FRACTAL ANALy SIS OF THE DISTRIBUTION OF CAVE LENGTHS IN SLOVENIA FRAKTALNA ANALIZA PORAZDELITVE DOLŽIN JAM V SLOVENIJI Timotej VERBOVŠEK 1 Izvleček UDK 551.435.84:51-7 Timotej Verbovšek: Fraktalna analiza porazdelitve dolžin jam v Sloveniji Dolžina jam v Sloveniji je porazdeljena po potenčnem zakonu, ki je značilen za fraktalne objekte. Fraktalna dimenzija jam se giblje okoli vrednosti 1.07 in se spreminja glede na tektonsko in �idrogeološko okolje. Odstopanja od idealne premice nastanejo zaradi podcenjenega števila jam, saj je krajši� jam več, kot ji � je dejansko zabeleženi�. Analiza tektonskega in �idrogeološkega okolja kaže, da so najvišje vrednosti fraktalne dimenzije značilne za kamnine s kraško-razpoklinsko in razpoklinsko poroznostjo ter najnižje za slabo prepustne kamnine. Bližina tektonski� struktur zelo vpliva na porazdelitev dolžin jam, vpliv pa je večji pri jama�, ki ležijo bližje prelomom in narivom. Vrednosti di - menzij jam so manjše kot dimenzije mrež razpok ali prelomov, najverjetneje zaradi koncentriranja tokov (kanalski� efektov) po mreža� razpok, kar posledično zmanjša fraktalno dimen - zijo. Fizikalni vzroki, ki povzročajo potenčno odvisnost in vari- acije fraktalni� dimenzij (eksponentov potenčnega zakona), so še vedno delno nepojasnjeni. Vseeno pa la� ko nastanek mrež razpok pripišemo fraktalni fragmentaciji kamnin, ki deluje neodvisno od merila, jame pa nato ob nastajanju podedujejo določene fraktalne lastnosti razpok. Ključne besede: dolžina jam, fraktalna dimenzija, Slovenija, kraška �idrogeologija. 1 University of Ljubljana, Faculty of Natural Sciences and Engineering, Department of Geology, Aškerčeva 12, Ljubljana, Slovenia p� one: +386 1 4704644, fax: +386 1 4704560, e-mail: timotej.verbovsek@guest.arnes.si Received/Prejeto: 01.10.2007 COBISS: 1.01 ACTA CARSOLOGICA 36/3, 369-377, POSTOJNA 2007 Abstract UDC 551.435.84:51-7 Timotej Verbovšek: Fractal analysis of the distribution of cave lengths in Slovenia The lengt� s of t� e Slovenian caves follow t� e power-law distri- bution t�roug � several orders of magnitude, w �ic � implies t � at t� e caves can be considered as natural fractal objects. Fractal dimensions obtained from distribution of all caves are about 1.07, and vary wit�in different tectonic and � ydrogeological units. Some deviations from t� e ideal best fit line in log-log plots (i.e. lower and upper cut-off limits) can be explained by underestimation, as many very s� ort caves are not registered. The study of tectonic and � ydrogeological setting indicates t� at t� e greatest dimensions occur in t� e rocks wit� karstic-fracture and fracture porosity and t� e lowest in low-permeability rocks. Proximity to major tectonic structures s� ows a detectable effect on t� e cave lengt� distribution, and t � e influence is greatest for t� e caves closer to t� e faults and t�rust fronts. Dimensions are lower t� an t� ose of fracture networks and faults, w�ic � can be most probably explained by flow c� anneling along t� e fracture networks, w�ic � causes t � e decrease of fractal dimension. The p� ysical causes of power law scaling and variations in fractal dimensions (power law exponents) are still poorly understood, but t� e be� aviour of fracture networks is believed to be caused by a scale-independent fractal fragmentation of t� e blocks, and during t� e process of forming t� e caves in� erit some fractal geometrical properties of t� e networks. Key words: cave lengt�, fractal dimension, Slovenia, karst � y- drogeology. ACTA CARSOLOGICA 36/3 – 2007 370 Fractals are defined as geometric objects wit� a self-simi - lar property, w�ic � implies t � at t� ey do not c� ange t� eir s� ape wit� scale (Feder, 1988). This statement is valid only for strictly self-similar mat� ematical fractals, like Koc� curve or