Math Geosci (2014) 46:445–481DOI 10.1007/s11004-013-9508-8

Accuracy Analysis of Digital Elevation Model Relatingto Spatial Resolution and Terrain Slope by BilinearInterpolation

Wenzhong Shi · Bin Wang · Yan Tian

Received: 28 March 2013 / Accepted: 7 November 2013 / Published online: 23 January 2014© International Association for Mathematical Geosciences 2014

Abstract This study investigates the accuracy analysis of the digital elevation model(DEM) with respect to the following two major factors that strongly affect the interpo-lated accuracy: (1) spatial resolution of a DEM and (2) terrain slope. Unlike existingstudies based mainly on a simulation approach, this research first provides an analyti-cal approach in order to build the relationship between the interpolated DEM accuracyand its influencing factors. The bi-linear interpolation model was adopted to producethis analytic model formalized as inequalities. Then, our analytic models were verifiedand further rectified by means of experimental studies in order to derive a practicalformula for estimating the DEM accuracy together with an optimization model for cal-culating the required resolution when a prescribed upper bound to the DEM accuracyis given. Moreover, this analytic approach can cope with either a grid-based DEM or arandomly scattered scenario whose efficacies have been validated by the experimentsusing both synthetic and realistic data sets. In particular, these findings first establishthe rules for directly correlating the horizontal resolution of DEM data with verticalaccuracy.

Keywords DEM · Analytic inequality · Accuracy assessment · Terrain complexity ·Bilinear interpolation method

W. Shi (B) · B. Wang (B)Department of Land Surveying and Geo-Informatics, The Hong Kong Polytechnic University,Hung Hom, Kowloon, Hong Konge-mail: john.wz.shi@polyu.edu.hk

B. Wange-mail: wbcumt@163.com

Y. TianDepartment of Electronic and Information Engineering,Huazhong University of Science and Technology, Wuhan, Chinae-mail: tianyan2000@126.com

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1 Introduction

The digital elevation model (DEM) has been widely used in many applications, includ-ing digital terrain representation and geo-spatial analysis. A DEM is also used assource data to derive terrain features such as slopes, aspects, profiles and curvatures.The accuracy of a DEM is especially critical for situations where engineering accu-racy is essential, such as hydraulic engineering applications. DEM accuracy can beaffected by many factors, such as the interpolation model, the quality of the sourcedata (acquisitions error, native scale, data type, etc.), the resolution (grid size) of theDEM, and degree of terrain complexity. Terrain feature is one of the prominent fac-tors affecting soil erosion, and slope information is subsequently one of the importantparameters for predicting soil erosion (Armstrong and Martz 2003; Wu et al. 2013).The scientific and effective representation of the relationship between terrain featureand hydrological landscape ingredients thus becomes the foundation of hydrologyand soil erosion research. According to the geomorphologic principle, Hutchinson putforward a hydrologically correct DEM algorithm (Hutchinson 1988; Hutchinson andGallant 1999, 2000) and used corresponding software ANUDEM for characterizingthe hydrological landscape after exploring various interpolation methods and the datastructure. Although many techniques are available to produce DEMs (Moore et al.1991; Hutchinson 2011), ANUDEM has become one of the most well-known, reli-able and computationally efficient tools for generating hydrologically sound DEMsfrom various data sources, including contour lines, spot heights and stream lines.(Yang et al. 2007) performed a study in the Coarse Sandy Hilly Catchments of theLoess Plateau, China, which demonstrated the effectiveness of a hydrologically cor-rect digital elevation model algorithm for improving the DEM quality by identifyingand correcting source topographic data errors and optimizing ANUDEM algorithmparameters. For this reason, ANUDEM has already been integrated into ArcGIS asone functional interpolation module invoked by Topogrid or Topo to Raster.

A DEM surface is normally generated by applying interpolation models (bilinear,bicubic, B-spline, Kringing, etc.) based on a given set of node elevations. Severaleffective interpolation approaches have been reported (Pumar 1996; Caselles et al.1998; Almansa et al. 2002; Shi and Tian 2006; Maune 2007). Of these, the bilinearinterpolation method (Maune 2007) is the simplest, while other higher order inter-polations (for example the Hermite method) may require more terrain informationthan just elevations, such as slope. Comprehensive studies have been conducted onevaluating DEM interpolation methods. Rees (2000) showed that simple bi-linear orbi-cubic convection was an adequate approach for DEM interpolation according tosome case studies. However, based on different case studies, Kidner (2003) arguedthat high-order interpolation techniques were more accurate than the bilinear algo-rithm. Florinsky (2002) presented a new interpretation of the following three classesof digital terrain model (DTM) errors: (a) errors in interpolation of DEMs causedby the Gibbs phenomenon, (b) errors in DTM derivation from DEMs with enhancedresolution due to noise increase following DEM differentiation and (c) errors in DTMderivation caused by displacement of a DEM grid. Several preventative measures forhandling these DEM errors have been recommended. Sinha and Schunck (1992) pro-posed a two-stage algorithm for discontinuity-preserving surface reconstruction using

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a weighted bicubic spline as a surface descriptor. By using zonal analysis, Shan etal. (2003) showed that the standard deviation of the 1-degree DEM can be largelyapproximated by a linear function of the slope change. Based on Grid-based Con-tours, Gousie and Franklin (2005) improved thin plate DEM approximations by theaddition of tension into the biharmonic interpolation operator to counter the effectsof Gibbs phenomena. The contour-based approach has also been employed for clodsidentification and characterization on a soil surface (Taconet et al. 2010).

Because terrain factors are closely related to the applied terrain analyses and theaccuracy of these factors is closely related to the accuracy of the original DEM,research on the relationships between DEM accuracy and terrain factors is very neces-sary. Extensive research on this subject has been reported in the literature (Holmes etal. 2000; Zhou and Liu 2004; Raaflaub and Collins 2006). In addition, the accuracy ofa derived terrain slope is dependent on the accuracy of the original DEM, with largererrors associated with the areas of steepest slope (Toutin 2002; Wu et al. 2012). Thetendency for slope accuracy to vary with grid size has also been reported (Chang andTsai 1991; Gao 1997; Tang et al. 2001). Shi and Tian (2006) combined the bilinear andbi-cubic methods and proposed a hybrid interpolation method for DEM generation.His experimental results demonstrate that the hybrid method is effective for interpo-lating DEMs for various types of terrain. In those studies, it was assumed that theoriginal data sources from which a DEM was generated were error-free and that themain error source affecting the accuracy of the interpolated DEM was the interpolationmodel error itself. Li and Zhu (2003) derived the mean error for a regular grid DEMinterpolated by bilinear interpolation methods. Shi and Tian (2005) provided a modelfor estimating DEM propagation errors resulting from high-order interpolation algo-rithms, with the simplification that the node derivatives are replaced approximatelyby first order differentials. Zhu et al. (2005) derived the average DEM accuracy of thetriangulated irregular network (TIN) model interpolated by linear methods with theassumption that the errors associated with the given nodes are independent of eachother. (Kyriakidis and Goodchild 2006) further derived average line accuracy, the TINmodel and a rectangle model, using bilinear interpolation methods where the errorsat given nodes are supposed to be independent. In a practical study, Yamazaki et al.(2012) developed a new algorithm for adjusting a space-borne DEM in floodplainhydrodynamic modeling using drainage network information. A robust multiquadricmethod (Chen and Li 2013) was put forward to reduce the impact of outliers on theaccuracy of DEM construction. In addition, the support vector machine (SVM) clas-sifier is adopted for the adaptive selection of appropriate terrain modeling methodsbased on the terrain complexity (Jia et al. 2013).

Generally, methods for characterizing those factors (for example, resolution andterrain slope) which affect DEM accuracy can be classified into two categories. Thefirst is based on simulation while the second is analytical. Many studies have beenperformed using the simulation approach. Focusing on the TK-350 stereo-scenes ofthe Zonguldak test field in northwestern Turkey, Büyüksalih et al. (2004) showed thatDEM accuracy depends mainly on the surface structure and slope of the local terrain.By analyzing different levels of terrain detail, Tang et al. (2001) built a linear model forDEM error estimation, DEM resolution and DEM mean profile curvature. By utiliz-ing a quadratic polynomial model, Wang et al. (2004) provided a modified version of

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Tang’s model. Vaze et al. (2010) found that both DEM accuracy and resolution affectthe quality of DEM-derived hydrological features as do the significant differencesbetween the elevation and slope values derived from high resolution LiDAR DEMand their counterparts from coarse resolution contour derived DEM. However, theseresearchers conducted experimental studies without constructing analytical modeling.The principle of the simulation approach is straightforward and easy to understand, butthe approach cannot provide predictive analytical expressions for identifying the rela-tionships between DEM accuracy and the corresponding related factors. Furthermore,simulation results cannot be guaranteed as valid for all possible cases that might arise.

The second approach for studying relationships between DEM accuracy and theabove related factors is analytical in nature and based on theoretical analysis and math-ematical inference. Theoretically, this approach is more rigorous, and the mathematicalrelationships of the analytical models are more generic in their applicability. For exam-ple, elevation data for floodplain mapping, published by the National Research (2007),investigated the relationship between DEM accuracy and terrain slope. However, therelationships between DEM accuracy and other related factors based on theoreticalanalyses have not received much attention. In addition, these analytic findings firstestablish the rules for directly correlating the horizontal resolution of DEM data withvertical accuracy, which has always been regarded as an unresolved issue in the DEMmanual (Maune 2007).

