1978). Unlike field surveys that obtain beddinginformation at single locations (point measurements), the photo-geological techniques allow for the detection and mapping of thegeneral trend of the geological structures, including beddingplanes. This ability proves particularly important to define andmap bedding attitude domains i.e. portions of terrain character-ized by a similar bedding attitude or by similar geometrical rela-tionships between the bedding attitude and the local terrain slope(Cardinali et al. 2001b).
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In this work, we first propose a method implemented in ageographical information system (GIS) to extract quantitative in-formation on bedding planes from the analysis of informationcaptured through the visual interpretation of stereoscopic aerialphotographs. Next, we use the geometrical information on thebeddings, in combination with terrain information obtained froma digital elevation model (DEM), to determine bedding domains ina study area in Umbria, Central Italy. We then exploit the morpho-structural zonation, in combination with landslide information forthe same area, to investigate the role of beddings in controlling thedistribution and abundance of landslides in the study area. Weconclude discussing the limitations of the proposed method.
MethodWe have developed a semi-automatic method to extract quantita-tive information on the orientation of bedding planes from theanalysis of information captured through the visual interpretationof stereoscopic aerial photographs. We implemented the methodin the Geographic Resources Analysis Support System (GRASS)GIS environment (Neteler et al. 2012, GRASS Development Team2013) using the Python scripting language. The method consists oftwo main steps. First, a value for the dip direction and the incli-nation (dip angle) of the bedding plane is determined (§2.2) usinggeometrical information captured by a detailed visual interpreta-tion of stereoscopic aerial photographs (§2.1). Next, the informa-tion on the attitude of individual bedding planes is interpolated(§2.3) to obtain two geographically continuous raster layers show-ing values for the dip direction and the dip angle of the modelledbedding plane. This spatially continuous information is used, incombination with information on local terrain slope and aspect, toclassify the slopes (§2.4) and to prepare a morpho-structuralzonation of the study area (§2.5).
Bedding tracesWhere layered rocks crop out, individual layers or groups of layersintersecting at an angle the topographic surface leave discerniblelinear signatures that can be recognized and mapped, in the fieldor through the interpretation of remote imagery (e.g. aerial photo-graphs, very high resolution satellite images) (Fig. 1). In this work,we name “bedding trace” (BTs) the linear signatures left by layeredrocks on the topographic surface. Visual evidence of a BT on theremote imagery depends on multiple factors, including (1) thetype, mechanical and hydrological characteristic of the rocks, (2)the contrast in resistance to erosion of the different rock layers, (3)the attitude of the bedding planes, (5) the steepness of the slopeand the complexity of the topography and (5) the presence (orabsence) of vegetation that may emphasize or hide individualbeddings. Figure 1 shows examples of BTs and of their visualsignatures on aerial photographs in different geological settingsin Umbria, Central Italy.
Bedding attitudeThe orientation (or attitude) of a bedding plane is determined bythe strike and the dip of the geometrical plane that describes thebedding plane. The strike line is the line that represents theintersection of the bedding with a horizontal plane. The strikeangle (i.e. the azimuth) is the angle made by the strike line to areference point, most commonly the north, N. The dip is 90° off
the strike angle, and the angle of dip is the angle between the dipdirection to the horizontal direction.
To determine the attitude of a bedding plane, we adopt astepwise approach (Fig. 2) that exploits information on the geo-graphical location of a BT, including coordinates and elevation ofindividual points along the BT, to define strike and dip values thatdetermine univocally the orientation of the BT. First, the BT isdrawn on a base map in projected coordinates (Fig. 2a). Next, wedraw the unique segment joining the two end points of the linerepresenting the BT (Fig. 2b). The orientation of the segment isdetermined by the position of the two end points of the BT and isindependent of (and does not represent, necessarily) the strike ofthe bedding. The BT and the new added segment encompass apolygon. Then, we generate a set of equally spaced points along thecombined line (BT plus added segment) representing the bound-ary of the polygon (Fig. 2c), and for each point, we obtain theterrain elevation from the available DEM. We then use the nnlibrary (Sakov 2012) implemented in the GRASS GIS add-on mod-ule r.surf.nnbathy to interpolate geographically the elevation val-ues measured at the single points. The result is a raster surface thatshows the modelled bedding plane in 3D (Fig. 2d).
Next, values for the local slope and the local aspect are com-puted for each grid cell in the modelled 3D bedding surface, anddescriptive statistics are calculated, including the mean and thestandard deviation of the dip angle, and the mean, the angularstandard deviation and the circular variance of the aspect (Davis1990; Butler 1992; Nichols 2009). The latter values measure theuncertainty associated to the geometrical definition of the orien-tation of the bedding (strike and dip) and provide a quantitativeevaluation of the quality of the orientation measurementsobtained by fitting a plane through a set of 3D points. With thisrespect, the method is alternative to the method proposed byFernández (2005). Lastly, the computed values, and informationon the polarity (normal or reversed) of the bedding plane, areassociated to a point located in the centre of a rectangle boundingthe BT and stored in the GIS (Marchesini et al. 2013).
Bedding interpolation procedureInterpolation of bedding data is not a trivial task (de Kemp1998; Meentemeyer and Moody 2000; Günther 2003; Grelle et al.2011), chiefly because the strike and the dip directions areangular measurements in the range 0°–360°, with 0°=360°=N,and 0°–90°. Problems arise when beddings dipping in oppositedirections are interpolated. As an example, for two beddingsdipping to the NNE (N10) and to the NNW (N350), averagingof the two strike values will provide a grossly erroneous result(N180, instead of N360). In addition, simple interpolation of dipangles does not consider the polarity of the bedding, introduc-ing an additional source of error. To overcome these problems,the dip direction and the dip angle must be interpolated simul-taneously. This is achieved considering the single bedding meas-urements as unit vectors and combining them using analyticgeometry. To accomplish this task, we adopt an approach basedon directional cosines i.e. the cosines of the angles between theunit vector bn perpendicular to the bedding plane and the threecoordinate axes (de Kemp 1998; Günther 2003; Cencetti et al.2009) pointing to the E (x), to the N (y) and upward (z). Forbedding planes with a normal polarity, the three components ofthe unit vector bn are given by (Fig. 3a):
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nx ¼ sin αð Þ � sin βð Þny ¼ cos αð Þ � sin βð Þnz ¼ cos βð Þ
ð1Þ
where α is the dip direction and β is the dip angle of thebedding plane.
