a new method of surface modeling and its application to ... · classical methods. the results show...

2007) 161–172www.elsevier.com/locate/geomorph

Geomorphology 91 (

A new method of surface modeling and its application toDEM construction

Tian-Xiang Yue a,b,⁎, Zhen-Ping Du a, Dun-Jiang Song a, Yun Gong c

a Institute of Geographical Sciences and Natural Resources Research, 100101 Beijing, Chinab Agroscope Reckenholz-Taenikon Research Station ART, 8046 Zurich, Switzerland

c Xi-An Science and Technology University, 710054 Xi-An, China

Received 24 June 2006; received in revised form 2 February 2007; accepted 11 February 2007Available online 16 February 2007

Abstract

A new method of surface modelling based on the fundamental theorem of surfaces (SMTS) is presented. Eight different testsurfaces are employed to comparatively analyze the simulation errors of SMTS and the classical methods of surface modeling in GIS,including TLI (triangulated irregular network with linear interpolation), SPLINE, IDW (inverse distance weighted) and KRIGING.Numerical tests show that SMTS is much more accurate than the classical methods. SMTS theoretically gives a solution to the errorproblem that has long troubled DEM construction. As a real-world example, SMTS is used to construct a DEM of the Da-Fo-Si coalmine in Shaan-Xi Province, China. Its root mean square error (RMSE) is compared with those of DEMs constructed by the fourclassical methods. The results show that although SMTS also has a higher accuracy in the real-world example, the improvement ofaccuracy is less distinct than that expected from the numerical tests. The accuracy loss seems to be caused by location differencesbetween sampling points and the central points of lattices of the simulated surfaces. Two alternative ways are proposed to solve thisproblem.© 2007 Elsevier B.V. All rights reserved.

Keywords: Surface modeling; Fundamental theorem of surfaces; Numerical test; Error analysis; Digital elevation model

1. Introduction

A digital elevation model (DEM) is a representationof terrain elevation as a function of geographiclocation. Discrete DEMs sampled on a regularlyspaced lattice, known as gridded DEMs, can be easilystored and manipulated in digital form. They provide

⁎ Corresponding author. Institute of Geographical Sciences andNatural Resources Research, 100101 Beijing, China. Tel.: +86 1064889633; fax: +86 10 64889630.

E-mail address: [email protected] (T.-X. Yue).

0169-555X/$ - see front matter © 2007 Elsevier B.V. All rights reserved.doi:10.1016/j.geomorph.2007.02.006

basic information required to characterize the topo-graphic attributes of terrain (Hutchinson and Gallant,2000; Pike, 2000). Any DEM, even a very high qualityone, is an approximation to a real-world continuoussurface (Carter, 1988). The sources of DEM errorsinclude the quality of source data, data-capturingequipments, the transformation method of controlpoints, the mathematical model for constructing thesurface, and grid resolution and orientation (Zhou andLiu, 2002; Zhu et al., 2005). Such DEM errors can bepropagated through simulation processes and affect thequality of final products (Huang and Lees, 2005;Oksanen and Sarjakoski, 2005). Although there are

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162 T.-X. Yue et al. / Geomorphology 91 (2007) 161–172

many types and sources of error and uncertainty ingeographical data and their processing, the problem isnot simply technical and may arise from an evidentinability of GIS (Unwin, 1995).

A large number of studies have focused on thesolutions of DEM errors. For instance, Goodchild(1982) introduced fractal Brownian process as a terrainsimulation model for improving DEM accuracy. Walshet al. (1987) found that the total error may beminimized by recognizing both inherent errors withininput products and operational errors created bycombinations of input data. Hutchinson and Dowling(1991) introduced a drainage enforcement algorithmthat tried to remove spurious pits in order to yield aDEM that would reflect the natural drainage structure.Unwin (1995) suggested that it would be obviouslyuseful if general tools for examining how errorspropagate through GIS operations were to be available,after he reviewed relevant work by Heuvelink et al.(1989) and Heuvelink and Burrough (1993); Wise(2000) argued that a clear distinction needs to be madebetween the lattice and pixel models when current GISis used; the value stored in a pixel relates to the wholeextent of the grid, while the value stored in a latticerelates solely to the central point of the grid. TheUnited States Geological Survey (1997) adoptedstatistical accuracy tests, data editing, verification oflogical and physical formats of data files, and visualinspection as the principle items of its DEM qualitycontrol system (Lo and Yeung, 2002). Florinsky (2002)suggested four alternative methods to reduce DEMerrors caused by the Gibbs phenomenon. Shi et al.(2005) investigated high-order interpolation algorithmsto reduce DEM errors. Podobnikar (2005) proposed toproduce high quality DEMs by using all available datasources, even lower quality datasets and datasets withouta height attribute. Chaplot et al. (2006) evaluated theperformance of various classic techniques for DEMgeneration. However, more studies are needed to attackthe error problem at the root.

