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分形与多重分形及其在地学中的简单应用谢淑云s地大IAMG学生分会活动2005.11.10MandelbrotandFractal:1973年，Mandelbrot在法兰西学院讲学期间首次提出分形学的思想。1975年他在写专著的过程中，碰到法文动词frangere(破坏、破碎)的形容词fractus，联想到英文中的同根词fracture(断裂)和名词fractaion(分数),在此基础上创造了fractal一词。Fractal本意是不规则的、破碎的、分数的。Mandelbrot是想用此词来描述自然界中传统欧几里德几何学所不能描述的一大类复杂无规的几何对象。例如，弯弯曲曲的海岸线、起伏不平的山脉，粗糙不堪的断面，变幻无常的浮云，九曲回肠的河流，纵横交错的血管，令人眼花僚乱的满天繁星等。它们的特点是，极不规则或极不光滑。直观而粗略地说，这些对象都是分形。分形的定义：♣部分与整体以某种形式相似的形，称为分形。Fractalsareshapesthatlookalmostthesameonvariousscalesofmagnification(Mandelbrot,1975;Cheng,1994),inthesensethateachpiece(howeversmall)isidenticaltothewholeaftersomerescalingandtranslation.Theyareneitherentirelyregular,norentirelyrandomanditmaybesaidthatobjectsareself-similarorofself-similarity.KochcurveKochcurveAfractalKochcurve([Koch,1904]),reproducedfrom[AfractalKochcurve([Koch,1904]),reproducedfrom[WelanderWelander,,1955]toillustratethemixingofatwodimensionalfluid.1955]toillustratethemixingofatwodimensionalfluid.Log-logplot尺度次数甲10m7乙1m85丙5mm2200斜率＝-1.0853（3次）S=-1.2618（反复）=-log4/log3power-lawandfractaldimension分形维数——定量刻画分形特征的参数♣相似维数（Similaritydimension）♣信息维数（Informationdimension）♣关联维数（Correlationdimension）♣集团维数（Clusterfractaldimension）♣盒子维数（Boxdimension）♣罗盘维数（Compassdimension）♣容量维数（Capabilitydimension）盒子维数:εεε)()(NL=2)()(εεεNA=对海岸线应满足下列关系：DN−∝εε)(N（ε）曲线或曲面相交的盒子数εεFractal&multifractal¾Coastline:howlongisthecoastline?¾Island:howfluctuantistheisland?6507007508008509009501000105010701100GeochemicalLandscapeWaterS1S2……S水的分布（waterdistribution）金的分布（golddistribution）Manyquantitiesexhibitsuchkindofbehaviour;thatis,thequantityu=theamountofgroundwaterbelowSisanexampleofameasurewhichisirregularatallscales.Multifractal:Evertsz和Mandelbrot于1992年指出，当我们所研究的度量（measure）在不同的尺度上均相同，或者至少在统计意义上是一样的，就可以说我们所研究的度量是自相似的，这一度量就是多重分形。Whentheirregularityisthesameatallscales,oratleasestatisticallythesame,onesaysthatthemeasureisself-similar,orthatitisamultifractal.Asierpińskigasketisaself-similarset,inthesensethateachpiece(howeversmall)isidenticaltothewholeaftersomerescalingandtranslation;somethingsimilarholdsformultifractalmeasure.Multifractal:多重分形是许多个单一分形在空间上的相互缠结(intertwined)、镶嵌，是单一分形的推广，单一分形可以看作是多重分形的一种特例。多重分形与单一分形一样，也是自相似的，与尺度无关.多重分形通常所描述的是定义在某一面积（二维）或体积（三维）中的一种度量（u).通过这种度量值或数值的奇异性可将所定义的区域分解成一系列空间上镶嵌的子区域，每一个子区域均构成单个分形.这样形成的分形除具有分形维数外，还具有各自度量的奇异性(singularity).一系列的分形维数和奇异性将构成所谓的维数谱函数f(α)----multifractalspectrum.定量描述多重分形的参数－－多重分形频谱(MultifractalSpectrum)计算方法：在各个科学领域，已经有了多种计算维数谱函数的方法，如矩方法（Momentmethod）、直方图法（Histogrammethod）、小波方法（Waveletmethod）、乘数法（Multipliermethod），以及二次维矩方法（Doubletracemoment(DTM)method）。