© Acta hydrotechnica 23/38 (2005), Ljubljana ISSN 1581-0267 19 UDK / UDC: 502.5:519.61/.64:551.3 Prejeto / Received: 12. 1. 2005 Pregledni znanstveni članek – Review scientific paper Sprejeto / Accepted: 10. 7. 2006 MODELIRANJE GIBANJA SKALNIH PODOROV – PREGLED MODELLING OF ROCKFALL MOTION – A REVIEW Urška PETJE, Matjaž MIKOŠ, Bojan MAJES Pomemben element rizi čnega menedžmenta v gorskem svetu je analiza naravnih nevarnosti, ki jih povzro čajo tudi skalni podori (skupni izraz za odlome in prosto padanje kamenja, skal in ve čjih skalnih blokov ter podobne oblike masnega gibanja pod vplivom težnosti). Skalni podori so zaradi svoje energije in hitrosti gibanja skalnih gmot še posebej nevarni dejavniki tveganja. Zato jim širom sveta namenjajo veliko pozornost in jih v raznih oblikah tudi modelirajo – napovedujejo njihovo pot in doseg. V prispevku so prikazane glavne zna čilnosti najpomembnejših enostavnih modelov obnašanja skalnih podorov, pripravljene na osnovi pregleda literature. Dispozicijski modeli so tisti, ki nam povedo, kje lahko pride do skalnih podorov. Procesni modeli simulirajo dinamiko podornega procesa. Glede na pristop k obravnavani procesa jih lahko delimo na empiri čne modele in analiti čne modele. Empirični procesni modeli na splošno temeljijo na povezavi med topografskimi faktorji in obmo čjem odlaganja skalnega podora. Analiti čni procesni modeli pa so sestavljeni iz modela trajektorij in modela trenja. Analiti čni modeli opisujejo in simulirajo v dveh ali treh dimenzijah gibanje podorne mase in se lo čijo glede na na čin, kako upoštevajo podorno maso (masna to čka, oblika togega telesa) in kako simulirajo gibanje po pobo čju (poskakovanje, kotaljenje, drsenje). Modeli na osnovi geografskega informacijskega sistema izkoristijo prednosti tega sistema in potekajo v treh korakih: dolo čitev obmo čij izvora podorov, določitev trajektorij posameznih skalnih blokov in dolo čitev obmo čij izteka (odlaganja) podorne mase. Glavni namen pregleda modeliranja gibanja skalnih podorov je, da bi strokovnjakom olajšali izbor ustreznega modela za lokalno in regionalno merilo. Klju čne besede: skalni podori, skalni odlomi, modeliranje, geografski informacijski sistemi, naravna tveganja, rizi čni menedžment, prostor An analysis of natural hazards caused by rockfalls (common expression for falling stones and boulders; and other similar forms of gravitational mass movements) is an important element of risk management in mountainous regions. Due to their energy and velocity rockfalls represent an especially dangerous hazard factor. Because of that rockfalls are given much attention all over the world and they are modelled in different ways – simulating their paths and run-out distances. In this paper, a literature review of the main characteristics of the most important non-comprehensive rockfall models is presented. The dispositional models are those that tell us where a hazardous process may occur. The process-based models simulate rockfall process dynamics. They can be classified in relation to the process approach into empirical models and into analytical models. Empirical process-based models are generally based on the relationship between topographic factors and rockfall run-out zone. Analytical process-based models are composed of a trajectory model and a friction model. They describe and provide 2-D or 3-D simulation of the movement of the rockfall masses and can be differentiated regarding the way how the rockfall mass (lumped mass, rigid body shape) and the movement on the slope (bouncing, rolling, sliding) are described, respectively. The GIS-based models use the advantages of this system and work in three steps: the determination of rockfall source areas, the determination of trajectories of single boulders, and the determination of run-out distances and run-out zones. The main aim of the review on modelling of rockfall motion is to make it easier for the professionals to choose an adequate rockfall model at local and regional scales. Key words: rockfalls, rockslides, modelling, geographical information systems, natural hazards, risk management, environment Petje, U., Mikoš, M., Majes, B.: Modeliranje gibanja skalnih podorov – pregled – Modelling of Rockfall Motion – A Review © Acta hydrotechnica 23/38 (2005), 19–38, Ljubljana 20 1. UVOD Skalni podori razli čnega dosega so predvsem v gorskem svetu pogosta in obenem zaradi svoje energije obi čajno zelo intenzivna oblika naravnih tveganj (Brilly et al., 1999). Tipi čen primer skalnega podora v ozki alpski dolini je npr. podor Osojnik v dolini Trente (slika 1). 1. INTRODUCTION Rockfalls with different run-out distances are of frequent occurrence especially in mountainous areas and due to their energy they normally represent a severe form of natural hazards (Brilly et al., 1999). As a typical example of a rockfall in a narrow alpine valley, on Fig. 1 the Osojnik rockfall in the Trenta valley, W Slovenia, is shown. Slika 1. Skalni podor Osojnik v dolini Trente (foto: avtorji, 2004). Figure 1. The Osojnik rockfall in the Trenta valley (photo: authors, 2004). Izraz skalni podor se v tem prispevku uporablja skladno z definicijo Kienholz et al. (1998) za odlom in gibanje posameznih skal (kamenja in blokov; Steinschlag v nemš čini) kakor tudi za skalno maso, ki lahko razpade v posamezne bloke (Felsturz v nemš čini), ki delujejo eden na drugega le v manjši meri. Tako kot druga masna gibanja, kot so kamninski zdrsi, kamninski in zemljinski plazovi ter pobo čni drobirski tokovi, se skalni podori prikazujejo v kartah nevarnosti. Te karte so razli čnega merila, izdelane na razli čnih osnovah, med katere spada tudi modeliranje nastanka in gibanja skalnih podorov (Petje, 2005). Rezultati takega modeliranja se lahko koristno uporabijo pri analizah tveganja pred skalnimi podori, za kar v Sloveniji od leta 2002 (sprejet nov Zakon o vodah) obstaja zakonska podlaga in obveza ( Đurovi ć & Mikoš, 2004). The term rockfall in this paper will be used according to the definition by Kienholz et al. (1998) for detachment and motion of single rocks (stones and boulders; Steinschlag in German) as well as of a rock mass that can disintegrate into single blocks (Felsturz in German) that interact with each other only to a minor extent. Like other mass movements, such as rockslides, landslides and slope debris flows, rockfalls are shown on hazard maps. These maps are of different scales and made on different bases, which include modelling of rockfall initiation and dynamics (Petje, 2005). The results of such a modelling can be usefully applied in risk analyses of rockfalls for what in Slovenia since 2002 (new Water Act) legal basis and obligation exist ( Đurovi ć & Mikoš, 2004). Petje, U., Mikoš, M., Majes, B.: Modeliranje gibanja skalnih podorov – pregled – Modelling of Rockfall Motion – A Review © Acta hydrotechnica 23/38 (2005), 19–38, Ljubljana 21 V prispevku so prikazani na čini modeliranja gibanja skalnih podorov, razdeljeni v skupine modelov. Namen prikaza je pomagati izbrati model skalnega podora, ki se lahko uporabi za analizo tveganja pred skalnimi podori v lokalnem ali regionalnem merilu. Primer uporabe dvodimenzijskega modela v lokalnem merilu je prikazan drugje (Petje, 2005). 