Proceedigs, 11 th FIG Symposium o Deformatio Measuremets, Satorii, Greece, 2003. DETECTION OF LANDSLIDE BLOCK BOUNDARIES BY MEANS OF AN AFFINE COORDINATE TRANSFORMATION Michaela Haberler, Heribert Kahme Istitute of Geodesy ad Geophysics, Departmet of Applied ad Egieerig Geodesy, Viea Uiversity of Techology, Gusshausstrasse 27-29, 1040 Viea, Austria Abstract For a ew approach i ladslide moitorig, it is importat to detect the boudaries betwee blocks with differet directios ad rates of movemet, so that at these boudaries the chages of movemet ca be moitored with high precisio geotechical sesors. The idea is to use the displacemet vectors (which ca be foud by a deformatio aalysis) to split the moitored poits ito the several blocks. The assumptio is that poits lyig o oe block will have a similar patter of movemet. With the help of the results of a over-determied affie trasformatio, you ca distiguish if all the used poits are lyig o oe commo block or if oe poit of a eighbourig block was take ito accout by mistake. The residuals of the over-determied trasformatio are a good idicator for this distictio. The aalysis of the residuals is doe by a fuzzy system which must decide i each step of the iterative algorithm if the searchig algorithm should be stopped. The iput parameters of the fuzzy system are e.g. the rage of the residuals, the chage of the stadard deviatio compared to the last step, ad strai parameters. Output of the fuzzy system is a value represetig the probability that all the used poits are formig oe commo block. A eample will be give to preset the capabilities of this approach. 1. Itroductio Ladslides are oe of the major types of atural hazards i the world. Especially i Europe, due to the growig tourism i alpie regios, people are livig or workig more ad more i areas with ustable slopes. So the umber of people ad ifrastructure affected by ladslides is growig. E.g. i Italy, i the last decade (1990 1999) 263 people were killed by ladslides. Due to the compleity of this topic, aswers ca oly be foud by a combiatio of several disciplies, e.g. geology, geodesy, geomechaics, geomorphology, hydrology. Our approach, OASYS (Itegrated Optimisatio of Ladslide Alert Systems) is a efficiet geodetic moitorig system cosistig of classical geodetic etworks, improved by high precisio geotechical sesors i relevat parts of the slopes. These relevat parts are the boudaries of differet blocks of the ladslide. Most of the ustable slopes cosist of several blocks movig i differet directios with differet velocities. The istallatio of geotechical sesors across these block boudaries gives importat iformatio o the relative movemet of the blocks. These permaet observatios ca be used i a kowledge-based system together with the geodetic measuremets to assess (almost i realtime) the behaviour of the slope. This paper deals with oe part of these ivestigatios, the detectio of the block boudaries.
2. Affie coordiate trasformatio We assume that o the ustable slope ad the surroudig stable area a geodetic etwork has bee istalled ad measured for at least two epochs. If we restrict to 2D, the umber of observed poits should be at least four per block. The measuremets of the two epochs (GPS ad/or tacheometric observatios) are used i a geodetic deformatio aalysis to get the displacemet vectors for each observed poit. The basic idea behid the algorithm is to use a over-determied affie coordiate trasformatio to map the coordiates of the first epoch o the coordiates of the same poits of the secod epoch. If a certai group of poits is lyig o oe commo block the the movemet patter of these poits will be similar ad the affie coordiate trasformatio will give good results (i.e. a good stadard deviatio). If the group of poits is lyig o differet blocks the the stadard deviatio (ad other idicators eplaied later) will be sigificatly larger. For 2D, the trasformatio ca be writte as follows: y + 1 + 1 = a y = d y + b + e + c + f where y, coordiates of epoch y +1, +1 coordiates of epoch +1 a,..,f trasformatio parameters The si parameters (a,..,f) ca be iterpreted as two traslatios (t, t y ), two rotatios (w, w y ) ad two scale parameters (m, m y ). a = m cos w b = m si w c = t d = m si w e = m cos w f = y y y y y t Usually, oly oe scale parameter is used. Here the two scale parameters are ecessary due to the property of ladslides that a slidig block will be more distorted i the directio of movemet tha i ay other directio. The secod scale parameter has to couterbalace this aisotropy. For 2D, at least four poits have to be used to get a over-determied equatio system. Importat results out of these trasformatios are the residuals v ad the stadard deviatio s 0. The trasformatio parameters themselves are ot very sigificat due to the small displacemets. Fig. 1 shows oe of the scale parameters, m, take from the eample give i sectio 5. All possible blocks cosistig of 4 poits (170 differet combiatios) were ivestigated. These combiatios ca be divided ito 3 groups: the correct blocks ( 4:0, all 4 poits lyig o oe block) with 31 combiatios, ad two groups of icorrect blocks ( 3:1, 3 poits o oe block ad 1 poit o aother block with 81 combiatios; 2:2, 2 poits o two differet blocks with 58 combiatios). It ca be see that a clear distictio betwee the differet groups ca ot be made out of this parameter m.
