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Exploring scaling laws in surface topography M.J. Abedini a, * , M.R. Shaghaghian b a Dept. of Civil and Environmental Engineering, University of Waterloo, Waterloo, ON, Canada N2L 3G1 b Dept. of Civil and Environmental Engineering, Shiraz University, Shiraz, Iran article info Article history: Accepted 30 March 2009 abstract Surface topography affects many soil properties and processes, particularly surface water storage and runoff. Application of fractal analysis helps understand the scaling laws inher- ent in surface topography at a wide range of spatial scales and climatic regimes. In this research, a high resolution digital elevation model with a 3 mm resolution on one side of the spectrum and large scale DEMs, with a 500 m spatial resolution on the other side were used to explore scaling laws in surface topography. With appropriate exploratory spatial data analysis of both types of data sets, two conventional computational procedures – vari- ogram and Box Counting Methods (BCM) – address scaling laws in surface topography. The results respect scaling laws in surface topography to some extent as neither the plot treat- ment nor the direction treatment has a significant impact on fractal dimension variability. While in the variogram method, the change in slope in Richardson’s plots appears to be the norm rather than the exception; Richardson’s plots resulting from box counting implemen- tation lack such mathematical behavior. These breaks in slope might have useful implica- tions for delineating homogeneous hydrologic units and detecting change in trend in hydrologic time series. Furthermore, it is shown that fractal dimension cannot be used to capture anisotropic variabilities both within and among micro-plots. In addition, its numerical value remains insignificant at the 5% level in moving from one direction to another and also from one spatial scale to another while the ordinate intercept could dis- criminate the surface roughness variability from one spatial scale to another. Ó 2009 Elsevier Ltd. All rights reserved. 1. Introduction Surface topography controls many transport processes on and/or across the soil surface boundary at a range of spatial scales [1]. From hydrological point of view, almost all processes within the hydrologic cycle either directly or indirectly are affected by surface topography. The earth’s surface topography is extremely variable over a wide range of spatial and temporal scales. It strongly varies from one location to another and from one spatial scale to another, making it hard to be tackled with traditional available tools. Exploring the self-organization inherent in surface topography and providing a realistic descriptions and models of its variations over a wide range of scales have long been a basic challenge not only to hydrologists [2], but also to physicists and mathematicians [3–5], hydraulic engineers [6], soil physicists [7,8], geomorphol- ogists [9], environmental scientists [10,11], Material scientists [12,13], geographers [14,15], geoscientists [16] and many more. An accurate description of surface topography could be used to better explain many aspects of surface water hydrol- ogy. In practical terms, such a description could also be used as input in various applications involving topography/bathym- etry as a boundary condition. 0960-0779/$ - see front matter Ó 2009 Elsevier Ltd. All rights reserved. doi:10.1016/j.chaos.2009.03.121 * Corresponding author. On sabbatical leave from: Department of Civil and Environmental Engineering, Shiraz University, Shiraz, Iran. Tel.: +98 711 6474604; fax: +98 711 6473161. E-mail address: [email protected] (M.J. Abedini). Chaos, Solitons and Fractals 42 (2009) 2373–2383 Contents lists available at ScienceDirect Chaos, Solitons and Fractals journal homepage: www.elsevier.com/locate/chaos

Exploring scaling laws in surface topography

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Page 1: Exploring scaling laws in surface topography

Chaos, Solitons and Fractals 42 (2009) 2373–2383

Contents lists available at ScienceDirect

Chaos, Solitons and Fractals

journal homepage: www.elsevier .com/locate /chaos

Exploring scaling laws in surface topography

M.J. Abedini a,*, M.R. Shaghaghian b

a Dept. of Civil and Environmental Engineering, University of Waterloo, Waterloo, ON, Canada N2L 3G1b Dept. of Civil and Environmental Engineering, Shiraz University, Shiraz, Iran

a r t i c l e i n f o a b s t r a c t

Article history:Accepted 30 March 2009

0960-0779/$ - see front matter � 2009 Elsevier Ltddoi:10.1016/j.chaos.2009.03.121

* Corresponding author. On sabbatical leave from6474604; fax: +98 711 6473161.