Sierpinski carpet. One s� ould note t� at natural fractals differ from t� e ideal ones, as alt� oug� t � ey ap- pear self-similar or self-affine at some scales, t� ere always exist a natural lower and upper cut-off scale, and frac- tal analyses of t� ese objects are valid only wit�in t � ese two values. Fractal approac� es are appropriate w� ere classical geometry is not suitable for describing t� e ir- regular objects found in nature. Generally t� ese cannot be modelled by easily-defined mat� ematical objects – for example t� e “clouds are not sp� eres, mountains are not cones, coastlines are not circles, and bark is not smoot�, nor does lig� tning travel in a straig� t line” (Mandelbrot, 1983). The fundamental property of fractals is t� eir frac- tal dimension (D), w�ic � represents t � e ability of an ob- ject to fill t� e space (in one, two or t�ree dimensions). It can occupy non-integer values, compared to t� e integer values c� aracteristic of Euclidean objects, suc� as 3-D cubes or 2-D planar surfaces. As an example, an object wit� a fractal dimension of 1.4 ex�ibits properties of bot � 1-D and 2-D objects, as it fills t� e more space t� an a line (D = 1) and less space t� an a surface (D = 2). The caves form during t� e selective enlargement of fractures, bedding planes, faults and ot� er discontinuities in t� e soluble rock and only a few presolutional openings develop in larger passages (Palmer, 1991, Ford & Wil- liams, 2007). The degree of a cave to fill t� e neig� bor- ing rocks can be described quantitatively wit� t � e fractal dimension D. Bot� caves (Curl, 1999) and consequently cave lengt� s (Laverty, 1987) � ave been found to ex�ibit fractal properties. A study of Curl (1966) was performed for distribution of cave lengt� s and t� e number of en- tranceless for t� e “proper caves” – t� ose of accessible size including t� ose wit� no entrances. However, t � e influ- ences of different lit� ologic properties, � ydrogeologic and tectonic settings on t� e distribution of cave lengt� s � ave not been yet discussed in detail. The goal of t�is paper is to analyze and discuss t � e distribution of lengt� s of t� e caves in Slovenia in differ- ent tectonic and � ydrogeological environments plus t� e influence of t� e distance of t� e caves to t� e most obvious tectonic structures. As already noted by Curl (1986), t� e fractal interpretations probably do not directly reveal any details about geomorp�ic processes responsible for t � e distribution of lengt� s of caves, but t�is distribution does contain information about t� e geometry of caves and possibly constrains ideas about geomorp�ic processes. INTRODUCTION MATERIALS AND METHODS Three different influencing factors on t � e cave lengt� dis - tribution were studied, as mentioned above (tectonic and � ydrogeological position plus t� e distance to t� e major tectonic structures). The data for 7552 caves were ana- lyzed (spatial coordinates in t� e national Gauss-Krueger system and cave lengt� s), as recorded in t� e national cave register. The lengt� s are based on survey lengt� s, as recorded in t� e register. There exist many ot� er ways of measuring cave lengt� s besides classical survey, includ- ing 3-D measurements wit� sp � erical linked modular elements (Curl, 1986; 1999) and measuring in 2-D plane (plan lengt�) instead of performing classical total survey lengt� s in all t�ree dimensions (Laverty, 1987). Never - t� eless, regardless on met� od used, cave lengt� s distribu- tion ex�ibits fractal properties. Also, as caves are usually long compared to passage breadt�, t � e classical approac� is acceptable. Unfortunately t� ere exists no data on sur- veying met� od in t� e register, so t� e lengt� values are taken directly from register. This approac� is similar to t� e one of Curl (1966), w� ere if t� e lengt� of a cave was only stated in t� e report, t�is value was used. An impor - tant factor w�ic � can affect t � e results of analyzed cave lengt� s is t� e number of entranceless caves, studied in detail by Curl (1966). The number of entranceless caves in Slovenia is not known, but probably it is �ig �, as pre - dicted by Curl. However, � e noticed t� at t� e average lengt� s of entranceless caves are more like t� ose of caves wit� one or more entrances t � an like t� e predicted aver- age lengt� of entranceless caves. Therefore t� e effect on t� e greater number of entranceless caves s� ould be uni- formly distributed along a complete cumulative curve of cave lengt� s and s� ould not affect t� e s� ape of t� e curve, but s� ould only s�ift it upwards. The register was imported into relational database program (MS Access) and t� e data was furt� er analyzed wit� GIS and statistical software. Some basic statistics were also calculated, suc� as minimum and maximum lengt� and median. The median was used instead of mean or geometric mean, as t� e data does not follow nei- t� er normal nor lognormal distribution. TIMOTEJ VERBOVŠEK ACTA CARSOLOGICA 36/3 – 2007 371 RESULTS TECTONIC SETTING Caves were grouped into seven tectonic units according to t� eir location in t� e structural-tectonic map (Placer, 1999; Poljak, 2000; Fig. 1). Wit� minor deviation in t � e left-� and side of t� e plot, cave lengt� s follow power law distribution (linear line in log-log plot), c� aracteristic for fractal be� aviour. The median values of lengt� s (Tab. 1) are quite similar, except for t� e group of Adriatic fore- land, and � ave t� e value around 23 m. The fractal dimensions enable more appealing in- sig� t into t� e cave lengt� properties t � an t� e classical sta- tistical approac� using t � e median or ot� er statistics, and t� ey vary among t� e tectonic units (Tab. 1). All results ex�ibit a very �ig � value of R 2 . Note t� at t� e values of D and R 2 in t� e table are valid only for t� e linear part, not for t� e complete curve. The lowest values can be found in t� e tectonic units of Periadriatic igneous rocks and Internal Dinarides, and t� e �ig � est in t� e unit of Exter- nal Dinarides and also in Sout� ern Alps. The discussion of t� e results is given in t� e next section. The number of analyzed caves (N=9) in t� e Adriatic foreland is too small to comment reliably, and deviations of t� e curve can be also seen in t� e plot (Fig. 2), so t� e D could not be calculated. Hy DROGEOLOGIC SETTING Similar be� aviour of cave lengt� distribution can be ob - served in t� e plot (Fig. 4) for t� e different � ydrogeologi- cal units (Fig. 3). The �ig � est values (Tab. 2) are found in aquifers wit� karstic and fracture porosity and t � ose wit� fracture porosity (D=1.06) and lowest in t� e aquifers and beds wit� intergranular porosity (D=0.87, D=0.86). De - viations occur only for t� e group “Beds wit� low poros - ity” , as D is greater t� an expected, about 1.08. This curve does not s� ow suc� a linear trend as t � e ot� ers, and t� e number of t� e data is muc� smaller. DISTANCE TO THE MAJOR TECTONIC STRUCTURES Caves were grouped into t�ree classes (±150m, ±250m and ±500m), w� et� er t� ey fell into t� e 300m, 500m or 1000m wide belt around t� e fault or t�rust front (Fig. 5), as s� own on t� e structural-tectonic map (Poljak, 2000). Similar be� aviour of general cave lengt� distribution as for t� e tectonic and � ydrogeological units can be ob- served in t� e plot for t� e t�ree groups, as t � e lengt� s fol- low a linear fit line in t� e log-log plots. The median values are similar, approximately 23 m. As for t� e tectonic units, t� e units wit� �ig � er D contain longer caves, w�ic � is reasonable for t� ose caves wit� fractal dimension