This research has focused on modeling DEM accuracy as a function of the DEMresolution and the average terrain slope. Unlike other studies, the research was mainlybased on an analytical approach with the aim of deriving quantitative model relation-ships, expressed in the form of inequalities, and derived using mathematical theories.Inequality relationships are then further rectified to equality relationships on the basisof simulation experiments. The common bilinear interpolation method is adopted asa first step in studying the relationship between DEM accuracy and the resolutionand terrain complexity. Accuracy analysis for nonlinear interpolation methods will beaddressed in future studies. To facilitate a better understanding of the relationships tobe studied, a one-dimensional case is first introduced and then our analysis is extendedinto two dimensions, which are of more concern. The rest of this paper is organized asfollows. In Sect. 2, the principle of the bilinear model is briefly introduced. In Sect. 3,theoretical findings of DEM accuracy in relation to DEM resolution and terrain slopefactors are concluded for both one-dimensional and two-dimensional cases for rasterand scattered DEM data. Sections 4.1–4.3 are devoted to conducting experimentalstudies based on both real DEM data and synthetic data aimed at verifying the validityof the analytical findings obtained in Sect. 3. Meanwhile, Sect. 4.4 further derives amore accurate quantitative model of the relationship between the sampling ratio andaverage terrain slope by fitting of the DEM RMSE error and its corresponding upperbound based on the experimental analysis on Gaussian surfaces. Conclusions drawnfrom this research are presented in the last section.

2 Bilinear Interpolation Model

The method applied for DEM interpolation is a significant factor affecting the accuracyof the DEM. In general, DEM interpolation methods can be classified as linear and non-

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linear. Each of these interpolation methods has its own characteristics. For instance,linear interpolation is applicable for areas with lower levels of terrain change andis easily implemented, while nonlinear interpolation seems to be more appropriatefor higher levels of terrain change. However, this latter method is normally time-consuming in implementation (Shi and Tian 2006). In this research, a bilinear linearinterpolation model was used as an example to study the impact of DEM resolution andterrain slope on DEM accuracy. In the following equations, the bilinear interpolationmethod is briefly introduced. The bilinear polynomial interpolation model for a regulargrid takes the following form

z = a1 + a2x + a3 y + a4xy, (1)

where ai (i = 1, 2, 3, 4) are coefficients that need to be determined. For the one-dimensional case, the above equation is simplified to

z = a1 + a2x, (2)

where ai (i = 1, 2) are coefficients that need to be estimated in advance. There are twoways of implementing the bilinear interpolation model. One is to provide four points inadvance and substitute these points into the bilinear formula to obtain a group of fourequations. Their solutions give the values of the four parameters ai (i = 1, 2, 3, 4).These linear equations can be solved using Cramer’s rule. The second method isto divide the bilinear interpolation into three one-dimensional linear interpolations.Stated more precisely, the first two linear interpolations are carried out along thevertical direction of the original cell. Based on these interpolations, the third linearinterpolation is then implemented along the horizontal direction of the original cellpassing through the point in question. This type of bilinear interpolation is frequentlyadopted due to its simplicity and practicality.

3 Analytical Modeling of DEM Accuracy

In this section, we analyze the impact of DEM resolution and terrain slope on theinterpolated DEM accuracy. The result of this analysis is an analytical model depictingthe relationships between the variables involved. To facilitate an understanding of thederivation process, our analysis started with the one-dimensional case and proceeded tothe two-dimensional case. The analysis procedure for both cases is as follows. First, theoriginal DEM with higher resolution is provided. Then, a re-sampled DEM with lowerresolution is generated on the basis of the original DEM by a down-sampling method.Next, the interpolated DEM is obtained by applying the linear interpolation model (forthe one-dimensional case) or the bilinear interpolation model (for the two-dimensionalcase) on the re-sampled DEM. Finally, the relationships between the accuracy of theinterpolated DEM and (a) DEM resolution and (b) terrain slope are modeled.

3.1 Accuracy Analysis for One-Dimensional, Uniformly Distributed Points

For a general re-sampling ratio r in the one-dimensional case, the original DEM and thecorresponding re-sampled DEM are denoted by f (ri + j) and f (ri + j), respectively.

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450 Math Geosci (2014) 46:445–481

As illustrated in Fig. 1, points represented as , such as x0, x4, x8, x12, denote thehorizontal coordinates of the sampling data, while the circled stars , such as x1, x2, x3,mark the coordinates of the true elevation values. The notation is explained as follows:n is the total number of terrain elevation items of the data, i denotes the region number,such as [x0, x4], to be interpolated, and j is the point index located within each region.Therefore, xk = xri+ j ; 0 ≤ i < n/r, i ∈ N ; 0 ≤ j ≤ r, j ∈ N .

The linear interpolation method for the one-dimensional case leads to the followingequations

f (ri + j) =(

1 − j

r

)f (ri) + j

rf (ri + r) . (3)

Hence, the difference between the true elevation value and the interpolated one is

f (ri + j) − f (ri + j) = f (ri + j) −[(

1 − j

r

)f (ri) + j

rf (ri + r)

]

=(

1 − j

r

)( f (ri + j) − f (ri))

+ j

r( f (ri + j) − f (ri + r)) . (4)

Therefore, the root mean square error (RMSE) between the original true elevation andthe interpolated DEM values can be calculated as

(RMSE)2 ≤ 2(r4 − 1

)15r4 · s2 E{(Slope)2}. (5)

See Appendix A (part 1) for the derivation of Eq. (5) in detail. In terms of this finding,if the terrain slope can be estimated beforehand, then the upper bound of the square ofRMSE by bilinear interpolation is directly proportional to the square of the re-sampledDEM resolution (re-sampling DEM post spacing) s2 and inversely proportional to theaverage value of the square of the DEM slope E{(Slope)2} with a scale multiple of

x4 x5x1 x2x0 x3 x9 x10x6 x7 x8 x

f

x11 xnx12

ii-1 i+1

Fig. 1 Schematic diagram of the linear interpolation of the DEM with re-sampling ratio r = 4 for one-dimensional case

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Math Geosci (2014) 46:445–481 451

2(r4−1)

15r4 . This conclusion is consistent with our intuitive observation of DEM accuracyin relation to the slope and resolution factors. Therefore, a mathematical model (theinequality of Eq. 5) has been provided to quantify the relationship between DEMaccuracy and (a) DEM resolution and (b) terrain slope.

3.2 Accuracy Analysis for Two-Dimensional Grid-Based DEM

For the general re-sampling ratio r in the two-dimensional case, the original DEMand the corresponding re-sampled DEM are denoted by f (ri + k, r j + l) andf (ri + k, r j + l), respectively. As illustrated in Fig. 2, the points represented as ,such as (x0, y0), (x0, y4), (x4, y0) and (x4, y4), denote the projection coordinatesof the sampling data, while the points represented by , such as (x1, y1), are thosecoordinate points of the true elevation values. The notation is as follows: m is thetotal number of data points in the x direction while n is the total number of datapoints in the y direction. Therefore, mn is the total number of points in this area.Meanwhile, i denotes the region number of points, such as [x0, x4], to be interpo-lated along the x direction, while j denotes the region number of points, such as[y0, y4], to be interpolated along the y direction, k is the point index located withineach region i and l is the point index located within each region j . Additionally, theoriginal DEM cell is assumed to be square, that is, the cell size is h along both thex and y directions. Therefore, 0 ≤ i < m/r, i ∈ N ; 0 ≤ k ≤ r, k ∈ N ,0 ≤ j < n/r, j ∈ N ; 0 ≤ l ≤ r, l ∈ N .

It is easy to obtain the following equations for the one dimensional linear interpo-lation case

x2x1x0 x3 x4 x

y2

y1

y0

y3

y4

y

f

i

j

Fig. 2 Schematic diagram of the linear interpolation DEM with re-sampling ratio r = 4 for two-dimensional case

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452 Math Geosci (2014) 46:445–481

f (ri + k, r j + l) =(

1 − k

r

)(1 − l

r

)f (ri, r j) +

(k

r

)(1 − l

r

)f (ri + r, r j)

+(

1− k

r

)(l

r

)f (ri, r j + r)+

(k

r

)(l

r

)f (ri + r, r j + r) .

(6)

Similarly, but by a more complex formula deduction process, the mean square error(RMSE)2 can be calculated as follows (a more rigorous derivation can be found inAppendix A (part 2) of this paper)

(RMSE)2 ≤ 2(8r2 − 9r + 4

) (r4 − 1

)h2

45r4 E{(Slope)2}

= 2(8r2 − 9r + 4

) (r4 − 1

)s2

45r6 E{(Slope)2}.

That is to say

(RMSE)2 ≤ 2(8r2 − 9r + 4

) (r4 − 1

)45r6 · s2 E{(Slope)2}. (7)

A more accurate model of the relationship between sampling ratio and average terrainslope is obtained by adjustment of the analytical model presented as Eq. (13) basedon the experimental analysis of Gaussian surfaces given below in Sect. 4.3. Note thatfor the two-dimensional case, the slope at point (i, j) is defined as ( f 2

x + f 2y )1/2,

which is the Euclidean norm for the gradient of the terrain function f at location(i, j). Therefore, the average slope squared for the entire terrain surface should bedefined as

E{(Slope)2} = 1

n2

∑i, j

(f 2x (i, j) + f 2

y (i, j)). (8)

Based on Eq. (7), it can be concluded that the smaller the slope value or the higherresolution of the original DEM (i.e., the smaller of s value), the corresponding RMSEof the bilinear interpolated DEM becomes smaller.