2. VRSTE MODELOV Ra čunalniško simuliranje naravnih procesov v geomorfologiji uporablja dva popolnoma razli čna pristopa: realisti čni pristop in funkcionalisti čni pristop (Howes & Anderson, 1998). Nasprotno pa se modele, ki se jih uporablja pri naravnih nesre čah, deli na dispozicijske in procesne modele (Hegg & Kienholz, 1995). Dispozicijski modeli (imenujemo jih tudi stati čni modeli) služijo za raziskavo možnih izvorov nevarnosti – povedo nam, kje lahko pride do nevarnega procesa. V dolo čenih primerih lahko dolo čimo pretekle podorne procese skozi daljše časovno obdobje na osnovi geoloških dokazov (neme pri če kot npr. posamezni skalni balvani, položaj in debelina odkladnin). Tako lahko dolo čimo doseg podora z neko povratno dobo. Vendar pa se povsod tega ne da dolo čiti, saj se lahko spremenijo razmere, pri katerih poteka preperevanje, ali pa se odkladnin skalnega podora ne more lo čiti od ledeniških ali drugih odkladnin (Evans & Hungr, 1993). Procesni modeli (znani tudi kot dinami čni modeli) simulirajo dinamiko procesa. Glede na pristop k obravnavani procesa jih lahko delimo na empiri čne procesne modele (funkcionalisti čni pristop po Howes & Anderson, 1998) in na analiti čne procesne modele (realisti čni pristop po Howes & Anderson, 1998). Empiri čni procesni modeli (imenujemo jih tudi statisti čni modeli) na splošno temeljijo na povezavi med topografskimi faktorji in obmo čjem odlaganja skalnega podora. Analiti čni procesni modeli so sestavljeni iz: • modela trajektorij in • modela trenja. Tako analiti čni procesni modeli dolo čijo In this paper, ways of modelling rockfall motion are shown, divided into separate groups of models. The aim of the review is to help select a rockfall model that may be used for rockfall risk analysis at local or regional scale. A case study using a two-dimensional model at local scale is given elsewhere (Petje, 2005). 2. GROUPS OF MODELS Computer simulation of natural processes in geomorphology uses two fundamentally different approaches: realist approach and functionalist approach (Howes & Anderson, 1998). On the contrary, the models associated with natural disasters can be divided into dispositional models and process-based models (Hegg & Kienholz, 1995). Dispositional models (also called statical models) are used for the research of possible source areas of hazards – they tell us, where a hazardous process may occur. In some cases recent or paleo rockfall processes in longer geological periods can be determined on the basis of geologic proofs (silent witnesses as i.e. single boulders, position and thickness of deposits). Thus a rockfall run-out distance with a return period can be determined. This cannot be determined under all circumstances, due to changes in field conditions regarding weathering, or rockfall deposits that cannot be differentiated from glacial or other deposits (Evans & Hungr, 1993). Process-based models (also known as dynamic models) simulate process dynamics. They can be classified in relation to the process approach into empirical process-based models (functionalist approach after Howes & Anderson, 1998), and into analytical process- based models (realist approach after Howes & Anderson, 1998). Empirical process-based models (called also statistical models) are generally based on the relationship between topographic factors and rockfall run-out zone. Analytical process-based models are composed of: • a trajectory model and • a friction model. They determine possible pathways, along Petje, U., Mikoš, M., Majes, B.: Modeliranje gibanja skalnih podorov – pregled – Modelling of Rockfall Motion – A Review © Acta hydrotechnica 23/38 (2005), 19–38, Ljubljana 22 možne poti, po katerih se proces odvija (trajektorije pri podorih), ter za njih predvidijo hitrosti, kineti čno energijo in mesta odlaganja (doseg procesa). S to lo čitvijo na dva dela dosežemo boljše strukturiranje procesnega modela skalnega podora za uspešno rešitev problema. Ra čunalniški (simulacijski) program je tako sestavljen iz modulov, kar poenostavi verifikacijo modela. Modeli trajektorij ne morejo simulirati realnih poti za pobo čne procese, če ne vsebujejo podrobnega digitalnega modela višin. V nadaljevanju bodo prikazane zna čilnosti dveh glavnih vrst procesnih modelov (empiri čni modeli, analiti čni modeli) brez navajanja vseh modelnih podrobnosti. 3. EMPIRI ČNI PROCESNI MODELI Scheidegger (1973) je ugotovil, da je dolžina dosega odvisna od kota med horizontalno črto in črto, ki povezuje za četno in končno težiš če skalne mase, in je enak kotu trenja φ (0,57–0,83), ki kontrolira gibanje, ter je približno enak kotu gibanja (slika 2). which the process takes place (trajectories with rockfalls) and determine velocities, kinetic energy, and areas of deposition (reach of a process). With this division into two parts a better process-based rockfall model structure for a successful solving of the problem is achieved. The computer (simulation) program thus consists of modules, what makes the model validation easier. The trajectory models cannot simulate true pathways for slope processes without incorporating a detailed digital elevation model (DEM). Next in the paper, characteristics of the two main groups of the process-based models (empirical models, analytical models), will be shown without too many details. 3. EMPIRICAL PROCESS-BASED MODELS Scheidegger (1973) stated that run-out distance is a function of angle between the horizontal line and the line that connects the starting point and the centre of the deposited mass and is equal to the angle of friction φ (0.57–0.83) that controls the movement, and approximately equals to the “travel angle” (Fig. 2). A B D dz E F C β βe f Slika 2. Zasnova empiri čnih modelov. To čki A in B sta težiš či skalnega podora pred in po premiku mase; C ozna čuje pot težiš ča skalne mase; E ozna čuje energijsko črto; F ozna čuje kot gibanja. Figure 2. The concept of empirical models. Points A and B denote the centre of gravity of the rockfall before and after the mass movement, respectively; C is the pathway of the center of gravity of the rockfall mass; E denotes the energy line; F denotes the travel angle. Petje, U., Mikoš, M., Majes, B.: Modeliranje gibanja skalnih podorov – pregled – Modelling of Rockfall Motion – A Review © Acta hydrotechnica 23/38 (2005), 19–38, Ljubljana 23 Onofri in Candian (1979) ter Toppe (1987) so predlagali princip “kota gibanja” (Heim, 1932) za dolo čitev cone izteka. V slovenš čini še nimamo uveljavljenega izraza za nemški izraz “Fahrböschung”, zato predlagamo izraz “kot gibanja”, ki je neposredni prevod angleškega izraza. Horizontalna prepotovana razdalja x je tako izražena z višino padanja h: Onofri and Candian (1979) and Toppe (1987) proposed to take this “travel angle” (Heim, 1932) for determining the run-out zone. In the Slovenian language there is no established term for the German term “Fahrböschung”, so here a term “kot gibanja” is proposed, which is a direct translation from English. The horizontal travel distance x is thus expressed by the vertical free fall height h: φ tan h x = (1) Razni avtorji (npr. Hsü, 1975; Onofri & Candian, 1979; Domaas, 1985; glej pregled v Mei βl, 1998) navajajo, da je za ve čino podorov ta kot > 32 °, vendar pa se za volumne podora nad 10 6 m 3 kot zelo hitro zmanjša. Kot, ki povezuje za četno in kon čno težiš če mase, predstavlja translatorno komponento celotnega gibanja in se po tej dinami čni zna čilnosti tudi razlikuje od kota gibanja. Vendar pa ravno tako ne vsebuje informacije o razširjanju materiala med gibanjem, ki je klju čno za dolo čitev prepotovane razdalje. Scheideggerjev model (1973) upošteva trenje med drse čo maso in tlemi in ne upošteva deformacije mase med gibanjem. Konvencionalni model predpostavlja, da je horizontalna hitrost v h enaka vertikalni hitrosti v v , ki jo doseže padajo ča masa tik pred trkom s tlemi, vendar pa se horizontalna hitrost zmanjša zaradi izgub pri trku. Tako lahko zapišemo: Different authors (e.g. Hsü, 1975; Onofri & Candian, 1979; Domaas, 1985; see the overview in Mei βl, 1998) quote that for the majority of rockfalls this angle is > 32 °, but for rockfalls with volumes > 10 6 m 3 this angle very quickly decreases. The angle that connects the starting point and the centre of gravity of the deposited mass represents the translational component of the whole movement and