To find a solution for the DEM error problem, anew method of surface modeling based on thefundamental theorem of surfaces (SMTS) is developed.The fundamental theorem of surfaces assures that asurface is uniquely defined by the first and secondfundamental coefficients of a surface (Somasundaram,2005). The performance of SMTS is analyzed bycomparing its absolute and relative errors with those ofthe widely used classical methods of surface modeling,including inverse distance weighted (IDW), triangu-lated irregular network with linear interpolation (TLI),KRIGING and SPLINE.

2. SMTS

A surface model including a DEM is the mathema-tical representation of a surface. Various techniques ofsurface modeling have been used for compiling DEMs(Etzelmuller, 2000; Schneider, 2005), and DEMs havebeen used for identifying and classifying landforms(Prima et al., 2006; Hayakawa and Oguchi, 2006),analyzing terrain characteristics (Gao, 1997; Ganaset al., 2005; Lin and Oguchi, 2006; Glenn et al.,2006), and studying geomorphic processes (Bolongaro-Crevenna et al., 2005; Mitasova et al., 2005). Therefore,establishing a basicmethod of surfacemodeling is highlyimportant.

According to the fundamental theorem of surfaces, asurface is uniquely defined by the first and the secondfundamental coefficients (Henderson, 1998). If a surfaceis a graph of a function z= f(x,y) or r=(x,y,f(x,y)), thefirst fundamental coefficients, E, F and G, can beformulated as,

E ¼ 1þ f 2x ð1ÞF ¼ fxfy ð2ÞG ¼ 1þ f 2y : ð3Þ

The second fundamental coefficients, L, M and N,can be formulated as,

L ¼ fxxffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1þ f 2x þ f 2y

q ð4Þ

M ¼ fxyffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1þ f 2x þ f 2y

q ð5Þ

N ¼ fyyffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1þ f 2x þ f 2y

q : ð6Þ

The first and second fundamental coefficients satisfythe Gauss–Codazzi equations (Somasundaram, 2005).The Gauss equation can be formulated as,

fxx ¼ G111fx þ G2

11fy þ L EG−F2� �−1

2 ð7Þ

fxy ¼ G112fx þ G2

12fy þM EG−F2� �−1

2 ð8Þ

fyy ¼ G122fx þ G2

22fy þ N EG−F2� �−1

2: ð9Þ

The Codazzi equation can be formulated as,

BLBy

−BMBx

¼ L G112 þM G2

12−G111

� �−N G2

11 ð10Þ

163T.-X. Yue et al. / Geomorphology 91 (2007) 161–172

BMBy

−BNBx

¼ L G112 þM G2

22−G112

� �−N G2

12 ð11Þ

where

G111 ¼

12

GEx−2FFx þ FEy

� �EG−F2� �−1 ð12Þ

G112 ¼

12

GEy−FGx

� �EG−F2� �−1 ð13Þ

G122 ¼

12

2GFy−GGx−FGy

� �EG−F2� �−1 ð14Þ

G211 ¼

12

2EFx−EEy−FEx

� �EG−F2� �−1 ð15Þ

G212 ¼

12

EGx−FEy

� �EG−F2� �−1 ð16Þ

G222 ¼

12

EGy−2FFy þ FGx

� �EG−F2� �−1

: ð17Þ

Our previous studies (Yue et al., 2004; Yue and Du,2005, 2006) show that seven different surface modelscould be developed using different combinations of Eqs.(7)–(11). Errors and computing times of these surfacemodels differ from each other, amongwhich the one withthe least error and less computing time (i.e. SMTS) couldbe formulated as (Yue and Du, 2006),

fxx ¼ G111fx þ G2

11fy þ L EG−F2� �−1

2 ð18Þ

fyy ¼ G122fx þ G2

22fy þ N EG−F2� �−1

2: ð19Þ

If 1) {(xi, yj)} is a network created by orthogonaldivision of a computational domain Ω, 2) hxi=xi − xi − 1

and hyj=yj − yj − 1 are respectively grid spacing at a gridcell (xi,yj) in directions x and y, 3) f̃ i,j is interpolationapproximate value at the same grid cell in terms ofsampling data (xi,yj, f