3.3 Accuracy Analysis for Two-Dimensional Randomly Scattered DEM

In this section, consideration has shifted to the two-dimensional, randomly scatteredDEM data. Connection of the scattered data points into triangles forms a triangulatedirregular network (TIN) rather than a regular square grid. Thus, a straightforwardnumerical difference method cannot be applied for estimating the information of thetopographic slope. Therefore, one solution for this dilemma is the conversion of therandomly scattered DEM elevation data into the regular square grid data. The similarresult presented in Eq. (7) certainly remains valid after such conversion because theglobal DEM interpolation error indicated by RMSE is inherently determined by the

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Math Geosci (2014) 46:445–481 453

Fig. 3 Convex combination ofthe triangular derivatives aroundnode P in the triangulation

1

2

3

45

P

π

π

ππ

π

specific topographic character, interpolation method and resolution size. However,such a new directly transformed DEM may inevitably contain additional errors, forinstance, arranging extra temporary sampling locations when converting TIN to newmesh grids, as well as further numerical differences for estimating the partial deriv-atives along the x and y directions. To this end, we first adopt one method of localderivative estimation by using a convex combination of all derivatives on related tri-angular planes Goodman et al. (1995). Pick up a node P from the triangulation ofthe scattered DEM data and let πi , i = 1, . . . , k be the triangles taking P as its onevertex as illustrated in Fig. 3. We denote gi as the gradient of the linear interpolationto the DEM data at vertices of πi . One method for estimating the gradient at P isgiven by a convex combination of the gradients of its located triangles, g1, . . . , gk ,that is

DP =k∑

i=1

λi gi/

k∑i=1

λi , (9)

where the weights λi are selected to be the reciprocal of the base triangle’s area �i ofthe corresponding triangle πi , that is, λi = 11/�i . In addition, the gradient gi and thearea �i can be obtained according to the following computational procedure. Supposethat three vertices (x j , y j ) with corresponding DEM elevation values z j , j = 1, 2, 3,produce a triangle plane πi given by

ax + by + cz + d = 0,

where (a, b, c)T constitutes the plane’s normal vector

a = y1 (z3 − z2) + y2 (z1 − z3) + y3 (z2 − z1)

b = x1 (z2 − z3) + x2 (z3 − z1) + x3 (z1 − z2)

c = x1 (y3 − y2) + x2 (y1 − y3) + x3 (y2 − y1) .

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454 Math Geosci (2014) 46:445–481

Therefore, the gradient for this triangle plane πi is

gi =(

∂z

∂x,∂z

∂y

)T

=(

−a

c,−b

c

)T

,

and the corresponding base area is �i = 12 |c|.

As mentioned above, Eq. (7) remains valid for the newly generated regular squaregrid from the randomly scatted DEM elevation data. Meanwhile, the global average ofthe terrain slope square (indicator for estimating the degree of DEM rugged bumpiness)can be approximated by averaging all local derivative estimations by means of theabove generalized numerical finite difference calculation method

E{(Slope)2} = 1

n

n∑i=1

‖Di‖22, (10)

where n is the number of the total nodes constituting TIN and Di represents the convexcombination of the gradients at node i according to Eq. (9). In addition, the squareof the resolution size is adjusted to be s2 = 1

n

∑i∈T �i , where T is the triangulation

coverage of the concerned district, based on the fact that the area cannot be changed bysuch conversion from TIN into Rectangle. Therefore, the formula for characterizingthe relationship between the RMSE and the sampling ratio, the terrain slope for TIN,becomes

(RMSE)2 ≤ 2(8r2 − 9r + 4

) (r4 − 1

)45r6 · 1

n

∑i∈T

�i · 1

n

n∑i=1

‖Di‖22. (11)

Suppose that there are N original randomly scattered DEM data and n sampling pointsmaking the sampling ratio r = √

n/N . It should be noted that here the sampling ratiois supposed to represent the reciprocal of the sampling proportion in terms of thenumber of DEM points. However, introducing the Sqrt operation is nothing less thanmaintaining the consistency with the previous sampling ratio as employed in Sects. 3.1and 3.2 referring to the sampling proportion in terms of the one-dimensional intervallength. Indeed, it may still introduce some tiny DEM errors after such conversion fromscattered data into the regular values. However, at the outset, our manuscript (Sect. 3.2)only provides the initial inequality relationships between the RMSE indicator (DEMinterpolation error estimation) and the resolution and terrain slope. Apparently, theseintrinsically qualitative relationships between the DEM error and its influencing fac-tors (resolution and topographic relief represented by the terrain slope) would not beaffected after such conversion from TIN to Rectangle. Meanwhile, more precise quan-titative relationships can be rectified in Sect. 4.4 by experimental study of a specifiedgeomorphic district because these estimated relationships between RMSE, resolutionand terrain slope most likely vary with different resolutions and rugged bumpiness ofthe various landforms.

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Math Geosci (2014) 46:445–481 455

4 Experimental Analysis

4.1 Experimental Analysis for Realistic Grid-Based DEM Data Sets

To verify the validity of the findings and further adjust the results presented by Eq. (7),which depict the analytical relationships between DEM accuracy and (a) the DEMresolution and (b) terrain slope, an experimental study was carried out to validate thetheoretical findings using practical DEM data sets with different levels of complexity.The three test areas are located within Shannxi province (left part of Fig. 4), and thethree test areas/sample regions represent the three typical types of terrain: plain, hilland mountain.

Sample region A lies within 33.577◦–33.844◦N and 107.465◦–107.785◦E, situatedin the Qianlong Mountains, in the southern part of Shanxi province. Sample regionB is located within 34.231◦–34.497◦N and 108.135◦–108.455◦E and is situated onthe central Shaanxi plain. Sample region C is located within 36.065◦–36.322◦N and109.487◦–109.804◦E. It is a loess hill and gully area in northern Shannxi, a hillytopography. All three sample regions are illustrated in Fig. 4 and their characteristicsare described in Table 1.

Fig. 4 Three different test areas within Shannxi province representing mountain, hill and plain terrains

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456 Math Geosci (2014) 46:445–481

Table 1 Characteristics of thetopographic surfaces of thestudied area

All surfaces are grid DEMscomposed of 106 points andwith 30×30 m spacing

Terrain descriptive statistics Highmountain

Hill Plain

Min (m) 1,067 1,005 224

Max (m) 3,069 1,464 401

Average elevation (m) 2,002 1,266 338

Hmax − Hmin (m) 2,002 459 177

Std dev. of elevation (m) 386.70 78.28 11.26

Elevation coefficient ofvariation (%)

19.32 15.98 2.95

Min slope (◦) 0 0 0

Max slope (◦) 69.75 61.46 53.25

Average slope (◦) 26.38 17.78 7.10

Std dev. of slope (◦) 10.89 8.46 4.56

Slope coefficient ofvariation (%)

41.28 27.58 14.23

The newly released DEM data, advanced Spaceborne Thermal Emission and Reflec-tion Radiometer (ASTER) Global Digital Elevation Model Version 2 (GDEM2), pro-duced by The Ministry of Economy, Trade, and Industry of Japan and the UnitedStates National Aeronautics and Space Administration (NASA), were used for thestudy. Because of the 260,000 additional scenes, GDEM2 is reported to have a muchhigher spatial resolution than its previous version, GDEM1. Moreover, the negative 5-m overall bias detected in GDEM 1 has been removed. GDEM2 is in GeoTIFF formatwith geographic lat/long coordinates and a 1 arc-second (30 m) grid of elevation post-ings. GDEM2 is referenced to the WGS84/EGM96 geoid. Pre-production estimatedaccuracies for this global product were 20 m at 95% confidence levels for vertical dataand 30 m at 95% confidence levels for horizontal data. When calculating the terraingradient value, the projection was converted to UTM/WGS84. Three topographic sur-faces measuring 1,000 by 1,000 pixels were extracted from the original dataset to bethe test data. Based on our theoretical findings above, for the experiment a quarter ofthe original terrain data were first evenly selected (that is to say, a sampling ratio ofr = 2 in this experiment), and then interpolated by bilinear interpolation. Therefore,the RMSE can be calculated by comparison of the elevation differences between thesetwo DEM surfaces. Meanwhile, slopes for every point were extracted based on theoctagonal linear differences. Figure 5 displays the original DEM and the bilinear inter-polated DEM, as well as the corresponding DEM residuals of a 60 ×60-point area forthe top left corner of the three plain, hill and mountain districts. It is difficult to obtaina perfect picture if all 1000 × 1000 points are presented together in a single image foreach terrain case. The bilinear interpolation method was applied on the re-sampledDEM and the interpolated DEM accuracy was assessed by comparing the differencebetween the original DEM data and the interpolated data, that is, | f (x, y) − f (x, y)|,for all the points on the DEM.

A striking visual contrast was obtained by the above experiment as the terraincomplexity increased significantly in each of the three cases. Table 2 below presentsthe corresponding results for this experimental study.