larger t� an one compared to t� ose wit� D lower t � an one. Nevert� eless, a gap of number of caves occurs in t� e rig� t-� and side of all t�ree plots (Fig. 6), for example at L = 3000m (logL = 3.5) for t� e ±150m distance group. This indicates t� at t� e number of caves long about 3000m is muc� lower t � an in case w� ere all t� e caves are consid- ered regardless of distance to t� e faults. The influence of t� e tectonic structures is greater w� en t� e caves are clos- er to t� e structures, as t� e gap is more noticeable for t� e ±150m group and slowly disappears towards t� e ±500m group. For t� e determination of tectonic setting, t� e struc- tural-tectonic map of Slovenia (Poljak, 2000) was digi- tized into a GIS s� ape file and t� e tectonic unit names were assigned to polygons. Caves belonging to a selected polygon (i.e. tectonic unit) were consequently selected from t� e complete dataset. For t� e determination of � y- drogeologic setting, t� e s� ape file wit� t � e polygons of different � ydrogeological units was obtained from t� e Eu- roWaterNet project website (� ttp://nfp-si.eionet.eu.int/ ewnsi), and t� e process of grouping t� e caves was similar to t� e grouping into tectonic units. The major faults and t�rust fronts were digitized from t � e same structural-tec- tonic map (Poljak, 2000) and using t� e GIS software t� e caves were grouped into t�ree classes (±150m, ±250m and ±500m), w� et� er t� ey fell into t� e 300m, 500m or 1000m wide belt around t� e fault or t�rust front. Subsequently t� e relations�ip between t � e numbers of caves N in t� e specific setting wit� lengt � greater t � an L was establis� ed, and t� e correlations were inspected in t� e log-log plots. For example, caves belonging only to t� e tectonic unit of External Dinarides were selected as explained in t� e former paragrap�, and t � eir distribution was analyzed in t� e following way. According to equa- tion D = log N(s) / log L (Bonnet et al., 2001), t� e fractal dimension D was calculated as t� e negative slope of t� e linear regression best-fit line of log N–log L plot. The pro- cess of calculation of D was repeated for all ot� er caves belonging to different units or groups of distance to t� e major tectonic structures. The number of steps for t� e lengt� s interval was c� osen as t� e power of 2 (1, 2, 4, 8 ...), wit� some major additional steps in between (10, 50, 100 etc). FRACTAL ANALy SIS OF THE DISTRIBUTION OF CAVE LENGTHS IN SLOVENIA ACTA CARSOLOGICA 36/3 – 2007 372 Fig. 2: Log-log distribution plot for the number of caves (N) longer than a specific length (L) in different tectonic settings Tab. 1: Results for fractal dimension of cave lengths in different tectonic units (D=fractal dimension, R 2 =coefficient of determination, N=number of caves.The same notation is valid for the Tab. 2. Tectonic setting D R 2 N median min max Adriatic foreland - - 9 10.0 5 876 Southern Alps 1.00 0.9974 1744 21.5 1 10870 Internal Dinarides 0.74 0.9934 60 20.0 4 1726 External Dinarides 1.10 0.9970 5166 24.0 1 19555 Eastern Alps 0.92 0.9940 44 18.0 5 2057 Tc and Q sediments 0.89 0.9950 158 18.5 3 1300 Periadriatic igneous rocks 0.60 0.9741 13 20.0 7 205 Total 1.08 0.9993 7194 23.0 1 19555 TIMOTEJ VERBOVŠEK Fig. 1: Structural-tectonic setting of the caves ACTA CARSOLOGICA 36/3 – 2007 373 Fig. 3: h ydrogeological setting of the caves Fig. 4: Log-log distribution plot for the number of caves (N) longer than a specific length (L) in different hydrogeological settings Tab. 2: Results for fractal dimension of cave lengths in different hydrogeological environments Hydrogeologic setting D R 2 N median min max Aquifers with intergranular porosity 0.87 0.9957 263 20.0 2 8057 Aquifers with karstic-fracture porosity 1.06 0.9975 5872 23.0 1 19555 Aquifers with fracture