−ij) ∈ Φ, and 4) fi,j

0 = f̃i,j, then thefinite difference of the first fundamental coefficients canbe formulated as,

Ei;j ¼ 1þ f̃ iþ1;j− f̃ i−1;j2hxi

� �2

ð20Þ

Fi;j ¼ f̃ iþ1;j− f̃ i−1;j2hxi

� �f̃ i;jþ1− f̃ i;j−1

2hyj

� �ð21Þ

Gi;j ¼ 1þ f̃ i;jþ1− f̃ i;j−12hyj

� �2

: ð22Þ

The finite difference of the second fundamentalcoefficients can be formulated as,

Li; j ¼f̃ iþ1; j−2f̃ i; jþf̃ i−1; j

h2xiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1þ

�f̃ iþ1; j−f̃ i−1; j

2hxi

�2

þ�

f̃ i; jþ1 − f̃ i; j−12hyj

�2s ð23Þ

Mi; j ¼f̃ iþ1; jþ1−f̃ iþ1; j−1

4hxi hyj

� �− f̃ i−1; jþ1 − f̃ i−1; j−1

4hxi hyj

� �ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1þ f̃ iþ1; j−f̃ i−1; j

2hxi

� �2þ f̃ i; jþ1−f̃ i; j−1

2hyj

� �2r ð24Þ

Ni;j ¼f̃ i;jþ1−2 f̃ i;jþ f̃ i;j−1

h2yjffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1þ f̃ iþ1;j−f̃ i−1;j

2hxi

� �2þ f̃ i; jþ1−f̃ i;j−1

2hyj

� �2r

:

ð25Þ

The finite difference of Γkl (k=11, 12, 22; l=1, 2) can

be formulated as,

G112

� �i;j¼

Gi;j Ei;jþ1−Ei;j−1� �

hxi−Fi;j Giþ1;j−Gi−1;j� �

hyj

4 Ei;jGi;j−F2i;j

� �hxihyj

ð26Þ

G212

� �i;j¼

Ei;j Giþ1;j−Gi−1;j� �

hyj−Fi;j Ei;jþ1−Ei;j−1� �

hxi

4 Ei;jGi;j−F2i;j

� �hxihyj

ð27Þ

G111

� �i;j¼

Gi;j Eiþ1;j−Ei−1;j� �

hyj−2Fi;j Fiþ1;j−Fi−1;j� �

hyj þ Fi;j Ei;jþ1−Ei;j−1� �

hxi

4 Ei;jGi;j−F2i;j

� �hxi hyj

ð28Þ

G122

� �i;j¼

2Gi;j Fi;jþ1−Fi;j−1� �

hxi−Gi;j Giþ1;j−Gi−1;j� �

hyj−Fi;j Gi;jþ1−Gi;j−1� �

hxi

4 Ei;jGi;j−F2i;j

� �hxi hyj

ð29Þ

G211

� �i;j¼

2Ei;j Fiþ1;j−Fi−1;j� �

hyj−Ei;j Ei;jþ1−Ei;j−1� �

hxi−Fi;j Ei;jþ1−Ei;j−1� �

hxi

4 Ei;jGi;j−F2i;j

� �hxi hyj

ð30Þ

G222

� �i;j¼

Ei;j Gi;jþ1−Gi;j−1� �

hxi−2Fi;j Fi;jþ1 þ Fi;j−1� �

hxi þ Fi;j Giþ1;j−Gi−1;j� �

hyj

4 Ei;jGi;j−F2i;j

� �hxi hyj

:

ð31Þ


The finite difference of the SMTS can be formulatedas,fiþ1;j−fi;jhxiþ1

− fi;j−fi−1;jhxi

0:5 hxi þ hxiþ1

� � ¼ G111

� �i;j

fiþ1;j−fi−1;jhxi þ hxiþ1

þ G211

� �i;j

fi;jþ1−fi;j−1hyj þ hyjþ1

þ Li;jffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiEi;j þ Gi;j−1

p ð32Þ

fi;jþ1−fi; jhyiþ1

− fi; j−fi; j−1hyi

0:5 hyj þ hyjþ1

� � ¼ G122

� �i; j

fiþ1; j−fi−1; jhxi þ hxiþ1

þ G222

� �i; j

fi; jþ1−fi; j−1hyj þ hyjþ1

þ Ni; jffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiEi; j þ Gi; j−1

p:

ð33Þ

Based on the relevant method of numerical mathe-matics (Quarteroni et al., 2000), the iterative formula-tion of SMTS could be expressed as,

f nþ1iþ1;j−f

nþ1i;j

hxiþ1−

f nþ1i;j −f nþ1

i−1;jhxi

0:5 hxi þ hxiþ1

� � ¼ G111

� �ni;j

f niþ1;j−f ni−1;jhxi þ hxiþ1

þ G211

� �ni;j

f ni;jþ1−fni;j−1

hyj þ hyjþ1

þ Lni;jffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiEni;j þ Gn

i;j−1p ð34Þ

f nþ1i;jþ1−f

nþ1i;j

hyjþ1−

f nþ1i;j −f nþ1

i;j−1hyj

0:5 hyj þ hyjþ1

� � ¼ G122

� �ni;j

f niþ1;j−fni−1;j

hxi þ hxiþ1

þ G222

� �ni;j

f ni;jþ1−fni;j−1

hyj þ hyjþ1

þ Nni;jffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

Eni;j þ Gn

i;j−1p ð35Þ

where

Eni;j ¼ 1þ f niþ1;j−f

ni−1;j

hxi þ hxiþ1

� �2

ð36Þ

Fni;j ¼

f niþ1;j−fni−1;j

hxi þ hxiþ1

� �f ni;jþ1−f

ni;j−1

hyj þ hyjþ1

� �ð37Þ

Gni;j ¼ 1þ f ni;jþ1−f

ni;j−1

hyj þ hyjþ1

� �2

ð38Þ

Lni;j ¼f niþ1;j

hxiþ1− 1

hxiþ1þ 1

hxi

� �f ni;j þ

f ni−1;jhxi

hxiþhxiþ1

2

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1þ f niþ1;j−f

ni−1;j

hxiþhxiþ1

� �2þ f ni;jþ1−f

ni;j−1

hyjþhyjþ1

� �2r ð39Þ

Nni;j ¼

f ni;jþ1

hyjþ1− 1

hyjþ1þ 1

hyj

� �f ni;j þ

f ni;j−1hyj

hyjþhyjþ1

2

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1þ f niþ1;j−f

ni−1;j

hxiþhxiþ1

� �2þ f ni;jþ1−f

ni;j−1

hyiþhyjþ1

� �2r ð40Þ

G111

� �ni; j¼

Gni; j En

iþ1; j−Eni−1; j

� �−2Fn

i; j Fniþ1; j−F

ni−1; j

� �þ Fn

i; j Eni; jþ1−E

ni; j−1

� �hxi þhxiþ1ð Þhyj þhyjþ1

� �Eni; jG

ni; j− Fn

i; j

� �2� �

hxj þ hxiþ1

� � ð41Þ

G122

� �ni;j¼

2Gni;j Fn

i;jþ1−Fni;j−1

� �−Gn

i;j Gniþ1;j−Gn

i−1;j

� �hyj þhyjþ1

� �hxi þhxiþ1ð Þ −F

ni;j Gn

i;jþ1−Gni;j−1

� �Eni;jG

ni;j− Fn

i;j

� �2� �

hyj þ hyjþ1

� � ð42Þ

G211

� �ni;j¼

2Eni;j Fn

iþ1;j−Fni−1;j

� �hyj þhyjþ1

� �hxi þhxiþ1ð Þ −E

ni;j En

i;jþ1−Eni;j−1

� �−Fn

i;j Eni;jþ1−En

i;j−1

� �Eni;jG

ni;j− Fn

i;j

� �2� �

hyj þ hyjþ1

� � ð43Þ

G222

� �ni;j¼

Eni;j Gn

i;jþ1−Gni;j−1

� �−2Fn

i;j Fni;jþ1−F

ni;j−1

� �þ Fn

i;j Gniþ1;j−G

ni−1;j

� �hyj þhyjþ1

� �hxi þhxiþ1ð Þ

Eni;jG

ni;j− Fn

i;j

� �2� �

hyj þ hyjþ1

� � : ð44Þ

3. Numerical tests

3.1. SMTS simulation process