For beddings with a reverse polarity, the three components ofthe unit vector bn are given by (Fig. 3b):
nx ¼ −sin αð Þ � sin βð Þny ¼ −cos αð Þ � sin βð Þnz ¼ − cos βð Þ
ð2Þ
The three components of the unit vector perpendicular to thebedding plane can be interpolated using different approaches. Inthis work, we adopt the regularized spline with tension andsmoothing (RST) method (Mitášová et al. 2005; Neteler andMitášová 2008). The analytical interpolation results in three rasterGIS layers showing the components of the unit vector br : rx; ry; rzcomputed for each grid cell in the modelling domain. In thefollowing, we use boldface to show raster-based GIS layers. Thespatially distributed geometrical information is used to calculatethree GIS layers showing for each grid cell (1) the magnitude of theunit vector br obtained by combining analytically br : rx; ry; rz , (2)the dip angle of the bedding plane β and (3) the dip direction ofthe bedding plane α. In the Appendix, we list a pseudo-code tocalculate the br , α and β GIS layers. The equations listed in thepseudo-code exploit standard, grid-based “map algebra” operators(Neteler and Mitášová 2008) available in raster-based GIS.
Slope typesBased on the geometrical relationship between the orientation ofthe bedding planes and the geometry of the slopes, slopes can beclassified as (Grelle et al. 2011) (Fig. 4) the following: (1) anaclinalslopes, where bedding dips into the slope (a in Fig. 4), (2) ortho-clinal slopes, where the dip direction is orthogonal to the azimuthof the slope direction (b in Fig. 4), and (3) cataclinal slopes, wherebedding dips towards the slope free face. Cataclinal slopes can befurther subdivided into (4) cataclinal over-dip slopes, where thebedding dip angle is less steep than the local terrain gradient (c inFig. 4), (5) cataclinal dip slopes, where the bedding dip anglecoincides (within a tolerance) with the local terrain gradient (ein Fig. 4) and (6) cataclinal under-dip slopes, where the beddingdip angle is steeper than the local terrain gradient (d in Fig. 4).
Morpho-structural domainsMorpho-structural domains are geographical areas characterizedby homogeneous geometrical relationships between the attitude ofthe bedding planes and the geometry of the slopes (Cardinali et al.2002). To identify these homogeneous areas, a set of classificationrules must be defined. For the purpose, we adopted theTOpographic Bedding plane Intersection Angle (TOBIA) indexapproach (Meentemeyer and Moody 2000). The TOBIA index Tis a function of the topographic slope and aspect, and of thebedding dip direction and inclination, and is given by
T ¼ cos βð Þ � cos φð Þ þ sin βð Þ � sin φð Þ � cos α−γð Þ; 0≤T ≤1; ð3Þ
where β is the bedding dip angle (0°–90°), α is the bedding dip
Fig. 1 Examples of bedding traces in different lithological settings in Umbria, Central Italy. All images taken from Google Earth®. Scale bars show approximate scale ofthe images. a Heterogeneous, hard- and soft-layered rocks. b Homogeneous, hard-layered rocks. c Heterogeneous, soft continental deposits. d Heterogeneous, hard- andsoft-layered rocks with vegetation
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direction (0°–360°), φ is the local terrain gradient (0°–90°) and γ isthe local terrain aspect (0°–360°).
Where α=γ i.e. for a pure cataclinal slope, the equation reducesto
T ¼ cos βð Þ � cos φð Þ þ sin βð Þ � sin φð Þ; 0≤T ≤1: ð4Þ
Where β=φ,T=1 i.e. a pure cataclinal dip slope.
Fig. 3 Geometrical schemes showing unit vector bn normal to the bedding planeand its components along the three axes (nx, ny, nz). Angles α and β used inEq. (1) and Eq. (2) for the computation of nx, ny and nz are shown. a Beddingwith regular polarity. b Bedding with reverse polarity
Fig. 2 Scheme of the method to determine the attitude of a bedding plane from abedding trace. a Bedding trace (red line) is drawn on a base map. b End nodes of theline representing the BT (red line) are joined with a single segment (blue line) to forma polygon. c Equally spaced points (white dots) are selected along the boundary of thepolygon, and their elevation is obtained from the DEM. d Elevation values along the BTare interpolated to obtain a 3D model of the bedding plane. Faint shades of colourshow terrain elevation values. Solid shades of colour show elevation of the interpolatedbedding plane
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In Eq. (3), the term cos(α−γ) defines the cosine of the differencebetween the local terrain aspect α and the bedding dip direction γ i.e.the directional cosine. This difference can be used to separate slopescharacterized by different bedding geometries, including
& Cataclinal slopes, where 0°≤ |(α−γ)|≤45°i.e.,cos(α−γ)>0.707,& Anaclinal slopes, where 135°≤ |(α−γ)|≤225°i.e.,cos(α−γ)<−
0.707 and
& Orthoclinal slopes, where 45°≤ |(α−γ)|≤135°i.e.,cos(α−γ) and225°≤ |(α−γ)|≤315°i.e.,−0.707<cos(α−γ)<0.707.
Cataclinal slopes can be further subdivided into (Meentemeyerand Moody 2000)
& Cataclinal over-dip slopes, where T<0.99 and β<φ,
& Cataclinal under-dip slopes, where T<0.99 and β>φ and
& Cataclinal dip slopes, where T≥0.99.