differs in this dynamical characteristic from the travel angle. However, it also does not incorporate information on spreading of material during motion, which is a key factor for determining the travel distance. The Scheidegger model (1973) takes into account friction between the sliding mass and the ground and does not take into account mass deformations during travel. A conventional model assumes that the horizontal velocity v h equals the vertical one v v , which is achieved by the falling mass immediately before the impact with the ground. The horizontal velocity is reduced due to losses at impact. So one can state that: φ tan 2 h r x = , (2) kjer je r koeficient odboja, njegova vrednost je pod 1: where r is coefficient of restitution and its value is below 1: v h v r v ⋅ = (3) Corominas (1996) je v raziskavi 204 plazov, od tega 47 skalnih podorov in skalnih plazov, ugotovil, da se z naraš čanjem prostornine kot gibanja zmanjšuje in da to ne velja le za velike plazove. Zmanjšanje kota naj bi bila posledica u činka ovir in topografskih ovir. Pri ve čjih volumnih na odboje ne Corominas (1996) examined 204 slides, out of them 47 rockfalls and rock avalanches, and found that with increasing volume the travel angle decreases and that this is not only valid for large falls and slides. The decreasing of the angle should be the consequence of obstacles and topographic constraints. With larger Petje, U., Mikoš, M., Majes, B.: Modeliranje gibanja skalnih podorov – pregled – Modelling of Rockfall Motion – A Review © Acta hydrotechnica 23/38 (2005), 19–38, Ljubljana 24 vplivajo ovire in vegetacija. Kot gibanja je odvisen od materiala, prostornine podorne mase in mehanizma gibanja in ne od potencialne energije oziroma višine padanja. Višina padanja ima vpliv na horizontalno prepotovano razdaljo, vendar ni nujno, da daljša horizontalna razdalja ustreza manjšemu kotu gibanja, lahko ustreza le ve čji relativni mobilnosti. Davies in McSaveney (1999) sta z laboratorijskimi eksperimenti opazovala obnašanje skalnih podorov in plazov. Ugotovila sta, da je za skalne podore velikosti 1.000 do 10.000 m 3 geometrija odlaganja na pobočjih z naklonom 35 ° do 45° v mo čni povezavi s prostornino. Rezultati se ujemajo z meritvami na terenu. Za ve čje skalne plazove s prostorninami ve čjimi od 10 7 m 3 pa so zna čilni zelo veliki dosegi. Vzrok naj bi bil verjetno povezan s faktorji, kot sta drobljenje med gibanjem ter erodibilna podlaga. Keylock in Domaas (1999) sta testirala tri empiri čne modele (imenujeta jih statisti čni modeli) in en preprost dinami čni model glede njihove sposobnosti napovedovanja maksimalne dolžine dosega z uporabo preprostih topografskih parametrov. Ugotovila sta, da ima statisti čni model prednost, če želimo hitro in u činkovito dolo čiti tveganje pred podori za ve čja obmo čja. Okura et al. (2000b) je z eksperimenti iskal zvezo med dosegom in prostornino. Iz eksperimentov in simulacij sledi, da obstaja jasna pozitivna povezava med dosegom in prostornino ter negativna povezava med razdaljo težiš ča mase in prostornino. Podori z naraš čanjem prostornine težijo k uteko činjenju. Naklon pobo čja in doseg sta v obratnem sorazmerju. Topografija je eden od zelo pomembnih faktorjev za dolo čevanje dosega. 4. ANALITI ČNI PROCESNI MODELI Analiti čni modeli opisujejo ali simulirajo gibanje v dveh ali treh dimenzijah. Modeli v dveh dimenzijah se obi čajno uporabljajo v lokalnem merilu (posamezno pobo čje), modeli v treh dimenzijah pa so primernejši za volumes the obstacles and vegetation have no influence on rebounds. The travel angle depends on material, mass volume and mechanism of motion and not on potential energy or fall height. The height of falling has some influence on the horizontal travel distance but it is not necessarily true that larger travel distances correspond with smaller travel angles – it can correspond with higher relative mobility. Davies and McSaveney (1999) performed laboratory experiments and observed behaviour of rockfalls and rock avalanches. They found out that for rockfalls of the size between 1,000 and 10,000 m 3 deposition geometry on slopes with a gradient between 35° and 45° is strongly related to the volume. The results agree with field measurements. For large rock avalanches with volumes larger than 10 7 m 3 extraordinarily long run-out distances are typical. The cause may well be connected to factors such as fragmentation during motion and the erodible base. Keylock and Domaas (1999) tested three empirical models (called statistical models) and a simple dynamical model for their capability of forecasting the maximum run-out distance using simple topographic parameters. They found that the statistical model has an advantage, if rockfall risk in larger areas has to be determined quickly and effectively. Okura et al. (2000b) searched a connection between the run-out and the volume. From experiments and simulations it follows that a clear positive correlation exists between the run-out distance and volume as well as a negative correlation between distance of gravitational center of mass and volume. Rockfalls with increasing volume tend to fluidization. The slope gradient and run-out distance are in inverse proportion to each other. Topography is one of important factors for determining the run-out distance. 4. ANALYTICAL PROCESS- BASED MODELS Analytical models describe and simulate motion in two or three dimensions. Two- dimensional models are usually applied at local scale (individual slope), and three- dimensional models are more appropriate for Petje, U., Mikoš, M., Majes, B.: Modeliranje gibanja skalnih podorov – pregled – Modelling of Rockfall Motion – A Review © Acta hydrotechnica 23/38 (2005), 19–38, Ljubljana 25 obravnavo v regionalnem merilu (cela dolina, občina, posamezni kartni list). Pred obravnavo zna čilnosti analiti čnih modelov v dveh oziroma treh dimenzijah bodo prikazani glavni modelni parametri ter ob čutljivost in zanesljivost teh modelov. 4.1 MODELNI PARAMETRI ANALITI ČNIH MODELOV Za izra čun trajektorij padanja skal potrebujemo naslednje podatke: • lokacijo potencialnih mest izpadanja skal, izvor (za četna to čka, angl. orig. source area, detachment point); • obliko in geometrijo skal; • možno velikost skal, maso (interval od– do); • mehanske lastnosti skal in pobo čja; • odbojni koeficient; • zna čilne vzdolžne profile in topografijo pobočja; • drobljenje skal ob padcu. Eden izmed vhodnih parametrov v modelih je masa skal. Pri tem upoštevamo pravilo, da imajo ve čje skale manjšo verjetnost pojavljanja. Vpliv geometrije brežine na rezultat izra čuna dolo čimo s spreminjanjem oblik terena med dvema to čkama na brežini (dolo čitev ob čutljivosti na obliko brežine). Za vpliv, ki ga ima lastnost hribine, delamo izra čun z vpeljavo velike standardne deviacije odbojnega koeficienta R n . Za mesto izvora padanja lahko v izra čunu predvidimo številna mesta, ponavadi pa so najbolj kriti čna na vrhnjem delu brežine. V naravi so oblika in velikost skale, mehanske lastnosti in natan čna lokacija izvora težko dolo čljive. Geometrija pobo čja (gradient, dolžina, hrapavost materiala) se spreminja po pobo čju in je ne moremo popolnoma zajeti. Ve čina programov uporablja profile, ki jih je dolo čil uporabnik in zahtevajo podrobnejše informacije o materialu. Zna čilni vzporedni profili po padnici brežine so navadno dobljeni iz topografskih kart ali pa jih izdelamo s terestri čnim terenskim snemanjem v podrobnem merilu. Litološke zna čilnosti in rabo tal se dolo či s terenskimi raziskavami, iz geoloških kart in kart rabe tal an analysis at regional scale (whole valleys, communities, single cartographic sheets). Before dealing with characteristics of analytical models in two and three dimensions the main model parameters will be shown as well as models’ sensitivity and realibility. 