Mineral resource analysis by parabolic fractals

Available online at w.cencedirnct.comMININGScienceDirectSCIENCE AND。王。TECHNOLOGYEL SEVIERMining Science and Technology 19 (2009) 0091-0096www.elsevier.com/ocatejcumtMineral resource analysis by parabolic fractalsXIE Shu-yun1223, YANG Yong guol34, BAO Zheng-yu12,KE Xian-zhong, LIU Xiao-long2'State Key Laboratory of Geological Processes and Mineral Resources, China University of Geosciences,Wuhan, Hubei 430074, China'Faculty of Earth Science, China University of Geosciences, Wuhan, Hubei 430074, China'Department of Geography, York Universit, Toronto, ON, M3J IP3, Canada'School of Resources and Earh Sciences, China University of Mining & Technology, Xuzhou, Jiangsu 211166 ChinaAbstract: Elemental concentration distributions in space have been analyzed using dffrent approaches. These analyses are ofgreat signifcance for the quantitative characterization of various kinds of distribution patterns. Fractal and multi-fractal methodshave been extensively applied to this topic. Traditionally, approximately linear-fractal laws have been regarded as useful tools forcharacterizing the self-similarities of element concentrations. But, in nature, it is not always easy to find perfect linear fractal laws.In this paper the parabolie fractal modelisusbolic fractal model is used. First a two dimensional multiplicative multi-fractal cascade model is used to studyS ofthe concentrations and the validityof no-linearthe concenration pttrns. The resuts show the parabolic fractal (PF) propertes of the concentatins and the validityfractal analysis. By dividing the studied area into four sub-areas it was possible to show that each part follows a non-linear para-bolic fractal law and that the dispersion within each part varies. The ratio of the polynomial cofficients of the fited paraboliccurves can reflect, to some degree, the reltive concentration and dispersal distribution patterms. This can provide new insight intothe ore-forming potential in space. The parabolic fractal evaluations of ore- forming potential for the four subareas are in goodagreement with field investigation work and geochemical mapping results based on analysis of the original data.Keywords: parabolic fractal; multiplicative mulifractal cascade modeling; ore-forming potential; geochemical mappingfrequencies7. This model has been used to separate1 Introductionanomalies from the background in metallic geo-Regional geochemical information plays a signifi-chemical exploration, in oil/gas prospecting and incant role in providing avenues for mineral resourceenvironmental studies'". When the slope of a lineassessment. More and more geochemical data proc-fitted to a log-log plot of A(C > C})vs. C is calcu-essing techniques have been developed to better un-lated, it is found that major rock-forming elementsderstand how to evaluate the data and how to elimi-usually have larger slopes. Smaller slopes usuallyate outliers and analyze geochemical structures.correspond to ore-forming elements; this shows theThese techniques include the Kriging method, condi-association of slope value to the mineral containingtional simulation, Fourier filtering and wavelet flter-potential of the structure, to a certain degreeo. Theing techniquest- 3. The purpose of all these methodsP-A fractal model", the concentration-distance frac-is to make the geochemical structures clearer and total model", the S-A model and multi-fractal IDWexplore new indexes capable of quantitatively char-interpolation and fractal filtering techniques have alsoacterizing such structures.been extensively applied. The correspondingSince fractal concepts emerged, fractal and multi-fractal dimension, and multi-fractal parameters, hasfractal methods have been extensively applied in dif-been extracted for quantitative geochemical asss-ferent fieldst46. Techniques based on fractal andment. Different sorts of multi-fractal spectrum curves,multi-fractal models have been introduced to deline-ind the consequent muli-fractal parameters, wereate the spatial structures associated with elementalalso investigated for their relevance to spatial elementconcentrations. Cheng used the CA fractal model toconcentration distribution patterns.display the relationship between concentrations and中国煤化工Received 05 April 2008; accepted 15 July 2008Projects 40502029, 40472146 and 40373003 supported by the Naural Science FoundaJIYTHC N M H Gkey Lab of GologjcalCorresponding