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Math Geosci (2014) 46:445–481 457

2.37

2.37

5

2.38

2.38

5

x 10

5

3.79

3.79

05

3.79

1

x 10

6

550

600

2.37

2.37

5

2.38

2.38

5

x 10

5

3.79

3.79

05

3.79

1

x 10

6

550

600

3.64

5

3.65

3.65

5 x 10

5

3.99

05

3.99

1

3.99

15

3.99

2

x 10

6

1250

1300

1350

1400

3.64

5

3.65

3.65

5 x 10

5

3.99

05

3.99

1

3.99

15

3.99

2

x 10

6

1250

1300

1350

1400

7.29

7.29

5

7.3

x 10

5

3.71

75

3.71

8

3.71

85

x 10

6

1600

1800

2000

7.29

7.29

5

7.3

x 10

5

3.71

75

3.71

8

3.71

85

x 10

6

1600

1800

2000

Fig

.5C

ompa

rativ

evi

sual

izat

ion

ofth

eor

igin

alan

din

terp

olat

edte

rrai

nsu

rfac

efo

rm

ount

ain,

hill

and

plai

nar

eas

with

inSh

annx

ipro

vinc

e

123

458 Math Geosci (2014) 46:445–481

Table 2 The experimental results for three different terrain types

Terrain type s0 = 2h0 E{(Slope)2} (RMSE)2 3/16s2 E{(Slope)2}Plain 57.4824 0.0227 9.0416 14.0885

Hill 56.2585 0.1364 15.2410 80.9341

Mountain 57.7782 0.3372 25.9608 211.0785

Table 3 The experimental results for rectangle with different levels of terrain complexity (r = 2)

Terrain parameters s0 = 2h0 E{(Slope)2} (RMSE)2 3/16s2 E{(Slope)2}A B C

9 30 1 300 0.0005 0.8208 8.6449

50 150 9 300 0.0512 82.0424 863.9056

200 500 25 300 0.1538 249.4699 2594.7536

500 1,500 50 300 1.3087 2109.714 22084.4387

For the plain region, the values of E{(Slope)2}, {RMSE}2 and 316 s2 E{(Slope)2} are

0.0227, 9.0416 and 14.0885, respectively. The value 316 s2 E{(Slope)2} was calculated

for the re-sampling DEM with a resolution twice that of the original cell size. Inthis case, 9.0416 < 14.0885; that is to say inequality of Eq. (7) and (RMSE)2 <316 s2 E{(Slope)2} is valid according to the experiment. In fact, similar experimentalfindings were obtained for the hill and mountain cases. As observed from Table 3, thelevels of terrain complexity steadily increase from the top to the bottom in Fig. 5, as dotheir corresponding RMSEs and their upper bounds in the column 3

16 s2 E{(Slope)2}.This result nicely demonstrates the validity of the findings represented by Eq. (7). Itshould be noted that these 3

16 s2 E{(Slope)2} values are often larger than the RMSEsfor the introduction of inequality techniques and the approximation of terrain slopewith its mean value for estimating the DEM accuracies.

4.2 Experimental Analysis for Synthetic Grid-Based DEM Data Sets

This section is devoted to verifying the validity of the proposed model with respect todifferent levels of simulated terrain complexity using mathematical surfaces. Actually,the terrain complexity is difficult to fully quantify due to the many factors involved,such as the frequency of changes in terrain surface orientation, the average slopeand the amplitude of the surface. Below only the surface amplitude considered forcharacterizing the terrain complexity, as also adopted by (Zhou and Liu 2002, 2003)in generating terrain surfaces of different levels of complexity. The Gaussian surfacefunction is expressed by

z = A(

1 − x

m

)2e−( x

m )2−( y

n +1)2 − B

(x

5m−

( x

m

)3 −( y

n

)5)

e−( xm )

2−( yn )

2

− Ce−( xm +1)

2−( yn )

2, (12)

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Math Geosci (2014) 46:445–481 459

Fig. 6 Comparative visualization of the original and interpolated Gaussian surfaces with different com-plexity levels (rectangle)

where A, B and C are parameters determining the amplitude and m and n are the para-meters which control the spatial extent of the surface. The true values of the elevationand slope can be computed using the above expression. For the sake of convenience,m = n = 500 was set, meaning the terrain was located within x, y ∈ [−1,000, 1,000].In this study, four different levels of terrain complexity were generated by assigningthe Gaussian surface function with four groups of parameter values. In addition, thesampling ratio r is still set at 2 for this experimental study.

The visualization of the interpolation results is shown in Fig. 6, including acomparison between the original and the interpolated surfaces together with theresiduals. As observed in the above table, the levels of terrain complexity increasein turn from the top to the bottom in Fig. 6, as do their corresponding RMSEs andtheir upper bounds in the column 3

16 s2 E{(Slope)2}. This result fully demonstrates thevalidity of Eq. (7). It should be noted that the 3

16 s2 E{(Slope)2} values are often muchlarger than the RMSEs for the introduction of inequality techniques and the approxi-mation of terrain slope with its mean value for estimating the DEM accuracies. Moreexperiments with real data sets are presented in the next section expressing DEMRMSE values in terms of both the average terrain slope and the re-sampling ratio.

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460 Math Geosci (2014) 46:445–481

4.3 Experimental Analysis for Randomly Scattered DEM Based on MathematicalSurfaces

This section is devoted to verifying the theoretical findings from Sect. 3.2. TheGaussian surface together with its configuration is adopted again for carrying outexperiments on randomly scattered DEM data. Still, the terrain district was restrictedwithin x, y ∈ [1,000, 1,000] and the same four groups of parameter values generatefour different Gaussian surface topographies. Figure 7 illustrates the comparison ofthe original image and the bilinear interpolation image, together with their interpolatedresiduals placed in the middle column.

For each surface, 400 locations are randomly selected out from the above exper-imental district to create a triangulation mesh, among which only 100 locations andtheir elevations are randomly chosen as the seed points for spanning the interpolationsurface, thereby approximating the elevations of the other 300 locations. It shouldbe noted that only these 100 DEM seed points as well as their elevations can beadopted to calculate the upper bound of the DEM error, the right term in Eq. (11),while in practical applications usually no surplus check points are available for exam-

Fig. 7 Comparative visualization of the original and interpolated Gaussian surfaces with different com-plexity levels (TIN)

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Math Geosci (2014) 46:445–481 461

Table 4 The experimental results for TIN with different levels of terrain complexity (r = 2, 4)

Terrain parameters r = √n/N (RMSE)2 2(8r2−9r+4)(r4−1)

45r6

· 1n∑

i∈T �i · 1n∑n

i=1 ‖Di ‖22

Multiple

A B C

9 30 1 2 0.8740 4.8818 5.586

4 0.8378 6.6595 7.949

50 150 9 2 22.299 128.17 5.748

4 21.617 171.64 7.940

200 500 25 2 257.10 1553.4 6.042

4 253.38 2020.2 7.973

500 1,500 50 2 2222.1 12697 5.714

4 2148.7 17043 7.932

ining the real DEM interpolation error. Although the theoretical findings in Sect. 3.2cannot characterize the realistic DEM error in terms of RMSE, they still provide theupper bound estimation as a function of the sampling ratio, resolution size and terrainslope.

Table 4 lists the experimental results corresponding to four different levels of terraincomplexity with the sampling ratio at 2 and 4. From this table, it can be observed thatthe upper error bounds fully confine the square of the RMSE. Moreover, an apparentfinding emerges that a steady constant multiple exists between the square of the RMSEof the DEM error and its upper bound, which is related to the sampling ratio, butis independent of terrain complexity. The following subsection further extends thediscussion of these findings for the scenario of regular square grids.

4.4 Extension of the Study and Analysis of the General Re-Sampling Ratio CaseBased on Gaussian Surfaces

Functionalities of the experimental studies in Sects. 4.1 and 4.2 are mainly to val-idate the theoretical findings using a re-sampling ratio of 2 for regular mesh grids.In this section, attention is turned to the general re-sampling ratio r . Furthermore, asmentioned above, a large gap exists between the DEM interpolation error in termsof the RMSE and its upper bound as formulated by Eqs. (5) and (7). To this end,the following subsection is devoted to narrowing this gap by the data fitting method.17 groups of terrain parameters for the Gaussian surface were selected for similarexperiments (see Tables 5, 8, 9 in Appendix B) to obtain a more precise relationshipdescription between DEM accuracy (RMSE) and its upper bound with re-samplingratios from 2 to 4 and even larger. Because the inequality of Eq. (7) leads to a large gapbetween the RMSE and its upper bound, the RMSE upper bound does qualitativelyaccount for the DEM accuracy. However, by conducting a large number of numericalsimulation experiments, we find that it is reasonable to assume a linear relationshipbetween the DEM accuracy expressed by the (RMSE)2 indicator and its upper bound2(8r2−9r+4)(r4−1)

45r6 s2 E{(Slope)2} consisting of the re-sampling ratio and the average

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462 Math Geosci (2014) 46:445–481

Table 5 The experimental results for different levels of terrain complexity with r = 2, k = 10.4125

Terrain parameters s0 = 2h0 E{(Slope)2} (RMSE)2 3/16s2 E{(Slope)2}A B C

3 10 1/3 300 0.0001 0.0912 0.9605

9 30 1 300 0.0005 0.8208 8.6449

30 100 3 300 0.0057 9.1158 95.9895

300 750 30 300 0.0228 36.4633 383.9581

50 150 9 300 0.0512 82.0424 863.9056

60 200 6 300 0.0910 145.8531 1535.8322

90 300 9 300 0.1427 228.5045 2408.8325

120 400 12 300 0.3454 560.5679 5829.2307

400 900 45 300 0.5194 843.6800 8765.2211

150 500 24 300 0.0132 21.1793 222.0079

200 500 25 300 0.1538 249.4699 2594.7536

450 1,200 50 300 0.8658 1402.4979 14611.0981

500 1,500 50 300 1.3087 2109.7141 22084.4387

600 1,600 60 300 1.5382 2491.9384 25957.5923

700 1,800 80 300 1.9719 3197.3737 33275.0838

800 2,000 90 300 2.4583 3988.8640 41483.6484

1,000 2,000 100 300 2.7194 4407.5113 45889.5424

terrain slope. In other words, the analytical inequality (Eq. 7) does not quantify DEMvertical errors (by RMSE2) with an explicit function of the terrain slope and DEM hor-izontal resolution. However, it has actually clarified this issue with only one multipleto be determined.