For pure cataclinal slopes, the T value can be larger than 0.99only where |(β−φ)|≤8.1°. A T value larger than 0.99 exists for non-pure cataclinal slopes, with a related reduction of the difference|(β−φ)|. At the limit, where |(β−φ)|=0 i.e. where the bedding dipangle equals the local terrain gradient, a T value of 0.99 isexceeded where |(α−γ)|<14.1°. The area marked as F in Fig. 4represents the zone where T≥0.99.
This set of classification rules can be applied spatially in a GISexploiting raster-based map algebra operators. In particular, usingGRASS GIS, the r.mapcalc module can be used. The terrain slope (φ)and terrain aspect (γ) raster layers are readily calculated from theDEM.
Study areaWe have tested the method for the semi-automatic extraction ofquantitative information on the orientation of bedding planes in
the Collazzone area, Umbria, Central Italy (Fig. 5). This is aportion of a larger area studied by Guzzetti et al. (2006) to maplandslides and ascertain landslide hazard and by Guzzetti et al.(2009) to map landslides and determine landslide mobilizationrates. Our study area covers 50.5 km2 of hilly terrain, with eleva-tions in the range between 145 and 380 m (mean=236 m, standarddeviation=45.2 m) and terrain gradient in the range from 0° to 52°(mean=10.1°, standard deviation=5.6°). In the area, landscape iscontrolled by lithology and the attitude of bedding planes, whichalso control the shape and the extent of the slope failures. Valleysoriented N–S are shorter, asymmetrical and parallel to the maindirection of the bedding planes. The W-facing slopes are longerand less steep than the E-facing slopes, which are shorter andsteeper. Valleys oriented E–W are longer, symmetrical and mostlyperpendicular to the direction of the bedding planes. Sedimentaryrocks crop out in the area, including continental gravel, sand andclay, Plio-Pleistocene in age (Conti and Girotti 1977; Barchi et al.1991). The Plio-Pleistocene continental sediments are arranged in amonocline dipping gently towards W, the result of extensionaltectonic active in the Central Apennines (Ambrosetti et al. 1987;Martini and Sagri 1993; Barchi et al 2001). Landslides are abundantin the area and range in type and volume from very old, and partlydismantled, large deep-seated slides to shallow slides and flows(Guzzetti et al. 2006).
Digital topographyFor the study area, a digital elevation model (DEM) with a groundresolution of 10×10 m was available to us (Fig. 5). The DEM wasprepared through the interpolation of 5- and 10-m contour linesshown on topographic base maps (Guzzetti et al. 2006).
BeddingsWe obtained information on the geographical distribution of thebedding traces (BTs) in the study area through the visual interpre-tation of 1:33,000 scale, black and white aerial photographs takenin 1954 (red lines in Fig. 5). The geographical information on theBTs was obtained in three steps, adopting a procedure commonlyused to acquire digital geomorphological information from ste-reoscopic aerial photographs (Cardinali et al. 2001a; Antonini et al.2002). First, the individual BTs were identified visually on theaerial photographs and drawn on a transparent plastic sheetsuperimposed on the stereoscopic aerial photographs. Second,the individual mapped features were transferred visually on non-deformable plastic sheets superimposed on 1:10,000 scale topo-graphic base maps. In this phase, care was taken to accuratelylocate the BTs with respect to the local topography shown on thebase maps. Third, the information drawn on the non-deformableplastic sheets was scanned (at 300 dot-per-inch (dpi)),corresponding to a resolution of 0.84 m on the ground), geo-referenced, transformed into vector (line) format and stored in adedicated geo-database (Marchesini et al. 2013). Fourth, the posi-tion of the mapped BTs was checked on a digital stereo modelobtained from a stereoscopic GeoEye satellite image to improvetheir positional accuracy. The stereo model was obtained usingERDAS IMAGE® Leica Photogrammetry Suite SW, and the 3Dvisualization was obtained using a PLANAR StereoMirror™ HW(Ardizzone et al. 2013). This technology allows for the digitizationof 3D vectors on the 3D stereo model, which reduces the errors
Fig. 4 Possible bedding plane–slope relationships (Grelle et al. 2011). Half sphereshows synthetic topography and white dotted lines show homogeneous beddingsdipping N90E 45°. Legend a anaclinal, b orthoclinal, c cataclinal over-dip, d cataclinalunder-dip, e cataclinal dip, f cataclinal dip
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associated to the manual drawing of the interpreted geomorpho-logical features, including the BTs.
LandslidesA multi-temporal landslide inventory in digital format was avail-able for the study area. The map was prepared at 1:10,000 scalethrough the interpretation of five sets of aerial photographs takenbetween 1941 and 1997 and field surveys between 2003 and 2005(Guzzetti et al. 2006; Fiorucci et al. 2011). The informationobtained in the field or through the interpretation of the aerialphotographs was digitized, organized and stored into a geograph-ical database. In the multi-temporal inventory map, landslideswere classified according to the type of movement and the esti-mated age, activity, depth and velocity. Landslide type was defined
according to Cruden and Varnes (1996) and the WP/WLI (1990).Landslide age, activity, depth and velocity were determined basedon the type of movement, the morphological characteristics andappearance of the landslides on the aerial photographs and in thefield, the local lithological and structural setting and the date ofthe aerial photographs or the field surveys.
