1 123 LVA 2.1 LVA D {Z(u) u D} Kriging γz(h) γz( h ) [1] h h h 2 (2) 1 γz(h) ; (Stream 2 h [4] Networks) 1 (Moving- 2.2 averagefunction) ( D {Z(u) u D

29 1 2014 2 REMOTESENSING TECHNOLOGY AND APPLICATION Vol.29 No.1 Feb.2014 LiuFeng GuoJianwen.HeterogeneityAnalysisandItsCalculation MethodforStructuredGeographic Objects[J].RemoteSensingTechnologyandApplication201429(1)122-129.[. [J]. 201429(1)122-129.] doi10.11873/j.issn.1004-0323.2014.1.0122 刘丰 12 郭建文 1 (1. 中国科学院寒区旱区环境与工程研究所 甘肃兰州 730000; 2. 中国科学院研究生院 北京 100049) 将具有明显空间结构的非连续性地物数据的结构分析研究 归纳为空间数据的结构化异质性 分析问题在此概念的基础上引用曲线论的方法分析了结构化异质性的变异函数的构成 得出变 异距离受样本点间地物弧长及其空间复杂度共同影响的结论 由此推出变异距离函数的解析表达 公式之后给出了结构化异质性的实验变异函数的一种近似计算方法 该方法在样本合理分布的 前提下一致收敛于真实变异值最后介绍了结构化异质性模型在青藏铁路路基沉降数据的结构分 析计算中的一个应用实例 变异函数 ; 结构化异质性 ; 结构分析 ; 曲率 ; 三次样条插值 P208 A 1004-0323(2014)01-0122-08 ) [2] (1) LVA (LocalyVaryingAnisotropy Field) ; (VariogramorSemi- Chris [3] variogram) (Sil) LVA [1] (Nugget) (Range) Kriging ( ) [3] ( ) ( ) ( ) ( [2] Boisvert [2] LVA 1 2012-10-29; 2013-01-02 973 (2012CB026106) (1983-) GIS E-mailliufeng@lzb.ac.cn (1970-) GIS E-mailguojw@lzb.ac.cn

1 123 LVA 2.1 LVA D {Z(u) u D} Kriging γz(h) γz( h ) [1] h h h 2 (2) 1 γz(h) ; (Stream 2 h [4] Networks) 1 (Moving- 2.2 averagefunction) ( D {Z(u) u D} ) D 0 N(γZ(Δh) ;2 D 0 )= 1 Δh 0 d 0 ( 0π] D (Stream 0 +d 0 Distance) ;3 N(γZ(Δh)D)dD =0 D D 0 -d 0 ( ) Δh u (u+δh) 烄 N(γZ(Δh)D)= 1 γz(δh) D 烅烆 0γZ(Δh) D (3)Curriero [5] 0 isometricembedding 1 1 D u u Δh u (u+δh) D u U(u;δ)D r(t)= (x(t)y(t)z(t))t U(uδ) Δh 0 Δh u [2] r(t+δt)-r(t) r (t)=lim Δh Δt 0 Δt r (t)δt not-a- knot 1 Fig.1 Anexampleofstructuredheterogeneity

124 29 κ τ κi = r i (1) τi = ( r i r i r i) r i 2 ζ [8] [6-7] 4 ; ζ κ τ C 3 Δh 0 D κ τ r (t) Δh D ζ κ τ C [8] ; C Δh D u u+δh h( (2) D ζ κτδh) u χ =χ ( ζ κτδh) C h( ζ κτδh)=ζ ( 1+χ ) (3) ξ 0 γ(h)=γ[ ζ (1+χ )] χ ξ D 1 {Z 1 (u) u D 1 } D 2 {Z 2 (u) u D 2 } ξ = R Z1 Z 2 u 2 ) R Z1 u 2 ) ( RZ 1 Z u 2 1 u 2 )= E[Z 1 )Z 2 (u 2 )] Z 1 (u) Z 2 (u) R Z1 u 2 )= E[Z 1 )Z 1 (u 2 )] Z 1 (u) u 1 u 2 = Φ ξ 0 D1 D 1 D 2 θi c i c i-1 ζ D 1 D 2 D 1 i D 2 D 1 D 1 K i = κids c i -Δh 0 0 K -Δh =0 K C =K- K -Δh =K = Ki+ Lagrange θi K i = κi 珔 ζ i κi 珔 c i κimax ; 1 κimin Lagrange c i K i ζ i 3 K C K = 1 u u+δh D K/K s K C K s 烇烋 uu+δ h S S C = {c i c i C 3 0<i< } C ζ= ζ i ζ i c i c i r i =r i ( ζ ) c i C -Δh C -Δh K C K -Δh Fenchel (1) [9] ζ K = κds+ θi = 0 ζ i κids+ θi 0 = Ki + θi (1) r(t)= (acostasintbt)a = Δh 2 b= Δh 2 2ζ [ ] t 0 2 ζ (2) Δh

