applications of a laser scanner to quantify soil microtopography

Applications of a Laser Scanner to Quantify Soil MicrotopographyChi-hua Huang* and Joe M. Bradford

ABSTRACTMany transport processes on or across tbe soil surface boundary

are controlled by surface microtopograpby, or roughness. How rough-ness affects the transport process depends on the length scale of theprocess. The most commonly used method of expressing soil surfaceroughness, the roughness length or random roughness, is constrainedby the measurement technique and does not embody the concept ofscale. The structural function, or variogram, plotted on a log-log scalewas used in this study to express the surface roughness at differentscales. With the aid of a laser scanner, surface topography was mea-sured down to 0.5-mm grid spacing. Data collected from a variety ofsurface conditions showed that soil roughness can be quantified by acombination of fractal and Markov-Gaussian processes at differentscales. Potential applications of the roughness quantification were alsodiscussed.

SOIL SURFACE ROUGHNESS controls many transferprocesses on and across the surface boundary, for

example, infiltration, runoff, soil detachment by waterand wind, gas exchange, evaporation, and heat flux.Despite the recognized roughness effects, there are arelatively small number of published studies focusingon this subject. Romkens and Wang (1987) has at-tributed the lack of publications on soil surface rough-ness to the complex and seemingly random nature ofsurface roughness and the difficulty of its mathemat-ical description. We believe that the problem of quan-tifying soil roughness was mainly due to laboriousfield techniques that produced only low-resolution datasets. Consequently, the development of analytic pro-cedures was further hindered by the poor-quality data.

Quantification of soil surface roughness consists oftwo steps: (i) the collection of surface-elevation data,and (ii) the analysis of elevation data sets. A portablelaser scanner capable of measuring soil surface topog-raphy at submillimeter grids was recently reported byHuang and Bradford (1990a). Other scanning systemsbased on similar optical principles are also reportedin the literature, e.g., Romkens et al. (1988), Khor-ashahi et al. (1987), and Rice et al. (1988). The de-velopment of surface-measurement technology hasbrought forth the availability of large topographic datasets normally in the millimeter grids. Questions oftenraised are why such high-resolution data sets are neededand what can be learned from these data sets.

The overall objective of this study was to demon-strate the concepts evolved from analyzing elevationdata sets on millimeter-scale grids and their potentialapplications. The specific objectives were to (i) pro-pose a methodology to express soil roughness as afunction of scale, and (ii) use statistical random processmodels to quantify the roughness function. Applica-tions of the roughness analysis will also be discussed.

National Soil Erosion Research Lab., Purdue Univ., Bldg. SOIL,West Lafayette, IN 47907. Contribution from Agronomy Dep.,Purdue Univ. and USDA-ARS. Purdue Agric. Exp. Stn. no. 12823.Received 19 Mar. 1990. 'Corresponding author.

Published in Soil Sci. Soc. Am. J. 56:14-21 (1992).

EXISTING MODELS OF SOIL SURFACEROUGHNESS

Soil roughness is a measure of variation in surfaceelevation. Different elements, ranging from individualgrains, aggregates, clods, tillage marks, and land-scape features, contribute to roughness at their re-spective scales. Reported works on soil roughness aremostly concerned with the contribution from tillageimplements and soil clods. Tillage often creates a sys-tematic pattern, called oriented roughness, which canbe easily quantified. This discussion is limited toroughness from soil clods and smaller elements, al-though the proposed methodology to quantify soil to-pography is, applicable to the entire spectrum of scales.

Random roughness, or roughness index, is probablythe most commonly employed method for reportingsoil surface roughness. It uses a length scale that isequivalent to or related to the standard deviation ofelevations from a mean surface (Kuipers, 1957; All-maras et al, 1966; Currence and Lovely, 1970). Thistechnique implicitly assumes that the surface has arandom roughness with no spatial correlation. Mostdata sets analyzed in this fashion were collected bymultiple-pin-type microrelief meters with a pin spac-ing ranging from 5 to 50 mm.