E-mail address: [email protected] (M.J. Abed

Surface topography affects many soil properties and processes, particularly surface waterstorage and runoff. Application of fractal analysis helps understand the scaling laws inher-ent in surface topography at a wide range of spatial scales and climatic regimes. In thisresearch, a high resolution digital elevation model with a 3 mm resolution on one side ofthe spectrum and large scale DEMs, with a 500 m spatial resolution on the other side wereused to explore scaling laws in surface topography. With appropriate exploratory spatialdata analysis of both types of data sets, two conventional computational procedures – vari-ogram and Box Counting Methods (BCM) – address scaling laws in surface topography. Theresults respect scaling laws in surface topography to some extent as neither the plot treat-ment nor the direction treatment has a significant impact on fractal dimension variability.While in the variogram method, the change in slope in Richardson’s plots appears to be thenorm rather than the exception; Richardson’s plots resulting from box counting implemen-tation lack such mathematical behavior. These breaks in slope might have useful implica-tions for delineating homogeneous hydrologic units and detecting change in trend inhydrologic time series. Furthermore, it is shown that fractal dimension cannot be usedto capture anisotropic variabilities both within and among micro-plots. In addition, itsnumerical value remains insignificant at the 5% level in moving from one direction toanother and also from one spatial scale to another while the ordinate intercept could dis-criminate the surface roughness variability from one spatial scale to another.

� 2009 Elsevier Ltd. All rights reserved.

1. Introduction

Surface topography controls many transport processes on and/or across the soil surface boundary at a range of spatialscales [1]. From hydrological point of view, almost all processes within the hydrologic cycle either directly or indirectlyare affected by surface topography. The earth’s surface topography is extremely variable over a wide range of spatial andtemporal scales. It strongly varies from one location to another and from one spatial scale to another, making it hard tobe tackled with traditional available tools. Exploring the self-organization inherent in surface topography and providing arealistic descriptions and models of its variations over a wide range of scales have long been a basic challenge not only tohydrologists [2], but also to physicists and mathematicians [3–5], hydraulic engineers [6], soil physicists [7,8], geomorphol-ogists [9], environmental scientists [10,11], Material scientists [12,13], geographers [14,15], geoscientists [16] and manymore. An accurate description of surface topography could be used to better explain many aspects of surface water hydrol-ogy. In practical terms, such a description could also be used as input in various applications involving topography/bathym-etry as a boundary condition.

. All rights reserved.

: Department of Civil and Environmental Engineering, Shiraz University, Shiraz, Iran. Tel.: +98 711

ini).

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2374 M.J. Abedini, M.R. Shaghaghian / Chaos, Solitons and Fractals 42 (2009) 2373–2383

Modeling surface topography using scaling laws dates back to late 60 with influential and remarkable work of [3]. Later,he invented an analytical tool called fractional Brownian surfaces (fBs) to model surface terrain [17]. Subsequently, manydirect and indirect fractal analysis of surface topography were made for a variety of purposes to see if topography respects‘‘fractal” statistics. A variety of estimators are available to compute fractal parameters from topographic surfaces including 1-D and 2-D variogram methods [6,7,18,19]; spectral method [5,13,20]; Box Counting Method (BCM) [12,19,21] and manymore. Interested readers may want to consult with a few review papers including [9]; [10] and [15] for further detail regard-ing method of computations and current research agenda and directions. A quick glance at existing literature on fractal anal-ysis of surface topography showed that in contrast to indirect monofractal-based computation of fractal parameters, directestimates of fractal dimensions of topography and bathymetry (i.e., BCM) – where the raw data have the final say – receivedscant attention [5,15].

In addition to method-induced variations and algorithmic-induced variations within each method, a number of other fac-tors such as the choice of input parameter values and digital elevation data used may also affect computed Df values. Due toavailability of a variety of estimators and conflicting results obtained, there is considerable uncertainty regarding the natureand extent of variations in computed Df values which somehow may limit the utility of fractal dimension as an index of dis-crimination. As such, a few investigators question the utility of describing surface topography in fractal terms and suggestedto resolve these issues with further research on estimator comparison in the context of a variety of real data in different cli-matic regimes and at a range of spatial scales [9,10,15].

Usefulness of fractal analysis to address scaling laws in surface topography lies in the fact that features corresponding tomicro and macro-topography broadly manifest themselves at a range of spatial scales. Different elements, ranging from indi-vidual grains, aggregates, clods, tillage marks and landscape features contribute to fluctuations in surface topography at theirrespective scales. Data sets with very fine spatial resolution at the size of a grain or less could resolve changes in elevation atthe grain level; while data sets at higher spatial resolutions simply dismiss fluctuations in elevation at the grain size level.This nested structure of surface topography at different spatial scales could be considered as a good cause to collect DEMs ata range of spatial resolutions and climatic regimes, subject them to rigorous fractal analysis and then monitor the change infractal dimension while moving from one spatial resolution to another.