1 125 K S C Δh Z Δh S r XY (t)= 4 (acostasint0) r Z (t)= (00bt) ζ S =at= ζ ; ZΔh Z(u)=μ ( u)+v(u)+ε(u) (4) S Z Δh bt= Δh μ (u)=e(z(u)) D [8] κ ε(u) σ 2 ε V(u) 1 S 2 Δh K S = 1 2 Δh ζ ε(u) V(u) D V(u) Δh c i K i = K i /K si = 2 κi/δh 珔 R Z(uR)= Z(u) μ (u) ε(u) S Z(uR)=μ ( ur)+v(ur)+ε(ur) Lagrange =μ ( u)+γv( ωi ζ (1+χ )R)+ε(u) i i θ θ max(θ)=π θ =θ/π K = K i + θ i = 2 珔 κi Δh i + θi π ( 3) T T T C χ = K +T γ[ ζ (1+K γz(h) +T )] u u+δh D γz(δh)= 1 2 E {[Z(u)-Z(u+Δh)] 2 }= 1 2 E [(V(u)- V(u+Δh))+ (ε(u)-ε(u+δh))] 2 = h i =ζ (1+χ ) i i ζ i χ i 1 h i 2 E [V(u)-V(u+Δh)] 2 + 1 2 E [ε(u)- ωi ε(u+δh)] 2 =γv (Δh)+σ 2 ε (6) γ(h)=γ( ihi )=γ( i ζ i (1+χ i )) ωi χ i ζ i γv( ωi ζ i (1+χ i )R)=γv( ωi ζ i (1+χ i )) Z(uR)=μ ( u)+γv( ωi ζ i (1+χ i ))+ε(u)=z(u) (5) 5 ihi min (h i ) C τ=0 T T 0 γ[ ωi ζ (1 i u 0 = uu 1 +K i )] C u n-1 u n =u+δh C ζ i u i-1 u i ( c i ) S( ζ )

126 29 (9) (1.3.2) (1.1) S i ( ζ )=r i ( ζ ) ζ (0 ζ i); Δζ 0 Si ( ζ ) r i ( ζ ) Si ( ζ +σ) (1.2) S (k) i ( S ζ )=lim (k-1) i ( ζ +Δζ ) -S (k-1) i ( ζ ) = r i ( ζ +σ) Si ( ζ -σ)= r i ( ζ -σ) θi < Δζ 0 Δζ [8] θsi θi ζ ( ζ i-1 ζ i)k=12; (1.3) Δζ=max( ζ ) j 0 (1.3.1)S i ( ζ ) (1) Lagrange κsi 珋 κi;(1.3.2)s 珋 i-1 ( ζ ) S i ( ζ ) θsi θi (1.1) (1.2) (2) S i (x i yi) S(x) (2.1) S(x i )=yi; (2.2) S (k) S (x)=lim (k-1) i (x+δζ ) -S (k-1) i (x) < Δx 0 Δx x (x i-1 x i )k =12; ( θi) (3.1) S(x i )=yi; (3.2) S (k) (x i-0)=s (k) (x i+0)k=012; (3)(1.3) Δζ 0 S( ζ ) S(x) 3 K S K DeBoor not-a-knot [10-11] K S K Taylor S i ( ζ +σ)=s i ( ζ )+ S i ( ζ )σ+ σ2 2 S i ( ξ ) (7) r i ( ζ +σ)=r i ( ζ )+ r i ( ζ )σ+ σ2 2 ri ( ξ ) (8) 0<σ< Δζ σ=o(δζ ) ξ Lagrange Si ( ξ )= κsi r( 珔 ξ )= κi 珔 (7) ~ (8) 珋 κsi - 珋 κi = S( ξ )- r i ( ξ )= 2 σ 2 {[ S i ( ζ +σ)-r i ( ζ +σ)]+ [r i ( ζ )-S i ( ζ )]+ [ r i ( ζ )- S( ζ )]σ} [12] max S (k) i ( 0 ζ ζ ζ )-r (k) i ( ζ ) =O(Δζ 3-k )0 k 2(9) i (9) (1.3.1) 珋 κsi - 珋 κi 2 σ 2 [ S i ( ζ +σ)-r i ( ζ +σ) + S i ( ζ )- r i ( ζ ) + S i ( ζ )- r i ( ζ ) σ] 2 σ 2[ O(Δζ 3 )+O(Δζ 3 )+O(Δζ 2 )σ] = O(Δζ ) (3.3) S (x i-0)=s (x i+0)i=1n-1 (3.1) (3.2) (3.3) S( ζ ) S(x) S( ζ ) i (1.3) θi =0 l i = ζ i j=1 S i ( ζ ) r i ( ζ ) ζ Lagrange S i ( ζ ) S i (l) m i =S i (l i ) 珔 κsi = mi -mi-1 ζ i S i (l) m i m i (10) [1113] 烄 λ0 1 λ1 2 μ 1 O 烆 λ2 2 μ 2 λn-2 2 μ n-2 O λn-1 2 μ n-1 1 μ λ1 = ζi+1 ζ i + μ i = ζi ζ i+1 ζ i + ζ i+1 烌烄 烌烄 烌 m 1 b 1 (10) n 烎 m 0 m n-1 烆 m n 烎 b 0 b n-1 烆 b n 烎 ( ) S(l i )-S(l i-1 ) S(l i+1)-s(l i ) b i =3λi +μ ζ i i ζ i+ 1 i=12 n-1 λ0 = ζ2 ζ 1 + μ n = ζn-1 ζ 2 ζ n + ζ n-1