Romkens and Wang (1986) introduced a dimen-sionless parameter, MIF, which is the product of MIand F. The value of MI is equivalent to the meanabsolute deviation of measured elevation from a ref-erence plane, and F is the number of elevation peaksper unit length of a transect. This parameter was usedto examine different tillage effects (Romkens and Wang,1986) and to quantify roughness changes due to rain(Romkens and Wang, 1987). Lehrsch et al. (1987,1988) further analyzed how soil properties contributeto variation of the MIF parameter and identified MIFas the most sensitive roughness parameter among aset of eight different measures for describing surfaceroughness and its change after simulated rain.

The major problem associated with the single-val-ued roughness scale, including the MIF parameter ofRomkens and Wang (1986), is that it quantifies soilroughness at a single length scale that is normallycontrolled by the measurement device. In addition, theMIF parameter contains a fallacy. Since MIF is theproduct of an amplitude term (MI) and a frequencyterm (F), a surface with a small number of large clodscan have the same index as a surface with a largenumber of small clods. In addition, the use of a non-dimensional definition further implies that soil surfaceroughness is scale independent.

Linden and Van Doren (1986) proposed a meanabsolute elevation difference model and showed a lin-ear relationship between 1/AZ and l/h where AZ isthe absolute elevation difference at horizontal distanceAbbreviations: MIF, product of microrelief index and peak fre-quency; MI, microrelief index, F, peak frequency; D, fractal di-mension; pdf, probability density function; Cov, covariance; Smv,semivariance; Psd, power spectral density; Bm, Brownian motion;fBm, fractional Brownian motion; MG, Markov-Gaussian.

14

HUANG & BRADFORD: QUANTIFYING SOIL MICROTOPOGRAPHY 15

h apart. Their model expressed roughness as a func-tion of separation length scale. The model was derivedfrom a regression procedure biased heavily toward datasets at small separations since 1/AZ and l//t were clus-tered together at large AZ and h values.

Correlation and spectral analyses were also used toquantify soil surface roughness. A detailed discussionof their utilities are given by Currence and Lovely(1970) and Merva et al. (1970). Podmore and Muggins(1980) used spectral-analysis techniques to examinesurfaces of a laboratory flume and concluded that theirtest surfaces were similar to random-roughness sur-faces with no characteristic frequency. Dexter (1977)used both the autocorrelation and spectral-densityfunctions to examine roughness changes due to rainfor different tillage operations and concluded that thesefunctions, when compared with a variance term alone,did not yield additional information about rain effects.

Apparently, correlation and spectral analyses wereonly used to identify the major scale of periodic var-iation or oriented roughness. In the absence of ori-ented roughness, the surface was considered randomand a variance-related single roughness index was ad-equate. The definition of random roughness deservesfurther discussion because both roughness and its ran-domness are scale dependent.

Recently, the concept of fractal geometry (Man-delbrot, 1983) has been used to quantify topographicvariations ranging from landscapes with an areal lengthscale of 10 km (Mark and Aronson, 1984) to small-scale (5-10 m) surfaces such as glacial tills (Elliot,1989) and gravel-bed channels (Robert, 1988). As acommon belief, the three studies expressed surfaceroughness by a single parameter called fractal dimen-sion, D. We will show, in the following section, thata fractal roughness model requires two parameters andD alone is not sufficient.

One additional issue that has been ignored in quan-tifying soil surface roughness is dimensionality. Mostroughness data sets were collected by profile metersalong a transect and, in many cases, only a smallnumber of transects were measured. Analysis basedon a single profile produces biased statistics of thesurface topography. The cause of this inherent erroris discussed below.