The SMTS uses existing data, as points or contours,to globally fit a surface through several iterativesimulation steps. This surface is then used to interpolatea height value at an unknown point. The unknownpoints are located on a regularly spaced lattice and theend result is a gridded digital terrain model. The iterativesimulation steps are summarized as follows: 1) normal-izing the computational domain Ω and the normalizedcomputational domain Ω ⊂ [0,1]× [0,1]; 2) performinginterpolation on the normalized computational domainin terms of sampling data (xi, yj, f

_i,j), from which we can

get interpolated approximate values {f̃ i,j} at point (xi,yj);3) letting fi,j

0= f̃ i,j and calculating the first fundamentalcoefficients Eij

n, Fijn and Gij

n and the second fundamentalcoefficients Lij

n and Nijn as well as the coefficients of the

SMTS equations in terms of {fi,jn}; 4) for n≥ 0, we can get

{fi,jn+1} by solving the SMTS equations; and 5) the iterative

process is repeated until simulation accuracy is satisfied.

3.2. Comparative analysis of simulation errors

In addition to the newly developed SMTS, there aremany other methods of surface modelling, which arewidely used in various GIS applications (Isaaks andSrivastava, 1989; Cressie, 1993), such as TLI, IDW,KRIGING and SPLINE. These four classical surface

Fig. 1. The eight test surfaces used for model evaluation. (a) f1(x,y),(b) f2(x,y), (c) f3(x,y), (d) f4(x,y), (e) f5(x,y), (f) f6(x,y), (g) f7(x,y),(h) f8(x,y).


models were employed to comparatively analyze SMTSerrors. All the classical interpolation methods wereperformed using the module of 3D Analyst in ARCGIS9.0, and the default parameters of the software wereemployed. For IDW, the power is 2, the search radius isvariable, and the maximum number of the searched points is12. For SPLINE, the REGULARIZED option is used, theweight is 0.1, and the number of the searchedpoints is 12. ForKRIGING, ordinary method is selected, the model of

Table 1RMSE of simulation methods under uniform sampling

Test surface SMTS SPLINE

f1(x,y) 1.2463×10−3 3.9862×10−3

f2(x,y) 2.3406×10−4 1.9878×10−3

f3(x,y) 4.3798×10−5 6.5524×10−4

f4(x,y) 5.8282×10−5 4.3721×10−4

f5(x,y) 2.8501×10−5 1.0962×10−4

f6(x,y) 1.8963×10−5 1.0445×10−4

f7(x,y) 8.6858×10−6 1.5837×10−4

f8(x,y) 5.7448×10−6 6.5652×10−5

semivariogram is spherical, the search radius is variable,and the maximum number of the search points is 12.

Eight mathematic surfaces were taken as the test sur-faces so that the ‘true’ value can be pre-determined toavoid uncertainty caused by uncontrollable data errors.Weselected mathematical surfaces which represent differenttypes of shapes (Fig. 1) with the following formulas,

f1 x; yð Þ ¼ cos 10yð Þ þ sin 10 x−yð Þf g ð45Þ

f2 x; yð Þ ¼ e − 5−10xð Þ2=2f g þ 0:75e − 5−10yð Þ2=2f gþ 0:75e − 5−10xð Þ2=2f g þ e − 5−10yð Þ2=2f g ð46Þ

f3 x; yð Þ ¼ sin 2kyð Þd sin kxð Þ ð47Þ

f4 x; yð Þ ¼ 0:75e − 9x−2ð Þ2þ 9y−2ð Þ2f g=4½ �þ 0:75e − 9xþ1ð Þ2=49− 9yþ1ð Þ=10f gþ0:5e − 9x−7ð Þ2þ 9y−3ð Þ2f g=4½ �−0:2e − 9x−4ð Þ2− 9y−7ð Þ2f g

