ErrorAnalysisonGrid-BasedSlopeandAspectAlgorithms

Qiming Zhou and Xuejun Uu

AbstractSlope and aspect are the most frequently used surfacegeomorphic parameters in terrain analysis. While derivedfrom grid DEM. the parameters often display noticeable errorsdue to errors {aJ in data. {bJ inherent in data structure. and{c} created by algorithms. It has been observed that somecontroversial results were reported in evaluating the resultsby various slope and aspect algorithms. largely because ofthe variety in assessment methodology and the difficultiesin separating errors in data and those generated by thealgorithms. This paper reports the study that assesses andcompares the results from numerous grid-based slope andaspect algorithms using an analytical approach. Tests weremade based on artificial polynomial surfaces which can bedefined by mathematical formulae. with control/able "added"data errors. By this approach. different algorithms werequantitatively tested and their error components wereanalyzed. Thus. their suitability and tolerance related toDEM data characteristics can be described.

IntroductionSlope and aspect have been regarded as two of the mostimportant geomorphic parameters. as they not only efficientlydescribe the relief and structure of the land surface. but arealso widely applied as vital parameters in hydrologicalmodels (Band. 1986; Moore et al.. 1988; Quinn et al.. 1991;1995; Jenson. 1994). landslide monitoring and analysis (Ouanand Grant. 2000), mass movement and soil erosion studies(Dietrich et al.. 1993; Desmet and Govers. 1996a; Mitasovaet al.. 1996; Biesemans et al.. 2000) and landuse planning(Desmet and Govers. 1996b; Stephen and Irvin. 2000).

In most applications today. slope and aspect are typicallyderived from OEM that is based on raster data structure. Todate. there have been numerous mathematical models andalgorithms that calculate slope and aspect from elevation data(e.g.. Sharpnack and Akin. 1969; Fleming and Hoffer. 1979;Horn. 1981; Unwin. 1981; O'Callaghan and Mark. 1984;Zevenbergen and Thorn. 1987; Wood, 1996). Although themathematical definition of slope and aspect is quite clear. itsimplementation based on grid-based OEM may vary. sincesome assumptions must be made on how the continuoussurface is approximated by discrete sample points (i.e., gridcells). The variation in such implementation would only pre-sent minor problems in applications such as surface visualiza-tion and classification. but its impact on terrain analysis thatis based on quantitativp. models could be very significant(Zhou et al.. 1998; Tang. 2000; Zhou and Liu, 2002). It waspointed out that selection of algorithms could be a critical

Q. Zhou is with the Dp.partment of Geography. Hong KongBaptist University. Kowloon Tong, Kowloon. Hong Kong.China (qiming@hkbll.edll.hk).

X. Liu is with the School of Geographical Science. NanjingNormal University. Nanjing 210097. China.

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factor that might create a grp.at impact on the analytical results(Moore. 1996; Burrough and McDonnell, 1998).

Studies have been conducted to analyze the errorscreated by the slope and aspect algorithms with a variety ofapproaches and methodologies. One approach emphasizeserrors in OEM itself while paying little attention on models.For example. Skidmore (1989) and Florinsky (1998) analyzedslope and aspect errors using the actual survey data. Anotherapproach focused on mathematical models while ignoringOEM errors (e.g.. Hodgson. 1995; Jones. 1998). The conclu-sions from these studies have been quite different. sometimescontroversial. For example. Skidmore (1989) concluded thatthird-order finite difference method (Horn 1981) derivedbetter results than the second-order finite difference method.His finding was also confirmed by Florinsky (1998). On theother hand. Hodgson (1995) and Jones (1998) stated that thelatter performed better than the former in computing slopeand aspect values.

To make a fair comparison between different slope andaspect algorithms. three pre-conditions must be satisfied:

1. The source of error should be identified;2. The kinds of errors must be independent. identifiable and

controllable in the test so that their impact can be quantified:and

3. r\ reference (or true value) must be established to make theresults comparable.