When profile data is examined, a peak shown alonga soil profile is generally not the local summit but itsshoulder. The same analogy applies to valleys andlocal depressions. To quantify soil surface roughness,one would like to include these peaks and troughsbecause, for many physical processes, the greatestsurface effects come from these elevation extremes.Thus a variance-related term, such as random rough-ness, calculated from a single profile would under-estimate the roughness of the surface. To illustratethis, we showed in Fig. 1 a plot of standard deviationscalculated from individual profiles and compared themwith the standard deviation for the entire two-dimen-sional data set. The surface-elevation data set, col-lected by the laser scanner, consisted of 280 500-mm-long profiles spaced 2 mm apart and, within each pro-file, soil elevation was digitized at a 0.5-mm grid. Inthis particular case, 67% of profile-based standard de-viations were smaller than the value calculated from

4.5

4.0 -

3.5 -

|«:

"2.5-

2.0 -

1.5 -

1.0

-fr-/

8 af

a profile SDSurface SD

100 200 300 400 500 600Profile Position, mm

Fig. 1. Standard deviations (SD) from individual profiles, spaced2 mm apart, compared with the value based on the entiresurface data set.

the entire two-dimensional data set. If surface topog-raphy follows a Gaussian distribution, relationshipsbetween profile-based statistics and those of the sur-face can be derived theoretically (Longuet-Higgins,1957; Nayak, 1971).

ROUGHNESS MODELS AND STRUCTURALFUNCTIONS

The term random roughness has been widely usedto describe soil microtopography without reference toa specific spatial scale. This type of usage could bemisleading because surface roughness might appearrandom at one scale and in an orderly fashion at an-other scale. Fundamental to the quantification of soilroughness is the concept of scale. Soil surface rough-ness is a function of areal length scale. Qualitatively,one can expect a higher variation in surface topogra-phy as the area under inspection is increased. Thus adescriptive model for soil roughness should have scale-dependent characteristics.

Statistical models of two-dimensional randomprocesses were used to describe surface topography(Longuet-Higgins, 1957; Nayak, 1971). To define atwo-dimensional random process, one needs to knowthe pdf and the spatial-correlation structure. The spa-tial structure can be expressed by one of the followingfunctions: Cov, Smv, or Psd. Journel and Huijbregts(1978) discussed conditions based on the existence ofa global variance for a proper usage of Cov or Smv.We used the Smv function, or equivalently the var-iogram, to quantify soil roughness to avoid the con-straint of an existing global variance. Since soilroughness is a measure of variation in surface eleva-tion, the variogram gives a direct relationship betweenan elevation-difference term and the length scale asgiven by the lag separation.

The semivariance, y(h), is defined as

y(h) = 0.5 E{(ZCx) - ZCx+htf}

16 SOIL SCI. SOC. AM. J., VOL. 56, JANUARY-FEBRUARY 1992

where Z/^ and Z(~+A) are elevations at positions h hor-izontal distance apart, and E{ } is the expectation op-erator. Here we give examples of several randomprocess models, related to surface roughness descrip-tion and their characteristic structural functions. Thesemodels are (i) completely random; (ii) random walk;(iii) fractal and (iv) Markov-Gaussian.

A completely random process is represented by awhite-noise-type elevational variation with no spatialcorrelation. The structural function is approximatedby a constant equal to the variance, cr2, at all scales.The completely random model can be characterizedby a single parameter, cr.

The random-walk or Brownian-motion (Bm) modelis characterized by its increment function,

AZ(h) a h0-5

where AZ(/z) is the absolute elevation difference at hdistance apart. The structural function is thus a linearfunction of spatial separation, h, e.g.,

j(h) a h.The variogram, plotted in log-log scales, is a straightline with slope 5 = 1. The spectral-density function,also plotted in a log-log fashion, is a straight line ofs = -2. Podmore and Huggins (1980) found thatslopes of Psd functions from their test surfaces hadvalues around — 2 and concluded that their test surfacehad random elevation. Note the difference between arandom-walk and a white-noise-type roughness model.In theory, these two processes are related and theBrownian motion is the integral of a Gaussian whitenoise.