Table 1 lists descriptive statistics for the multi-temporal land-slide inventory available for the study area. For our experiment,the 1785 landslides in the multi-temporal inventory were groupedin three classes as follows: (1) old deep-seated landslides (LO) (2)deep-seated landslides (LD) and (3) shallow landslides (LS)(Fig. 6). For the subdivision, we used the information on theestimated age, depth and type of movement stored in the geo-graphical database. For the estimation of landslide age, we adop-ted the morphological criteria indicated by Turner and Schuster(1996) to distinguish between (1) dormant old or relict landslides,(2) dormant young and dormant mature landslides and (3) activeor recently active landslides. Within this framework, the old deep-seated landslides correspond to the dormant old or relict land-slides of Turner and Schuster (1996), whereas the deep-seated andthe shallow landslides can be both dormant mature or active orrecently active slope failures. The class of the old deep-seatedlandslides (LO) comprised 16 large, mostly relict slides and com-pound failures (slide-earthflows) in the range 6.6×104<AL<5.4×105 m2, with an average area AL=1.7×10
4 m2. These landslides arecontrolled by lithology, structure and the attitude of beddingplanes and are located preferentially at the head of the valleyswhere relative relief is larger. The largest landslides have modifiedthe local morphology, and the attitude of the bedding planes aredismantled by erosion processes and host younger and smallerfailures that have further modified their morphological appear-ance (Turner and Schuster 1996). The class of the deep-seatedfailures (LD) comprised 258 slides, slide-earthflows and complexmovements ranging in size from AL=2.5×10
2 to AL=1.7×105 m2,
wi th an average lands l ide area of AL = 2 .5 × 104 m2 .Morphologically, the largest deep-seated failures (AL>10
5 m2) aresimilar to the relict landslides, but are less dismantled by erosionand are less modified by more recent failures (Turner 1995). Theclass of the shallow landslides (Ls) comprised 1,511 failures, mostlyslide and flow type movements, with areas in the range 1.0×102<AL<2.8×10
4 m2. Investigations conducted immediately followinga rapid snowmelt event (Cardinali et al. 2001a) or rainfall eventsor periods (Guzzetti et al. 2009; Fiorucci et al. 2011) revealed thaterosion, other landslides and agricultural activities (chieflyploughing and tilling) cancelled the shallow landslides easily. Forthis reason, the number of shallow landslides in the multi-tempo-ral inventory underestimates the number of shallow landslidesthat have occurred in the area (Malamud et al. 2004; Fiorucci etal. 2011).
The old deep-seated landslides collectively cover 3.89 % of thestudy area, corresponding to a density of 0.03 landslides/km2. Thedeep-seated landslides cover 10.71 % of the area, a density of 5.10landslides/km2, and the shallow landslides cover 9.45 % of the area, adensity of 29.90 shallow failures/km2. Deep-seated landslides that donot overlap (totally of partially) old (relict) landslides cover 9.88 % ofthe study area, for a density of 4.55 landslides/km2 and shallowlandslides that do not overlap pre-existing, old deep-seated ordeep-seated failures cover 4.58 % of the area, equivalent to a densityof 17.73 shallow failures/km2.
Fig. 5 Shaded relief image for the Collazzone study area, Umbria, Central Italy,obtained from a 10×10-m DEM. Red lines show bedding traces obtained from thevisual interpretation of stereoscopic aerial photographs. Dashed rectangle showslocation of Fig. 11. UTM zone 33, datum ED50 (EPSG:23033)
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Definition of the morpho-structural domainsUsing the available digital terrain (aspect Fig. 7a and slope Fig. 7b)and bedding (dip direction Fig. 7c and dip Fig. 7d) information,and using the method presented in the “Method” section, we
prepared the map shown in Fig. 8. In this directional cosinemap, the anaclinal slopes have − 1<cos(α−γ)<−0.707, the
Deep-seated landslide not overlapping old landslides 230 252 24,865 173,518 23,344 4.55
Shallow landslide not overlapping deep-seated orold deep-seated landslides
896 123 3,089 28,271 3,229 17.73
Number number of landslides,AMINminimum landslide area,AAVGmean landslide area,AMAXmaximum landslide area,ASTD standard deviation of landslide area,DL landslide density
Fig. 6 Landslide inventory map for the Collazzone study area, Umbria, CentralItaly. Histogram shows legend and total landslide area (in km2) for three landslideclasses, namely LS shallow landslide, LD deep-seated landslide and LO old deep-seated landslide. UTM zone 33, datum ED50 (EPSG:23033)
Fig. 7 Maps showing terrain morphology and attitude of bedding planes for theCollazzone study area, Umbria, Central Italy. Histograms show legends and counts ofcells. a Terrain aspect. b Terrain slope. c Bedding dip direction (strike). d Bedding dip.UTM zone 33, datum ED50 (EPSG:23033)
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cataclinal slopes have 0.707<cos(α−γ)<1 and the remainingslopes are orthoclinal and have −0.707≤cos(α−γ)≤0.707. Usingthe values of the TOBIA index T (Meentemeyer and Moody 2000)obtained from Eq. (3), the cataclinal slopes were further subdi-vided into dip slopes, over-dip slopes and under-dip slopes. Theresult is shown in Fig. 9 that portrays a morpho-structural mapshowing bedding domains in the study area i.e. geographical sub-divisions characterized by homogeneous geometrical relationshipsbetween the attitude (strike and dip) of the bedding planes and thegeometry of the slope (terrain gradient and aspect).
Quantitative analysis of the morpho-structural map (Fig. 9)revealed that orthoclinal slopes occupy 49.8 % of the study area.In these areas, bedding dips at a significant angle from the terrainslope. The large proportion of orthogonal slopes was expected. In
the model (Fig. 4), orthoclinal slopes include grid cells with 45°<|(α−γ)|≤135° and 225°<|(α−γ)|≤315°, which encompass 50 % ofthe total angular range (0°–360°). The large proportion of ortho-clinal slopes suggests a mature landscape, where the incision of ariver network has carved the regional anaclinal–cataclinal setting,resulting in a large proportion of orthoclinal slopes. Further in-spection of the map revealed that cataclinal slopes characterize asignificant proportion of the study area (32.8 %), including 24.8 %of dip slopes, 5.2 % of over-dip slopes and 2.8 % of under-dipslopes. The significant proportion of cataclinal slopes is related tothe regional setting of the study area, arranged in a large, W-dipping monocline. Cataclinal over-dip slopes concentrate wherebeddings parallel to the slope intersect steep terrain formed by amore resistant rock layer. Cataclinal under-dip slopes are most
Fig. 8 Map showing cosine direction for the Collazzone study area, Umbria, CentralItaly. Histogram shows legend and proportion of cells. Shades of blue show anaclinalslopes, shades of green to brown show orthoclinal slopes, shades of orange showcataclinal slopes. UTM zone 33, datum ED50 (EPSG:23033)
Fig. 9 Map showing morpho-structural domains for the Collazzone study area,Umbria, Central Italy. Pie chart shows legend and counts of cells. a Anaclinal domain.b Orthoclinal domain. c Cataclinal over-dip domain. d Cataclinal under-dip domain. eCataclinal dip domain. UTM zone 33, datum ED50 (EPSG:23033)
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common in the eastern portion of the study area, where bedding issteeper. Anaclinal slopes cover 17.4 % of the study area and arepresent chiefly in the western part of the area.