Zhou and Liu (2002) reported a method that is based onmathematical surfaces to evaluate the errors generated byflow routing algorithms in digital terrain analysis. Using thismethod. errors created by an algorithm can be isolated fromOEM errors so that the quantitative measures can be made tocompare a derived. specific catchment area with the truevalue defined by the polynomial surfaces.

Using an analytical approach. Florinsky (1998) developedtest for the precision of four methods for computing partialderivatives of elevations. and produced formulae for RootMean Square Error (RMSE)of four local topographic variablesincluding slope and aspect as focused by this paper. Florinskyalso used a real-world OEM to map the error distribution forvisual analysis. but the contribution from the OEM data errorlargely remained unknown.

This paper reports the study that attempts to extend theabove methods into the test of slope and aspect calculation. Inthis study. we extended Florinsky's method to general termsand designed tests on artificial polynomial surfaces with con-trollable data errors. Two representative surfaces. namely. theinverse ellipsoid and Gauss synthetic surface. were used totest six commonly-employed slope and aspect algorithms

Photogrammetric Engineering & Remote SensingVol. 70. No.8, August 2004. pp. 957-962.

0099-1112/04/7008-0957/$3.00/0@ 2004 American Society for Photogrammetry

and Remote Sensing

August 2004 9S7

{Non.We;gl1!ed (Sharpnack and Akin. 1969)

)"orderFlI'IiteDifference Werghled(UnwlO1981)

Frame Finite Difference (Jones. 1998)

Maximum Downhill Slope (O'Callaghan and Mark. 1984)

~ order Finite Difference (Fleming and Hoffer. 1979. Unwin. 1981)Quantitative

AnalysiS

Mathematical

models of slope

andaspectcomputationlrom OEM

linear regression plane (Shalpl1actl and MGri:: 1969)

{Weoghled (Wood 1996)

ConstraIned

{Non-Weoghled (Wood. 1996)

Ouadraticsurface .

{Weoghled (Horn. 1981)

Non-constr arned

Non-Wetghted (Evans. 1980)

Trend Surface

Partial Quadratic surface (Zel/enbergen and Thorn. 1981)

Vec1or-based (Ritter. 1987)

Fast FourierTransforms (Papo and Gelbman, 1984)

Figure 1. Algorithms deriving slope and aspect from OEM.

found in literature and GIS software. The derived slope andaspect values were then compared with the true value derivedby mathematical inference. Quantitative analysis can then beconducted for the comparison of the algorithm accuracies.

Slope and Aspect AlgorithmsAt a given point on a surface z = I(x. y), the slope (5) andaspect (A) is defined as a function of gradients at X and Y(Le.. WoE and N-S) directions. Le.;

5 = arctan VI; + IJ

(Ix)/,

A = 270° + arctan /, - 90° 1/,1

where Ix and Iy are the gradients at WoEand N-S directions.respectively.

From the above equations. it is clear that the key for slopeand aspect computation is the calculation of /, and Iy- Using agrid-based OEM. the common approach is to use a moving3 X 3 window to derive finite differential or local surfacefitting polynomial for the calculation. Figure 1 shows variousmethods found in the literature and GIS software.

Considering the popularity and the use of different algo-rithms. we have selected six commonly-employed algorithmsfor test (Table 1):

. Second-order Finite Difference (2FD)(Fleming and Hoffer.1979:Zevenbergen and Thorne. 1987;Ritter. 1987).

. Third-order Finite Difference (3FO)(Sharpnack and Akin.1969; Horn, 1981; Wood. 1996).

. Third-order Finite Difference Weighted by Reciprocal ofSquared Distance (3FDWRSO) (Horn. 1981).

. Third-order Finite Difference Weighted by Reciprocal ofDistance (3FDWRO) (Unwin. 1981).

. Frame Finite Difference(FFO)(Chu and Tsai 1995).and

. Simple Difference(SIMPLE-D)(Jones 1998).