Mandelbrot and van Ness (1968) further expandedthe concept of the Brownian random-walk model andintroduced a special class of fractal process, calledfractional Brownian motion (fBm) where the exponentof the increment function of a Bm model is allowedto vary from its generally accepted value of 0.5 withina range from 0 to 1. The elevation variation of an fBmmodel is given by

AZ(fc) « h", 1 > H > 0with the following structural function:

y(h) « h2".Plotted in log-log scales, the variogram of an fBmmodel is a straight line with 2 > s > 0. For surfacetopography, the H parameter is related to fractal di-mension, D, by D = 3 — H. A thorough discussionof the fractal geometry was presented by Mandelbrot(1983). Burrough (1983a, 1983b) and Brown andScholz (1985) also presented introductory remarks onthe basic concepts of fractal process and its applicationto quantification of soil variations and rock surfaceroughness.

The fBm process is characterized by two parame-ters, D and crossover length /. The / parameter wasmodified from the original definition proposed by Wong(1987) for pore structure and used by Brown (1987)for rock surface topography. Wong (1987) defined thecrossover length based on a power-law-type variancestructural, e.g.,

o-2 (X) = b2-1" Vwhere o2(X) is the variance from local regions withsize scale X, and b is the crossover length. The termcrossover was used because, when b — X, or = b.The b parameter was also called topothesy, K, by Saylesand Thomas (1978) for the specific case when m =1 (i.e., the Bm process) and in a more general fashionby Berry and Hannay (1978).

Here we defined / based cm the structural function,y(h), such that

y(h) = P-2" h2".In this case, when I = h, y(h) = h2.

The variance function, cr^X), is related to the struc-tural function, y(h), by

where c is a constant. For an fBm process, the aboveintegration produced

CT^X) a .y(X).

Thus the crossover-length scale defined by Wong (1987)and Brown (1987), the topothesy, K, and the struc-tural-function-based / parameter were functionallyequivalent for an fBm-type fractal process.

Here we emphasis that a fractal roughness modelrequires two parameters, D and /, instead of the com-monly known single parameter D. This is analogousto the slope and intercept parameters required to quan-tify a straight line. Thus two parallel lines in the log-log variogram will have an identical D but different/. Below, we will show that a difference in soil rough-ness is reflected in both D and / parameters.

Another class of random process is the Markov-Gaussian (MG) model, with an exponential-type cor-relation structure. The Smv of the MG model is givenby

y(h) = a2 (1 - e-**)where cr2 is the variance and L the correlation lengthscale. The MG process is characterized by two lengthscales, cr and L. The log-log variogram of the MGmodel showed a varying slope between 1 and 0. Atsmall scales, h « L, the variogram can be approx-imated by a straight line of s = 1, similar to the Bmmodel.

Figure 2 shows the variograms of these four models,e.g., completely random, Bm, fBm, and MG. Of thesefour models, only the completely random and MGprocesses have variances. The conventional random-roughness calculation assumed the existence of a var-iance at local scale. In fact, the random roughnessreported in the literature was the roughness at the larg-est scale, as dictated by the measurement device afterthe removal of the oriented roughness.

Here we propose a combination model of fBm andMG processes at different scales to quantify the rough-ness function. The fractal regime was used to describethe sloping straight-line portions of the variogram,whereas the MG process was used to describe theregion where the slope varied gradually toward 0 atan apparent local homogeneous state.


MATERIALS AND METHODSSurface elevations from a range of field and laboratory

conditions were digitized by a portable laser scanner. Thedesign and operation of the laser scanner were reported indetail by Huang and Bradford (1990a). The digitized sur-faces were from (i) natural fallow plots, (ii) field rain-sim-ulator plots, and (iii) laboratory soil pans.

Both natural fallow and rain-simulator plots were locatedat the North Carolina Upper Piedmont Agricultural Exper-iment Station near Reidsville, NC. Two sites, one withPacolet (clayey, kaolinitic, thermic Typic Kanhapludult)soil series and the other one with Rion (fine-loamy, mixed,thermic Typic Hapludult) series were studied. The naturalfield plots were fallow during 1988, tilled in late fall (De-cember 1988), and remained fallow during the winter monthsof 1988-1989. Surface topographies of randomly selectedlocations were measured immediately after the fall tillagein December 1988 and before disking in April 1989. Po-sitional markers were placed in the scanned area to ensurethat the same location was scanned in the spring. At eachselected location, an area of 0.9 by 0.9 m was digitizedwith a 1-mm grid along the scan line and a 10-mm spacingbetween scan lines. Approximately 82 000 elevation datawere collected from each test surface.