4.1 ANALYTICAL MODEL PARAMETERS So as to compute the trajectories of falling rocks the following data are required: • positions of detachment points or source areas; • the shape and geometry of rocks; • possible rock size and their mass (range from–to); • mechanical properties of the rocks and the slope; • coefficient of restitution; • typical longitudinal profiles and slope topography; • crushing of rocks at impacts. One of the input parameters in the models is rock mass. In doing so the rule is obeyed, which defines that larger rocks have lower probability to detach. The influence of slope geometry on the computational results can be determined by changing the slope shape between two points on the slope (sensitivity analysis on the slope form). The influence of rock characteristics can be determined by large standard deviation of coefficient of restitution R n . For the source area of falling stones numerous points can be determined and the most critical ones are usually in the upper part of the slope. The shape and size of rocks, mechanical properties and the precise location of sources are hard to determine in the field. The slope geometry (gradient, length, material roughness) changes along the slope and cannot be fully captured. The majority of computer programs use profiles, determined by users, and ask for detailed material properties. Characteristic longitudinal profiles along the slope gradient are usually obtained from topographic maps or produced by terrestrial field surveys in precise scale. The lithological properties and soil use are determined by field research, from geological maps and soil use maps or by the interpretation of aerial Petje, U., Mikoš, M., Majes, B.: Modeliranje gibanja skalnih podorov – pregled – Modelling of Rockfall Motion – A Review © Acta hydrotechnica 23/38 (2005), 19–38, Ljubljana 26 ali pa z interpretacijo aerofoto posnetkov. Ve čina programov dela zadovoljivo dobro samo na majhnih površinah, za katere so na voljo podrobnejše (tudi topografske) informacije. Rezultate napovedi zelo izboljšamo z natan čno terensko analizo padanja kamnov na obravnavanem terenu. Koli čina izgub energije ob trku ali kotaljenju je v veliki meri odvisna od oblike, velikosti, hitrosti in dinamike bloka (translatorna in kotna hitrost), geotehni čnih lastnosti pobo čja (granulometri čna sestava, elasti čni modul, vsebnost vode …), geometrije, topografije in hrapavosti površine ter kota trka. Te parametre je težko to čno dolo čiti. Napovedovanje trajektorije je zato kompleksna operacija, ki vsebuje veliko negotovosti. Razmerja med energijskimi izgubami in drugimi spremenljivkami niso to čno dolo čena. V ve čini primerov se vsi u činki zaradi plasti čnih deformacij podlage in geometri čne konfiguracije trka upoštevajo s »kontaktnimi funkcijami«, ki opisujejo kinematiko skale (hitrost) ali dinamiko (energijo) pred in po trku. Te funkcije so izražene kot koeficient odboja in koeficient trenja. Koeficient odboja je tista spremenljivka, s katero umerjamo model padanja skal. V izra čunu za poljuben odsek profila vnesemo dva odbojna koeficienta, normalni in tangencialni (preglednica 1). Normalni koeficient odboja R n , ki se ponavadi spreminja med vrednostma 0,3 0,5 n R << , se uporabi za primere, ko kamen udari na tla blizu kota 90°. Tangencialni koeficient odboja R t , ki se ponavadi spreminja med vrednostma 95 , 0 8 , 0 < < t R , pa je primeren za padce skal, ki padejo na površino pod ostrim kotom. Mehke cone zemljine in vegetacija zavzemajo spodnje dele obsega odbojnega koeficienta, odkrita hribina in asfalt pa zgornje. Toda že majhna sprememba koeficienta odboja prinese popolnoma druge rezultate. Na primer, če skala pade od trde hribine (R n = 0,50) le 10 cm stran v mehko cono (R n = 0,35), se lahko udarec popolnoma absorbira – v nasprotju z velikim odbojem v primeru udarca ob trdo hribino. Medtem ko imajo inženirji dober ob čutek photographs. The majority of computer programs only perform well in small areas, for which detailed (also topographic) information is available. The prediction results are greatly improved by precise field analysis of falling rocks in the area under consideration. The energy consumption at impacts or rolling is to a large extent a function of shape, size, velocity, and block dynamics (translational and angle velocity), geotechnical slope properties (granulometric composition, module of elasticity, water content …), geometry, topography and surface roughness as well as impact angle. These parameters are hard to determine precisely. The trajectory prediction is therefore a complex task, incorporating large uncertainties. The correlations between energy losses and other variables are not precisely determined. In most cases, all influences due to plastic deformations of the base and due to impact geometrical configurations are considered using “contact functions”, which describe block kinematics (velocity) or dynamics (energy) before and after the impact. These functions are in the form of the coefficient of restitution and the friction coefficient. The coefficient of restitution is the variable that helps validating a model of falling rocks. In the computation of a chosen profile section, two coefficients of restitution are given, the perpendicular and the tangential ones (Table 1). The perpendicular coefficient of restitution R n , usually located between the limits 0.3 0.5 n R < < , is used for cases when a rock hits the ground near the angle of 90°. The tangential coefficient of restitution R t , usually located between the limits 95 . 0 8 . 0 < < t R , is used for rocks that hit the ground at low angles. Zones of soft soils and vegetation occupy the lower values of the coefficient of restitution, while bare rock and asphalt cover the upper values. However, only a minor change in the coefficient of restitution causes completely different results. For example, if a stone hits a soft zone (R n = 0.35) that is only 10 cm away of hard rock (R n = 0.50) a hit can be fully absorbed – instead of a large rebound in the case of a hit against hard rock. While engineers in general have a good Petje, U., Mikoš, M., Majes, B.: Modeliranje gibanja skalnih podorov – pregled – Modelling of Rockfall Motion – A Review © Acta hydrotechnica 23/38 (2005), 19–38, Ljubljana 27 za kot notranjega trenja, pa tega ne moremo re či za koeficient odboja. Kot smo že prej nakazali, je to mogo če rešiti tako, da z ra čunalniškim programom opravimo povratno analizo, dokler ne dobimo pravih (izmerjenih) rezultatov padanja skal (tj. dejansko ugotovljena kon čna mesta padca skale). feeling for the angle of repose, this is not true for the coefficient of restitution. This can be solved, as shown, by using a computer program and performing a back analysis until true (measured) results for falling rocks are obtained (i.e. stopping places of falling rocks that have been recognized in the field). Preglednica 1. Tangencialni R t in normalni koeficient odboja R n ter koeficient trenja µ – prikazan za razli čne pokrovnosti tal (Dorren, 2003). Table 1. Tangential coefficient of restitution R t , perpendicular coefficient of restitution R n , and friction coefficient µ – given for different soil cover (Dorren, 2003). pokrovnost tal / soil cover R t R n µ klif, strme stene 60–90 ° / cliff, steep rock faces 60–90 ° 0.95 0.45 0.25 strma gola pobo čja 40–60 ° / steep bare slopes 40–60° 0.90 0.40 0.45 gruš čnata pobo čja 30–40 ° / talus slopes 30–40° 0.88 0.32 0.60 gola pobo čja 0–30 ° / bare slopes 0–30° 0.87 0.35 0.50 travnik / meadows 0.87 0.30 0.55 alpsko grmovje / alpine bushes 0.85 0.30 0.60 grmovje / bushes 0.83 0.30 0.65 gozd (200 dreves/ha) / forest (200 trees/ha) do / up to 0.85 povpre čno / mean 0.67 0.28 1.00 gozd (300 dreves/ha) / forest (300 trees/ha) do / up to 0.85 mean 0.57 0.28 1.50 gozd (500 dreves/ha) / forest (500 trees/ha) do / up to 0.85 mean 0.38 0.28 2.00 gozd (700 dreves/ha) / forest (700 trees/ha) do / up to 0.85 mean 0.27 0.28 2.20 4.2 OB ČUTLJIVOST IN ZANESLJIVOST ANALITI ČNIH MODELOV Na zmanjšanje zanesljivosti izra čuna z analiti čnimi modeli vplivajo: • neznano mesto izvora padanja; • spremenljive lastnosti hribine (vzdolž profila ni mogo če opredeliti vseh sprememb v lastnostih materiala, ker so odvisne od lokalnih sprememb v vzorcu razpokanosti, stopnji preperelosti …); • spreminjanje oblike brežine; • problem izbora kriti čnih profilov za ra čunsko analizo; 4.2 ANALYTICAL MODELS’ SENSITIVITY AND RELIABILITY The following causes for decreased reliability of results of analytical models are recognized: • unknown source area; • variability of rock properties (along the slope profile it is not possible to take into account all variations in material properties, which are caused by local changes in fracturing pattern, weathering stage …); • variability in slope shape; • the problem of selecting critical profiles for computational analysis; Petje, U., Mikoš, M., Majes, B.: Modeliranje gibanja skalnih podorov – pregled – Modelling of Rockfall Motion – A Review © Acta hydrotechnica 23/38 (2005), 19–38, Ljubljana 28 • lokalne nepravilnosti, ki jih pri snemanju profila ni možno zajeti. Osnovni inženirski pristop k izra čunu je previdnost in konzervativnost pri izbiri vhodnih podatkov. Odlo čitev, koliko bomo konzervativni, pa je odvisna tudi od zunanjih faktorjev, kot je na primer gostota prometa. Pri cestah z malo prometa bomo privzeli 95 % mejo v diagramu porazdelitve možnih trajektorij, za avtoceste z gostim prometom pa bomo upoštevali vse možne trajektorije padajo čih skal. Za opredelitev nevarnosti padanja skal uporabljamo posebne izra čune, ki temeljijo na statisti čnih metodah, med njimi se pogosto uporablja metoda Monte Carlo (Vose, 1996). Za vsak vplivni faktor po tej metodi dolo čimo spodnjo in zgornjo mejo in statisti čno porazdelitev vrednosti med obema mejama. Ra čunalniški program nato s slu čajnim izbiranjem vrednosti spremenljivk med obema mejnima vrednostma ve č tiso čkrat ponovi izra čun in izra čunava trajektorije gibanja skal. S spreminjanjem koeficienta odboja izra čun umerimo glede na v naravi ugotovljene posamezne padce kamnov. Kot rezultat dobimo ovojnico trajektorij padanja skal. Ko izvajamo izra čun, ponavadi vse vrednosti vplivnih faktorjev držimo enake, razen ene spremenljivke, kateri ra čunalnik slučajno izbira vrednosti. Tako spremenljivko eno za drugo testiramo in ugotavljamo, kakšna je njena “pomembnost” v kon čnem izra čunu. Kljub takemu pristopu pa se ne moremo izogniti vsem neznankam. Nujne so razli čne poenostavitve, ki se poznajo v manjši kakovosti kon čnega izra čuna. Realisti čno napovedovanje podorov je nadalje komplicirano s tridimenzijsko naravo dejanske geometrije pobo čja. Dvodimenzijski programi ne upoštevajo tridimenzijskega učinka topografije na trajektorije (Agliardi & Crosta, 2003; Azzoni et al., 1995). Najpomembnejši 3D-u činek je lateralna disperzija trajektorij (Crosta & Agliardi, 2003). Lateralna disperzija je deviacija trajektorij od smeri z najve čjim gradientom in predstavlja klju čni problem pri modeliranju, saj ima velik vpliv na na čin modeliranja dinamike, projektiranje ukrepov in dolo čevanja nevarnosti. Crosta in Agliardi • local irregularities which cannot be represented during field survey of profiles. The basic engineering approach to computations is caution and conservation when selecting the input parameters. The decision, how conservative to be, is also related to outside factors such as traffic density. With roads of low traffic volumes, the 95 % boundary limit in the diagram of possible trajectories will be taken, and for highways with large traffic volumes all possible trajectories of falling rocks will be taken into consideration. For assessing the hazard of falling rocks special computations are used, based on statistical methods, among those especially the Monte Carlo method (Vose, 1996). For each relevant factor using this method the lower and the upper limit and the statistical density function between these limits are determined. The computer program takes random values of variables between the given limits and repeatedly computes trajectories of falling rocks as often as several thousand times. By changing the coefficient of restitution the computation is validated by using stopping points of falling rocks as observed in the field. The result of this procedure is a trajectory envelope of falling rocks. When doing computations, usually all relevant factors are kept constant, apart from one variable, whose values are randomly selected by the computer. Doing so, all variables are tested one after another and the procedure gives their relative “relevance” for the final result. Despite such a procedure, all uncertainties cannot be avoided. Simplifications that result in the lower quality of final results are needed. Realistic rockfall forecasting is furthermore complicated by a three-dimensional (3-D) nature of true slope geometry. Two- dimensional (2-D) programs do not take into account the three-dimensional effect of topography on trajectories (Agliardi & Crosta, 2003; Azzoni et al., 1995). The most important 3-D effect is the lateral dispersion of trajectories (Crosta & Agliardi, 2003). The lateral dispersion is the deviation of trajectories from the slope gradient and represents the key modelling problem, as it has a major influence on the way how dynamics is modelled, on designing of measures and hazard determination. Crosta and Agliardi (2003) researched the influence of weight, Petje, U., Mikoš, M., Majes, B.: Modeliranje gibanja skalnih podorov – pregled – Modelling of Rockfall Motion – A Review © Acta hydrotechnica 23/38 (2005), 19–38, Ljubljana 29 (2003) sta raziskovala vpliv teže, hitrosti, naklona terena in mikrotopografije na disperzijo trajektorij. Najve čja slabost dvodimenzijskih programov je, da so neprimerni za dolo čitev tveganja na širšem obmo čju (regionalno merilo), kjer podrobnejše tematske informacije niso dosegljive. Zaradi prisotnosti lateralne disperzije je težko a priori dolo čiti trajektorijo v 2D- pristopu. 3D-u činek topografije je tako pomemben kot vpliv geometrije na dinamiko padajo če skale. Zato opis celotne topografije in stohasti čni pristop zagotavljata možnost modeliranja ve čjega števila trkov skale in tal, saj se upošteva prostorska spremenljivost parametrov in možnost manj pogostih in redkih trajektorij. Pomembno je poudariti tudi koncept »nepri čakovanega dogodka« – najbolj nevarni dogodki se bodo manj verjetno zgodili. Zato je konservativni pristop, ki upošteva najbolj verjetno trajektorijo, lahko včasih nezadovoljiv. Vpliv 3D-topografije je ve čji pri ve čjih dolžinah trajektorij, saj se pove čujejo napake zaradi variabilnosti parametrov zaradi naraš čanja števila trkov in morfoloških vplivov. V inženirski praksi se navadno skalne podore simulira v dveh dimenzijah vzdolž profilov, ki smo jih v naprej definirali. Čeprav je 2D-pristop najbolj razširjen zaradi uporabe komercialnih ra čunalniških programov pa je interpretacija rezultatov in njena razširitev na sosednja podro čja lahko zelo subjektivna. 