ErrorComponentsThe accuracy of slope and aspect computation is directly re-lated to the partial derivatives at X and Y direction. /, and Iv.which are estimated by numerous proposed methods. Takingthe second-order finite difference as an example. let (x. y)denote the coordinates of the center cell in a 3 x 3 windowand g denote the OEM spatial resolution (Le.. grid cell size),the partial differential can be expressed as (King. 2000):

d'f = gZ (r;(tx'y) + 1:'('Yx'y))dz. - dzz

x 6 2 +(3.1)

(2.1)

The first term represents the errors caused by the uncertaintyof mathematical model implementation and tx' t,.. 'Yxand 'Yvaredependences off"'. gx E (x. x + g), gv E (y, Y + g), 'Yx E (x --g. x)and 'YyE (y - g. y). Since the relationships between these vari-ables and x and yare usually not clear. it is difficult to definetheir values in application. Thus it is common to set the upperlimits for f'" instead. Let Mxand My as the upper limits of f'" interms of x and y, respectively; Equation 3.1 can be altered as:(2.2)

gZ dZ8 - dzz

dlx = r;M, + 2g

gZ dZ6 - dZ4

dly = r;My + 2g

The second term of the equation represents OEM data error(including data precision).

(3.2)

TABLE 1. MATHEMATICALMODELS OF SLOPE AND ASPECT COMPUTATIONFROM DEM

2ndorder finite difference (2FD). Vector-based algorithmDPartial quadratic equation

3,,1order finite difference (3FD). Linear regression plane. Unconstrainedquadratic surface, constrained quadratic surface

3"1order finite difference weighted by reciprocal of squared distance (3FDWRSD).Weighted constrained quadratic surface. Weighted unconstrained quadratic surface

3"1order finite difference weighted by reciprocal of distance (3FDWRD)

I, = (Z6 - z4)/2g.fy = (Z8- z2)/2gfx = (Z3 - z, + Z6- Z, + Z9 - z7)/6gfy = (Z7 - z, + Z8- Z2+ Z9- z3)/6gfx = (z" - z, + 2(Z6 - z,) + Zq- z7)/8gf. = (Z7 - ZI + 2(Z8 - Z2) + Zq - z,,)/8gI, = (Z3 - z, + \I2(Z6 - Z4)+ Z9- z7)/(4+2\12)gfv = (Z7 - z, + \I2(z. - zzJ + 29 - z,,)/(4+2\12)g

I, = (zJ - z, + Z9- zN4gfy = (Z7 - z, + Zq- 23)/4g

I, = (Z5 - Z4)/g.[y = (25 - 22)/g

Frame tlnite difference (FFD)

Simple difference (Simple-D)

958 August 2004

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7 I 8 I 9 5 = arctan Vf; + f

4 I 5 I 6A = 2700 + arctan(t) - 900Tf'r

2 I 3 g = spatial resolution (i.e.. grid cell size)

In Equation 3.2, Mx and Myare estimated based on the'worst scenario', usually represent much larger estimatesthan the actual case, with a given distribution of probability.Let the RMSE of M, and Myequal to M, and m denote the RMSEof the OEM, the result is:

(2

)2

m2 = m2,

= fL M2 + m2J. J. 6 2rf

Deriving the differentials of slope and aspect equations(Equations 2.1 and 2.2), and considering 5 = arctanVI~ + I~and tan2 5 = I~ + f~; the result is: .

Referring to Equation 3.3, the RMSE of slope (ms) and aspect(mA) can therefore be expressed as:

m~ = [(~2r M2 + *]cos4 5

[(

2

)2 2

]2 1 g 2 m

mA = tan25 '6 M + zgr

Note that Equation 3.5 is for the second-order finite differencemethod. Let a = f;g2and b = Jzg-1; Equation 3.5 can then beexpressed in a general form:

ms = Va2~ + b2m2cos2 5

m = ~Va2~ + b2m2A tan 5

Similar to the above procedure and referring to Table 1, wecan deri ve the RMSEfor each selected algorithm as shown inTable 2.