This paper intends to address these issues by undertaking a fully two-dimensional approach to fractal characterization,achieved by using fully three-dimensional digital elevation models obtained using both laser scanner and digital photogram-metry in totally two different continents at six order of magnitude spatial resolution variations. This characterization isundertaken to assess whether or not a two-dimensional approach will respect fractal statistics under contrasting scenarios,how the conflicting results in current literature can be explained and to consider what, if any, additional information such anapproach can provide. Thus, the objectives of this paper are to (a) identify appropriate tools for the two-dimensional fractalanalysis of topographic surfaces that can yield reliable estimates of Df ; and (b) assess the various implications of preliminaryresults. The next section provides a brief account of the basic concepts of fractal geometry, followed by a description of studyarea and two computational methods used in this study. Extensive analysis of DEMs at a range of spatial scales and climaticregimes are then presented. Major issues encountered in fractal analysis of surface topography are discussed and the paper isconcluded with some general remarks.

2. Fractals, self-similarity, self-affinity and the fractal dimension

Many important features and patterns of nature are so irregular that classical Euclidean geometry is hardly of any help indescribing their form. It was this inability of classical geometry to describe the real world that led [22] to invent the conceptof ‘‘fractal” to fill the void caused by the absence of suitable geometric representation for a family of shapes that are contin-uous but not differentiable. A fundamental characteristic of fractal objects is that their measured metric properties, such aslength or area, are a function of the scale of measurement. In fractal geometry, dimension is treated as a continuum insteadof a crisp and even number. A curve’s dimension, for example, can take on any non-integer value between 1 and 2, dependingon the degree of irregularity of its form. Similarly, a surface’s dimension may be a non-integer value between 2 and 3. In asense, fractal dimension can be thought of as a measure of an object’s ability to ‘‘fill” the space in which it resides. More gen-erally, the more irregular an object becomes, the more space it fills, and the higher its Df value. While fractal dimension couldbe found analytically for regular fractals, for random fractals, fractal dimension can only be found via numerical methods.

Self-similarity is another key property of fractals. Formally, self-similarity is defined as a property where the small objectsare essentially reduced versions of the large objects, and the large objects are enlarged versions of the small ones. In a sense,The property of self-similarity or for that matter, scale invariance implies that the form of an object is invariant with respectto spatial scale.

Self-similar objects – either in a strict or statistical sense – are isotropic (or rotation invariant) upon rescaling. If rescalingof an object is anisotropic, then the object is said to be self-affine. Formally, with self-affine fractals, the variation in onedirection scales differently than the variation in another direction [4]. The fractal dimension (Df ) can be used to characterizeboth self-similar and self-affine features extracted from complex natural topography. It could be an interesting observationto see if anisotropy in variogram parameters such as range or sill anisotropy could lead to anisotropy in fractal dimensionand vice versa. If anisotropy in fractal dimension can be linked to anisotropy in variogram parameters, then fractal analysiscan be considered as a useful tool to address and delineate anisotropy in geostatistics.

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3. Materials and methods

3.1. The study area

Central to the application of two-dimensional fractal analysis is the availability of high quality and two-dimensionalDEMs. Given that the focus of this research involved exploring scaling laws in surface topography, it was necessary to usespecially collected DEMs from both the field and the laboratory at a range of spatial scales and climatic regimes. For this pur-pose, two sets of data from two totally different continents at six orders of magnitude spatial scales were collected descrip-tion of which will be summarized in the following paragraphs:

3.1.1. Micro-plot data setsThe micro-plots selected for the present study involved runoff plots located on the northeastern slope of a drumlin

(extending from the northwest to the southeast) on the campus of the University of Guelph, Guelph, Ontario. Ten runoffplots, each 202 m2 in area, with a length of 44.2 m, a width of 6.4 m and a relatively uniform slope of 7–9% along the plotlength, had been established on the drumlin in 1963 and have been extensively studied by other researchers for various pur-poses over the years, including the characterization of surface depressions [23,24].

At the time of collecting DEMs in this area, all runoff plots had received the same surface treatment with corn residualsleft over the ground. After visual inspection of the plot surfaces, three micro-plots having relatively similar slopes were se-lected for studying scaling laws in surface topography. Surface elevation for the three micro-plots was digitized by using aportable laser scanner [25]. An area of 1:0� 1:0m was digitized for each micro-plot with a 3-mm grid spacing both along thescan-line and between the scan-line. Approximately 110,000 elevation data were collected from each micro-plot. Fig. 1shows the histograms of elevation data along with shaded-relief map of the three micro-plots.