According to Eq. (7)

2(8r2 − 9r + 4

) (r4 − 1

)45r6 · s2 E{(Slope)2} = k (RMSE)2 . (13)

For each fixed re-sampling ratio r , a linear coefficient k can be calculated based on theexperimental results for the 17 terrain types by means of the least squares method. Forexample, the calculated result for a re-sampling ratio of 2:1 is presented below in Table5. Figure 8 is the corresponding fitted relationship line represented by Eq. (13). Twoother comparison figures are provided in Appendix B as Figs. 10 and 11, correspondingto Tables 8 and 9 for each re-sampling ratio from Eqs. 3 and 4, respectively.

The experiments show that the parameter k becomes ever larger with the increaseof the re-sampling ratio r . Further experiments were then conducted to more explicitlyestablish their relationship. For this reason, all re-sampling ratios below 30 were tested,leading to a series of linear coefficients k for different re-sampling ratios as shown inTable 6.

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Math Geosci (2014) 46:445–481 463

Fig. 8 The fitted relationship between {RMSE}2 and its upper bound with r = 2, k = 10.4125

Table 6 The experimental coefficient k corresponding to each re-sampling ratio r

r 2 3 4 5 6 7 8 9 10 11

k 10.41 12.40 13.55 14.30 14.82 15.20 15.50 15.74 15.93 16.09

r 12 13 14 15 16 17 18 19 20 21

k 16.23 16.34 16.44 16.53 16.61 16.68 16.74 16.79 16.84 16.88

r 22 23 24 25 26 27 28 29 30

k 16.92 16.96 17.00 17.03 17.06 17.08 17.11 17.13 17.16

The following expression can be the optimized curve providing the relationshipbetween the adjusted coefficients k and each corresponding re-sampling ratio r

k = 6.48 ln(r) − 2.77 × 10−4r3 + 2.30 × 10−2r2 − 0.86r + 7.59. (14)

The curve of Fig. 9 demonstrates visually the closeness of fit.Thus far, by combining Eqs. (13) and (14), the exact relationship between DEM

accuracy and both re-sampling ratio (resolution) and average terrain slope for a DEMthat is generated by bilinear interpolation can be given as follows

(RMSE)2 = 1

k· 2

(8r2 − 9r + 4

) (r4 − 1

)45r6 · s2 E{(Slope)2}. (15)

By deformation of Eq. (15), this model can also be adopted to determine the necessaryDEM resolution for practical applications when a prescribed RMSEp value is givensupposing the fluctuation of the concerned terrain indicated by the average squared

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464 Math Geosci (2014) 46:445–481

Fig. 9 The fitted relationship between the linear coefficients k and re-sampling ratio r

slope has been approximated. To be exact, the precise re-sampling ratio r can beobtained by solving its implicit function (Eq. 15). In other words, a suitable re-samplingratio r should be determined by means of Eq. (15); it must be an integer and also alarger ratio r corresponding to a lower resolution because s = rh leads to higher DEMerrors in terms of (RMSE)2. What concerns us, therefore, is the maximum re-samplingratio r , which can be expressed as an integer optimization problem as follows

max r

s.t.

{RMSE � RMSEpr ∈ N

, (16)

in which RMSE is calculated according to Eq. (15). The optimized re-sampling ratiocan be found by computing several re-sampling ratios below a large prescribed integervalue, such as 20, as an approach to solving the above optimization problem. Thisfinding should have great potential in DEM data generation and applications. For aspecified topographic region, its average value of the global terrain slope would becertain and can be calculated by numerical finite differences of all DEM data or partialsamplings. In this case, the suitable re-sampling ratio r can be calculated in terms ofEq. (16) in order to meet the prescribed accuracy requirement in terms of RMSEp.Actually, this formula has often been urgently required in many practical engineeringDEM applications.

The result given by Eq. (15) first establishes the rules for directly correlating thehorizontal resolution of DEM data with vertical accuracy, which has always beenregarded as an unresolved issue in the DEM manual (Maune 2007). As a response tothis issue, Eq. (15) directly correlates the horizontal resolution s of the digital elevationdata with the vertical accuracy RMSEp. Thus, it could be tabulated for comparisonand reference with regard to the common re-sampling ratios for the requirement ofpractical implementation shown below in Table 7. The second column indicates the

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Math Geosci (2014) 46:445–481 465

Table 7 The practical DEM accuracy indicator and resolution with sampling ratio 2 ≤ r ≤ 5

Sampling ratio r DEM accuracy (RMSE)2 Re-sampling ratio r

2 (RMSE)2 = 0.0180s2 E{(Slope)2}max r

s.t.

{RMSE ≤ RMSEpr ∈ N

3 (RMSE)2 = 0.0193s2 E{(Slope)2}4 (RMSE)2 = 0.0196s2 E{(Slope)2}5 (RMSE)2 = 0.0197s2 E{(Slope)2}

DEM accuracy (RMSE)2 interpolated by the bilinear method when the resolution sizeand the average terrain slope are given. The last column lists the required re-samplingratio r when the prescribed RMSEp and the average terrain slope are given.

5 Conclusions

In this research, the relationship between the accuracy of an interpolated DEM and(a) the DEM resolution and (b) slope of terrain has been modeled analytically fromthe theoretical analysis. For both one-dimensional and two-dimensional DEMs, rela-tionship models were built and presented first in the form of two inequalities. Theserelationship models reveal the fact that the accuracy of an interpolated DEM dependson (a) its resolution and (b) the slope of the terrain, an indication of terrain complexitylevel. When the slope value is smaller and the DEM resolution is greater, the generatedRMSE for the bilinear interpolated DEM will become smaller. Two types of experi-mental study were conducted based on both real DEM data sets and mathematicallygenerated DEM surfaces. The experimental results demonstrate that the inequalityexpressions developed for the relationships between DEM accuracy, DEM resolutionand terrain slope are valid. Experimental studies verified the validity of the analyt-ical relationship models. Furthermore, a more precise expression of DEM accuracyin terms of RMSE with respect to both DEM resolution and average terrain slopewas obtained. In addition, a practical optimized model was derived for calculating therequired re-sampling ratio r when a prescribed DEM accuracy upper bound in RMSEis given.

The proposed theoretical models providing the relationships between the interpo-lated DEM accuracy and its resolution and terrain slope can be used for determiningDEM resolution in practical DEM applications. This new knowledge might be includedin future DEM manuals for determining DEM resolution or to estimate DEM accuracyin engineering applications involving DEM. For a given terrain with a fixed slope andDEM accuracy level, the DEM resolution needed can be deduced using the relationshipmodel developed in this study. On the other hand, if both DEM resolution and terrainslope are known, an accuracy estimation for the interpolated DEM can be provided interms of the maximum possible RMSE for the interpolated DEM using the developedrelationship models, that is, the inequalities of Eq. (7), together with the more preciseestimated expression using Eq. (15).

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466 Math Geosci (2014) 46:445–481

As mentioned in the DEM manual (Maune 2007), no established criterion existsto depict the vertical accuracy of digital elevation data by its horizontal resolution.The work presented here can be viewed as an attempt to provide a solution to thisissue. Indeed, our findings do directly correlate the horizontal resolution s = rh of thedigital elevation data with the vertical accuracy RMSE as expressed by Eq. (15). Futuredevelopment of this study will investigate the accuracy of interpolated DEMs for awider range of terrain types using many other interpolation models, for instance, thebi-cubic model, as a step towards the bilinear interpolation-based model. A possibleapproach is to first transform the nonlinear model into a linear one and then to applyto that linear model the techniques developed in this study related to linear models.

Acknowledgments The research presented in this paper is support by Ministry of Science and Technology,China (Project No. 2012AA12A305).

Appendix A: Derivations of Eqs. (5) and (7) for the General Re-Sampling Ratio

A.1 Formula Derivations for One-Dimensional Uniformly Distributed Points

According to the elevation difference Eq. (4), together with 1st order Taylor expansionfor the elevation function f at the sampling nodes

(RMSE)2 = 1

n

∑i, j

(f (ri + j) − f (ri + j)

)2

= 1

n

∑i, j

[(1 − j

r

)( f (ri + j) − f (ri)) + j

r( f (ri + j) − f (ri + r))

]2

≤ 2

n

∑i, j

[(1 − j

r

)2

( f (ri + j) − f (ri))2 +(

j

r

)2

( f (ri + j) − f (ri + r))2

]

= 2h2

n

∑i, j

[j2

(1 − j

r

)2 ( f (ri + j) − f (ri)

jh

)2

+ ( j − r)2(

j

r

)2 ( f (ri + j) − f (ri + r)

( j − r) h

)2]

≈ 2h2

n

∑i, j

{j2

(1 − j

r

)2 [f ′ (ri)

]2 + ( j − r)2(

j

r

)2 [f ′ (ri + r)

]2

}

= 2h2

n

∑i, j

{j2

(1 − j

r

)2 ([f ′ (ri)

]2 + [f ′ (ri + r)

]2)}

= 2h2

n

r−1∑j=1

[j2

(1 − j

r

)2] n/r−1∑

i=0

{[f ′ (ri)

]2 + [f ′ (ri + r)

]2}

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Math Geosci (2014) 46:445–481 467

=(r4 − 1

)h2

15r2

∑n/r−1i=0

{[f ′ (ri)

]2 + [f ′ (ri + r)

]2}

n/r

= 2(r4 − 1

)h2

15r2 E{(Slope)2} = 2(r4 − 1

)s2

15r4 E{(Slope)2}.