author. Tel: +86 27-67883033; E mail adres: tinaxic2006@gmail.comMining Science and TechnologyVol.19 No.1to characterize the distribution patterns of elements ingrid box within the space is defined as the metalspace. The paper consists of two major parts. The firstconcentration within the box. A weighted sum over. describes generation of standard muli-fractal pattermnsall the boxes gives the function:based on a two dimensional multiplicative multi-X,(e)x e*(9)fractal cascade model and the testing of that model.The second is a case study to characterize mineralFor certain values ofq the exponents t(q) haveresource potentials using the parabolic fractal model.physical significance. Thus, the scaling exponentsT(q) reveal different aspects of the heterogeneity of2 Parabolic fractal analysisspatial distribution patterns. For multi-fractal meas-Many natural geometric objects are well depictedIres t(q) is a nonlinear function: t(q) =by fractals, e.g; coastlines and fractures 4. Fractalaxq- f(a), wherea=dr/dq is not constant. Theself-similarity is regarded as a particularly powerfulfractal dimension, f(a), introduced earlier, is relatedconcept that has been investigated in recent years'*.to t(q)through a Legender transformation:However, self-simiarity is not always expected tohold perfectly in nature. As a remedy, differentf(x)=aq-q .straight-line segments have been used to interpretTwo parameters of the f(a)- a curves, namelyelf-similarity properties on varied scales, such as inAa,-0apthe C-A fractal model'. In 1996, Jean Laherrere putthe asymmetry index R= 10and the rangeforward the concept of the Parabolic Fractal LawOar+Sap，(PFL), which shows a non-linear parabolic deviation.of the curve Oa = Oar + Oap , have been proposedCompared to single self- similarity, usually describedand used extensivelyl17-19. In these formulae △arby linearization techniques, the PFL method gives aclear result.and Oap are thea range of the left part of theIt has been found that petroleum field sizes listedcurve and the right part, respectively.in order of decreasing size and plotted in a log-logformat show parabolic fractals'The PFL is re-4 Multiplicative multi-fractal cascade simu-garded as an imperfect self-similar law in petroleumlation resultswork so a quadratic term has been added:log(S()) = a+ bxlog() + cX(log())2 (1)Multiplicative cascades were first suggested bywhere S(i) is the oil field size of rank i and c is calledKolmogorov in the modeling of turbulencethe PFL curvature. If the distribution pattern followsrecent years, different kinds of multiplicative cascadeperfect self-similarity c will be 0. The curvature formodeling have found application in the modeling ofthe world's oil reserves has been estimated as c =different kinds of phenomena, such as biological-0.07. At present there is no proof showing the actualmodeling or element distribution modeling'esl. Onone hand, it is easy to visualize one multiplicativeapplication of curvature values.Different segments of fited-lines are usually usedcascade process and construct the correspondingfor the analysis of element concentration distributionmultiplicative cascade structures. On the other hand,pattermns based on log-log plots of N(C> C) Vs.the structures are very complicated and their mathe-matical properties are not simple. In a two dimen-C(C is the concentration range and N(C>C) issional coordinate system the original geochemicalthe number of samples having concentration greaterdata can be simulatedasr, r(1-r), (1-r)r andthan c). In the following sections, simulation(1-r). This multiplier, r , can be a random vari-data and one case study will be used to establishable at each stage of the cascade simulation. Conse-whether the element distribution patterms followquently, this process can be iterated indefinitely andparabolic fractal laws and whether the results awill result in a multiplicative cascade structure.suitable for mineral resource assessment.Without loss of generality, in this study we take theiteration time as seven, since this is enough to ana-3 Multi-fractal analysislyze the structures of simulated geochemical datfollowing the multiplicative cascade processes.To perform multi-fractal analysis we must first de-Two similar simulations have been performed infine a spatial measure. This measure relates to localthis study. First, the multiplier, r，was changed atconcentration values in space. Without loss of gener-each中国煤化工m three