ð48Þ

f5 x; yð Þ ¼ 19

tanh 9y−9xð Þ þ 1f g ð49Þ

f6 x; yð Þ ¼ 1:25þ cos 5:4yð Þ6 1þ 3x−1ð Þ2n o ð50Þ

f7 x; yð Þ ¼ 13e− 81=16ð Þ x−0:5ð Þ2þ y−0:5ð Þ2f g ð51Þ

f8 x; yð Þ ¼ 13e− 81=4ð Þ x−0:5ð Þ2þ y−0:5ð Þ2f g: ð52Þ

In the normalized computational domain, 14,641(121×121) lattices were created by an orthogonaldivision of the computational domain. Therefore, in

KRIGING TLI IDW

1.4295×10−1 2.8509×10−2 1.4030×10−1

4.7567×10−3 9.0201×10−3 4.6162×10−2

2.4732×10−3 4.1618×10−3 3.9796×10−3

1.2908×10−3 2.4557×10−3 1.3953×10−2

5.0507×10−4 9.9612×10−4 4.8667×10−3

2.5823×10−4 4.9668×10−4 3.9796×10−3

4.2607×10−4 8.7978×10−4 4.6944×10−3

2.3417×10−4 4.0412×10−4 3.7144×10−3

Table 2Ratio of RMSE of a classical method to that of SMTS under uniformsampling

Test surface SPLINE KRIGING TLI IDW Average

f1(x,y) 3.20 115 22.9 113 63.3f2(x,y) 8.49 20.3 38.5 197 66.1f3(x,y) 15 56.5 95 90.9 64.3f4(x,y) 7.5 22.1 42.1 239 77.8f5(x,y) 3.85 17.7 35 171 56.8f6(x,y) 5.51 13.6 26.2 210 63.8f7(x,y) 18.2 49.1 101 540 177f8(x,y) 11.4 40.8 70.3 647 192On average 9.15 41.8 53.9 276

Table 4Ratio of RMSE of a classical method to that of SMTS under randomsampling

Test surface SPLINE KRIGING TLI IDW Average

f1(x,y) 6.21 146 34.4 1.34 47.1f2(x,y) 7.91 39.7 47.4 154 62.3f3(x,y) 5.67 35.8 36.1 233 77.7f4(x,y) 9.56 54.9 63.1 305 108f5(x,y) 7.48 31.2 36.1 152 56.8f6(x,y) 7.99 50 47.3 325 108f7(x,y) 20.1 108 122 574 206f8(x,y) 10.1 90.4 72.1 621 198On an average 9.37 69.6 57.4 296


simulation processes using SMTS and the classicalinterpolation methods, the grid spacing (or simulationstep length) was set as hxi ¼ hyj ¼ h ¼ 1

120 : Twosampling methods, uniform sampling and randomsampling, were adopted to comparatively analyze errorsof SMTS and the classical interpolation methods. Forthe uniform sampling, 5h was selected as the samplinginterval to have 625 sampling points. For the randomsampling, 1296 points were randomly sampled. The rootmean-square error (RMSE) in this case is expressed as,

RMSE ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

1121d 121

X121i¼1

X121j¼1

fij−Sfi;j� �2

:

vuut ð53Þ

where Sfi,j is the numerical solution of f(x,y) at point (xi,yj); fij is true value of f at the lattice (i,j).

Under uniform sampling (Tables 1 and 2), theaccuracy of SMTS compared with the other methodsis the lowest for the test surface f5(x,y) on average.However, the SMTS accuracy is still 3.85, 17.7, 35 and171 times as high as the accuracy of SPLINE,KRIGING, TLI and IDW respectively. Test surface f8(x,y) gives SMTS the highest accuracy, which is 11.4,40.8, 70.3 and 647 times as high as that of SPLINE,KRIGING, TLI and IDW, respectively. On average, theaccuracy of SMTS simulation is 9.15 times as high as

Table 3RMSE of simulation methods under random sampling

Test surface SMTS SPLINE

f1(x,y) 7.7636×10−4 4.8223×10−3

f2(x,y) 2.0898×10−4 1.6529×10−3

f3(x,y) 1.0532×10−4 5.9686×10−4

f4(x,y) 4.0324×10−5 3.8536×10−4

f5(x,y) 2.8466×10−5 2.1296×10−4

f6(x,y) 1.0014×10−5 8.0028×10−5

f7(x,y) 7.0024×10−6 1.4075×10−4

f8(x,y) 5.2814×10−6 5.3117×10−5

that of SPLINE, 41.8 times of KRIGING, 53.9 times ofTLI, and 276 times of IDW.