3.1.2. Watershed scale data setsThe study area selected for the large scale part of this research involved the mountainous region of Kohkiloyeh and Bou-

yerahmad province in the South Western part of Iran with a total area of about 15,463 km2. Physiography within the studyarea varies from the near-horizontal depositional surfaces of the Gachsaran and Dehdasht regions to the rugged escarpmentsand steep slopes formed by fault scarps and erosional surfaces in the surrounding mountain ranges. Elevations in the studyarea range from about 4287 m in the headwater of Khersan watershed to as low as 32 m in the flat land of Behbahan. Forareas with elevation greater than 2500 m, the precipitation occurs in the form of snow. For the purpose of this study, tworectangular scenes were extracted from the original province DEMs (spatial resolution ¼ 500 m). The first scene locatedbetween longitude 51�5301500 and 53�3801200 east and between latitude 28�4704900 and 30�4005400 north. The second scene

Freq

uenc

y

Elevation (mm)28.0 38.0 48.0 58.0 68.0 78.0 88.0 98.0 108.0

0.000

0.040

0.080

0.120

Adjusted elevation data, Plot INumber of Data 111870

mean 57.3891std. dev. 7.2941

coef. of var 0.1271maximum 109.1000

upper quartile 61.9000median 57.0000

lower quartile 52.9000minimum 28.0000

(a) Micro-plot I

Freq

uenc

y

Elevation (mm)24. 44. 64. 84. 104. 124.

0.0000

0.0100

0.0200

0.0300

0.0400

0.0500

0.0600

0.0700

0.0800

Adjusted elevation data, Plot IINumber of Data114570

mean62.0024std. dev.15.7849

coef. of var0.2546maximum128.0000

upper quartile72.4000median61.9000

lower quartile51.6000minimum24.3000

(b) Micro-plot II

Freq

uenc

y

Elevation (mm)14. 54. 94. 134.

0.000

0.020

0.040

0.060

0.080

0.100

Adjusted elevation data, Plot IIINumber of Data 113886

mean 43.6711std. dev. 11.3683

coef. of var 0.2603maximum 134.6000

upper quartile 50.2000median 42.7000

lower quartile 35.6000minimum 14.4000

(c) Micro-plot III

(d) Micro-plot I (e) Micro-plot II (f) Micro-plot III

Fig. 1. Frequency histogram and surface-shaded relief display for three micro-plots.

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located between longitude 52�2504700 and 54�5902200 east and between latitude 27�5903600 and 29�1801400 north. Fig. 2 showsthe histograms of elevation data along with shaded-relief map of the two rectangular scenes.

3.2. Computational methods

A large number of methods have been proposed to compute the fractal dimension of natural objects. These methods differin the ways they measure the quantities of the object under consideration using various step sizes. They all share the follow-ing three procedural steps in calculating fractal dimension:

� First, measure the quantities of the object under consideration using various step sizes.� Second, plot log (measured quantities) versus log (step sizes) and fit a least squares regression line through the data

points. The log–log plot is often referred to as the Richardson plot.� Third, use the slope of the regression line to derive the Df of the object.

Broadly speaking, there are two basic approaches for computing the Df of objects. The first is to directly estimate Df fromthe surfaces being analyzed. In the second method, fractal dimension will be computed in some indirect way. While fractalanalysis via direct methods (i.e., box counting method) works with original elevation data without making any restrictiveassumption, the indirect methods start by hypothesizing a priori that the surface under consideration resembles fractionalBrownian surfaces, and then exploit special monofractal relations to find the unique fractal dimension from structural func-tions (variograms), power spectra or other statistical exponents. In either case (i.e., direct or indirect methods), one couldtake an approach so-called dimensionality-reduction technique whereby the Df of a surface is estimated by first calculatingthe Df of contours or profiles extracted from the surface and, then, simply adding 1 to account for the topological dimension.In this study, two estimators namely; variogram and box counting methods were selected from each group for comparisonpurposes.

3.2.1. Variogram methodStructural analysis (variogram method) is an effective tool to study the effect of scale on topographic organization and its

possible scaling because the variance of topographic properties is treated as a function of scale. The range of scales wherespatial dependence is present can be identified from a plot of semi-variance against the lag distance (called variogram),

Rel

ativ

e Fr

eque

ncy

Elevation (m)610. 1610. 2610. 3610.

0.000

0.050

0.100

0.150

0.200

0.250

Hitogram for DEM of Window 1Number of Data 141192mean 1989.13std. dev. 414.02coef. of var 0.21maximum 3352.80upper quartile 2273.92median 1958.18lower quartile 1828.80minimum 609.60

(a) Window I

Rel

ativ

e Fr

eque

ncy

Elevation (m)305. 1305. 2305. 3305.