Equation 3.6 and Table 2 show that the overall errors ofderived slope and aspect come from three sources:

1. Algorithm errors:caused by approximation and sampling ofcontinuous surfaces (VariableM in Equation 3.6),

2. DEMdata errors:caused by DEMdata capture and generation(Variablem in Equation 3.6), and

3. DEMspatial resolution (i.e., the grid cell size-Variables a andb in Equation 3.6, defined in Table 2).

(Note that the coefficient b of m for 2FDand 3FDconfirms theresults reported by Florinsky (1998), which represent specialcases for the selected algorithms.)

TABLE 2. THE RMSE FOR SELECTEDSLOPE AND ASPECT ALGORITHMS

RMSE of slope

RMSEof aspect

Algorithm

mA = Va 2M2 + b2m2/tgS

Coefficient a of M Coefficient b of m

2ndorder Finite Difference ~-tylZg--.L - tvi5g1

V5.33g-t

3'd order Finite Difference

3,d order Finite DifferenceWeighted by Reciprocal of

Squared Distance

3,d order Finite DifferenceWeighted by Reciprocal

Distance

1V5.83g-'

frame finite Difference

Simple Difference

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(3.3)

Figure 2. Surface defined by an inverse ellipsoid (A = 400,B = 300 and C = 300).

(3.4)

(3.5)

TestDesignBy examining Equations 3.5 and 3.6, we understand that for agiven OEM spatial resolution (g), the accuracy of derivedslope and aspect is related to the errors by the algorithm thatderives Ix and Iy (M), and OEM error (m). The overall accuracyof slope and aspect is dependent upon which of the M and IIIdominates the analysis. While analyzing the algorithms usingthe real-world OEM (e.g., Skidmore, 1989; Chang and Tsai,1991; Florinsky, 1998; Bolstad and Stowe, 1994), the OEMerror (m) would cause much greater error than that by thealgorithm, so that the method would be more appropriate foranalyzing the impact of data error on derived slope and aspectvalues. On the other hand, using OEM defined by mathemati-cal surfaces (Hodgson, 1995; Jones, 1998; Carter, 1992) wouldeliminate data error, thus the observed errors would only becaused by algorithms. Since the error sources could not bedefined, the results of the studies appeared inconclusi ve.

Our approach is to employ mathematical surfaces, in asimilar way to Hodgson (1995) and Jones (1998), but witha complexity that represents a closer approximation to theactual land surface. For this purpose, we have selectedinverse ellipsoid (Equation 4.1) and Gauss synthetic surface(Equation 4.2) to define the surfaces and generated OEM for agiven resolution (Figure 2 and Figure 3).

(3.6)

(z < 0) (4.1)

Z = A[ 1 - C~ YJe-(;~;)'-(*+1)'- B[ 0.2C~) - (;~J

- (*rJ e -(~r-(M - Ce-((~)+,r-(H

(4.2)

where A, Band C are parameters determining surface relief,and m, n in Equation 4.2 are the parameters controlling the

Figure 3. A Gauss synthetic surface (A = 3, B = 10, C =1/3, -500 ~ X, Y ~ 500).

August 2004 959

spatial extent of the surface. The true values of slope and as-pect can be computed using the above equations and Equa-tions 2.1 and 2.2.

On the OEM generated from the above surfaces. the se-lected algorithms have been applied to compute slope and as-pect values and statistics were then generated to compare theRMSEbetween results derived by different algorithms. In orderto analyze the impact of OEM data error. we added some noise(Le.. random errors) into the generated OEM to simulate OEMdata error.

Resultsand DiscussionTables 3 and 4 summarize the RMSEcomparison results betweenthe selected six algorithms on two surfaces (inverse ellipsoidand Gauss synthetic surface) with no added error (Le.. OEMdata error = 0). Figures 4 and 5 shows the RMSEof derived slopeand aspect on OEM with various added data errors.