porosity 1.06 0.9954 510 24.5 4 5800 Beds with intergranular & fracture por. 0.86 0.9943 404 23.0 3 2780 Beds with low porosity 1.08 0.9852 77 25.0 7 1159 Total 1.07 0.9991 7126 23.0 1 19555 FRACTAL ANALy SIS OF THE DISTRIBUTION OF CAVE LENGTHS IN SLOVENIA ACTA CARSOLOGICA 36/3 – 2007 374 Fig. 5: Settings of the caves according to distance to the major faults and thrust fronts Fig. 6: Log-log distribution plot for the number of caves (N) longer than a specific length (L) in three groups of distance to the major tectonic structures TIMOTEJ VERBOVŠEK ACTA CARSOLOGICA 36/3 – 2007 375 Cave lengt� distribution can be described as fractal. Re - markably similar be� aviour of curves in t� e plots is ob- served, as a linear plot of number of caves, longer t� an specific lengt� in t � e log-log plots. The fractal approac� provides a better insig� t into t� e cave geometry by ana- lyzing t� e fractal dimension D instead of median or ot� er common statistics values. The fractal dimension calculated from t� e distribu- tions can not be directly interpreted as a fractal dimension of t� e caves t� emselves, i.e. used as a direct measurement of t� e geometry of t� e caves, as t� ese two dimension are obtained in a different way. The first one is calculated as a negative slope of t� e distribution of cave lengt� s, and t� e second one is usually obtained by a Ric� ardson’s (yard- stick) or box-counting met� od (Feder, 1988). However, t� ese distributions probably � ave a natural source, and t� e differences between t� e fractal dimensions are clearly observable, as discussed below. The lowest values can be found in t� e tectonic units of Periadriatic igneous rocks and Internal Dinarides, w�ic � are comprised mostly of low-porosity and especially of low-permeability rocks. The �ig � est fractal dimensions (D=1.10) appear in t� e unit of External Dinarides. This unit is represented mostly by carbonates of Dinaric car- bonate platform, w�ic � are intensely fractured and karst - ified. Similar explanation is valid for t� e unit of Sout� ern Alps (D=1.00), also consisting of karstified and fractured carbonates. The number of analyzed caves (N=9) in t� e Adriatic foreland is too small to comment reliably, and deviations of t� e curve can be also seen in t� e plot (Fig. 2), so t� e D could not be calculated. The rocks represented in t�is unit are clastic (flysc �) sediments, and caves occur in t� e relatively t�in-bedded layers of calcarenite. Value of D in Tertiary and Quaternary sediments is lower t� an one, w�ic � can indicate t � at t� e caves formed in t�is unit could resemble objects wit� geometries between a point and a line, and not t� e branc�ing c � annels wit� D �ig � er t� an one. The fractal dimension closer to zero resembles point-like objects, t� e one closer to one linear objects and t� e one closer to two planar-filling objects. Values of D lower t� an one are t� erefore possible, as dimension is ob- tained from t� e distribution and not from t� e geometric properties of t� e caves. Anot� er explanation for t� e low- er values of D, alt� oug� less possible, could be found in t� e surveying met� od, as t� e caves are usually surveyed by classical linear met� od. One s� ould be t� erefore very careful w� en applying t� e results for fractal dimension obtained from t� e lengt� distribution to geometric prop - erties of t� e caves. Nevert� eless, t� e value of dimension less t� an one clearly indicates t� at t� ese cave lengt� s are different from t� e ones wit� t � e �ig � er dimension, and interpretation of t� ese values is still possible by fractal met� ods. The fractal dimension is lower in less soluble and less erodable rocks, like igneous rocks (D=0.60) or rocks of Internal Dinarides (D=0.74), w�ic � were af - fected by lower degree of