0.000

0.050

0.100

0.150

0.200

Hitogram for DEM of Window 2Number of Data 149985mean 1462.39std. dev. 474.50coef. of var 0.32maximum 3048.00upper quartile 1828.80median 1463.77lower quartile 1219.20minimum 304.80

(b) Window II

(c) Window I (d) Window II

Fig. 2. Frequency histogram and surface-shaded relief display for large scale scenes.

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the length of the range (i.e., the distance over which the elevations are no longer correlated), and the general form of spatialvariation of surface topography can be readily visualized from this plot [18]. For many natural phenomena including surfacetopography, the variogram tends to increase with sampling intervals, reaching a maximum value (i.e., a sill), then leveling off[26]. Basic relationships used for the variogram and fractal analyses can be found elsewhere [10,27].

3.2.2. Box counting methodGenerally speaking, Box Counting Method amounts at computing the number of cells (i.e., pixel for 2-D and voxel for 3-D)

required to entirely cover an object, with grid of cells of varying size working directly with original digital elevation data.Practically, this is performed by superimposing regular grids over an object and by counting the number of occupied cells.The logarithm of NðdÞ, the number of occupied cells, versus the logarithm of 1=d, where d is the size of one cell, gives a linewhose gradient corresponds to the box dimension.

1 Due

Df ¼ � limd!0

log NðdÞlog d

ð1Þ

3.3. Application of methods

The methods embodied in variogram analysis and BCM were applied to all DEMs collected for this study. For all applica-tions, and following recommendations made by [9] and [10], a distance of reliability was adopted, with the maximum lagconsidered taken as one third of the maximum separation distance in each case. Number of lags, unit separation distanceand other variogram parameters were selected in such a way to have uniform distribution of number of data points contrib-uting to each lag.

GSLIB [28] was utilized to analyze the spatial structure of the elevation data sets. Within the framework of GSLIB, it ispossible to estimate range and sill anisotropy of the elevation data sets [29]. In order to investigate the possibility of eitherrange and/or sill anisotropy [26, p. 385] in the elevation data sets, three different directions were selected: namely, x; y andx ¼ y directions. For the sake of comparing results to BCM, omnidirectional variograms were also prepared for all DEMs. Fur-thermore, in order to investigate the impact of overall slope on fractal dimension, all five DEMs went through three types ofdetrending process (i.e., linear, quadratic and cubic) and both computational procedures were also applied on detrendeddata. The graphical representations of the omnidirectional and directional experimental variogram, which were determinedfor all DEMs were shown in Figs. 3 and 4.1 All variograms reveal a general trend of sloping straight lines linked at longer lagseparations to a region where the slope decreases gradually toward an asymptote.

By visualizing graphical representation of cðhÞ versus h in log–log space, a bi-fractal behavior was identified for a majorityof variogram plots regardless of spatial resolution and climatic regimes. In order to reduce the subjectivity and uncertaintyinvolved in extracting useful information from variogram plots in log–log space, a segmented linear regression was con-ducted by performing separate linear regressions to the data with h values smaller and greater than a certain separation va-lue, the breakpoint. Segmented linear regression applies linear regression to ½h; cðhÞ� pairs that does not have an overall linearrelation but behaves linearly in segmented regions called bands. It introduces one or more breakpoints, whereby separatelinear regressions are made for the linear segments. Thus, the non-linear relation is approximated by linear segments. A crit-ical element is to delineate the breakpoint. [30] have presented a method for calculating confidence intervals of the break-points so that the breakpoint with the smallest interval, i.e., the optimum breakpoint, can be selected. This enables one toclassify non-linear relations into various linear bands suitable for delineation of fractal dimension, ordinate intercept foreach band and breakpoints separating two neighboring bands. The SegReg computer program uses such criteria whichwas used extensively in this study.

Graphical representation of Richardson’s plots with regard to applying BCM on original as well as detrended version ofDEMs were summarized in Figs. 5 and 6. The values calculated for Df , ordinate intercepts and breakpoints for all directionsand for all DEMs were listed in Tables 1 and 2. The range of Df values from 2 to 3 falls within the range of values reported byother researchers at different spatial scales and physiographic units [9,19,27,20,18].