4.3 DVODIMENZIJSKI ANALITI ČNI MODELI Najprej bodo obravnavani dvodimenzijski modeli, ki se omejujejo na gibanje skale v vertikalni ravnini ter zato ne simulirajo stranskega gibanja skal. Nadalje je trajektorija skale v gibanju v teh modelih sestavljena iz ravnih odsekov z naklonom pobo čja enakim merjenim srednjim gradientom na dolo čenem odseku pobo čja. In nazadnje je gibanje skal simulirano s fazami padanja in fazami stika s podlago. Faza padanja je simulirana z ena čbo parabole, za četno hitrostjo v x- in y-smeri in težnostnim pospeškom. To čka trka s tlemi se ra čuna s se čiš čem parabole in ravnim odsekom pobočja. velocity, slope gradient, and microtopography on trajectory dispersion. The most important weak point of 2-D models is that they are not appropriate for hazard assessment of larger areas (regional scale), where detailed thematic information are not reachable. Due to existence of lateral dispersion it is hard to a priori determine a trajectory in a 2-D approach. The 3-D topography effect, thus, is important as a geometry effect on the falling rock dynamics. Therefore, only the complete topography description and stochastic approach ensure the possibility of modelling numerous impacts of stones on the ground, since the spatial variability of parameters is taken into account and thus also the possibility of less frequent or rare trajectories. It is important to stress the concept of the so-called “unexpected event”: the most hazardous events will happen with lower probability. Thus, a conservative approach, which takes into account the most probable trajectory, can be unsatisfactory. The influence of a 3-D topography is higher with longer trajectory lengths, since the errors due to parameters variability increase due to increase in the number of impacts and due to morphological influences. In engineering practice, rockfalls are normally simulated in two dimensions along the pre-defined longitudinal profiles. Even though a 2-D approach is most frequently used due to the use of commercially available computer programs, the interpretation of the results obtained and their extrapolation to neighbouring areas may be very subjective. 4.3 TWO-DIMENSIONAL ANALYTICAL MODELS Firstly, two-dimensional models will be discussed that are limited to the rock motion in the vertical plane. As a consequence, lateral motion cannot be simulated. Furthermore, a rock trajectory in these models is composed of straight reaches with a slope gradient equal to the measured mean gradient in the single slope reach. And lastly, rock motion is simulated by phases of falling and phases of contact with the ground. The falling phase is simulated by the equation of the parabola, the initial velocity in the x and y directions and the gravitational acceleration. The point of impact with the ground is computed by the intersect between the parabola and the straight reach of the slope. Petje, U., Mikoš, M., Majes, B.: Modeliranje gibanja skalnih podorov – pregled – Modelling of Rockfall Motion – A Review © Acta hydrotechnica 23/38 (2005), 19–38, Ljubljana 30 Kirkby in Statham (1975) sta razvila model za gibanje skal prek vršaja s predpostavko, da skale drsijo po površini vršaja. Rezultati so bili primerljivi z rezultati laboratorijskih eksperimentov. Model je najprej izra čunal hitrost padajo če skale v ob vznožju stene višine h: Kirkby and Statham (1975) developed a model for motion of rocks on a talus slope using an assumption that rocks slide across the slope surface. The results were comparable to the results from laboratory experiments. The model firstly computes the velocity of a falling rock v at the toe of the rock face with the height h: gh v 2 = (4) Na osnovi hitrosti padajo če skale v je bila izra čunana komponenta hitrosti tangencialno na pobo čje s predpostavko, da se ta hitrost med prvim udarcem s tlemi ni spremenila. Lokacija, kjer se skala ustavi, je bila dolo čena z deležem med hitrostjo padanja in silo upora, ki je bila dolo čena z dinami čnim kotom upora. Dvodimenzijski modeli upoštevajo specifi čen algoritem za ra čunanje hitrosti kotaljenja in drsenja z uporabo Coulombovega zakona upora: Using the velocity of a falling rock v, the velocity component tangential to the slope was computed under the assumption that this velocity did not change during the first impact with the ground. The position where a rock stops was determined by a ratio between the falling velocity and the resistance force, which was determined by the dynamic resistance angle. The two-dimensional models take into account a specific algorithm for computing rolling and sliding velocity using the Coulomb friction law: β µ cos ⋅ ⋅ ⋅ = g m F f t (5) kjer je F t sila upora (tangencialno na pobo čje, kgm/s 2 ), f µ je koeficient upora, m je masa skale (kg), g je pospešek sile teže in β je srednji gradient pobo čja (º). Izra čunana sila upora se lahko uporabi za ra čun hitrosti drsenja in kotaljenja skale. Koeficient upora je najpomembnejši faktor za dolo čitev hitrosti. Za dolo čitev hitrosti pred in po odboju se uporabljata dva principa. Oba principa ra čunata hitrost pred in po trku na osnovi izgube energije. Prvi princip definira izgubo energije s koeficientom u činka trka, ki je delež skupne kineti čne energije skale pred in po trku. Drugi princip ra čuna energijsko izgubo na osnovi tangencialnega koeficienta odboja, ki deluje vzporedno s pobo čjem, in normalnega koeficienta odboja, ki deluje pravokotno na pobo čje. Dvodimenzijski modeli so primerni za lokalno merilo (pregled v preglednici 2). where F t is friction force (tangential to slope surface, kgm/s 2 ), f µ is friction coefficient, m is rock mass (kg), g gravity, and β is average slope gradient (º). The friction force can be used to calculate the sliding and rolling velocity of rocks. The friction coefficient is the most important factor when determining velocity. Two principles are used for determination of velocity prior and after a rebound. In both cases the velocities are calculated based on energy loss. The first principle defines the energy loss with the coefficient for the efficiency of collision, which is the ratio of total kinetic energy of the rock prior and after the impact. The other principle calculates energy loss based on the tangential coefficient of restitution, which acts parallel to the slope and normal coefficient of restitution acting perpendicular to the slope. Two-dimensional models are adequate at local scale (overview in Table 2). Petje, U., Mikoš, M., Majes, B.: Modeliranje gibanja skalnih podorov – pregled – Modelling of Rockfall Motion – A Review © Acta hydrotechnica 23/38 (2005), 19–38, Ljubljana 31 Preglednica 2. Nekateri dvodimenzijski analiti čni procesni modeli, uporabni za lokalno merilo, in njihove zna čilnosti. Table 2. Some two-dimensional analytical process-based models, applicable at local scale, and their characteristics. model Model opis podorne mase rock mass description opis gibanja movement description Bozzolo & Pamini (1986) togo telo / rigid body poskakovanje, kotaljenje & drsenje bouncing, rolling & sliding Bozzolo et al. (1988) togo telo / rigid body trki & poskoki / impacts & bounces Hungr & Evans (1988) masna to čka/lumped mass poskakovanje, kotaljenje & drsenje bouncing, rolling & sliding Pfeiffer & Bowen (1989) masna to čka/lumped mass trki & poskoki / impacts & bounces Kobayashi et al. (1990) masna to čka/lumped mass poskakovanje, kotaljenje & drsenje bouncing, rolling & sliding Zinggeler et al. (1991) masna to čka/lumped mass trki & poskoki / impacts & bounces Evans & Hungr (1993) masna to čka/lumped mass poskakovanje, kotaljenje & drsenje bouncing, rolling & sliding Azzoni et al. (1995) togo telo / rigid body poskakovanje, kotaljenje & drsenje bouncing, rolling & sliding Chau et al. (1998) togo telo / rigid body poskakovanje, kotaljenje & drsenje bouncing, rolling & sliding 4.4 TRIDIMENZIJSKI ANALITI ČNI MODELI Tridimenzijske modele so med drugimi razvili Descoudres in Zimmermann (1987), Scioldo (1991), Gascuel et al. (1998), Guzzetti et al. (2002) in Aglairdi & Crosta (2003). Le redki tridimenzijski modeli upoštevajo interakcijo med padajo čimi skalami. V zadnjem desetletju je modeliranje napredovalo pri dolo čitvi koordinat, hitrosti, kotne hitrosti v 3D-prostoru. Tako