In April 1989, a rain-simulator study was conducted onthe fallow plots. The plots were disked to form a visuallyuniform surface. After the machine preparation, erosionplots, measuring 1 by 1 m, were laid out and the test sur-faces were further prepared by a hand trowel to break uplarge (> 100 mm) clods. Artificial rain, produced by a rainsimulator using Vee-Jet nozzles (Spraying Systems Co.,Wheaton, IL), was applied to the erosion plots at a pre-determined duration and intensity sequence, resulting in atotal of 216 mm of rain for the 3-h rain period. Soil topog-raphy was measured before and after the rain events. Thescan area and grid spacings were similar to those used onthe natural fallow plots.

For the laboratory soil-pan study, surface soil from Manor(coarse-loamy, micaceous, mesic Typic Dystrochrept) se-ries in Howard County, MD, was air dried, sieved through20-mm openings, and packed under vibration into an ero-sion pan. The erosion pan had a total area of 1.2 by 1.2m, with a center test area of 0.54 by 0.6 m surrounded by

buffer zones. The soil surface was prepared to have a ho-mogeneous roughness relative to the scale of the pan. Thesoil pan was adjusted to 9% slope and rained on at 63 mm/h for 1 h. Approximately 24 h after the first rain event, anadditional 92 mm of rain was applied. Soil surface topog-raphy from the center test area was digitized before andafter both rain events with the soil pan set at a level posi-tion. Grid spacings were 0.5 mm along the scan line and 2mm between scan lines. Each elevation data file contained=300 000 points.

For the soil-pan study, the soil moisture condition wasset to either air dry or prewetted from the bottom with thesurface maintained at 50-mm ( —0.5-kPa) suction for 3 dprior to and during the rain. Differences in soil microto-pography resulting from the contrasting initial moistureconditions after the rain events were used to demonstrate apossible measure of aggregate stability and crust strengthunder the erosive forces of rain.

RESULTS AND DISCUSSIONSoil Surface Roughness

Variograms, plotted in a log-log fashion, were usedto display the scale-dependent roughness functions.Data collected from test surfaces were plotted in Fig.3 to 6. These variograms were presented using valuescalculated directly from the data set, with all possiblecombinations at each lag separation. Due to the largenumber of data pairs used in the variogram analysis,the variogram appeared smooth. We have not per-formed any smoothing on these graphs.

These plots showed a general trend of sloping straightlines linked by a region where the slope varied grad-ually toward an asymptote. Close examination of theseroughness functions revealed that it would be difficultto fit one universal model to these curves. It is pos-sible, however, to use a combination of fractal andMG models at different scales to fit the roughnessfunction. The straight-line portion of the log-log var-iogram is characteristic of a fractal process and theregion of gradual leveling off toward an asymptote

100q

0)oc

10-

1-

0.1 -

0.01

Completely Random

fBm

1000i

0.01 0.1 1Length Scale, h

10 100

Fig. 2. Structural functions, or variograms, for completelyrandom, Brownian motion (Bm), fractal or fractionalBrownian motion (fBm), and Markov-Gaussian (MG) models.

E100-

X

EE

o 10

0>(/) 1 '

0.1

Natural PlotReidsville, NC

Soil :Pacolet

---- Rlon

Dec. 1988

Apr. 1989

0.1 1 10 100Length Scale, mm

1000

Fig. 3. Roughness functions from fallow plots after fall tillage(measured in December 1988, two top lines) and in the spring(April 1989) on Pacolet and Rion soils.


1000q

E

EEo>oc

<D(/>

100-

10-

1-

0.1

Field Interrlll PlotRaldsvllle, NC

Soil :Pacolat

---- Rlon

Before ,Rain

Rain216 mm


1000

Fig. 4. Roughness functions from 1 by 1 m interrill-erosionplots before (two top lines) and after simulated rain on Pacoletand Rion soils.