Note, for the above derivation, when j = 0, the results should be zero for the eleva-tion values of those interpolation nodes that are exactly the same as the original truevalues. Because forward difference quotients have been introduced, the sampling sizes should be restricted within a very small range; otherwise, the average slope can-not be approximated in this fashion and could not represent the terrain map in actualapplications.

A.2 Formula Derivations for Two-Dimensional Grid-Based DEM

According to the linear interpolation method for the two-dimensional case, the fol-lowing equation can be easily obtained as follows

f (ri + k, r j + l) =(

1 − k

r

)(1 − l

r

)f (ri, r j) +

(k

r

)(1 − l

r

)f (ri + r, r j)

+(

1− k

r

)(l

r

)f (ri, r j + r)+

(k

r

)(l

r

)f (ri + r, r j + r) .

Therefore,

f (ri + k, r j + l) − f (ri + k, r j + l)

=(

1 − k

r

)(1 − l

r

)( f (ri + k, r j + l) − f (ri, r j))

+(

k

r

)(1 − l

r

)( f (ri + k, r j + l) − f (ri + r, r j))

+(

1 − k

r

)(l

r

)( f (ri + k, r j + l) − f (ri, r j + r))

+(

k

r

)(l

r

)( f (ri + k, r j + l) − f (ri + r, r j + r)) .

According to the Cauchy–Schwarz inequality 〈x, y〉 ≤ ‖x‖ ‖y‖, if y = ( 1 1 · · · 1 )T,then the following formula will hold

(n∑

i=1

xi

)2

≤ nn∑

i=1

x2i .

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468 Math Geosci (2014) 46:445–481

The final square of the RMSE can be divided into two types of series summations asfollows

(RMSE)2 = 1

mn

∑i, j,k,l

( f (ri + k, r j + l) − f (ri + k, r j + l))2

= 1

mn

∑i, j

⎛⎝∑

kl �=0

( f (ri + k, r j + l) − f (ri + k, r j + l))2

+∑kl=0

( f (ri + k, r j + l) − f (ri + k, r j + l))2

).

and

∑kl �=0

( f (ri +k, r j +l)− f (ri +k, r j +l))2

=∑kl �=0

⎛⎜⎜⎜⎜⎜⎜⎝

(1− k

r

) (1− l

r

)( f (ri +k, r j +l)− f (ri, r j))

+ ( kr

) (1− l

r

)( f (ri +k, r j +l)− f (ri +r, r j))

+ (1− k

r

) ( lr

)( f (ri +k, r j +l)− f (ri, r j +r))

+ ( kr

) ( lr

)( f (ri +k, r j +l)− f (ri +r, r j +r))

⎞⎟⎟⎟⎟⎟⎟⎠

2

≤ 4∑kl �=0

⎛⎜⎜⎜⎜⎜⎜⎜⎝

(1− kr )2

(1− l

r

)2( f (ri +k, r j +l)− f (ri, r j))2

+ ( kr

)2 (1− l

r

)2( f (ri +k, r j +l)− f (ri +r, r j))2

+(1− kr )2

( lr

)2( f (ri +k, r j +l)− f (ri, r j +r))2

+ ( kr

)2 ( lr

)2( f (ri +k, r j +l)− f (ri +r, r j +r))2

⎞⎟⎟⎟⎟⎟⎟⎟⎠

≈ 4h2∑kl �=0

⎛⎜⎜⎜⎜⎜⎜⎜⎝

(1− k

r

)2 (1− l

r

)2((k ∂

∂x +l ∂∂y ) f (ri, r j))2

+ ( kr

)2 (1− l

r

)2(((k−r) ∂

∂x +l ∂∂y ) f (ri +r, r j))2

+(1− kr )2

( lr

)2((k ∂

∂x +(l−r) ∂∂y ) f (ri, r j +r))2

+ ( kr

)2 ( lr

)2(((k−r) ∂

∂x +(l−r) ∂∂y ) f (ri +r, r j +r))2

⎞⎟⎟⎟⎟⎟⎟⎟⎠

≤ 8h2∑kl �=0

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

(1− kr )2

(1− l

r

)2(k2( fx (ri, r j))2+l2( fy(ri, r j))2)

+ ( kr

)2 (1− l

r

)2((k−r)2( fx (ri +r, r j))2+l2( fy(ri +r, r j))2)

+(1− kr )2

( lr

)2(k2( fx (ri, r j +r))2+(l−r)2( fy(ri, r j +r))2)

+ ( kr

)2 ( lr

)2((k−r)2( fx (ri +r, r j +r))2

+(l−r)2( fy(ri +r, r j +r))2)

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

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Math Geosci (2014) 46:445–481 469

= 8h2

r4

∑kl �=0

⎛⎜⎜⎜⎜⎜⎜⎝

(k2(r −k)2(r −l)2( fx (ri, r j))2+l2(r −k)2(r −l)2( fy(ri, r j))2)

+(k2(r −k)2(r −l)2( fx (ri +r, r j))2+k2l2(r −l)2( fy(ri +r, r j))2)

+(k2l2(r −k)2( fx (ri, r j +r))2+l2(r −k)2(r −l)2( fy(ri, r j +r))2)

+(k2l2(r −k)2( fx (ri +r, r j +r))2+k2l2(r −l)2( fy(ri +r, r j +r))2)

⎞⎟⎟⎟⎟⎟⎟⎠

,

= 8h2

r4

∑kl �=0

⎛⎜⎜⎜⎜⎜⎜⎝

k2l2(r −k)2(( fx (ri, r j +r))2+( fx (ri +r, r j +r))2)

+k2l2(r −l)2(( fy(ri +r, r j))2+( fy(ri +r, r j +r))2)

+k2(r −k)2(r −l)2(( fx (ri, r j))2+( fx (ri +r, r j))2)

+l2(r −k)2(r −l)2(( fy(ri, r j))2+( fy(ri, r j +r))2)

⎞⎟⎟⎟⎟⎟⎟⎠

= 8h2

r4

⎛⎜⎜⎜⎜⎜⎜⎝

∑kl �=0 k2l2(r −k)2(( fx (ri, r j +r))2+( fx (ri +r, r j +r))2)

+∑kl �=0 k2l2(r −l)2(( fy(ri +r, r j))2+( fy(ri +r, r j +r))2)

+∑kl �=0 k2(r −k)2(r −l)2(( fx (ri, r j))2+( fx (ri +r, r j))2)

+∑kl �=0 l2(r −k)2(r −l)2(( fy(ri, r j))2+( fy(ri, r j +r))2)

⎞⎟⎟⎟⎟⎟⎟⎠

.

Because these following equations hold up

∑kl �=0

k2l2(r − k)2 =r−1∑k=1

k2(r − k)2r−1∑l=1

l2 = (r − 1)(2r − 1)r2(r4 − 1)

180,

∑kl �=0

k2l2(r − l)2 =r−1∑k=1

k2r−1∑l=1

l2(r − l)2 = (r − 1)(2r − 1)r2(r4 − 1)

180,

∑kl �=0

k2(r − k)2(r − l)2 =r−1∑k=1

k2(r − k)2r−1∑l=1

(r − l)2 = (r − 1)(2r − 1)r2(r4 − 1)

180,

∑kl �=0

l2(r − k)2(r − l)2 =r−1∑k=1

(r − k)2r−1∑l=1

l2(r − l)2 = (r − 1)(2r − 1)r2(r4 − 1)

180.

therefore, we have

∑kl �=0

( f (ri + k, r j + l) − f (ri + k, r j + l))2

= 2(r −1)(2r −1)(r4−1)h2

45r2

⎛⎜⎜⎜⎜⎜⎜⎝

( fx (ri, r j))2 + ( fy(ri, r j))2

+( fx (ri, r j + r))2 + ( fy(ri, r j + r))2

+( fx (ri + r, r j))2 + ( fy(ri + r, r j))2

+( fx (ri + r, r j + r))2 + ( fy(ri + r, r j + r))2

⎞⎟⎟⎟⎟⎟⎟⎠

.