data setsality, we consider a grid box of sizeE that covers ahavingr the sevenpart of the whole space under study. Let the averagemultipYHC N M H Ghe saistial pa-concentration in the box be C. The measure of the ithrameters of the multipliers are shown in Table 1. AsXIE Shu-yumn etalMineral resource analyis by parabolic fractals93shown clearly in Fig. 1, such generated data, ofand wider, which reflects that the structures of thecourse, follow a perfect muli fractal distribution pat-generated datasets are becoming more and moretern. As the variance cofficients increase from 0.3 tocomplicated17].0.6 and 0.9 the multi-fractal spectrum becomes widerTable 1 Multipliers r used in the simulationsrat each cascade stageDatsMcansdv0.20680.32760.41880.18580.21260.29730.28290.2760 .0.08220.29770.48070.03600.27670.14600.42900.3400.26810.59990.00790.00820.09500.29350.02880.18380.31570.13330.1324 .0.99392.pcoefficients of the multipliers increase from 0.3 to 1.0the absolute curvature values gradually decrease. As.0is well known, the bigger the curvature the more.5abrupt the curve will be, which means the concentra-tions are more localized in such a case. On the con-trary, a smaller curvature and a flat parabolic curve0swill cover a wide range of concentrations havingmore dispersed distribution patterns. Also listed in0.0L十23456789Table 2 are the ratios of c/b, which are decreasingwith the increasing variance coefficients of the multi-Fig. 1 Plots of multi-fractal spectral functions f(a)pliers. ,versus a as the multiplier r changes randomly in theTable 2 Basic descriptive parameters for simulatedmultiplicative multi-fractal cascade modelingconcentrations with different 1The parabolic fractal displays from the multiplica-Data MeanMin_ Max Std_ VIbtive multifractal cascade modeling data are shown in0.0001.000 0.0123 0.0003 4.3678 0.039Fig.2. It is no coincidence that all the concentration0.000 0.0000 0.0058 0.0003 4.8400 0.033data generated by this simulation work show para-0.0001 0.0000 0.2448 0.0024 39.0664 0.010bolic fractal distribution patterns. As the variance10002|lo' |10'10~10310-310-10log (octratioa)logCocentaio)log Concentation)()ogM)=- 060055151(5-1.3737 (6) log(Nr - 0000 -.0910(C-0.93 (<)log(N)= r 00080C 01810(C-0.58clb-0.039104c16-0.033621c/b-0.010349Fig. 2 Parabolic fractal display of concentrations based on two dimensional mutiplicative multi-fractalcascade simulations with random mulipliersThe second set of simulations assumes the multi-plier r is constant during the multiplicative multi-2r-0.45fractal cascade processes. When r=0.45 and 1-r=0.55the generated data are concentrated in a limited range1:and the variance coefficients of the data are, conse-quently, small. As displayed in Fig. 3 the multi-frac-tality of the generated data with r=0.45 is much0.weaker. Its multi-fractal spectrum is short and narrow.But a smaller r corresponds to a largerl -r. The data中国煤化工一5。generated with smaller r and larger1 - r fluctuatesaround a wider range with a more dispersed distribu-.MH. CNM H G fcio(a)，otion pattern. The data generated with r=0.15, plottedversus a as the multiplier r changes from 0.15 to 0.454Mining Science and TechmologyVol.19 No.1as dotted circles in Fig. 3, show a much strongermedium level. Parabolic fractal analysis results (Fig.multi-fractality with a relatively long and wide4) imply that both the curvature and the absolute ratiomulti-fractal spectrum curve. The mutli-fractality ofc/b of the fitted parabolic curve increase gradually asthe data generated with a medium r-0.30 is also ofr increases.. 100°p10g 10ofo'f.1q0 1010* 10~10° 102log (Concentration)log (Concentnation)log (Concentation)(a)~ -0.15(b)r-0.30()r-0.45log(N)= 00140g(C - 067og(C-1.067log(M)= 020102C-.23log(C)-1.52log(N)= 0.891og(C) +0.7310g(C-0.28clb 0.051362Fig. 4 Parabolic fractal display of concentrations as the multiplier, r , changes from 0.15 to 0.455 Case studysub-area I. The highest anomalous concentrationshappen to occur in sub-area II where there are twoThe study area is located in a district within theapparent accumulation centers. Similar to those inQulong region, Tibet, China. There are a total of 1632sub-area I, the accumulated anomalies in sub-area IIIgeochemical samples collected in a two-dimensionalare ditributed along the diagonal centerline of thespace within this area. Concentrations of Ag havewhole area. On the contrary, in sub-area IV the an-been measured and are used to analyze the