1 127 b 0 = ( S(l 1 )-S(l 0 ) S(l μ 1 +2)λ1 +μ 2 2 )-S(l 1 ) 1 ζ 1 ζ 2 S(l b n =λn-1 2 n-1 )-S(l n-2 ) + ζ n-1 S(l n )-S(l n-1 ) (λn-1 +2) μ n-1 ζ n m i S i (l) C γ(h * )= 1 2N(h * ) [ Z(u)-Z(u+h * )] 2 (11) h * = ζ [ ] i 1+2 mi -mi-1 m i (10) Δhζ i [11] ( 5 ) 6 ( ) (11) 1118km forqinghai-tibetanrailway 632km 3 50% 40% 35.47km 0.22 0.20 h * 35.47km [14-15] ; 0.5 ; h * 3 80 (HoleEfect) [16] (Drift) 33 5.51~50.43km 2 3 5 not-a-knot Fig.3 SemivariogramfortheEmbankment etlementdataofqinghai-tibetanrailway 2 Fig.2 Distribution MapofMonitoringSections

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1 129 HeterogeneityAnalysisandItsCalculation Method forstructuredgeographicobjects LiuFeng 1 2 GuoJianwen 1 (1.Cold and Arid RegionsEnvironmentaland Engineering ResearchInstitute Chinese Academyof SciencesLanzhou730000China; 2.Graduate Universityof Chinese Academyof SciencesBeijing100049China) AbstractThestructuralanalysisproblemofthenon-continuitygeographicobjectswithsignificantspatial structureshasn'tbeenadequatelyaddressed.becauseofthenon-linearitycharacteristicofstructuredgeo- graphicobjectstheclassicspatialinterpolation methodssuchasordinarykrigingwhichisbasedoneu- clideandistancecannotcompletethismission.tosolvethisissueaconceptofstructuredheterogeneityof thespatialdatawasputforward.afterintroducingthecurvetheoryofdiferentialgeometrythesemivario- gram modelofthestructuredheterogeneitywasanalyzedwhichindicatesthatthelagdependsonthearc lengthbetweensamplepointsandthecomplexityoftheresearchregion.inthenextsectiontheanalytic formulasofthelagwasdeduced.theabovestepswerefolowedbyanapproximationcalculation method forexperimentalsemivariogram ofstructuredheterogeneitywhichisabletouniformlyconvergetotrue semivariogramintheconditionofthereasonabledistributionofsamplepoints.finalyanapplicationusing theembankmentsetlementdataofqinghai-tibetanrailwaywasprovided. KeywordsSemivariogram;Structuredheterogeneity;Structuralanalysis;Curvature;Cubicsplineinterpola- tion