123

470 Math Geosci (2014) 46:445–481

Meanwhile

∑kl=0

(f (ri + k, r j + l) − f (ri + k, r j + l)

)2

=∑

k(l=0)

( f (ri + k, r j + l) − f (ri + k, r j + l))2

+∑

l(k=0)

( f (ri + k, r j + l) − f (ri + k, r j + l))2

=∑

k

(f (ri + k, r j) − f (ri + k, r j)

)2 +∑

l

(f (ri, r j + l) − f (ri, r j + l)

)2

=∑

k

(f (ri + k, r j) − (

(1 − k

r

)f (ri, r j) +

(k

r

)f (ri + r, r j))

)2

+∑

l

(f (ri, r j + l) −

((1 − l

r

)f (ri, r j) +

(l

r

)f (ri, r j + r)

))2

=∑

k

((1− k

r

)( f (ri + k, r j)− f (ri, r j))+

(k

r

)( f (ri +k, r j)− f (ri +r, r j))

)2

+∑

l

((1− l

r

)( f (ri, r j +l)− f (ri, r j))+

(l

r

)( f (ri, r j + l)− f (ri, r j +r))

)2

≈ h2∑

k

(k

(1 − k

r

)fx (ri, r j) +

(k

r

)(k − r) fx (ri + r, r j)

)2

+h2∑

l

(l

(1 − l

r

)fy(ri, r j) +

(l

r

)(l − r) fy(ri, r j + r)

)2

= 2h2∑

k

(k2

(1 − k

r

)2

( fx (ri, r j))2 +(

k

r

)2

(k − r)2( fx (ri + r, r j))2

)

+2h2∑

l

(l2

(1 − l

r

)2

( fy(ri, r j))2 +(

l

r

)2

(l − r)2( fy(ri, r j + r))2

)

= 2h2r−1∑k=1

(k2

(1 − k

r

)2)(

( fx (ri, r j))2 + ( fx (ri + r, r j))2)

+2h2r−1∑l=1

(l2

(1 − l

r

)2)(

( fy(ri, r j))2 + ( fy(ri, r j + r))2)

= (r4 − 1)h2

15r

{[( fx (ri, r j))2 + ( fx (ri + r, r j))2

]

+[( fy(ri, r j))2 + ( fy(ri, r j + r))2

]}

123

Math Geosci (2014) 46:445–481 471

Note that in this case, 0 ≤ i ≤ m/r, i ∈ N ; 0 ≤ j ≤ n/r, j ∈ N .Therefore,

(RMSE)2 = 1

mn

∑i, j,k,l

( f (ri + k, r j + l) − f (ri + k, r j + l))2

= 1

mn

∑i, j

(∑kl �=0( f (ri + k, r j + l) − f (ri + k, r j + l))2

+∑kl=0( f (ri + k, r j + l) − f (ri + k, r j + l))2

)

≤ 1

mn

∑i, j

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

2(r−1)(2r−1)(r4−1)h2

45r2

⎛⎜⎜⎜⎜⎜⎜⎝

( fx (ri, r j))2 + ( fy(ri, r j))2

+( fx (ri, r j + r))2 + ( fy(ri, r j + r))2

+( fx (ri + r, r j))2 + ( fy(ri + r, r j))2

+( fx (ri + r, r j + r))2 + ( fy(ri + r, r j + r))2

⎞⎟⎟⎟⎟⎟⎟⎠

+ (r4−1)h2

15r

{[( fx (ri, r j))2 + ( fx (ri + r, r j))2

]+ [

( fy(ri, r j))2 + ( fy(ri, r j + r))2]}

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

= 1

mn

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

2(r−1)(2r−1)r2(r4−1)h2

45r4

∑m/r−1i=0

∑n/r−1j=0

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

( fx (ri, r j))2 + ( fy(ri, r j))2

+( fx (ri, r j + r))2 + ( fy(ri, r j + r))2

+( fx (ri + r, r j))2 + ( fy(ri + r, r j))2

+( fx (ri + r, r j + r))2

+( fy(ri + r, r j + r))2

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

+ (r4−1)h2

15r

{∑m/r−1i=0

∑n/rj=0

[( fx (ri, r j))2 + ( fx (ri + r, r j))2

]+∑m/r

i=0

∑n/r−1j=0

[( fy(ri, r j))2 + ( fy(ri, r j + r))2

]}

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

≈ 1

mn

{2(r −1)(2r −1)(r4−1)h2

45r2 (4mn

r2 E{(Slope)2}) + (r4 − 1)h2

15r(

2mn

r2 E{(Slope)2})}

=(

8(r − 1)(2r − 1)(r4 − 1)h2

45r4 E{(Slope)2} + 2r(r4 − 1)h2

15r4 E{(Slope)2})

= 2(8r2 − 9r + 4)(r4 − 1)h2

45r4 E{(Slope)2}.

Appendix B: Numerical Results Involved in Sect. 4.4

See Tables 8 and 9, and Figs. 10 and 11.

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472 Math Geosci (2014) 46:445–481

Table 8 The experimental results regarding different levels of terrain complexity with r = 3, k = 12.4001

Terrain parameters s0 = 3h0 E{(Slope)2} (RMSE)2 3/16r2 E{(Slope)2}A B C

3 10 1/3 300 0.0001 0.1007 1.2639

9 30 1 300 0.0005 0.9066 11.3755

30 100 3 300 0.0059 10.0688 126.3093

300 750 30 300 0.0136 23.3973 292.1342

50 150 9 300 0.0235 40.2752 505.2372

60 200 6 300 0.0529 90.6192 1136.7836

90 300 9 300 0.0940 161.1007 2020.9487

120 400 12 300 0.1474 252.4158 3169.7007

400 900 45 300 0.1587 275.6253 3414.4029

150 500 24 300 0.3566 619.3115 7670.6090

200 500 25 300 0.5362 932.3065 11534.1245

450 1,200 50 300 0.8939 1549.3610 19226.4759

500 1,500 50 300 1.3511 2330.3482 29060.3081

600 1,600 60 300 1.5880 2752.8284 34157.1181

700 1,800 80 300 2.0357 3532.3778 43786.1728

800 2,000 90 300 2.5379 4406.9707 54587.7962

1,000 2,000 100 300 2.8075 4871.9846 60386.4213

Table 9 The experimental results regarding different levels of terrain complexity with r = 4, k = 13.5462

Terrain parameters s0 = 4h0 E{(Slope)2} (RMSE)2 3/16r2 E{(Slope)2}A B C

3 10 1/3 300 0.0001 0.1042 1.4276

9 30 1 300 0.0005 0.9374 12.8486

30 100 3 300 0.0060 10.4101 142.6657

300 750 30 300 0.0138 24.1914 329.9654

50 150 9 300 0.0239 41.6405 570.6629

60 200 6 300 0.0537 93.6912 1283.9914

90 300 9 300 0.0955 166.5621 2282.6514

120 400 12 300 0.1498 260.9816 3580.1618

400 900 45 300 0.1613 284.9758 3856.5941

150 500 24 300 0.3624 640.3104 8664.0109

200 500 25 300 0.5450 963.9659 13027.9458

450 1,200 50 300 0.9084 1601.8853 21716.3895

500 1,500 50 300 1.3730 2409.3180 32823.5924

600 1,600 60 300 1.6138 2846.1302 38580.6148

700 1,800 80 300 2.0688 3652.1598 49456.7532

800 2,000 90 300 2.5791 4556.4362 61657.3305

1,000 2,000 100 300 2.8531 5037.7686 68207.6695

123

Math Geosci (2014) 46:445–481 473

Fig. 10 The fitted relationship between {RMSE}2 and its upper bound with r = 3, k = 12.4001

Fig. 11 The fitted relationship between {RMSE}2 and its upper bound with r = 4, k = 13.5462

Appendix C: Supplementary Explanation for Sect. 3

This self-contained supplementary material is provided for the special case with resam-pling ratio r = 2 for a better understanding of the detailed derivation process of Sect. 3.

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474 Math Geosci (2014) 46:445–481

C.1 Accuracy Analysis for One-Dimensional, Uniformly Distributed Pointswith Re-Sampling Ratio r = 2

For a one-dimensional case, the original DEM, the corresponding re-sampled DEMand the linear interpolated DEM are denoted by f (i), fD(i) and f (i), respectively. It isassumed that the original DEM is composed of a point set {xi }, (i = 0, 1, 2, . . . , n),the resolution of the original DEM is h, the re-sampling ratio is r , and the resolutionof the re-sampled DEM is s. A schematic diagram of the original DEM (Fig. 12a), there-sampled DEM (Fig. 12b), and the interpolated DEM (Fig. 12c) are given in Fig. 12where the re-sampling ratio is set as 2:1 (i.e., s = rh, r = 2) for the sake of simplicityto start the construction of our theoretical models.