By analyzing Tables 3 and 4. it is shown that on a OEMwith high accuracy. the error of derived slope and aspect issourced from the estimates of partial derivatives Ix and h.andsampling errors. In this case the second-order finite differencemethod provides a better result than the third-order finitedifference methods. For the tested six algorithms. from thebest to worst the order is Second-order Finite Difference (2FD),Third-order Finite Difference Weighted by Reciprocal ofSquared Distance (3FDWRSD),Third-order Finite DifferenceWeighted by Reciprocal of Distance (3FDWRO),Third-orderFinite Difference (3FD),Frame Finite Difference (FFO),andSimple Difference (SIMPLE0).

In Equation 3.6. if we ignore the algorithm error (Le.. thefirst term). we then derive the Florinsky's (1998) RMSEformu-lae in general terms:

m:; = bmcos"5 andbm

111A = tan S(5.1)

Equation 5.1 suggests that the influence of OEM data error (m)relates to slope (5) and coefficient b. which is determined by

960 August 2004 PHOTOGRAMMETRICENGINEERING& REMOTESENSING..

grid cell size g. Referring to Table 2. the third-order finitedifference methods appear to be less sensitive to data errorsfor given g and 5.

Algorithm error and OEM data error differently influencethe accuracy of derived slope and aspect. In general. all thethird-order finite difference methods have applied somesmoothing effect on the local data window, in order to avoidlocal relief extremes (Burrough and McDonnell. 1998) forbetter computation results of surface parameters. On the otherhand. the second-order finite difference and simple difference

0.6 !I 6.6 95 15I 19.5RMSLofl)l'M dala ,

m3FO

I!J3FOWRD I

fl3fUWRSO .~H.n

r<l2fD

EISinvIeD

Figure 4. Comparison between accuracyof derived slopeand aspect on inverse ellipsoid with increasing data errors.

----

TABLE3. STATISTICSOFDERIVEOSLOPEWITHNo DATAERROR(UNITS:DEGREES)

Frequency of Positive andRMSE Standard Error Mean Error Negative Errors ('Yo)- -

Gauss Gauss Gauss GaussSVllthetic Synthetic Synthetic Synthetic

Algorithm Ellipsoid Surface Ellipsoid Surface Ellipsoid Surface Ellipsoid Surface

2FD 0.216 0.003 0.189 0.002 0.104 -0.002 100/0 15/853FO 0.409 0,(J04 0.368 0.002 0.178 -0.003 100/0 9/913FDWRSD 0.358 0.003 0.320 0.002 0.160 -0.003 100/0 8/923FDWRO 0.384 0.004 0.345 0.002 0.170 -0.003 100/0 9/91FFO 0.507 0.004 0.459 0.003 0.214 -0.003 100/0 10/90StMPLEO 1.295 0.046 1.292 0.046 0.078 -0.002 51/49 49/51

TABLE4. STATISTICSOFDERIVEOASPECTON WITHNo DATAERRORS(UNITS:DEGREES)

Frequency of Positive andRMSE Standard Error Mean Error Negative Errors (%)-

Gauss Gauss Gauss GaussSvnthetic Svnthetic Synthetic Synthetic

Algorithm Ellipsoid Surface Ellipsoid Surface Ellipsoid Surface Ellipsoid Surface

2FO 0.133 0.117 0.133 0.117 0.000 -0.008 52/48 45/553FO 0.197 0.130 0.197 0.130 -0.001 0.000 52/48 51/493FDWRSO 0.122 0.118 0.122 0.118 -0.001 -0.002 52/48 48/523FOWRO 0.160 0.124 0.160 0.124 -0.001 -0.001 52/48 50/50FFD 0.342 0.167 0.342 0.167 -0.002 0.004 52/48 52/48SIMPLEO 20.76 15.13 20.63 15.128 2.347 -0.340 50/50 52/48

0.6 21 15.1 19.56.6 9.5

'.l~: dII:

.