distribu-omalously high concentrations distribute sporadicallytion patterns in space. The aim of this work is to es-in space and there is no obvious accumulation centertimate the Ag ore quality in different subareas underof Ag. Field work has demonstrated that the fracturesconsideration.and faults in the area are mainly along the NW-SEA geochemical mapping of Ag concentration baseddirection and that the structures located in this direc-on the traditional x土2σ techniquel,24] is showntion play a very important role in metal ore formation.in Fig. 5. To assess the accumulation centers of AgFrom this perspective, sub-areas II and II may be ofthe area has been divided into four sub areas (Fig. 5)high potential for containing Ag. The accumulatedfor simplicity. Only one half-ring anomalous center ofconcentration centers are closely associated with theconcentration exists close to the bottom boundary ofregional structures.(间) 1向) II(C)毋(d)IVFig. 5 Geochemical mapping ofAg based on original concentration dataTable 3 lists the basic descriptive parameters forTable 3 Basic descriptive parameters for concentrationsAg concentration in the different subareas, includingofAg in the different sub-areasthe mean, minimum and maximum values, the stan-Sub-areaMinSuVdard deviations and the variances. For sub-areas II0.0993 0.0330 0.2380 0.03730.3753and III the standard deviations and variances are0.1114 0.0310 0.2460 0.04650.4174higher than those in the other two sub-areas. The0.09730.02700.04410.4538standard deviation in sub-area IV is the smallestr0.0884 0.0230 0.24040.0335).3791whereas the variance in sub-areas I and IV are close中国煤化工to each other. In this point of view the concentrationsi-fractal spectrumin sub-areas II and II are more dispersed and thecurveYHC N M H Gfour subareas fol-concentrations in sub-area IV are much more concen-low multi-fractal properties since the difference betrated.XIE Shu-yun ctalMineral resource analysis by parabolic fractals95tween the minimum and the maximum singularitytrations of Ag in sub- areas I and IV are relatively de-exponents are fluctuating over relatively broad rangespleted but Ag concentrations in sub-areas II and III(as listed in Table 3). Obviously, the spatial concen-exist in more accumulated formations. These are welltration distribution patterns of Ag in sub-area I anddemonstrated by the positive values of R in sub-areassub-area IV are right-deviation multi fractalolII and II and the negative ones in sub-areas I and IV,whereas the multi-fractal spectrum curves, with theas listed in Table 4. The positive R values in sub-areasright parts broken a bit, in sub-area II and sub-area IIII and III imply a higher mineral potential for Ag.deviate to the left parts. From this aspect the concen-2.5[25p2.0-.co" 1.5-^, 1.5t、 1.s~1.0..5-0.:34(@) Suburea I(b)Subarea I(C) Subarea m(d) SubarealVFig. 6 Multi-fractal spectrum curves of Ag concentration in different sub-areasTable 4 Fractal and muli-fractal parameters for Agthe N(C>C,) versus C。 plots follow parabolicconcentration in different sub-areasfractals when plotted as log-log diagrams. In sub-areaSub-areasa0a,clbII, where the ore-forming potential for Ag is believed0.5815 0.3366 0.2449. 0.1577 -0.4668to be good, the absolute value of c/b is the lowest one,0.5034 0.2235 0.27990.1120-0.0977whereas the maximum absolute value is in sub-aream0.6397 0.1774 0.46230.4453. 0.1758IV where there is no obvious concentration anomaly.v0.4902 0.2665 0.2236-0.0875 -1.2475All four curvature values are negative and sub-areasII and II exhibit smaller absolute values: the biggerParabolic fractal analyses for Ag in the fourabsolute curvatures happen to be in sub- areas I and IV, ,sub- areas are displayed in Fig. 7. The correspondingwhich reflects that the mineral potential in the firstpolynomials of the fitted parabolic curves are alsoand fourth sub-areas is much weaker.listed in that figure. In different individual subareas10萝1021o04 105 10. 