fracturing and � ave generally lower permeability t� an t� e igneous rocks. The lowest values are found in Periadriatic group. The � ardness of t� ese rocks is greater compared to t� e ot� ers, and con- sequently t� ey are � ard to erode (Kusumayud� a et al., 2000), so t� e cave passages cannot develop in suc� extent as in more soluble carbonates or clastic rocks. Similar to t� e explanation of tectonic setting, t� e �ig � er D for hydrogeologic setting could correspond to t� e rocks � aving been affected by fractal fracturation and subsequent dissolution along t� e fracture networks. The �ig � est values (Tab. 2) are found in aquifers wit� karstic and fracture porosity and t� ose wit� fracture porosity (D=1.06) and lowest in t� e aquifers and beds wit� inter - granular porosity (D=0.87, D=0.86). Deviations occur only for t� e group “Beds wit� low porosity” , as D is great - er t� an expected, about 1.08. Possible explanation is t� at rocks wit� quite different � ydrogeological and lit� ologi- cal properties occur wit�in t �is group, w �ic � influences t� e fractal dimension. The vicinity of tectonic structures t� erefore � as a no- ticeable effect on cave lengt� distribution, and t �is can be most likely interpreted as tectonic dissection of lon- ger caves into s� orter ones, and t� e tectonic effects can be manifested by displacement or collapse of t� e caves. This effect is also seen on t� e middle part of t� e plot (to t� e left side of t� e gap), w� ere a lower slope indicates t� e greater number of s� orter caves, w�ic � are uniformly distributed along t� e line. Some points in t�is part lie �ig � er above t� e linear fit line t� an expected and t� ese represent t� e increased number of s� orter caves, w�ic � form by fragmentation of t� e longer ones. The deposited cave sediments can also influence t� e results, as t� ese obstruct t� e traversable passages and can t� erefore di- vide t� e cave into smaller segments. However, t�is pro - cess could � ardly be seen on t� e cumulative distribution plot for all caves, as t� e effect is more or less random and s� ould t� us be distributed along t� e complete plot and in addition it s� ould not be influenced by distance to t� e tectonic structures. The fractal dimension obtained from t� e distribu- tion of all caves is about 1.07 and varies among different tectonic and � ydrogeological units. The usual explanation of fractal dimension D �ig � er t� an 1 indicates t� at caves wit� suc � dimension fill more space t � an t� ose wit� ideal dimension of 1.00 (for example a straig� t line), and t� e geological constraints limit t� e dimension to be lower DISCUSSION AND CONCLUSIONS FRACTAL ANALy SIS OF THE DISTRIBUTION OF CAVE LENGTHS IN SLOVENIA ACTA CARSOLOGICA 36/3 – 2007 376 t� an 2. This is strictly true for dimensions calculated by Ric� ardson’s or box-counting met� ods, and possibly not directly applicable to t� e ones obtained by distribution analysis, t� oug� t � e results are in very good agreement wit� t � e ot� er studies, as follows. Kusumayud� a et al. (2000) obtained t� e dimension D = 1.04-1.08 ±0.01 for caves in different lit� ologic environments in Indonesia and � ave used t� e box-counting met� od. Šušteršič (1983) calculated t� e value of D = 1.08 for t� e cave Dimnice in Slovenia by Ric� ardson’s (yardstick) met� od and similar approac� was used by Laverty (1987), w � o noted t� at cave lengt� ex�ibits fractal be � aviour wit� dimensions between 1.0 and 1.5 for caves in Sarawak and Spain. Frac- tal dimension based on calculation from t� e distribution was determined by Curl (1986), w� o calculated a slig� tly �ig � er value D = 1.4 t� an in t�is study for caves in dif - ferent environments. The differences from t� e analyses of Curl (1986) can be attributed to t� e facts t� at in �is