.-

.

-

. D.. -00.01 ..

~~so::E !'" 0 . .. + . .

0.6 21 6.6 9.5 15.1 19.5

1!:13m

i!!3mWRD

1i131.nWRSIJ

6:!I-lU

02FD

EiSi""IeD

()---~ ~~~IIIii-:!) -.E:2§ 840 - --- ...~-00

&) -

Rgure 5. Comparison between accuracy of derived slope

and aspect on Gauss synthetic surface with increasing

data errors.

methods only utilize a part ofsamples in the local window, sothat they are more sensitive to the data error. This fact is alsoconfirmed by Figures 4 and 5.

The results also confirm the test results reported by Jones(1998). While using an error-free synthetic, trigonometricallydefined surface (Morrison's surface III), the 2FDmethod gavethe best results for both gradient and aspect. Using a real-world OEM, where data errors were unavoidable, the 3FDper-formed best.

ConclusionSlope and aspect are two most frequently utilized variablesin GIS-based terrain analysis and geographical modeling.There have been numerous analyses on the accuracy of thesevariables derived from grid-based OEM. The reported find-ings, however, did not always agree, and sometimes they arecontroversial. This study attempts to evaluate the issues andestablish a fair quantitative measure for assessing variousslope and aspect algorithms. With the findings of this study,we can draw the following conclusions:

1. It is important to identify the sources and nature of errors ofderived slope and aspect for evaluating algorithms and mathe-matic models.

2. Evaluation of algorithms and models must be based on an ob-jective, data-independent methodology. Our approach is basedon OEM defined by mathematical surfaces, thus the OEM dataerror can be controlled to make the fair comparison betweenselected algorithms.

3. On a OEM with high accuracy, the error of derived slope andaspect is sourced from the estimates of partial derivatives f,and 1;. and sampling errors. In this case the second-order finitedifference method provides a better result than the third-orderfinite difference methods.

4. In reality, the influence of OEM data error is in generalmuch larger than the algorithm errors. thus it suggests thatthe third-order finite difference method would be moreappropriate for applications since it is least sensitive to theOEM data error.

Further study will be focused on the impact of grid data5tructure and other related parameters such as resolution, pre-cision, orientation, and surface complexity Oil the results ofdigital terrain analysis. Mathematical models other than slope

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and aspect will also be analyzed for their algorithm accuracyand sensitivity to data errors. The real-world tests will beneeded to compare with the findings by the theoretical analy-sis. Based on these analysis, the ultimate goal is to set a con-clusive guideline for deriving geomorphic parameters fromOEM for a given application project.

AcknowledgmentsThis study is supported by Hong Kong Baptist UniversityFaculty Research Grant FRG/98-99/II-35, "3-0imensionalHydrological Modeling." The constructive comments andsuggestions for improving the quality of this paper fromanonymous referees are also acknowledged.

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To place your order. fill out thenecessary mailing and certifica-tion information. Cost is just $35for a stamp and $45 for a seal.these prices include domestic USshipping. International shippingwill be billed at cost. Pleaseallow3-4 weeksfor delivery

SIGNATURE DATE

SEND COMPLETED FORM WITH YOUR PAYMENT TO:

ASPRS Certification Seals[. Stamps. S410 Grosvenor Lane. SUite 210. Bethesda. MD 20814.2160

NAME PHONE.CERTIFICATION' EXPIRATION DATE

ADDRESS.COUNTRY

PLEASE SEND ME:O Embossing Seal $45 0 Rubber Stamp . $35

METHOD Of PAYMENT: 0 Check 0 Visa 0 MasterCard 0 American Express

CITY

.

STATE POSTAL CODE

CREDIT CARD ACCOUNT NUMBER

962 AuguSt 2004

EXPIRES

PHOTOGRAMMETRICENGINEERING& REMOTESENSING