Under random sampling (Tables 3 and 4), SMTSsimulation accuracies of the test surfaces f1(x,y), f2(x,y),f3(x,y), f4(x,y), f5(x,y), f6(x,y), f7(x,y) and f8(x,y) arerespectively 47.1, 62.3, 77.7, 108, 56.8, 108, 206, and198 times as much as the accuracies of the classicalinterpolation methods on average. The same method hasdifferent levels of power to simulate different testsurfaces. For instance, when IDW is used to simulate thesurface f8(x,y) under the random sampling, its RMSE is621 times as much as that of SMTS, but RMSE of IDWis only 1.34 times as much as that of SMTS for f1(x,y).

The sampling methods greatly affect the accuracy ofdifferent interpolation methods simulating the particulartest surface. For example, RMSE of IDW simulating thetest surface f1(x,y) is only 1.34 times as much as that ofSMTS under the random sampling, but 113 times underthe uniform sampling. For the simulation of the testsurface f3(x,y), the accuracy of SMTS under the randomsampling is 5.67, 35.8, 36.1 and 233 times as high as thatof SPLINE, KRIGING, TLI and IDW, respectively, but itis 15, 56.5, 95 and 90.9 times under the uniform sampling.

Many error assessments so far simply reported asingle number to express the accuracy without regard forthe location. If all errors balance out, non-site specificaccuracy assessments give misleading results (Lunetta

KRIGING TLI IDW

1.1371×10−1 2.6671×10−2 1.0430×10−3

8.3032×10−3 9.8964×10−3 3.2262×10−2

3.7706×10−3 3.7984×10−3 2.4583×10−2

2.2148×10−3 2.5441×10−3 1.2291×10−2

8.8910×10−4 1.0277×10−3 4.3340×10−3

5.0024×10−4 4.7327×10−4 3.2566×10−3

7.5926×10−4 8.5749×10−4 4.0161×10−3

4.7720×10−4 3.8095×10−4 3.2805×10−3

Fig. 2. Error distribution contours of simulations applied to f1(x,y). (a)SMTS, (b) SPLINE, (c) TLI, (d) KRIGING, (e) IDW, (f) shows thecontour chart of f1(x,y).

Fig. 3. Error distribution contours of simulations applied to f2(x,y). (a)SMTS, (b) SPLINE, (c) TLI, (d) KRIGING, (e) IDW, (f) showscontour chart of f2(x,y).

Fig. 4. Error distribution contours of simulations applied to f3(x,y). (a)SMTS, (b) SPLINE, (c) TLI, (d) KRIGING, (e) IDW, (f) showscontour chart of f3(x,y).

Fig. 5. Error distribution contours of simulations applied to f4(x,y). (a)SMTS, (b) SPLINE, (c) TLI, (d) KRIGING, (e) IDW, (f) contour chartof f4(x,y).


Fig. 10. Location and topography of the Da-Fo-Si coal mine.


et al., 1991). Therefore, we introduce an error matrix forevery method of surface modeling for each test surface,{Eij}. The error Eij at the lattice point (or grid cell) (i,j) isformulated as,

Eij ¼ Sfij−fij ð54Þwhere fij is true value of f and Sfij is simulated value of fat the lattice point (i,j).

Fig. 11. Location of height sampling points of the Da-Fo-Si coal mine.a) Points for DEM construction, b) points for inspecting DEMaccuracy.

Figs. 2–9 indicate that the errors of SMTS are biggeralong the boundaries of the resultant surfaces or in theareas near the peaks. If the test surfaces are symmetric,

Fig. 12. 3D shaded relief maps of the Da-Fo-Si coal mine, from DEMsconstructed by a) SMTS, b) KRIGING, c) TLI, d) SPLINE, and e)IDW. View from west. Light from NW, with a dip angle of 45°.


the error distributions of SMTS, IDW, TLI andKRIGING simulations are also symmetric, but theerror distribution of SPLINE is asymmetric and tends tobe more uniform.