The box plots in Fig. 10 summarize the statistics of terraingradient in the five morpho-structural domains identified in thestudy area. The statistics were obtained by computing terraingradient in a 3×3 moving window (i.e. a 30×30-m kernel).Inspection of the box plots reveals that the cataclinal over-dipdomain exhibits the largest mean terrain slope (16.2°), and theanaclinal and orthoclinal domains have the steepest slopes (50.5°and 51.9°, respectively). The mean (maximum) terrain gradient inthe cataclinal under-dip slopes is 7.5° (28.4°) and in the cataclinaldip slopes is 8.3° (20.9°). In general, the cataclinal slopes exhibit alower terrain gradient and the orthoclinal and anaclinal domainsthe steepest. This was expected, because the anaclinal and theorthoclinal settings generate steeper slopes than the cataclinalsetting (Guzzetti et al. 2006). Cataclinal over-dip slopes are anexception and show the largest mean value of terrain slope. In thiscase, local resistant layers of sand that form minor escarpmentscause the steep terrain.
Landslide abundance and types in the morpho-structural domainsFor a representative portion of the study area, Fig. 11 exemplifiesthe main geographical relationships between the morpho-structur-al domains and the different types of landslides in the study area.Inspection of Fig. 11 suggests that (1) the old, deep-seated land-slides (LO) and the deep-seated landslides (LD) (I in Fig. 11) areabundant in the cataclinal domain, and particularly in the cata-clinal dip and under-dip domains, and (2) the shallow landslides(LS) (II in Fig. 11) are present in all domains.
To investigate the influence of the bedding orientation on land-slide abundance and types, we compared in the GIS themap showingthe morpho-structural settings (Fig. 10) inside the slope failures
shown in the landslide inventory map (Fig. 6) with the morpho-structural settings in the entire study area. For the old deep-seatedlandslides (LO), we considered all the landslides shown in the inven-tory map (Fig. 6). For the deep-seated landslides (LD), we consideredonly failures that were not included in old deep-seated landslides,and for shallow landslides (LS), we considered only failures that werenot included in old deep-seated or in deep-seated landslides. Weapplied such criteria to all our following quantitative analyses.Results are summarized in Fig. 12 where three charts—one for eachof the considered landslide types (i.e. old deep-seated landslides,deep-seated landslides and shallow landslides)—show the following:(1) with coloured dots, the proportion of the five morpho-structuraldomains in the landslide areas and (2) with box plots, the proportionof the five morpho-structural domains in the entire study area. Thebox plots are slightly different for the three landslide types. This isbecause to prepare the box plots, we counted the number of land-slide pixels for a specific landslide type and we selected randomly
Fig. 10 Box-and-Whisker plots summarize statistics of terrain gradient in the fivemorpho-structural domains identified in the Collazzone study area, Umbria, CentralItaly. a Anaclinal domain. b Orthoclinal domain. c Cataclinal over-dip domain. dCataclinal under-dip domain. e Cataclinal dip domain
Fig. 11 Maps show examples of relationships between the morpho-structural domains(colours) and the landslides (white lines) in a portion of the Collazzone study area,Umbria, Central Italy. I Old deep-seated landslides (LO) and deep-seated landslides (LD).II Shallow landslides (LS). See Fig. 6 for complete landslide inventory map. a Anaclinaldomain. b Orthoclinal domain. c Cataclinal over-dip domain. d Cataclinal under-dip domain. e Cataclinal dip domain. UTM zone 33, datum ED50 (EPSG:23033)
Landslides
from the map showing the morpho-structural domains a number ofpixels equal to the number of landslide pixels. We repeated theprocedure 100 times to obtain a measure of uncertainty in thedefinition of the proportion of the morpho-structural domains foreach landslide type.
If landslides were not conditioned by the local morpho-struc-tural settings, the proportion of landslides in a specific morpho-structural domain would be the same or similar to the proportionof the same domain in the study area. In Fig. 12, the dots showingthe proportion of a single morpho-structural domain in the land-slide areas should coincide with the corresponding box plot show-ing the proportion of the same domain in the entire study area.Inspection of Fig. 12 reveals that with a few exceptions, the col-oured dots do not coincide with the corresponding box plots,confirming that beddings condition landslides in the study area.The differences between the individual coloured dots and thecorresponding box plots are large for the old deep-seated land-slides (LO), reduced for the deep-seated landslides (LD) and small
for the shallow landslides (LS). This indicates that the beddingscondition differently the different landslide types. The condition-ing is largest for the old deep-seated landslides, reduced for thedeep-seated landslides and weak for the shallow landslides.
Further inspection of Fig. 12 reveals that the proportion oflandslides in the cataclinal dip domain (red dots) is largest forthe old deep-seated landslides (∼36 %), intermediate for the deep-seated landslides (∼28 %) and reduced for the shallow landslides(∼19 %). The proportion of landslides in cataclinal dip slopes isalso significantly larger than expected (the box plot) for the olddeep-seated landslides, it is slightly larger than expected for thedeep-seated landslides, and it is smaller than expected for theshallow landslides. For cataclinal under-dip slopes (purple dots),the situation is similar. In this domain, the proportion of land-slides is significantly larger than expected (the box plot) forthe old deep-seated landslides, and it is expected for thedeep-seated and the shallow landslides (dots and box plotscoincide).