Fig. 1: Scale parameter m for the differet cases: correct block (4:0), icorrect blocks (3:1 ad 2:2). Welsch (Welsch, 1982) has show that the affie coordiate trasformatio is equivalet to a strai aalysis (assumig homogeeous ifiitesimal strai). So the si trasformatio parameters ca also be iterpreted as two traslatios plus 4 parameters of the strai tesor E: e E = e y e e y yy with e, e yy... rate of chage of legth per uit legth i directio of -ais resp. y-ais e y, e y rate of shear strai A better iterpretatio of the strai parameters ca be reached by the trasformatio ito the pricipal strai aes system, represeted by the strai ellipse (Tissot idicatri). The calculatio of the strai ellipse is aalogous to that of the geodetic poit error ellipse. So the si parameters ca be see as two traslatios ad oe rotatio of the block (rigid body movemet) plus the distortio represeted by the strai ellipse: e1, e2 (the semi-aes) ad θ (the orietatio of e1). Welsch metios that the traslatios should ot be used i a strai aalysis, but i the case of ladslide moitorig the model of a rigid body (with small ier distortios) movig dowwards is the most practicable oe. Without the two traslatioal parameters, the strai parameters would be falsified because they would have to couterbalace also the traslatioal part of the block movemet. 3. Algorithm The basic scheme of the algorithm ca be foud i fig. 2. Based o the residuals v ad the stadard deviatio s 0 of all possible trasformatios a miimal block cosistig of four eighbourig poits is chose. Additioally, the two semi-aes of the strai ellipse have to be withi a user-defied limit (to esure small ad rather isotropic distortios). I the followig step the most suitable oe of the eighbourig poits has to be icluded ito the miimal block. For all combiatios with 5 poits the trasformatios are calculated; the block with miimal s 0 (ad agai, e1, e2 withi some limits), is chose. At the same time, it has to be
checked if this block is still `correct`, i.e. if all poits are lyig o this block. This aalysis is doe by a fuzzy system (cf. later). If the block is `correct` the the algorithm is searchig for the et suitable poit withi all eighbourig poits, util the fuzzy system rejects the test of `correctess`. I this case the last poit has to be removed ad the et miimal block of four poits has to be foud out of the remaiig poits. Fig. 2: Scheme of the detectio algorithm 4. Short descriptio of the fuzzy system The fuzzy system was implemeted i MATLAB. Matlab provides a iitial fuzzy system, where may ecessary fuctios (membership fuctios, aggregatio ad defuzzificatio methods) are already implemeted. It is ecessary to choose the iput parameters ad the output with their most suitable membership fuctios ad the several calculatig methods. The iput parameters implemeted at the momet are: Rate of chage of the stadard deviatio betwee subsequet steps: If a poit from aother block is icluded ito a correct block, s 0 becomes larger. This rate of chage ca be tested. Semi-aes of the strai ellipse: e1, e2. Rate of chage of e1 ad e2 betwee subsequet steps. Iterquartile rage of the residuals (used i the eploratory data aalysis). Fig. 3 represets the variatio of the residuals. For the 170 possible cases (aalogous to fig. 1) the iterquartile rage was ivestigated. It ca be see, that there is a clear distictio betwee the correct cases ( 4:0 ) ad the icorrect cases ( 3:1 ad 2:2 ). The higher this iput parameter the higher the probability that the block is icorrect.