modeli simulirajo spremembo kineti čne energije delcev kot rezultat neelasti čnih trkov med sabo in s pobo čjem. Taki modeli temeljijo na metodi kon čnih elementov (Okura et al., 2000a) ali na analizi diskontinuitetnih deformacij DDA (Koo & Chern, 1998). RIG-DDA metoda (izboljšana DDA-metoda) vsebuje kinematiko vseh oblik gibanja in interakcijo med bloki. Možno je simulirati gibanje mase skal, ki vsebuje veliko skal vzdolž nepravilnega pobo čja v sprejemljivem ra čunskem času. Yang et al. (2004) predlagajo tridimenzijsko metodo DDA za simulacijo gibanja in napovedovanje trajektorije padanja. Model upošteva skalo kot sferi čno togo telo. 4.4 THREE-DIMENSIONAL ANALYTICAL MODELS The three-dimensional models were designed among others by Descoudres and Zimmermann (1987), Scioldo (1991), Gascuel et al. (1998), Guzzetti et al. (2002), and Aglairdi & Crosta (2003). Only rarely three- dimensional models take into consideration the interactions between falling rocks. Within the recent ten years, modelling has advanced in terms of determination of coordinates, velocity and angular velocity in a 3-D space. Thus, models simulate the change of kinetic energy of particles as a result of non-elastic impacts among themselves and with the surface. Such models are based on the finite element method (Okura et al., 2000a) or on discontinuous deformation analysis (DDA) (Koo & Chern, 1998). The RIG-DDA method (improved DDA method) contains the kinematics of all kinds of movement and interaction between blocks. It is possible to simulate the movement of rock mass, which includes rocks along the uneven slope within an acceptable calculation time. Yang et al. (2004) propose a 3-D DDA method for simulation of movement and prediction of falltracks. The model considers the rock as a spherical rigid body. Petje, U., Mikoš, M., Majes, B.: Modeliranje gibanja skalnih podorov – pregled – Modelling of Rockfall Motion – A Review © Acta hydrotechnica 23/38 (2005), 19–38, Ljubljana 32 5. MODELI NA OSNOVI GIS Geografski informacijski sistemi so u činkovita orodja za analizo prostorskih pojavov in za upravljanje prostorsko opredeljenih podatkov. S tem predstavljajo dragocen pripomo ček pri presoji tveganja. GIS-i so primerni tudi za upravljanje podatkov o katastrih in o škodnem potencialu. Naravne nesre če so ve čdimenzijske in predstavljajo interdisciplinarni pojav, ki ima mo čno prostorsko komponento, kadarkoli se pojavi. Razumevanje pojava in reševanje zahteva dostop do prostorsko orientiranih podatkov razli čnih izvorov, meril, resolucije, časovnega razvoja in analiz v štirih dimenzijah. Po ESRI (2002) lahko obseg dela razdelimo na: • zajem podatkov; • shranjevanje podatkov (v vektorski in rastrski obliki); • poizvedovanje (identifikacija obstoje čih podatkov); • analizo podatkov (analiza oddaljenosti, prekrivanje, mrežne analize); • prikaz podatkov (kartografija, priprava preglednic in poro čil); • izhod (karte, Internet, slike, dokumenti). Presoja nevarnih naravnih procesov zahteva informacije o površju, pokrovnosti tal in geoloških razmerah. Če naj presoja poteka na osnovi GIS, morajo biti te informacije na voljo v elektronski obliki. Prednost vektorskih podatkov je v enostavnejšem shranjevanju podatkov, krajšem ra čunskem času, boljši natan čnosti. Rastrska karta se sestoji iz matrike kvadratkov. Vsak piksel ima svojo vrednost. Med piksli ni logi čne povezave. Rastrski podatki imajo tudi svoje prednosti. Pri ra čunanju dosega podora je bistven relief, ki ga imamo navadno v rastrskem formatu, tako da rezultate dobimo ravno tako v rastru. Površje prikazuje digitalni model višin (DMV), ki je navadno v rastrski obliki. Za Slovenijo je uporaben DMV, izdelan na osnovi radarskih podatkov (Oštir et al., 2000). V splošnem se redko uporablja tudi trikotna nepravilna mreža (TIN). Iz podatkov o višinah se lahko izvedejo 5. GIS-BASED MODELS Geographic Information Systems are an efficient tool for analysis of spatial phenomena and management of spatial data. Hence, they represent a valuable tool in risk assessment. GIS are further applicable in managing the data on land register and damage potential. Natural hazards are multi-dimensional and represent an interdisciplinary phenomenon with a strong spatial component during each occurrence. Our understanding of the phenomenon and addressing the problem requires access to spatially referenced data of different origins, scales, resolutions, time course and analyses in four dimensions. According to ESRI (2002) the work can be divided into: • data capture; • data storage (vector and raster data); • querying (identification of existing data); • data analysis (analysis of distance, overlapping, network analysis); • displaying data (mapping, working with tables and reports); • output (maps, Internet, images, documents). The assessment of hazardous natural processes requires data on surface, soil cover and geological conditions. If the appraisal is based on the geographic information system, these data should be available in electronic form. The advantages of using vector data are easier data storage, shorter calculation time and better accuracy. The raster map is composed of grids. Every pixel has a value. There are no logical connections between pixels. Raster data also have their advantages. In calculating the run-out distance of a rockfall the key element is the relief, which is usually in the raster format, and the results are also in the raster format. The surface is represented by the Digital Elevation Model (DEM) which is usually in the raster format. For Slovenia, a DEM is useful that was prepared using radar data (Oštir et al., 2000). Generally, Triangulated Irregular Networks (TIN) are seldom used. Petje, U., Mikoš, M., Majes, B.: Modeliranje gibanja skalnih podorov – pregled – Modelling of Rockfall Motion – A Review © Acta hydrotechnica 23/38 (2005), 19–38, Ljubljana 33 drugi, za presojo nevarnosti pomembni parametri površja, npr. nagib površin, velikost prispevnega obmo čja. Digitalni model višin je tudi osnova za dolo čitev trajektorij nevarnih procesov. Ugodno je, če je dodana k digitalnemu modelu višin tudi hidrografska mreža v digitalni obliki. Podatki o pokrovnosti tal morajo povedati, ali so tla sestavljena iz skalovja (kamnin), ledu ali golega drobirja (zemljin) oziroma so porasla. Geološke informacije so nujne za ozna čitev podlage v bližini površine, povedo nam podatke o strukturnih zna čilnostih, lastnostih kamnin in o vrsti zemljin. Pomembno je, da so ti osnovni podatki na voljo po celi površini raziskovanega obmo čja v homogeni obliki in v ustrezni prostorski lo čljivosti. Minimalna lo čljivost je odvisna od procesa, ki se bo simuliral, kakor tudi od ciljnega merila. V splošnem velja pravilo, da lahko pri ploskovnih procesih delamo z bolj grobo lo čljivostjo kakor pri linearnih procesih. Pri uporabi GIS je velika nevarnost v tem, da je analize čisto tehni čno možno izvesti tudi z neprimernimi osnovnimi podatki, to pa ve činoma iz rezultatov ni takoj razvidno. Da se izognemo takšni situaciji, je nujno pri analiziz uporabljati karte primerljivih meril. GIS-modeli so tisti, ki te čejo pod GIS- okoljem, ali pa so rastrski modeli, katerih vhodni podatki so pridobljeni z GIS-analizami. GIS-modeli za podore se sestojijo iz treh postopkov: • identifikacija obmo čja izvora podora; • dolo čitev trajektorije; • ra čun obmo čja izteka. Število GIS-modelov stalno narašča (Carrara et al., 1991; Carrara, 1995; Guzzetti et al., 2002; Chau et al., 2004a; 2004b; Mayer et al., 2004; Rowbotham & Dudycha, 1998; Temesgen et al., 2001; van Westen & Getahun, 2003) – kljub temu ne morejo popolnoma nadomestiti terenskega dela. Prednosti, zaradi katerih se odlo čamo uporabljati GIS, so naslednje: • Možnost obdelave geografskih podatkov: geografski podatki se sestojijo iz