Fig. 12 Collazzone study area, Umbria, Central Italy. Coloured dots show proportion of the five morpho-structural domains in the landslide areas for LO old deep-seatedlandslides, LD deep-seated landslides, LS shallow landslides. Box plots show proportion of the five morpho-structural domains in the entire study area. a Anaclinal domain.b Orthoclinal domain. c Cataclinal over-dip domain. d Cataclinal under-dip domain. e Cataclinal dip domain
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Landslides
For the other morpho-structural domains, the situation is reversed.For cataclinal over-dip slopes (blue dots), the proportion of landslides issmaller than expected for the old deep-seated landslides and larger thanexpected for the deep-seated and the shallow landslides. Similarly, forthe orthoclinal domain (green dots), the proportion of landslides issignificantly smaller than expected for the old deep-seated landslides,it is marginally smaller than expected for the deep-seated landslides andis marginally larger than expected for the shallow landslides. For theanaclinal domain (yellow dots), the proportion of landslides is smallerthan expected for the old deep-seated and for the deep-seated landslidesand slightly larger than expected for the shallow landslides. From thisanalysis, we infer that the old deep-seated landslides and the deep-seatedlandslides occur preferentially in cataclinal dip slopes and in cataclinalunder-dip slopes. We further infer that orthoclinal slopes affect mostlythe old deep-seated landslides and that the local morpho-structuralsetting does not influence shallow landslides significantly.
To investigate further the influence (or the lack of influence) of themorpho-structural setting on landslides in the study area, we calculatedthe proportion of the mapped slope failures in the main beddingdomains (i.e. cataclinal, orthoclinal and anaclinal). Results are summa-rized in Fig. 13, where the rose diagrams show the proportion of terrainin the study area pertaining to the cataclinal (orange), the orthoclinal(green) and the anaclinal (blue) domains; the continuous thick blackline shows the circular density of the bedding classes for the entire studyarea (Fig. 13a); and the dashed black line shows the circular density of thebedding classes in the portion of the study area free of landslides(Fig. 13b). Inspection of Fig. 13a confirms that in the study area, thereis a prevalence of orthoclinal slopes followed by cataclinal slopes andthat anaclinal slopes are the least abundant. This is the result of theprevalent W-dipping monocline setting of the study area. The relativeproportions do not change significantly if only the landslide-free areasare considered (Fig. 13b). The other charts (Fig. 13c–f) show the propor-tions of landslide terrain in the different bedding domains (rose dia-grams) and the associated circular densities (continuous thin blacklines). In the charts, the pink and light blue areas outline differences inthe densities computed for the landslide-free area and for the differentlandslide types. The differences are largest for the old deep-seatedlandslides (LO) (Fig. 13e). In cataclinal slopes (orange in Fig. 13e), thedensity of old deep-seated landslides is significantly larger than thecorresponding density for the portion of the study area free of landslides(light blue area in Fig. 13e). Aminor positive deviation in the density forthe old deep-seated landslides is also present for orthoclinal slopes thatrotate into cataclinal slopes. The positive deviations in the density arecompensated by a reduced proportion (lack) of old deep-seated land-slides in the anaclinal and the orthoclinal domains (pink area in Fig. 13e).We conclude that in the study area, bedding controls the location andabundance of the old deep-seated landslides.
For shallow landslides (LS), Fig. 13c shows that the proportionof failures in the different morpho-structural domains (rose dia-gram) is similar, with a slight prevalence for the cataclinal (orange)and the orthoclinal (green) slopes. Inspection of Fig. 13 reveals thatonly minor differences are observed between the circular densitiescomputed for the landslide-free area and for the shallow land-slides. We conclude that in the study area, bedding does not exert asignificant control on the location and abundance of shallow land-slides. Deep-seated landslides (LD) (Fig. 13d) represent an inter-mediate condition between the old deep-seated landslides and theshallow landslides. The rose diagram in Fig. 13d shows that therelative proportion of deep-seated landslides is slightly larger in
the cataclinal slopes (orange) and reduced in the anaclinal slopes(blue). The difference between the circular densities computed forthe deep-seated landslides and for the landslide-free area is posi-tive (light blue area) for cataclinal slopes (orange) and for cata-clinal slopes that rotate to orthogonal slopes (orange to green),and it is negative (pink area) for anaclinal slopes. We conclude thatin the study area, bedding exerts some control on the location andabundance of the deep-seated landslides and that the control isweaker than for the old deep-seated landslides. This is a reason-able result, because the distinction between shallow and deep-
Fig. 13 Influence of the morpho-structural domains on landslides in theCollazzone study area, Umbria, Central Italy. Rose diagrams show the proportion ofterrain pertaining to the cataclinal (orange), the orthoclinal (green) and the anaclinal(blue) domains. Continuous thick black line shows circular density of beddingclasses for the entire study area. Dashed black line shows circular density of beddingclasses in the landslide-free portion of the study area. Continuous thin black linesshow circular density of bedding classes for different landslide types.Orange, cataclinaldomain. Green, orthoclinal domain. Blue, anaclinal domain. Light blue, positivedifference between the circular densities computed for a landslide-type andthe landslide-free area. Pink, negative difference between the circular densitiescomputed for a landslide-type and the landslide-free area
Landslides
seated landslide is faint and transitional. We acknowledge that theresult is, at least partly, due to possible inconsistencies in themapping of the landslides. In the landslide inventory map(Fig. 6), some of the small landslides may be erroneously classifiedas deep-seated failures.
DiscussionWe have proposed a GIS-based method to obtain spatially distrib-uted information on the 3D geometry of bedding planes from theanalysis of bedding traces (BTs). The reliability of the method andthe quality of the results depend on multiple factors, including thefollowing: (1) the completeness and accuracy of the mapping of theBTs, (2) the quality and resolution of the DEM used to capture theelevation information along the BTs, (3) how accurately the BTsrepresent the geometry of the bedding and (4) on the geologicaland morphological complexity of the investigated area.