4. Discussion of results

Digital elevation data with different surface roughness characteristics are expected to have different fractal dimensions.However, it is ironic to note that differences in surface roughness features are not the only factor affecting the computedvalue of Df . This raises an interesting question of what is actually captured by fractal dimension? A multitude of factorsincluding (1) spatial distribution of surface roughness features, (2) computational methods and estimation procedure; (3)input parameter specifications (4) different sources of DEM with various signal to noise ratio, could be considered as the ma-jor sources of variability and uncertainty in surface roughness characterization results based on fractal analysis. In the fol-lowing paragraphs, these factors will be assessed and evaluated in more details in reference to results obtained.

to space limitation, the graphical summary contains micro-plots variogams only.

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Fig. 3. Computation of Df via variogram analysis for different micro-plots in various directions – original data.

2378 M.J. Abedini, M.R. Shaghaghian / Chaos, Solitons and Fractals 42 (2009) 2373–2383

The self-similarity property underlying the fractal model predicts that for truly fractal surfaces, the computed Df

should be scale-invariance at all spatial scales, at all locations, and in all directions. Numerous studies have shown thatthe estimated Df values of most natural phenomena including surface topography are unstable with respect to scale,location, and/or orientation [10,15]. In our views, It is quite circular to interpret the results in either way. If the fractaldimension is quite stable, then one is inclined to confirm the scale-invariance notion, otherwise one can attribute thesource of variability to a variety of factors. Several factors may be responsible for the observed differences in estimatedDf obtained using different methods. First, some of the differences in computed Df may arise from the fact that fractalcomputational methods are not all measuring the same fractal quantity. That may be the reason why [4] called thedimension emerging from BCM as ‘‘Box dimension” instead of fractal dimension. Second, each method relies on totallydifferent assumption(s) while computing Df . As an example, for variogram analysis to be applicable, the topographic sur-face has to resemble fractional Brownian surfaces while there is no need to make such restrictive assumptions in usingBCM. Third, while variogram analysis could handle irregularly spaced data and could also compute directional fractaldimension, BCM could not handle neither of those capabilities attributed to variogram analysis. Fourth, the details ofthe estimation process such as the choice of input parameter values may also affect the resultant Df . As an example,proper execution of variogram analysis demands for identification of step size, number of lags, direction of computationand maximum separation distance. In fact, a number of studies showed that resulting variogram is quite sensitive tothese input parameter values [31]. While in BCM, the only input parameter values required, will be the beginningand ending step sizes and interval spacing (step size). Indeed, investigating the sensitivity of Df to various input param-eter values will be a fruitful and fertile field for further research.

In order to further explore the effect of plot to plot and direction variations on each key variable, a one-way anal-ysis of variance (i.e., ANOVA) was performed on each key variable. Table 3 summarizes the result of the analysis of

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Fig. 4. Computation of Df via variogram analysis for micro-plot II in various directions – various detrending.

M.J. Abedini, M.R. Shaghaghian / Chaos, Solitons and Fractals 42 (2009) 2373–2383 2379

variance for all variables. Differences from plot to plot and from one direction to another could be significant or non-significant depending on the variable considered. As seen in Table 3, differences in observed values for fractal dimen-sion were non-significant among both plot and direction treatments. Differences in observed values for ordinate inter-cept were highly significant among plot treatments, implying possibility of discriminating plots based on ordinateintercept values.

We have further noted that the Richardson’s plots constructed for a majority of experimental omnidirectional as well asdirectional variograms were nonlinear beyond a certain characteristic length scale, indicating breaks in the slope of theregression lines and hence the Df . These results suggest that most topographic surfaces and corresponding surface roughnesscharacteristics are not strictly self-similar; instead, they may be at most exhibit self-similarity in a statistical sense and suchstatistical self-similarity, when present, is applicable only in limited regions and over limited ranges of spatial scale [15]. Itseems that breaks in Richardson’s plots appears to be the norm rather than exception in most fractal techniques using vari-ogram method. However, such mathematical behavior cannot be detected when using BCM. It is quite surprising to note thatthe step size captures almost all variability attributed to number of cells in Richardson’s plot while using BCM as shown inFigs. 5 and 6. We preferred to use different nomenclature for results emerging from different type of fractal analysis as [4] didoriginally.

In a number of studies, it is argued that fractal analysis can be used effectively to delineate optimum spatial resolutionwhile collecting digital elevation data [9,15]. We simply do not share this view as optimum spatial resolution is objectivedependent [24]. Instead, we would like to argue that one could easily extract edge points by identifying breakpoints in esti-mated Df . Conceptually, such breakpoints could be considered as the boundaries between homogeneous regions with differ-ent textural features [32]. By observing the behavior of the fractal parameters obtained from windows of varying size and

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Fig. 5. Computation of Df via BCM in K & B province for various windows and detrending methods.