1027 108104 105 10 1027 10 109log (Concentration)(a) Subarea I(6) Subarea Ilo(-1.4lol(g.C3.210g(C-2.49log()-0.2710gC3- 27510og(C)+4.1010* [1o'0210210'104 105 106 107 108102 104 106 108log (Concentation)log (Concenraion)(C) Subara 1中国煤化工log(M-058log(C7 -3.3log(C)+1.93Fig. 7 Parabolic fractal distribution patternsYHCNMHG96_Mining Science and TechnologyVol.19 No.16 Conclusionsbons. Marine and Petroleum Geology, 2006(23): 529-542.Modeling elemental concentration distributions in[9] Lima A, De Vivo, Cicchella D, Cortini D, Albanese s.Multifractal IDW interpolation and fractal filteringspace should be carried out using several models. Onmethod in environmental studies: an application on re-a parabolic fractal display sub-areas II and III showgional stream sediments of (taly), Campania region.relatively concentrated accumulation distributionApplied Geochemistry, 2003(18): 1853-1865.patterms. The other two subareas are relatively de-[10] Xie S Y, Bao Z Y. Continuous multifractal models ofgeochemical fields. Geochimica, 2002, 31(2): 191-200.pleted. Both the multi-fractal spectrum analysis and(In Chinese)the geochemical mapping based on the original data[11] LiC J, Ma T H, Shi J P. Aplication of a fractal methoddemonstrate the results of parabolic fractal analysis.relating concentrations and distances for separation ofIn this perspective the polynomial functions of thegeochemical anomalies from background. Journal offited parabolic curve for the N(C>C) vs.ConGeochemical Exploration, 2003(77): 167-175.log-log plot provide a new avenue for quantitative[12] Xu Y G Cheng Q M. A fractal filtering technique forprocessing regional geocbemical maps for mineral ex-assessment of mineral resources.ploration. Geochemistry: Exploration EnvironmentAnalysis, 2001(1); 147-156.Acknowledgements13] Feder J. Fractals. New York: Plenum, 1988.141Zhang H, Wang Y J, Liu C z. Research on evaluation ofdegree of complexity of mining fault network based onThe work has been jointly supported by NaturalGIS. Journal of China University of Geosciences, 2007,Science Foundation of China (No.40502029,17(1): 6367.40472146 & 40373003) and the Open Foundation[15] Laherrere J, Somette D. Stretched exponential distribu-from the Key Lab of Geological Processes and Min-tions in nature and economy:“fat tails" with characteris-tic scales. The European Physical Journal B, 1998(2):eral Resources (No.GPMR2007-11).525- -539.[16] Laherrere J. Distribution of field sizes in a petroleumReferencessystem: parabolic fractal, lognormal or stretched expo-nential. Marine and Petroleum Geology, 2000(17): 539-[1] Hawkes H E, Webb J s. Geochemistry in Mineral Ex-546.ploration, New York: Harper, 1962.[17] Xie s Y, Bao z Y. Fractal and multifractal properties of2] LiF L, Pei T, Bao Z Y, Li C W. Principle and softwaregeochemical fields, Mathematical Geology, 2004, 36(7):847- -864.system of geological and geochemical processing. EarthScience-Journal[18] Kravchenko N A, Boast W C, Bullock G D. MultifractalScience~Jownal of China University of Geosciences,1999, 24(3); 712-715..analysis of soil spatial variability. Agronomy Jourmal,1999(91);: 1033-1041.3] Bao Z Y, Li F L, Jia x Q. Methodology of tempo-ral-spatial structure of geochemical fields. Earth Sci-19] Cheng Q M. Spatial and scaling modeling for geo-chemical anomaly separation. Journal of Geochemicalence-Journal of China University of Geosciences, 1999,24(3): 282- -286.Exploration, 1999(65);: 175- -194.4] Mandelbrot B B. Stochastic models for the Earth's relief,[20] Kolmogorov A N. A refinement of previous hypothesesconcemning the local structure of turbulence in a viscousthe shape and the fractal dimension of the coastlines, andthe number-area rule for islands. Proceedings of the Na-incompressible fluid at high Renolds number. J FluidMech, 1962(13): 82- -85.tional Academy of Sciences, 1975(72): 3825- 3828.5] Turcotte D. Fractals and Chaos in Geology and Geo-[21] Rangarajan G Ding M. Processes with Long -RangeCorrelations: Theory and Applications. New York:physics. Cambridge: Cambridge University Press, 1997.6] Dimri V P. Applcation of Fractals in Earth Sciences.Springer, 2003: 392.USA: Balkema Publishers, 2000.[22] Sugihara G Allan W, Sobel D, Allan K D. Proceedingsfor National Academic Sciences, USA, 1996(93): 2608.刀] Cheng Q M, Agterberg F P, Ballatyne S B. The separa-[23] Moran P A P Global stability of genetic systems gov-tion of geochemical anomalies from background bfractal methods. Journal of Exploration Geochemitry,emed by mutation and selection. Math Proc Camb Phil1994(51): 109-130.Soc, 1976(80): 331- 336.8] ZhangLP, Bai G Zhao K B, Sun C. Restudy of acid-24] Gahuszka A. A review of geochemical background con-extractable hydrocarbon data from surface geochemicalcepts and an example using data frorm Poland. Environ-survey in the Yimeng uplift of the Ordos basin, China:mental Geology, 2007(52): 861- -870.improvement of geochemical prospecting for hydrocar-中国煤化工MYHCNMHG