According to the interpolated maps, in the study area, beddinginclination ranges between 0° and 35°, with a mode in the range 3°–5°, while the more frequent dip directions are between N230–N270(WSW). We attempted at a validation of the interpolated data. Wesampled the values of dip direction and inclination at a locationwhere field data measurements exist (115 locations). Comparisonwas performed by computing the absolute value of the differencebetween measured and interpolated values. Results indicate that(1) the 65 % of the differences of the dip direction are lower than45°, and only for the 20 % of the data, the differences are greaterthan 90°; (2) the 47 % of the differences of bedding inclination islower than 5°, and only for the 10 % of the inclination data, thedifference is greater than 20°. Figure 14 shows a visual comparisonof the field measured and modelled bedding attitude data, for aportion of the study area. Visual inspection of this image confirmsa fair agreement of the bedding attitude data collected by these
two different methods. We believe that a perfect match betweenmodelled and measured data could not have been achieved.Indeed, since the field survey data represent local bedding infor-mation, they cannot be completely comparable with thoseobtained from bedding traces which, in turn, represent a moregeneral setting of inclination and dip direction. Essentially, webelieve that, from a quantitative point of view, nor the filed dataneither the bedding traces can be considered as a benchmark or asthe ground truth. They describe quantities that, though similarand correlated, may differ significantly among them.
To perform well, the method requires sufficient information onthe location of the BTs. Where BTs are not present in a study area,or are not visible in the remote imagery used to identify the BTs,the quality of the result will suffer. It is also important that the BTsare mapped accurately i.e. that they represent faithfully the geo-metrical relationships between the bedding and the terrain.Inaccurate mapping of the BTs will result in potentially erroneousgeometrical interpolations. The quality of the DEM used to cap-ture the elevation information along the BTs is also important. Notonly the elevation values have to be sufficiently accurate but theyshould also reflect the morphology of the terrain where the BTs arelocated. Mismatches between the DEM and the BTs affect nega-tively the result. For our experiment, we used a 10×10-m DEM. Wemaintain that use of a very high resolution DEM (e.g. a 1×1-mDEM captured by a Lidar survey) would have introduced noise inthe digital representation of topography, resulting in a similaroutcome with unnecessary additional data or in a poorer outcome.Conversely, use of a coarse resolution DEM (e.g. the 3-arc-secondelevation coverage provided by the Shuttle Radar TopographyMission) may not be sufficient to capture the relationships be-tween bedding and terrain elevation, also resulting in erroneousinterpolations. The method is also sensitive to a single (local) errorin the DEM, which can condition significantly the interpolation ofthe 3D geometry of the bedding plane. For the interpolation of thecomponents of the unit vector representing a bedding plane, weused the regularized spline with tension and smoothing (RST)method (Mitášová et al. 2005; Neteler and Mitášová 2008).However, different interpolation methods can be adopted obtain-ing slightly different results.
The method assumes that the bedding plane is planar, and thecurvature of the single bedding is not admitted. This is a limitationrelevant in highly deformed (folded) terrains. However, the meth-od calculates separately the 3D geometry of different BTs in thearea. Circular statistics (Davis 1990; Butler 1992; Nichols 2009) canbe used to identify variations in the interpolated plane that may beindicative of the presence of discontinuities, local folding or inter-polation errors. Where a sufficient number of BTs is available in anarea, and the different BTs capture geographical variations in theattitude of the bedding, the method is capable of reconstructingfolded structures. Interpolation of the bedding information (strikeand dip) in the GIS can be further improved locally defininginterpolation domains bounded e.g. by fault lines or fold axes.The method assumes that the geometrical information captured bya BT provides an accurate representation of the local geologicalsetting. In general, the geometrical information obtained by thinlylayered rocks is more accurate than the information obtained fromsingle, very thick layers. This is because the terrain elevation of athick layer (e.g. of sandstone or conglomerate) may vary signifi-cantly, depending on where the elevation is measured. Further, the
Fig. 14 Comparison of the field measured and modelled bedding attitude data fora portion of the study area. Green symbols indicate field measures, red symbolsindicate modelled data, black lines show the bedding traces
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Landslides
method performs better in areas where bedding dips gently tomoderately. In areas where bedding is very steep, or vertical, theBTs are linear, regardless of the local topographical settings, andlinear BTs are difficult to interpolate accurately (Fig. 2). Wherebedding is nearly horizontal, the method is capable of determiningthe attitude (strike and dip) of the representative plane. However,the circular variance or the strike will be large.
Acquiring bedding information from aerial photographspresents the advantage of collecting data where no detectablelandslides are present. The interruption of the lateral continuityof a bedding trace is one of the diagnostic features used by geo-morphologists to confirm the presence of a landslide. Hence, acheckpoint to evaluate a consistent mapping of bedding traces(and of landslides) is to overlay the landslide inventory and thebedding traces layers to check the degree of overlapping of thesetwo thematic layers. The less the overlapping, the higher theconsistency. Figure 15 shows an excerpt of the overlay of the entirelandslide inventory map (black polygons) and of the bedding tracemap (yellow lines). In our map, we reported a total 134.19 km ofbedding traces, 22.59 km of which are intersected with landslides(16.83 %). Analysis of these data shows a highly consistent map-ping of bedding traces and of landslides. Furthermore, inspectionof Fig. 15 reveals that a large part of the bedding traces intersectinglandslides falls in the escarpment area where it is often possible toobserve the geological structure. In the Collazzone area, the avail-able landslide inventories report a total landslide area of10.42 km2, which represent the 20.6 % of the entire study area
(50.5 km2). These figures suggest that the 83.17 % of the beddingtraces have been obtained in the 40.1 km2 of the landslide-freearea. Besides, the number of bedding attitude data per squarekilometre in the landslide-free area is 2.86 for the field measure-ments and 4.98 for the modelled data. This information allows usto affirm that collecting bedding attitude information startingfrom aerial photographs would prevent geologists from collectingdata in landslide-bearing areas, which would introduce bias in anysubsequent analysis. At the same time, this method helps geolo-gists to exploit as much as possible the landslide-free area tocollect information on the geological structure.