2380 M.J. Abedini, M.R. Shaghaghian / Chaos, Solitons and Fractals 42 (2009) 2373–2383

orientation one could classify the land surface into fractally homogeneous regions. Acknowledging that fractal dimension is ameasure of variability in attribute values, these breakpoints can be effectively utilized to delineate homogeneous hydrologicunits, detect change in trend in hydrologic time series, cluster data for regionalization purposes, etc. Of course, whether thebreakpoint remains invariant upon change in data support size and areal extent would be another avenue for furtherresearch.

Most if not all existing studies applying fractal techniques to surface roughness characterizations have focused on asingle fractal parameter, i.e., the fractal dimension. Certain computational methods, such as the variogram could producemore than just one parameter. As content of Table 1 clearly demonstrates, Variogram analysis provided not only the esti-mate of the slope but estimates for the breakpoint and the ordinate intercept of a variogram. While the breakpoints fortwo sets of data at two totally different spatial scales and climatic regions are significantly different, their intercepts areseemingly quite similar. Studies in other fields such as geomorphology have shown that the log–log plot ordinate inter-cept of a variogram seems to capture certain information that is not captured by Df [33]. Therefore, it seems desirable touse more fractal parameters to characterize surface roughness instead of using only Df . No doubt, not all methods includ-ing BCM can provide parameters other than Df which is considered as a basic disadvantage of this mono-parameter frac-tal approach.

The notion of mono versus multifractality demands further explanations with regard to dominant processes involved. Re-search on scaling in various branch of sciences including surface hydrology [34] confirmed the fact that different processesare responsible for various sources of variability at different spatial scales. As an example, change in spatial scale (i.e., movingfrom hill-slope to zero-order catchment) leads to change in dominant processes involved which subsequently gives rise tochange in hydrograph timing. This suggests that multi-fractal models appear to be more suited to characterization of surfaceroughness textures as different elements, ranging from individual grains, aggregates, clods, tillage marks and landscape fea-tures contribute to fluctuations in surface topography at their respective scales and different processes should be responsiblefor their generation [5]. It seems to us that the notion of monofractal is more the property of computational method used asopposed to nature of topographic surfaces. In our case, application of BCM on various data sets can be considered as a goodexample to confirm the above assertion.

In a few studies, nature of original data as regard to variability in Df received some attention. In our study, we tried toinvestigate the impact of data trending on fractal dimension. Results of fractal analysis on both original data and detrendeddata showed minimal impact. Indeed, Df remains invariant regardless of type of detrending (i.e., linear, . . .), method of anal-ysis and source of data used as content of Tables 1 and 2 clearly show. These results are quite consistent with results ob-tained by previous investigators [8].

It was anticipated that anisotropy in variogram parameters could eventually lead to anisotropy in fractal dimensionwhich did not happen in our study. Fractal dimension obtained for a particular data set in different directions did not dem-onstrate significant variability from one direction to another. At both spatial scales, results of fractal analysis using both BCMand omnidirectional variogram were quite consistent and comparable with results obtained by other investigators. Needless

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Fig. 6. Computation of Df via BCM for various micro-plots and detrending methods.

Table 1Fractal parameters for various data sets-variogram analysis.

Data set Fractal dimension, Df Ordinate intercept(m or mm)

Breakpoint(m or mm)

Coef. of determiniation R2

x–x y–y x–y Omni x–x y–y x–y Omni x–x y–y x–y Omni x–x y–y x–y Omni

K & B Window I 2.33 2.36 2.47 2.49 0.27 0.54 3.71 4.48 11048 11048 15678 11384 0.983 0.974 0.948 0.979DEMs Window II 2.32 2.36 2.46 2.51 0.20 0.54 2.94 6.55 11048 11048 15678 11384 0.986 0.977 0.950 0.983Micro-plot Plot I 2.51 2.52 2.46 2.47 0.53 0.74 0.17 0.26 112.2 91.8 97.5 105.6 0.998 0.993 0.993 0.995

Plot II 2.48 2.51 2.51 2.46 0.90 1.29 0.44 0.50 78.2 66.7 93.7 74.4 0.999 0.999 0.998 0.999DEMs Plot III 2.60 2.62 2.63 2.61 2.54 3.67 1.92 2.15 137 91.8 119.1 105.6 0.998 0.998 0.997 0.999Plot II Linear 2.48 2.52 2.51 2.48 0.54 0.97 0.45 0.53 84.8 66.7 93.7 90.0 0.999 0.997 0.998 0.999