这些方法的具体计算又被扩展了多种形式：格子方法(box-countingmethod)和活动格子方法(glidingbox-countingmethod)以及反格子方法（inversebox-countingmethod）、格子弯曲法（box-flexmethod）、格子旋转法（box-rotatemethod）等。矩方法是最常用的方法之一。3results&discussionProceduresofthemomentmethodtodeducemultifractalspectralfunctionforaDeWijsmodelofd=0.4(Agterberg,2001)多重分形的研究方法正演：DeWijs模型，蒙特卡罗模拟，自组织临界模型，分形生长模型反演：实际地质系统的分析1(1+d)2(1-d)2(1+d)(1-d)(1+d)(1-d)1+d1-d……Fig.1Constructingprocessesofa2-dimensionalDeWijsmodelwithenrichmentfactordafter1iteration模拟1：DeWijs模型模拟PartⅠΔaa0Fig.4a0,Daoff(a)ofDeWijsmodelsversustheirenrichmentfactorsdFig.3Plotofmultifractalspectralfunctionf(a)versusaasenrichmentfactordchangesafter7iterationsGlobaldivisionBasicmodelingFig.5If10,aconstant,dividedintoallconcentrationsofDeWijsmodelwithanyenrichmentfactord,themultifractalspectrumobtainedbymeansofthemethodofmomentsshowsnothingdifferentincomparisonwithFig.3(SpatialStructure)0.8-1.00.6-0.80.4-0.60.2-0.40.0-0.20.6-0.80.4-0.60.2-0.40.0-0.20.8-1.00.6-0.80.4-0.60.2-0.40.6-0.80.2-0.4012Fig.6Spectrumfigureofa-f(a)withdifferenttypes:RM--Right-deviatedmultifractal,LM--Left-deviatedmultifractal局部削减非对称指数局部叠加Tab.1ParametersoftheresultingofmultifractalspectraofsuperimpositionofDeWijsmodelofd1byanotherDeWijsmodd1d2Δαα0ΔαLΔαR1.16092.0460.59050.57040.41.51362.06450.95410.55940.61.92862.07281.37750.55120.20.82.29222.07971.74800.54422.48882.2431.25851.23030.62.74782.26081.53531.21260.40.83.10892.26731.90281.20624.08222.5922.07312.00910.60.84.25882.65562.27051.98830.86.42513.5973.25013.1750d2d1Fig.7Multifractalspectrumcurvesa-f(a)oflocalsuperimpositionofDeWijsmodelofd1byanotherDeWijsmodelofd2实例：金属矿产Tab.4-1GeneralInformationofGeochemicalDatausedforCaseStudyIDistrictNo.ofIndexesore-formingelementsSampleamountSamplecategorySamplearea(km2)Shaoguan25As,Bi,Mo,Sn,Sb,W,etc1448Rock4292SouthAnhui14548922000NorthAnhui14Cu,Fe,Pb,Zn,etc.4524Streamsediment18100实例1：粤北韶关金属成矿区Relationshipbetweenmultifractalspectrumvaluef(α)andsingularityexponentαInNanlingdistrictElementRElementRMo0.6782Ca0.1864W0.5499Ga0.1048Be0.5431Mg-0.0029As0.5138Cr-0.0838Zn0.4908Sn-0.1118Sb0.4521Sc-0.1482Ag0.4277Fe-0.1736Cu0.3003Ti-0.1916Co0.3099V-0.2130Pb0.3003Li-0.2710Bi0.2350Zr-0.3887Mn0.2324K-0.4465Ce0.2230MoWBeBeAs0.5ZnSbAg0.4CuCoPbBiMnCeCaGaMgCrSnScFeTiVLiZrKInfact,W,Mo,As,AgandPb,Znareallveryimportantinthisregion.实例2：安徽江南江北地区MultifractalSpectrumfunctionsofelementsinSouthandNorthAnhuiProvinceSomeparametersofelementsofstudyinSouthandNorthAnhuiΔαΔαLΔαΔαLSouthNorthSouthNorthSou