4. A real-world example

Five sets of DEMs for the Da-Fo-Si coal mine inShaan-Xi Province, China, were constructed using SMTS,KRIGING, TLI, SPLINE and IDW, and their errors werecomparatively analyzed. The coalmine (Fig. 10) is locatedatN35°05′ andE108°00′ in northwesternXian-YangCity,

Fig. 13. Comparisons between simulation results and observed dat

and its area is 1.4 km2. A great topographical variety of thearea is suitable for a study of DEM accuracy.

TwelveGPS (Global PositioningSystem) control pointswere allocatedwithin the coalmine. The Leica TC403 totalstation was used to collect 3856 elevation points, of which771 points (Fig. 11a) were randomly taken to constructDEMs, and the other 3085 points (Fig. 11b) were used toinspect the accuracy of the constructed DEMs.

DEMs of the coal mine with a spatial resolution of 1 mwere constructed by the five methods. 3D shaded reliefmaps of the DEMswere used to represent the terrain reliefof the coal mine (Fig. 12). The results show that the

a. a) SMTS, b) TLI, c) KRIGING, d) SPLINE, and e) IDW.


KRIGING simulation makes the DEM surface fragmental(Fig. 12b). The surface of TLI simulation within eachtriangle is a plane passing through the three vertices(Hugentobler et al., 2005), causing the so-called peak-truncation and pit-fill problem (Fig. 12c). The SPLINEsimulation has a higher oscillation on the boundary of theDEM (Fig. 12d), and the IDW simulation produces many“Bull's eyes” on the DEM surface (Fig. 12e). In contrast,the SMTS simulation provided a better result (Fig. 12a).

Fig. 13 shows the relationship between the heightfrom the simulated DEM and the corresponding height ofthe observed point altitude. This analysis also indicatesthat SMTS simulation has the highest accuracy. TLI,KRIGING and IDW tend to make lower elevation higherand higher elevation lower. Many SPLINE simulationpoints are relatively far from the straight line of y=x. Theresults of accuracy assessment show that the meanabsolute error of SMTS is 6.754 m, while the errors ofKRIGING, TLI, SPLINE and IDW are 7.523, 7.697,7.919 and 11.529 m, respectively.

5. Discussion and conclusions

Both the numerical tests and the real-world exampledemonstrate that the accuracy of SMTS is higher thanthat of the classical methods. In the numerical tests,various test surfaces, including symmetric, asymmetric,single-peak, multi-peak, gently changing and sharplychanging ones, were simulated by SMTS and theclassical methods. The comparative error analysesindicate that SMTS errors are much smaller than theones of the classical methods.

In the real-world example, although the accuracy ofSMTS was higher than the other four methods, thedifference was less distinct than the case of thenumerical tests. In other words, SMTS has accuracyloss in the practical application, which may have causedby location differences between the sampling points andthe corresponding central points of lattices of thesimulated surfaces. In the real-world example, most ofthe sampling points are not located at the centers of theircorresponding lattices, and the nearest neighbor methodwas used to accommodate the location differences. Thisproblem could be resolved by two alternative ways: 1)height sampling at the centers of the lattices through acareful design of investigation, and 2) establishing acontinuous Taylor expansion of a DEM, z= f(x,y), on thebasis of topographical characteristics.

SMTShas a huge computation cost because itmust usean equation set for simulating each lattice of a surface. If asurface consists of 2.25 million grid cells and SMTS isoperated on a typical personal computer available today,

the simulation process needs about 10 h, and thecomputation time is approximately proportional to thethird power of the total number of grid cells. A futureresearch focus is to shorten computing time, by applyingimproved SMTS equations, the domain decompositionmethod (Toselli and Widlund, 2005), and parallelcomputing technology (Petersen and Arbenz, 2004).

Acknowledgments

We would like to express our gratitude to Dr. TakashiOguchi, Dr. Balazs Szekely and an anonymous reviewerfor their useful comments. This work is supported byMajor Directivity Projects of Chinese Academy ofScience (KZCX2-YW-429), by the National High-TechR&DProgram of theMinistry of Science and Technologyof the People's Republic of China (2006AA12Z219), bythe National Key Technologies R&D Program of theMinistry of Science and Technology of the People'sRepublic of China (2006BAC08B04), and by the Sino–Swiss Cooperation Program.

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