1UU ;

EE 10,x :

E :

E ~eo 1 -3c :o~ :§E0)

<n 0.1 -:

0.01 -

Laboratory Pan StudySoil: Manor (Howard, MD)Moisture: Prewet

— --"1' — ~~"=^/^^"^

///•*/ ,r

/ */ J/ J/ ——— Before Rain

— ••- Rain 63 mm- — - Rain 155 mm


Fig. 6. Roughness functions from a laboratory soil pan beforeand after 63 and 155 mm of si initiated rain. The soil waspresetted for 3 d prior to rain.

EE

EE0)oca

10-

1-

0.1 -

0.01

Laboratory Pan StudySoil: Manor (Howard, MD)Moisture: Air Dry

——— Before Rain——— Rain 63 mm

— — — Rain 155 mm


Fig. 5. Roughness functions from a laboratory soil pan beforeand after 63 and 155 mm of simulated rain. The soil wasair dried prior to rain.

can be described by an MG model. The fractal andMG parameters obtained for these test surfaces aregiven in Table 1.

Procedures to identify the need for using a combi-nation of models to quantify the variogram were firstto examine a plot of derivatives (slopes) of the log-log variogram, or d(log -y)/d(log h) vs. h. From thisplot, the fractal model was identified by a scale rangedisplaying a constant value, and the MG process bya region of gradually decreasing slopes from 1 toward0 as h increased. Parameters for each model at theirdistinctive zones were obtained by least-squaring tech-niques. The separation point between models was de-termined by minimizing the error sum of square betweenpredicted and calculated variograms at the overlappingzone.

The fractal model used to describe soil roughnesswas only found appropriate within a limited range ofscale, contrary to the common belief that fractal processshould not have any scale limitations. Our findingsupported the concept of pseudofractal, or fractalprocess at a limited scale, discussed by Orford andWhalley (1983) and Whalley and Orford (1989). Otherevidences of the pseudofractal model for surface to-pography were also found from analyzing digitizedlandscapes (Mark and Aronson, 1984), glacial tills(Elliot, 1989), and gravel-bed channels (Robert, 1988)with areal length scales on the order of 5 m to 10 km.Here we have extended the evidence of pseudofractalprocess to soil microtopography within 1 by 1 m area.

The break in fractal scaling, or pseudofractal, im-plied that elevational variations at small scales do notscale up proportionally when the areal scale is in-creased. The pseudofractal-lype soil roughness isabundant in nature. Any flat surfaces observed by thenaked eye would constitute aa example of discontin-ued fractal scaling. Otherwise, the surface would nothave a scale range to be considered flat if its small-scale roughness is scaled upward proportionally.

At this point, we would argue for a limited-rangefractal scaling for soil topography, instead of a uni-versal fractal model for the entire scale spectrum. Wealso envision an extension of l:he log-log variogram toappear like a terraced hillslope profile, with slopingbanks joined by short plateaus. With this conceptualmodel in mind, it is conceivable to conclude a uni-versal fractal model if data sets were analyzed at lengthscales several orders of magnitude apart.

For a fractal-type surface roughness, / played a sig-nificant role in separating different degrees of rough-ness. The / parameter is related! to the intercept conceptin a linear model. A set of parallel lines in the log-log variogram implied identical D but different /. Inthis case, greater roughness is associated with larger/ values. In other words, the fractal dimension is anindex for the proportional distribution of different-sized


Table 1. Fractal (fBm) and Markov-Gaussian (MG) parameters from laser-scanned data sets.Fig. Soil Description Model Scale range Parameters'

3

4

5

6

Pacolet

Rion

Pacolet

Rion

Manor(air dry)

Manor(prewetted)