Using the linear interpolation method for the one-dimensional case, it is easy toobtain the following equations

f (i) − f (i) = 0 (i = 0, 2, . . . , 2k, k ∈ N , 2k ≤ n), (17)

f (i) − f (i) = f (i) − 1

2[ f (i + 1) + f (i − 1)]

= f (i) − f (i − 1) − 1

2[ f (i + 1) − f (i − 1)]

(i = 1, 3, . . . , 2k + 1, k ∈ N , 2k + 1 ≤ n). (18)

For the last term of Eq. (18), we notice that

f (i + 1) − f (i − 1) = f (i + 1) − f (i) + f (i) − f (i − 1). (19)

Therefore,

f (i) − f (i)

h= f (i) − f (i − 1)

h− 1

2

[f (i + 1) − f (i)

h+ f (i) − f (i − 1)

h

]

= f ′(i − 1) − 1

2( f ′(i) + f ′(i − 1))

= 1

2f ′(i − 1) − 1

2f ′(i). (20)

where f ′(i −1) and f ′(i) denote the forward difference quotients which approximatethe derivatives of function f (x) at the locations i − 1 and i , respectively. Becausef (x) is the terrain function, f ′(i) can be viewed as a depiction of the terrain slope atpoint i . Remembering that for the one-dimensional case, the error indicator, the rootmean square error (RMSE), has the form

RMSE =√∑

i ( f (i) − f (i))2

n. (21)

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Math Geosci (2014) 46:445–481 475

f

x4 x5x1 x2x0 x3 x9 x10x6 x7 x8 xn x

x4 x5x1 x2x0 x3 x9 x10x6 x7 x8 xn x

f

x4 x5x1 x2x0 x3 x9 x10x6 x7 x8 xn x

f

(a) The original DEM f(i)

(b) The re-sampled DEM fD(i)

(c) The linear interpolated DEM f (i)–

Fig. 12 Original one-dimensional DEM (a), the re-sampled DEM (b), and the linear interpolated DEM (c)on the re-sampled DEM (b)

123

476 Math Geosci (2014) 46:445–481

Additionally, considering Eqs. (17) and (18), we have

(RMSE)2 =∑

i ( f (i)− f (i))2

n=

∑i=even ( f (i)− f (i))2

n+

∑i=odd ( f (i)− f (i))2

n

=∑

i=odd ( f (i) − f (i))2

n. (22)

Combining Eqs. (21) and (22) yields

(RMSE)2 = h2

4n

∑i=odd

( f ′(i − 1) − f ′(i))2 ≤ h2

2n

∑i=odd

{[ f ′(i − 1)]2 + [ f ′(i)]2}.

(23)

Notice that the slope of point i − 1 is defined as | f ′(i − 1)|, and the average value ofthe square of the DEM slope can therefore be defined as

E{(Slope)2} = 1

n

n∑i=1

[ f ′(i)]2 = 1

n

∑i=odd

{[ f ′(i − 1)]2 + [ f ′(i)]2}. (24)

If the resolution of the re-sampled DEM is denoted by s, then obviously s = 2h forthis case. Therefore, Eq. (23) is simplified as

(RMSE)2 ≤ h2 E{(Slope)2}2

= 1

8s2 E{(Slope)2} (25)

In terms of Eq. (25) for the one-dimensional case, if the terrain slope can be estimatedbeforehand, then the upper bound of the RMSE of the bilinear interpolation resultcould be obtained. From Eq. (25), the upper bound for the square of the RMSE isdirectly proportional to the square of the re-sampled DEM resolution s2 and inverselyproportional to the average value of the square of the DEM slope E{(Slope)2} with ascale factor of 1/8. This conclusion is consistent with our intuitive observation of DEMaccuracy in relation to the slope and resolution factors. Thus, we have here provideda mathematical model (the inequality of Eq. 25) to quantify the relationship betweenDEM accuracy and (a) DEM resolution and (b) terrain slope.

C.2 Accuracy Analysis for the Two-Dimensional Grid-Based DEM with Re-SamplingRatio r = 2

For the two-dimensional case (i.e., two-dimensional DEM), it is assumed that theoriginal DEM, the re-sampled DEM, and the interpolated DEM are denoted byf (i, j), fD(i, j) and f (i, j) (i, j = 1, 2, . . . , n), respectively, as shown in Fig. 13.The resolution of the original DEM is also assumed to be h, the re-sampling ratio isdenoted by r , and the resolution of the re-sampled DEM is denoted by s. For the sakeof simplicity, the re-sampling ratio along each of the xand y directions is set to be 2:1,

123

Math Geosci (2014) 46:445–481 477

(a) The original DEM (b) The re-sampled DEM (c) The bilinear interpolated DEM

Fig. 13 An illustration of two-dimensional data sets: the original two-dimensional DEM, the re-sampledDEM, and the bilinear interpolated DEM

i.e., s = rh, (r = 2) to begin the two-dimensional case discussion of the constructionof the theoretical models.

For the bilinear interpolation method in the two-dimensional case,

f (i, j) =

⎧⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎩

f (i, j − 1) + 12 [ f (i, j + 1) − f (i, j − 1)], i = even, j = odd

f (i − 1, j) + 12 [ f (i + 1, j) − f (i − 1, j)], i = odd, j = even

14 [ f (i − 1, j − 1) + f (i + 1, j − 1) + f (i − 1, j + 1)

+ f (i + 1, j + 1)], i = odd, j = odd

f (i, j), i = even, j = even

(26)

Now we calculate the value | f (i, j) − f (i, j)|2 in the following four cases:

Case (1) i = even, j = odd

f (i, j) − f (i, j)

h= { f (i, j) − f (i, j − 1) − 1

2[ f (i, j + 1) − f (i, j − 1)]}/h

= { f (i, j) − f (i, j − 1) − 1

2[ f (i, j + 1) − f (i, j)

+ f (i, j) − f (i, j − 1)]}/h

= fy(i, j − 1) − 1

2

[fy(i, j) + fy(i, j − 1)

]

= 1

2[ fy(i, j − 1) − fy(i, j)].

By the inequality (a − b)2 ≤ 2(a2 + b2) (a, b ∈ R),

| f (i, j) − f (i, j)|2 = 1

4h2[ fy(i, j − 1) − fy(i, j)]2

≤ 1

2h2[| fy(i, j − 1)|2 + | fy(i, j)|2], (27)

where fy(·, ·) is the forward difference quotient with respect to the first variable y,which can be regarded as an approximation for the partial derivative of the functionf , that is, the terrain slope along the direction of coordinate y.

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478 Math Geosci (2014) 46:445–481

i, j i+1, j

i-1, j-1

i-1, j+1

i-1, j

i, j-1 i+1, j-1

i, j+1 i+1, j+1

Fig. 14 Explanation for the difference scheme

Case (2) i = odd, j = even

Following a similar approach to that of case (1),

| f (i, j) − f (i, j)|2 ≤ 1

2h2[| fx (i − 1, j)|2 + | fx (i, j)|2], (28)

where fx (·, ·) is the forward difference quotient with respect to the first variable x ,which could be viewed as an approximation for the partial derivative of the functionf , that is, the terrain slope along the direction of coordinate x .

Case (3) i = odd, j = odd

As denoted in Fig. 14 above,

f (i, j) − f (i, j) = f (i, j) − 1

4[ f (i − 1, j + 1) + f (i − 1, j − 1)

+ f (i + 1, j − 1) + f (i + 1, j + 1)]

= f (i, j) − 1

4{[ f (i − 1, j + 1) − f (i − 1, j)] + [ f (i − 1, j − 1)− f (i, j −1)]

+ [ f (i + 1, j − 1) − f (i + 1, j)] + [ f (i + 1, j + 1) − f (i, j + 1)]

+ f (i − 1, j) + f (i, j − 1) + f (i + 1, j) + f (i, j + 1)}= −1

4{[ f (i − 1, j + 1) − f (i − 1, j)] + [ f (i − 1, j − 1) − f (i, j − 1)]

+ [ f (i + 1, j − 1) − f (i + 1, j)] + [ f (i + 1, j + 1) − f (i, j + 1)]

+ [ f (i, j) − f (i − 1, j)] + [ f (i, j) − f (i, j − 1)] + [ f (i, j) − f (i + 1, j)]

+ [ f (i, j) − f (i, j + 1)]}

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Math Geosci (2014) 46:445–481 479

Therefore,

f (i, j)− f (i, j)

h= −1

4

[fy(i −1, j)− fx (i −1, j −1)− fy(i +1, j −1)+ fx (i, j +1)

+ fx (i − 1, j) + fy(i, j − 1) − fx (i, j) − fy(i, j)]

= 1

4[ fx (i, j) − fx (i, j + 1) − fx (i − 1, j) + fx (i − 1, j − 1)

+ fy(i, j) − fy(i, j − 1) − fy(i − 1, j) + fy(i + 1, j − 1)]

Thus,

| f (i, j) − f (i, j)|2

≤ 1

8h2

{[ fx (i, j) − fx (i, j + 1) − fx (i − 1, j) + fx (i − 1, j − 1)]2

+ [fy(i, j) − fy(i, j − 1) − fy(i − 1, j) + fy(i + 1, j − 1)

]2}

≤ 1

2h2

{[ fx (i, j)]2 + [ fx (i, j + 1)]2 + [ fx (i − 1, j)]2 + [ fx (i − 1, j − 1)]2

+ [fy(i, j)

]2 + [fy(i, j − 1)

]2 + [fy(i − 1, j)

]2 + [fy(i + 1, j − 1)

]2}

·(29)

where the meanings of fx (·, ·) and fy(·, ·) are the same as those of the previous twocases as explained above.

Case (4) i = even, j = even

By Eq. (26), obviously,

| f (i, j) − f (i, j)|2 = 0. (30)

For the two-dimensional case, the root mean square error (RMSE) is given as

RMSE =√∑

i, j ( f (i, j) − f (i, j))2

n × n.

Therefore,

(RMSE)2 = 1

n2

∑i, j

| f (i, j) − f (i, j)|2

= 1

n2

⎛⎝ ∑

i=even, j=odd

| f (i, j) − f (i, j)|2 +∑

i=odd, j=even

| f (i, j)− f (i, j)|2

+∑

i=odd, j=odd

| f (i, j) − f (i, j)|2 +∑

i=even, j=even

| f (i, j) − f (i, j)|2⎞⎠ .

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480 Math Geosci (2014) 46:445–481

By using Eqs. (27)–(30) with s = 2h (s is resolution of the re-sampled DEM),

(RMSE)2 ≤ 3

4h2 E{(Slope)2} = 3

16s2 E{(Slope)2} (31)

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