分形统计应用

1、2020/6/15,1,分形统计应用讲座,主讲人:孙洪泉教授,2020/6/15,2,主要内容,地质统计学,应用实例,分形几何,2020/6/15,3,分形几何,分形维数,分形插值,什么是分形,2020/6/15,4,分形起源1973年,法国数学家BenoitB.Mandelbrot在法兰西学院讲课时,首次提出了分维和分形几何的设想,创造了“分形(Fractal)”这个新术语英国的海岸线有多长?,2020/6/15,5,Area:Sfinitenumber(unitcircle),r=1,数学中的奇怪现象,2020/6/15,6,分形几何为研究自然界中广泛存在的复杂无序，不规则现象提供了一种定

2、量描述手段，为人们从局部认识整体、从有限认识无限提供了新的方法。分形：复杂结构自相似性分形(分数)维数,2020/6/15,7,自然界中的复杂形体欧氏几何与分形几何,云彩不是球体,2020/6/15,8,Mountainsarenotcones,2020/6/15,9,Barkisnotsmooth,2020/6/15,10,Lightningdoesnottravelinastraightline,2020/6/15,11,自然界中的自相似(分形体),花菜,2020/6/15,12,树木,2020/6/15,13,河流水系(密西西比河流),2020/6/15,14,龟裂纹,2020/6/15

3、,15,岩层的层理,2020/6/15,16,自相似性,严格自相似(数学方法生成),统计自相似(存在于自然界中),2020/6/15,17,strictself-similarity,Sierpinskitriangle,ThisideaiseasytoexplainusingtheSierpinskitriangle.NotethatSmaybedecomposedinto3congruentfigures,eachofwhichisexactly1/2thesizeofS.Seefigure(right).Thatistosay,ifwemagnifyanyofthe3piecesofSb

4、yafactorof2,weobtainanexactreplicaofS.Thatis,Sconsistsof3self-similarcopiesofitself,eachwithmagnificationfactor2.,oneself-similaritycase,S,2020/6/15,18,HowtomakeSierpinskiTriangle,2020/6/15,19,anotherself-similaritycase(TheKochSnowflake),2020/6/15,20,分形维数,整数维数,Apointhasnodimensions0,Alinehasonedimen

5、sion1,ASquarehastwodimensions2,ACubehasthreedimensions3,0,1,2and3areintegers.Thatistosay,dimensionsinEuclideanspaceareintegers.,Fractalscanhavefractional(orfractal)dimensions.Afractalmighthaveadimensionof1.6or2.4.,2020/6/15,21,howtocalculatefractaldimension,ReviewthedimensioninEuclideanspace(integer

6、dimension),box-counting,l=1,s=1,v=1,square,line,cube,integerdimensiond:,2020/6/15,22,分形维数,自相似物体的分形维数,SupposeN(r)isthenumberofr-spheres(spherewithdiameterr)neededtocoveraset.TheHausdorffdimensionofthesetisdefinedby:,2020/6/15,23,下列形体分形维数,KochSnowflake,D=ln(4)/ln(3)=1.262,Sierpinskitriangle,D=ln(3)/ln

7、(2)=1.585,SierpinskiCarpet,D=ln(8)/ln(3)=1.893,Afractaldimensionindicatesthedegreeofthecomplexityofthefigure,2020/6/15,24,分形体的构造1：,2020/6/15,25,分形体的构造2：,2020/6/15,26,分形体的构造3,2020/6/15,27,数学方法生成的分形体,2020/6/15,28,MandelbrotSet,2020/6/15,29,Somefractalpicturesmadebymathematics,2020/6/15,30,2020/6/15,31

8、,2020/6/15,32,2020/6/15,33,2020/6/15,34,分形的工程应用分形维数分形插值,2020/6/15,35,分形维数,1,r=1/3,r=1/6,n,D=theslopeoftheline,2020/6/15,36,N=小块体个数,2020/6/15,37,Differentproblemshavedifferent“boxes”inbox-counting,Box:asetofconcentriccircles,fractalporesandtheirfractalmeasurementsinrockmaterials,2020/6/15,38,Thechord

9、-lengthtechniqueisusedtomeasurefracturesurface,hi,r:hiN(r):N(hi),2020/6/15,39,Traditionalinterpolation,Fractalinterpolation,linearinterpolation,splineinterpolation,分形插值,2020/6/15,40,Fractalinterpolation,Thesimulatedprecisionismorethan90%,2020/6/15,41,Rockfracturesurfaces,ApplicationofFractalinterpol

10、ation(Case2),Fractalinterpolationofrockfracturesurfaces,2020/6/15,42,Statisticallyself-similarityofrockfracturesurfaces,2020/6/15,43,Empiricallymeasuredfracturesurfaces,Fractalinterpolatedfracturesurfaces,Theprecision95%,2020/6/15,44,Weobtained28elevationdataofafaultsurfacewithseismicreflectiondatai

11、nacoalfieldinsouthChina.,Linearinterpolation,ApplicationofFractalinterpolation(Case3),Fractalinterpolationoffaultsurfaces,Fractalinterpolation,Themorphologyofthissurfaceisclosetothenaturalfaultsurface,2020/6/15,45,地质统计学(空间信息统计学)与空间位置有关统计方法主要原理:方差最小;无偏估计估值精度高研究内容:克立格估值;变差函数研究不同方向上变化特征,地质统计学,2020/6/15

12、,46,地质统计学的主要优点,2020/6/15,47,Classicalstatistics:nodifferencesexpectationE(d1)=E(d2),variance2(d1)=2(d2)Geostatistics:canfindoutdifferenceswithsemivariogram(discusslatter),2020/6/15,48,2020/6/15,49,应用实例,2020/6/15,50,数据分布(云南),naturalconservationarea,2020/6/15,51,煤厚等值线图(七煤层),2020/6/15,52,自然保护区煤厚等值线图,(七煤层),2020/6/15,53,煤厚立体图,(七煤层),2020/6/15,54,底板等高线图(七煤层),2020/6/15,55,自然保护区底板等高线图(七煤层),2020/6/15,56,底板标高立体图(七煤层),2020/6/15,57,灰份含量等值线图,16煤层,3煤层,9煤层,2020/6/15,58,硫份含量等值线图,16煤层,3煤层,9煤层,2020/6/15,59,断层概率等值线图,2020/6/15,60,2020/6/15,