study only t� e caves in limestone, marble and magnesitic limestone were analyzed and t� ose in dolomite, insoluble rock and gypsum were excluded. The dimensions are valid for t� e caves situated in specific regions in t� e USA, and t� e two exceptions from t� ese values are found in t� e Austrian and Iris� limestones. The geological, � ydrologi- cal and tectonic settings certainly influence t� e distribu- tions, but t� ere is no available data to precisely compare t� e effects of t� e different environments. The fractal be� aviour of cave lengt� s distribution can be possibly explained as t� e dissolution occurs along t� e fractures, bedding planes, faults and ot� er disconti- nuities in t� e soluble rock. It is well known t� at fracture networks are fractal, and t� eir dimension in 2D varies from around 1.3 to 1.7 (Bonnet et al., 2001). Faults are also fractal objects wit� rat � er lower dimensions, around 1.0 – 1.5. Results of t�is study s � ow t� at t� e cave lengt� s distributions ex�ibit lower dimensions (D = 1.08) t � an t� e faults or t� e fracture networks. Alt� oug� t � e dimen- sions can not be directly compared, lower values can be explained by c� anneling of flow t�roug � t � e fracture networks and especially bedding planes, w�ic � serve as pat� ways for t� e water. It � as been observed t� at w� en a preferential way is dissolved t�roug � t � e network, t� e flow increases due to larger c� annels, t� e obliteration of irregular s� ape of t� e c� annel by erosion is faster and consequently t� e fractal dimension t� erefore decreases wit� larger flow rates (Kusumayud � a et al., 2000). The lower slope of t� e distribution curves on t� e left-� and side of t� e plots can be explained by unders- ampling (Villemin et al., 1995), as below some t�res � old values t� e number of caves is underestimated. Similar trends were observed by t�ree different studies. Curl (1966) analyzed t� e cave lengt� s, w� ere for t� e observed curves for natural data, t� e left part of t� e plots ex�ibited a lower slope and t� e modeled curves s� owed muc� uni - form slope. He also noted for �is data, t � at t� e cumula- tive distributions s� ould be smoot� er if enoug� accurate data were available and all caves were considered. Loucks (1999) observed t�is effect for t � e cave widt� s, w� ere deviations appeared for widt� below a t �res � old of few meters. Finally, Villemin et al. (1995) noticed t�is effect for fault lengt� s. The caves wit� lengt � s lower t� an few meters are merely not considered as caves (t� ey are not recorded in t� e register), and t� us t� eir number is muc� �ig � er in t� e nature t� an actually recorded. The problem of cave definition can be raised � ere and was already dis- cussed by Curl (1986). Generally t� e cave is regarded as suc� if it is traversable by � umans. Cave spaces evidently exist at all scales, but are not registered, and t� ese voids in t� e rocks are present from microns to � undreds of meters (Curl, 1999). The number of caves N wit� lengt � about 1 m s� ould t� us be muc� �ig � er, around 107,000 and not around 7,200 as seen from example of t� e “all units” in t� e Fig. 1. This number can be simply estimated by inserting t� e value of L = 1 m into t� e best linear-fit equation log N = 1.082 * log L + 5.029 for “all units” . This is only a quick estimation, as t� e entranceless caves are not considered in t�is study due to t � e lack of data in t� e register. The grap� could also be extended to a muc � low - er scale (fart� er to t� e left), and t� e rock porosity (disso- lution, fenestral, vug) can be also interpreted as a “cave” , but obviously not traversable by � umans. Extrapolation to t� e “longer” side is contrarily not possible, as in t�is case t� e number of caves becomes less t� an one, and t� e