Dec. 1988

Apr. 1989

Dec. 1988Apr. 1989Before rain

216-mm rain

Before rain

216-mm rain

Before rain63-mm rain

155-mm rain

Before rain

63-mm rain

155-mm rain

fBmfBmfBmMGfBmfBmMGMGMGfBmMGfBmMGfBmMGMGfBmMGfBmMGfBmfBmMGfBmMG

mm1-50

50-4005-100

100-4005-1605-1701-40

40-4001-40

40-4001-150

15^4001-80

80-4000.5-2000.5-9090-2000.5-110110-2000.5-9090-2000.5-5

5-1000.5-5

5-100

D= 2.60D = 2.72D = 2.34o = 11.4D = 2.68D = 2.45o = 6.32<r = 8.66o = 6.71D = 2.66a = 7.42D= 2.46a = 3.87D = 2.39o = 2.51a = 1.26D = 2.83<r = 1.34D = 2.68cr = 2.12D = 2.80D = 2.38a = 1.79D = 2.39a = 1.73

/ = 6.59/ = 9.48/ = 0.09

L = 75/ = 4.91/ = 0.78L= 15L = 80L= 115/ = 1.21

L = 28/ = 0.22

L = 23/ = 0.03L= 10L = 151 = 0.53

L = 9/ = 0.17

L = 12/ = 0.80/ = 0.12

L = 91 = 0.11

L = 9'Parameters /, <r, and L are shown in mm unit and D is nondimensional.

roughness elements in a relative scale, and / is thescaling parameter transforming the relative size to ac-tual scale. Our data, for example from fallow plots(Fig. 3) and the erosion plot for Rion soil (Fig. 4),showed that changes in soil roughness were reflectedmore significantly by the / parameter. Comparing Dalone would not be sufficient to separate the differ-ence. This may also explain the failure of Elliot (1989)to establish a correlation for different degrees of sur-face roughness from D alone.

For the portion of the log-log variogram where theslope has shown a gradual change toward zero, anMG-type model was used. The variance scale for theMG process, a2 or cr, is considered a local variancefor the apparent homogeneous scale. Thus, the MGmodel used here was only an approximation because,in theory, a2 is defined only at an infinite scale. Thecorrelation length scale, L, is a measure of how fastthe small-scale variation approaches the asymptoticvalue of a2. For the same cr2 value, surfaces withsmaller L would appear rougher than a surface withlarger L.

For the quantification of surface roughness, bothfractal (or pseudofractal) and MG models have beenadvocated in the literature separately. The MG modelwas proposed to quantify engineering material sur-faces where the roughness was limited to a confinedrange of scales (Whitehouse and Archard, 1970; Nayak,1971), whereas the fractal process has been used toquantify natural features like landscape or rock frac-tures resulting from a broader range of multiple-scaleinteractions (Brown and Scholz, 1985; Burrough, 1981;Mandelbrot, 1975). Our data indicated the coexistenceof these two models and their separation were entirelya matter of scale. For soil roughness, the fractal be-havior is the result of multiple-scale effects, with thesmall-scale-roughness elements being embedded inlarge-scale variations. At the plateau, region, the

roughness reached a spatially homogeneous state atthat particular local scale. For the examples shown,the large-scale variation was due to large clods, plotsideslopes, or erosion features such as rills.

The advantage of expressing surface roughness asa function of scale is that it provides a universal basefor its linkage to physical processes. Roughness func-tion alone is only a geometric description of the sur-face and it becomes meaningful only when related tophysical processes at their characteristic length scales.

Once roughness is expressed as a function of scale,the definition of random roughness becomes ambig-uous. Random roughness represents the largest rough-ness element that can be measured under themeasurement device after the removal of slope andoriented tillage marks. Since the random-roughnesscalculation was biased by the large-scale elements anddid not contain information in surface composition,one could show that two surfaces having similar ran-dom-roughness values might have totally differentsurface features. This is demonstrated by soil surfacesof a laboratory pan after 63 and 155 mm of simulatedrain. Figure 5 shows the variograms and Fig. 7 showssurface topographies from a 0.2 by 0.2 m section ofthe test surface before and after two rain events. After63 mm of rain, the soil surface was crusted, withaggregate remains sparsely distributed on the surface(Fig. 7b). An additional 92 mm of rain caused thedevelopment of microrills (Fig. 7c), and the surfaceappeared to have a slightly higher roughness. Hadthese two surfaces been quantified by the pin tech-nique at 10- or 20-mm grids, they would have similarrandom-roughness values. Differences in surface fea-tures are clearly depicted by the variogram.