kombinacije geometrijskih podatkov (položaj objekta v prostoru) in vsebinskih podatkov (lastnosti objekta). From the elevation data other surface parameters relevant for hazard assessment can be derived, such as surface slope, and size of the catchment area. The Digital Elevation Model is a basis for determination of trajectories of hazardous processes. It is beneficial if a digital hydrographic network is added to the DEM. The data on soil cover should reveal if the soil consists of rocks, ice or gravel, and whether the surface is vegetated. Geological data need to characterise the ground near the surface, and provide information on structure, rock features and soil type. It is important that these basic data are available for the entire study area in a homogeneous form and in an adequate spatial resolution. The minimum resolution depends on the process to be simulated and the targeted scale. In general, in areal processes rougher resolution is used than in linear processes. In using the geographic information systems the danger lies in the fact that analyses can (technically) be performed with inadequate basic data, which cannot be instantly deduced from the results. To avoid such a situation, maps in comparable scales should be used throughout the anaysis. The GIS models are the models working under the GIS environment, or raster models whose input data are acquired with GIS analyses. GIS models for rockfalls are composed of three procedures: • Identification of the areas of rockfall origin; • Determination of the falltrack; • Calculation of the run-out zone. The number of GIS models is in constant increase (Carrara et al., 1991; Carrara, 1995; Guzzetti et al., 2002; Chau et al., 2004a; 2004b; Mayer et al., 2004; Rowbotham & Dudycha, 1998; Temesgen et al., 2001; van Westen & Getahun, 2003) – they cannot completely replace field work. The advantages of using GIS are the following: • Possibility of processing geographic data: Geographic data consist of a combination of geometrical data (position of a structure in space) and content-related data (attributes). Such Petje, U., Mikoš, M., Majes, B.: Modeliranje gibanja skalnih podorov – pregled – Modelling of Rockfall Motion – A Review © Acta hydrotechnica 23/38 (2005), 19–38, Ljubljana 34 Takšni podatki potrebujejo posebno vrsto funkcij za obdelavo in analizo. GIS vsebuje obe vrsti funkcij, poleg tega je možna dodatna obdelava s programi CAD (Computer Aided Design). • Možnost vklju čitve digitalnih modelov reliefa: pri procesih gibanja mase ima relief eno od najbolj pomembnih vlog, predvsem naklon in izbo čenje pobo čja. • Nadaljnje prostorske analize: dolo čitev tveganja in s tem poznavanje infrastrukture in objektov na dolo čenem obmo čju. GIS je zelo primeren za izdelavo dispozicijskih modelov, druga če pa je pri procesnih modelih. Problemu se lahko izognemo tako, da vgradimo zunanji program, ki obdela podatke in jih nato vrne nazaj v GIS, kjer se jih lahko nadalje analizira. 6. ZAKLJU ČKI Poglavitni namen v prispevku podanega pregleda modeliranja gibanja skalnih podorov je strokovnjakom pomagati pri izboru ustreznega modela za lokalno in regionalno merilo. V prispevku so predstavljene glavne zna čilnosti izbranih modelov skalnih podorov, ki se trenutno uporabljajo v svetu. Gibanje skalnih podorov lahko raziskujemo in napovemo z uporabo empiri čnih procesnih in analiti čnih procesnih modelov ter z uporabo modelov, ki temeljijo na GISu. Empiri čni procesni modeli temeljijo na povezavi med topografskimi faktorji in obmo čjem izteka oziroma dolžino dosega, v časih jih imenujemo tudi statisti čni modeli. Analiti čni procesni modeli v dveh ali treh dimenzijah opisujejo in simulirajo gibanje podorne mase in se lo čijo glede na na čin, kako upoštevajo podorno maso (kot masno to čko; razne oblike togih teles) in kako simulirajo gibanje po pobo čju (poskakovanje, kotaljenje, drsenje). Modeli na osnovi GIS izkoristijo prednosti tega orodja in potekajo v treh korakih: dolo čitev obmo čij izvora podorov, dolo čitev trajektorij posameznih skalnih blokov in dolo čitev obmo čij izteka (odlaganja) podorne mase. Podro čje uporabe teh modelov je predvsem regionalno merilo. data require a specific kind of functions for processing and analysis. GIS incorporates both kinds of functions and enables further processing with CAD (Computer Aided Design) tools. • Possibility of inclusion of DTM: In processing that involves the movement of mass, the relief plays a significant role, especially regarding its slope angle and unevenness. • Further spatial analyses: Risk assessment and thus the knowledge of infrastructure and structures in a certain area. In contrast to process-based models, GIS is highly applicable in disposition models. The problem can be avoided by adding a software that processes the data which are then returned back to GIS, where they can be analysed further. 6. CONCLUSIONS The main aim of the review on modelling of rockfall motion in this paper is to help professionals to choose a rockfall model adequate at local and regional scales. In the paper, the main characteristics of the selected rockfall models that are currently used worldwide are presented. Rockfall motion can be studied and forecasted by using empirical process-based and analytical process-based models or by using GIS-based models. The empirical process-based models are based on the connection between topographic factors and the run-out distance (also called statistical models). The analytical process-based models describe and simulate movements of the rockfall masses in two or three dimensions. They can be differentiated regarding the way how the rockfall mass (as a lumped mass; different rigid body shapes) and the movement on the slope (bouncing, rolling, sliding) is described, respectively. The GIS-based models use the advantages of this tool and work in three steps: The determination of rockfall source areas, the determination of trajectories of single boulders, and the determination of run-out distances and run-out zones. The application of these models is especially at regional scale. Petje, U., Mikoš, M., Majes, B.: Modeliranje gibanja skalnih podorov – pregled – Modelling of Rockfall Motion – A Review © Acta hydrotechnica 23/38 (2005), 19–38, Ljubljana 35 ZAHVALA Avtorji se zahvaljujejo za finan čno pomo č Ministrstva za šolstvo, znanost in šport RS, Ministrstva za obrambo RS in Ministrstva za okolje, prostor in energijo RS, ki so financirali ciljni raziskovalni projekt (CRP) “Metodologija za dolo čitev ogroženosti pred zemeljskimi plazovi in na čin razvrš čanja zemljiš č v razrede ogroženosti”. Za pomo č pri izvedbi prakti čnega dela projekta se avtorji zahvaljujejo dr. Tomažu Podobnikarju iz Znanstvenoraziskovalnega centra Slovenske akademije znanosti in umetnosti iz Ljubljane. Poglobljen pregled prispevka sta opravila Hans Kienholz in Mihael Ribi či č. 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International Journal of Rock Mechanics & Mining Sciences 41, 476. Petje, U., Mikoš, M., Majes, B.: Modeliranje gibanja skalnih podorov – pregled – Modelling of Rockfall Motion – A Review © Acta hydrotechnica 23/38 (2005), 19–38, Ljubljana 38 Naslov avtorjev – Authors’ Addresses mag. Urška Petje Hidrosvet d.o.o. – Hidrosvet Ltd. Kunaverjeva 3, SI-1000 Ljubljana, Slovenia E-mail: urska.petje@lj.hidrosvet.si izr. prof. dr. Matjaž Mikoš Fakulteta za gradbeništvo in geodezijo – Faculty of Civil and Geodetic Engineering Univerza v Ljubljani – University of Ljubljana Jamova 2, SI-1000 Ljubljana, Slovenia E-mail: matjaz.mikos@fgg.uni-lj.si izr. prof. dr. Bojan Majes Fakulteta za gradbeništvo in geodezijo – Faculty of Civil and Geodetic Engineering Univerza v Ljubljani – University of Ljubljana Jamova 2, SI-1000 Ljubljana, Slovenia E-mail: bojan.majes@fgg.uni-lj.si