Where spatially distributed information on the orientation ofthe bedding planes is available, the individual slopes can be clas-sified based on the relationship between the bedding attitude andthe geometry of the slopes (Fig. 4). In this work, we have usedstandard criteria (Grelle et al. 2011) to separate anaclinal, orthocli-nal and cataclinal slopes and to classify the cataclinal slopesinternally in over-dip slopes, dip and under-dip slopes. With theadopted criteria, the class of the orthoclinal slopes covers 50 % ofthe possible angular range (−0.707≤cos(α−γ)≤0.707), and theanaclinal and the cataclinal slopes each cover 25 % of the angularrange i.e. −1<cos(α−γ)<−0.707 for the anaclinal slopes and 0.707<cos(α−γ)<1 for the cataclinal slopes. The separation criteria canbe changed e.g. assuming that each slope class (orthoclinal, ana-clinal and cataclinal) covers the same proportion of the totalangular range. This will result in a different zonation of themorpho-structural domains.
Fig. 15 Overlay of the entire landslide inventory map (black polygons) and of the bedding trace map (green lines) for a portion of the study area. Red symbols showthe bedding attitude modelled data
Landslides
ConclusionsIn landscapes dominated by layered rocks, the geometrical rela-tionships between the attitude of bedding planes and the attitudeof terrain condition the type, abundance and distribution of land-slides. We have proposed a method to obtain quantitative, spatiallydistributed information on the 3D geometry of bedding planesfrom the analysis of bedding traces (BTs), where a bedding traceis the geometrical intersection of individual beddings, or groups ofbeddings, with the local terrain described by a digital elevationmodel. The method uses map algebra operators available in araster-based GIS (Neteler and Mitášová 2008) to interpolate BTinformation captured from the visual interpretation of stereoscop-ic images. When the geometry of the beddings is determined, mapalgebra is used to establish the relationships between the beddingand the attitude of the terrain and to define morpho-structuraldomains i.e. areas characterized by homogeneous geometricalrelationships between the bedding and the geometry of the slopes.The method was tested in a study area in Umbria, Central Italy,where it proved capable of outlining accurately the morpho-struc-tural domains.
To investigate the influence of bedding on landslide abundance andtypes in the study area, themap of themorpho-structural domains wascompared in a GIS to a landslide inventory map, where the landslideswere classified as old (relict) deep-seated landslides, deep-seated land-slides and shallow (surficial) slope failures. The analysis revealed thatbeddings condition the location and abundance of the relict and thedeep-seated landslides, which are most abundant in cataclinal dip andunder-dip slopes and does not condition significantly the distributionand abundance of the shallow landslides.
We expect that the use of the method will facilitate the produc-tion of maps showing morpho-structural domains in layered geo-logical environments. Availability of accurate and spatiallydistributed information on morpho-structural domains for a largearea, which is now difficult and time-consuming to obtain, willcontribute to better understand landslide phenomena, to explaintheir distribution, abundance and patterns and to prepare moreadvanced landslide susceptibility and hazard models and the as-sociated terrain zonations.
Software availabilityTo perform the procedure described in the “Method” section, theauthor IM prepared two scripts for the GRASS GIS softwareenvironment (version 7). The first script calculates bedding atti-tude starting from a layer of bedding traces and a DEM. Thesecond script interpolates bedding attitude data and calculatesthe bedding attitude—slope relationships based on the TOBIAindex (Meentemeyer and Moody 2000). The scripts are avail-able for download at the following web address: http://geo-morphology.irpi.cnr.it/tools/gis-and-interpretation-of-aerial-photographs.
AcknowledgmentsMS and FB were supported by a grant of the Regione dell’Umbriaunder contract POR-FESR (Repertorio Contratti n. 861, 22/3/2012)and by a grant of the National Department of Civil Protection. FFwas supported by a grant of the the Regione dell’Umbria undercontract POR-FESR (Repertorio Contratti n. 793, 18/7/2011). Thecomments of two anonymous reviewers improved the quality andreadability of the manuscripts.
AppendixPseudo-code to calculate the magnitude of the unit vector repre-senting a bedding plane br , the dip angle of the bedding plane βand the dip direction of the bedding plane α.
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# Calculate the r layer = (rx
2 + ry2 + rz
2)1/2
# Calculate the m layer, the component of r in the horizontal plane m = (rx
2 + ry2)1/2
# For bedding with normal polarity ( rz > 0), and vertical beddingIF (rz ≥ 0)
{ # Calculate the bedding dip:
= ARCCOS(rz / r)# Calculate the dip direction layer, depending on quadrant # First quadrant [0°-90°)IF (rx ≥ 0) AND (ry ≥ 0) THEN { = ARCSIN (rx / m)}# Second quadrant [90°-180°)ELSE IF (rx < 0) AND (ry ≥ 0) THEN { = 360 – ARCSIN (-rx / m)}# Third quadrant [180°-270°)ELSE IF (rx < 0) AND (ry < 0) THEN { = 180 + ARCSIN (-rx / m)}# Third quadrant [180°-270°)ELSE { = 180 – ARCSIN (rx / m)}}
# For reversed bedding polarity ( rz < 0)ELSE
{ =180 - ARCCOS(-(rz)/r))
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M. Santangelo : I. Marchesini ()) : M. Cardinali : F. Fiorucci : M. Rossi : F. Bucci: F. GuzzettiIstituto di Ricerca per la Protezione Idrogeologica,Consiglio Nazionale delle Ricerche,via Madonna Alta 126, 06128, Perugia, Italye-mail: [email protected]
M. Santangelo : M. RossiDipartimento di Scienze della Terra,Università degli Studi di Perugia,piazza dell’Università 1, 06123, Perugia, Italy