Quadratic 2.48 2.52 2.52 2.48 0.55 0.98 0.45 0.53 84.8 66.7 93.7 90.0 0.999 0.997 0.998 0.999Detrending Cubic 2.52 2.54 2.52 2.49 0.78 1.16 0.46 0.57 99.5 66.7 93.7 101.5 0.996 0.995 0.998 0.997

M.J. Abedini, M.R. Shaghaghian / Chaos, Solitons and Fractals 42 (2009) 2373–2383 2381

to say, the preparation procedure regarding collecting DEMs at these order of magnitude spatial scales were totally different.These scale-invariance properties of DEMs using fractal analysis at a range of spatial scales and climatic regimes could besuccessfully utilized to derive various features of a fine-resolution DEMs by using only a coarse-resolution DEM as a fewinvestigators did it very recently [14,35].

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Table 2Fractal parameters for various data sets-box counting method.

Various data sets Original DEM Detrended DEM

Linear Quadratic Cubic

Window I 2.54 2.60 2.61 2.60K & B DEMs

Window II 2.64 2.63 2.63 2.63Micro-plot Plot I 2.44 2.45 – –

Plot II 2.49 2.47 2.47 2.49DEMs Plot III 2.44 2.44 – –

Table 3One-way analysis of variance of data for key variables.

Sources of variation Sum of squares Degree of freedom Mean square F-ratio F0:95 Significance level

(a) Fractal dimension, ðDf ÞLevel code = Plot treatments

Between means 0.0062 2 0.0031 2.657 5.14 0.1491Within samples 0.0070 6 0.0012Total 0.0132 8

Level code = Direction treatmentsBetween means 0.0045 2 0.0022 1.534 5.14 0.2896Within samples 0.0087 6 0.0014

Total 0.0132 8(b) Ordinate intercept

Level code = Plot treatmentsBetween means 0.4265 2 0.2132 8.268 5.14 0.0189Within samples 0.1547 6 0.0258Total 0.5812 8

Level code = Direction treatmentsBetween means 0.0729 2 0.0364 0.43 5.14 0.6691Within samples 0.5083 6 0.0847Total 0.5812 8

2382 M.J. Abedini, M.R. Shaghaghian / Chaos, Solitons and Fractals 42 (2009) 2373–2383

5. Concluding remarks

The appeal of fractal geometry as a statistical tool for characterizing and/or simulating surface topography features lies inits apparent simplicity. Fractal analysis using variogram technique offers three parameters, namely; Df , ordinate interceptand breakpoint which describes topography and its dependence on scale. Although it cannot be considered as a universalmodel, it does provide a fairly complete and consistent description for land surfaces considered in this study.

A number of methods have been proposed to calculate the fractal dimension of surface topographic features with seem-ingly conflicting results. In this paper, we tried to further address these same issues by studying and comparing two methodsfor the estimation of fractal dimensions of two-dimensional surfaces by using data sets at a range of spatial scales and cli-matic regimes. Our findings reveal several important points.

For small scale micro-plots, over short distances of approximately 70 mm, all surfaces analyzed do indeed resemble frac-tional Brownian surfaces with dimensions ranging between 2.45 and 2.65. For large scale scenes investigated in this study,again up to characteristic length scale of 11 km, all surfaces analyzed do imitate fractional Brownian surfaces with dimen-sions ranging between 2.30 and 2.60. At both scales, neither the plot treatment nor the direction has a significant impact onfractal dimension variability. We have shown that provided careful attention is given to methodological issues with regard toinput parameter selections, both the two-dimensional variogram and Box Counting Methods could be used to address appli-cability of scaling laws in surface topography. The results suggest that both methods could provide acceptable fractal dimen-sions for topographic surfaces consistent with results obtained by others. In particular, parameters such as breakpoint dataobtained from fractal analysis using variogram method could be effectively utilized to delineate homogeneous hydrologicunits, detect change in trend in hydrologic time series and data clustering for regionalization purposes. While break in slopeemerging from variogram analysis was quite distinctive, such break was not achieved while implementing BCM. This obser-vation leads us to consider the notion of monofractal to be a property of computational procedure used rather than the prop-erty of data.

Further research has to be performed to link self-similarity behavior observed in surface topography to physical processessuch as erosion which gives rise to this organization in surface topography. We would like to conclude our paper by just not-ing that individual differences should not only lead us to collective understanding of nature but also such insight from fractalgeometry eventually guides us to believe that the whole universe and its inherent determinism were not created in vain butfor a distinct objective.

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