curve also rapidly deviates from t� e linear fit line. Similar observations were made by Curl (1966), w� ere t� e ob- served (natural data) lengt� distributions ex�ibited more curvature on t� e plots t� an t� e modeled t� eoretical ones, so t� e proper basis for comparison of different cave set- tings is t� e use of all caves. Alt� oug� t � e exact values of D can not be interpret- ed directly by morp� ology of t� e caves, t� e larger fractal dimensions can be most probably interpreted by t� e abil- ity of t� e caves to form complex longer passages, most probably along t� e initial fracture networks and also bedding planes. The more soluble and fractured rocks ex�ibit greater fractal dimensions, larger t � an one, and rocks wit� intergranular porosity (generally t � ose wit� low porosity, low solubility and small degree of fractur- ing), s� ow D below one. These variations probably � ave a natural source, and t� e differences between t� e dimen- sions are clearly observable, Larger values of D could be expected in anastomotic or networks caves, and lesser values in branc� work or single-passage caves (Palmer, 1991). The p� ysical causes of power law scaling and varia- tions in fractal dimensions (power law exponents) are TIMOTEJ VERBOVŠEK ACTA CARSOLOGICA 36/3 – 2007 377 ACKNOWLEDGMENTS The aut� or t� anks all t� e cave explorers for t� e efforts encountered during t� e cave measurements, France Šušteršič for debate, David J. Lowe for smoot�ing t � e Englis� version of t � e text and Lee Florea for useful com- ments w�ic � improved t � e quality of t� e text. still poorly understood (Bonnet et al., 2001). The be- � aviour of fracture networks is believed to be caused by fractal fragmentation of blocks (Turcotte and Huang, 1995), w�ic � is scale-independent. Caves develop along t� e fractures and bedding planes, so t� ey in� erit t� e geometrical properties to some degree by dissolution of fractal networks. However, t� e processes w�ic � lead to t� e values of fractal dimensions of fracture networks and fractal be� aviour of distribution of cave lengt� s and t� eir dependence are still a c� allenge to be analyzed. REFERENCES Bonnet, E., Bour, O., Odling, N.E., Davy, P ., Main, I., Cowie, P ., Berkowitz, B., 2001: Scaling of fracture systems in geological media.- Reviews of Geop� y- sics, 39, 3, 347-383. Curl, R.L., 1966: Caves as a Measure of Karst. - Journal of Geology, 74, 5, 798-830. 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Mandelbrot, B., 1983: The Fractal Geometry of Nature.- W . H. Freeman & Co., p. 468, New york, Ny. Palmer, A. N., 1991: Origin and morp� ology of limestone caves.- Geological Society of America Bulletin, 103, 1-21. Placer, L., 1999: Contribution to t� e macrotectonic sub- division of t� e border region between Sout� ern Alps and External Dinarides.- Geologija 41, 191-221. Poljak, M., 2000: Structural-Tectonic map of Slovenia.- Mladinska knjiga & Geological Survey of Ljubljana, Ljubljana. Šušteršič, F., 1983: Determination of t� e unknown cave passages lengt� by means of fractal analysis.- In: Jančařik, A., ed.: Nove směri ve speleologii (New trends in speleology), 24.-28.10.1983 (Proceedings), 61-62. Šušteršič, F., 1992: Delovni seznam jam jugovz� odne Slo- venije.- Naše jame 34, 74-108. Turcotte, D. L. & Huang, J., 1995: Fractal Distributions in Geology, Scale Invariance, and Deterministic C� a- os. In: Barton, C. C. and La Pointe, P . R. (eds): The Fractals in the Earth Sciences, 1-40. Villemin T., Angelier, J., Sunwoo, C., 1995: Fractal Distri- bution of Fault Lengt� and Offsets: Implications of Brittle Deformation Evaluation – The Lorraine Coal Basin. In: Barton, C. C. & La Pointe, P . R. (eds): The Fractals in the Earth Sciences, 205-226. FRACTAL ANALy SIS OF THE DISTRIBUTION OF CAVE LENGTHS IN SLOVENIA