Fig. 3: The iterquartile rage of the residuals for the differet cases: correct blocks (4:0), icorrect blocks (3:1, 2:2). 5. Eample The simulated raw observatios for the eample ivestigated were published i (Welsch, 1983), where several methods for deformatio aalysis ad block detectio were tested o the same data. I our ivestigatio, epochs 1 ad 3b were chose ad a geodetic deformatio aalysis was calculated to get the displacemet vectors (see fig. 4). Fig. 4: displacemet vectors betwee epochs 1 ad 3b
I the first step of the algorithm all trasformatios of 4 eighbourig poits are calculated. The best suitable block is chose based o the aalysis of the stadard deviatio s 0 ad the strai ellipse parameters e1, e2. The idea is that a correct block will ot be very distorted, so e1 ad e2 should be withi some user chose limits (depedig o the actual geological coditios). I this eample the poits 3, 5, 11, 41 were chose because of the miimal stadard deviatio of 9.9 mm. Now the iterative algorithm starts to fid the et suitable poit. All trasformatios with 5 poits (i.e. the chose block of 4 poits + oe eighbourig poit) are calculated, ad the combiatio with the miimal s 0 is chose (tab. 1). Poit id s 0 [mm] Poit id s 0 [mm] 3 5 11 41 13 78.3 3 5 11 41 39 13 94.4 15 59.5 15 72.3 21 52.6 21 71.3 35 66.2 35 86.9 39 12.8 43 76.4 43 55.8 45 76.7 Tab. 1: Possible cadidates for the 5 th poit. Tab. 2: Cadidates for the 6 th poit. After choosig poit 39, the fuzzy system calculates all the ecessary idicators used for the assessmet of the block. The result of the aalysis i the fuzzy system is a probability of 30 % that the block is ot correct (i.e that the last poit take ito accout should be o aother block). So the et step i the iteratio is started, the sith poit of the block is searched for. The possible cadidates ca be see i tab. 2, ad due to the miimal stadard deviatio s 0 poit 21 is take ito this block. (Note the rate of chage of s 0 betwee the two steps of iteratio). Agai, the fuzzy system is used to aalyse the ew situatio. The output is 72 %, that meas that the block is ot correct ay loger, ad poit 21 has to be removed. So the correct block cosists of the poits 3, 5, 11, 41, 39. Out of the remaiig poits, the search for the et block ca be started. The miimal block cosists of poits 13, 15, 17, 35 with s 0 = 12.0 mm. The results of the several iteratios ad the output of the fuzzy system ca be foud i tab. 3. Number of iteratio Poits icluded i the actual block s 0 [mm] Probability of termiatio (fuzzy system) 1 13 15 17 35 12.0 2 13 15 17 35 47 11.9 0.30 3 13 15 17 35 47 45 11.0 0.30 4 13 15 17 35 47 45 37 17.1 0.39 5 13 15 17 35 47 45 37 43 24.3 0.48 6 13 15 17 35 47 45 37 43 21 31.6 0.50 Tab. 3: Results of the differet steps of iteratio startig with a miimal block of 4 poits (13, 15, 17, 35). The output of the fuzzy system shows that all the blocks are correct. Here the algorithm stops because all poits have bee used, so the secod block cosists of the poits 13, 15, 17, 35, 47, 45, 37, 43, 21. Comparig this result with the several methods preseted i (Welsch, 1983) it ca be see that all the algorithms give the same blocks.
6. Coclusio The huma mid is strogly iflueced by visual patter recogitio. Lookig at figures like fig. 4, it seems obvious to divide the situatio i two blocks. The compariso with the result of our detectio algorithm shows that the fuzzy system is able to reproduce the huma way of thikig. But a lot of work has to be doe, e.g. withi the fuzzy system. It is plaed to implemet some other parameters: The degree of freedom of the over-determied trasformatio, because the sigificace of the F-test i the backgroud is the better the bigger the umber of degree of freedom. Some geological parameters to assess the actual geological coditios will also be ecessary. This work was partially supported by the Europea Commissio, Research DG, Eviromet Programme, Global Chage ad Natural Disasters. Refereces Welsch W. (1982). Descriptio of homogeeous horizotal strais ad some remarks to their aalysis. Proceedigs of the Iteratioal Symposium o Geodetic Networks ad Computatios, DGK Reihe B, Heft Nr. 258/V, Müche, 1982. Welsch W. (1983) (ed.). Deformatiosaalyse 83. Beiträge zum Geodätische Semiar 22. April 1983. Schriftereihe Wisseschaftlicher Studiegag Vermessugswese Hochschule der Budeswehr Müche, Heft 9, Müche, 1983.