Examples shown here were selected from a databank of more than 500 digitized surfaces to cover awide range of surface conditions. All these variogramsthat we have analyzed can be scaled by the combi-


(a) Before rain

Fig. 7. Three-dimensional plots of a 0.2 by 0.2 m section from a soil pan (a) before rain, (b) after 63 mm, and (c) after 155 mmof simulated rain. The soil was air dried prior to rain.

nations of both fractal and MG models. We have mainlycentered on introducing the concept of scale relatedto roughness quantification. We are currently workingon further differentiation of the roughness parametersfor surfaces with different microgeomorphic featuressuch as crusts and rills.

Applications of Roughness QuantificationAs stated above, a statistical quantification of soil

topography is meaningful only when it can be relatedto some physical processes. Since many surfaceprocesses are directly controlled by soil topography,a statistical description of soil topography allowed usto examine the roughness effects.

For example, during rain, a protruded aggregate orclod would have been subjected to greater raindrop-impact effects than a local depressional area whereponding begins. Protruded large elements also changedthe flow pattern, and thus flow shear stress, on the

surface. On the other hand, a local depression has aholding capacity for detached sediments and runoffwater. A detailed surface topography enables us toanalyze the spatial distribution of these processes. Thus,information such as the relative portions of the surfacethat will be exposed to raindrop impact vs. those undersurface ponding are implicitly contained in the statis-tical roughness function. An example of using the sur-face roughness parameters to scale depressional storageis given by Huang and Bradford (1990b).

In addition to the scaling of surface effects basedon the roughness function, changes in soil roughnessduring rain also reflects the soil behavior under theerosive forces. During the rain, the soil surface wentthrough a series of transformations from an initialloosely aggregated condition to a consolidated crustand then toward the development of erosional featuressuch as rills (Fig. 5, 6, and 7). Thus soil roughnesscan either decrease or increase with an additional amountof rainfall, depending on botli the surface conditions


and the processes occurring on the surface. Since ero-sional processes would eventually produce a scarredlandscape with an increased roughness, the acceptedconcept of decreasing roughness with increasing amountof rainfall (Zobeck and Onstad, 1987) is not appro-priate for surfaces where erosion occurs.

The rate of change in surface topography under theerosive forces of rain is also a measure of surfacestability or strength. A stable surface shows little changeunder rain while an unstable surface would display arapid change. Data from the Manor soil pans showsthat the same amount of rain produced little changein surface topography for the prewetted case and dras-tic changes when the soil was air dried at the begin-ning of the rain. In this case, prewetting causedaggregates to maintain a higher stability against rain-drop impact. A similar finding was also reported byTruman et al. (1990). This example showed the pos-sibility of developing an in situ measurement of sur-face strength under the erosive forces of rain based onthe changes in surface topography.

CONCLUSIONSFrom analyzing laser-scanned surface roughness data

sets, we showed that surface roughness is a functionof scale and its quantification should include the scale-dependency characteristics. The variogram, or struc-tural function, was used to plot the roughness func-tion, and a combination of fBm and MG processeswas proposed to characterize roughness at differentscales.

Both processes required two separate parameterseach: D and / for the fractal process and cr and L forthe MG process. The fractal process used to quantifythe experimental variogram is applied to a limited rangeof scales, a process also referred to as pseudofracial.Here, we have emphasized the importance of the /parameter, in addition to the commonly known D, fora fractal-type roughness model in order to differentiatedifferent degrees of soil roughness.

Many surface boundary processes are controlled byroughness. A detailed surface microtopography pro-vides information on the spatial distribution of surfaceprocesses. For soil erosion, these processes are: rain-drop-impact effects, surface ponding pattern, flowmeandering and shear, sediment deposition, and windexposure and sheltering. Changes in roughness aftererosive events further reflects the soil's behavior againsterosive forces. A statistical model of soil roughnessfacilitated the scaling of these surface processes.

applications of a laser scanner to quantify soil microtopography

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