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HYDROLOGICAL PROCESSES, VOL. 5, 3-30 (1991)

DIGITAL TERRAIN MODELLING: A REVIEW OFHYDROLOGICAL, GEOMORPHOLOGICAL, AND BIOLOGICALAPPLICATIONSI. D. M O O R E

Centre fo r Resource and Environmental Studies, The Australian National University, Canberra, AC T , 2601,AustraliaR. B. G R A Y S O NCentre fo r Encironmental Applied Hyd rology , The University of Melbourne, Parkville, Victoria, 3052, AustraliaA N DA. R. L A D S O N

Department of Earth Sciences, Montana State University, Boieman, M ontana, 5971 7 , U.S .A .

ABSTRACTThe topography of a catchment has a major impact on the hydrological, geomorphological. and biological processesactive in the landscape. The spatial distribution of topogra phic attributes can often be used as a n indirect measure of thespatial variability of these processes and allows them to be mapped using relatively simple techniques. M any geographicinformation systems are being developed that store topographic information as the primary data for analysing waterresource and biological problems. Furthermor e, topography can be used to develop more physically realistic structuresfor hydrologic and w ater quality models that directly account for the impact of topography on the hydrology. Digitalelevation models are th e primary dat a used in the analysis of catchment topography. We describe elevation data sources,digital elevation model structures, and the analysis of digital elevation data for hydrological, geomorphological, andbiological applications. Some hydrologic models that make use of digital representations of topography are alsoconsidered.K E Y W O R D S Basin topography Digital elevation models Terrain analysis Hydrologic mod els

I N T R O D U C T I O NTh e dem and s placed on hydrologic models have increased considerably in recent times. Although i t was onc esufficient to model catch ment outflow, i t is now necessary to est im ate distributed surface and subsurface flowcharacterist ics, such a s flow dep th a nd flow velocity. These flow characteristics a re the driving me chanism sfor sediment and nutrient t ransport in landscapes and unless they can be predicted reasonably well , waterqual ity models cannot be expected t o adequate ly s imula te sed iment a nd nut r ien t t ransp ort . The cont ro lexerted by topograp hy on the movement of water w i th in th e landscape i s fundamenta l t o the predic t ion ofthese flow charac teristics. A grea t deficiency of man y hydrologic and water qual i ty models curren t ly in use istheir inability to represent the effects of three-dimensional terrain on flow processes and the spatialvariability of hydrologic processes without gross, an d often unrealistic, simplifications. In response t o thischallenge, metho ds a re being developed to digi tally represent terrain in hydrologic m odels.In addit ion t o these complex models, used largely for research into ca tch me nt processes, there is a d e m a n dfor simpler techniques to assis t with day-to -day land m anagement . Act ion agencies responsib le for land a ndwater management in ma ny part s of the world ar e being required t o identify those ar eas of lan d susceptible tovarious types of envi ronmenta l hazard and degra dat ion such a s erosion , sed imenta t ion , sa l in iza tion , non-point source pollut ion, an d water logging (H ern don , 1987)and to assess and man age b iological productiv i ty0885 -6087/91/010003 28$14.OO( 1991 by John Wiley & Sons, Ltd. Received and ac cepted 6 September 1990

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4 1. D. MOORE. R. B. GRAYSON AND A . R . LADSONand diversity within landscapes. A number of hydrologically-based, topographically-derived indices appearto be particularly pow erful and useful for determining this susceptibility t o hazard (M oo re and Nieber, 1989).Many geographic information systems and resource inventory systems are being developed storingtopographic information as primary data for use in analysing water resource and biological problems.This paper reviews the availability of digital elevation da ta and its accurac y, digital represen tation oftopograp hy, analysis of the digital da ta for hydrological, geomorphological, and biological applications, an ddescribes some models that make use of digital representations of topography. We use examples fromAustralia and the United States of America in ou r discussions because of our familiarity with hydrologicresearch and practice in these two countries.

DIGITAL ELEVATION MODELSA Digital Elevation Model (D E M ) is an ordered arra y of numbers that represents the spatial distribution ofelevations above some arbitrary datum in a landscape. I t may consist of elevations sampled a t discrete pointsor the average elevation over a specified segment of the landscape, although in most cases i t is the former.DE Ms are a subset of Digital Terrain Models (DT M s) which can be defined as ordered arra ys of numbersthat represent the spatial distribution of terrain attributes.The acquisition. storage, and presentation of topographic information is an area of active research.Generally, raw elevation data in the form of stereo photographs o r field surveys, and the equipment toprocess these data, are not readily available to potential end users of a D E M . Therefore, most users must relyon published topographic maps or DEMs produced by government agencies such as the United StatesGeological Survey (U.S.G.S.) o r the Australian Surveying and Land Information G ro up (AUSLIG -for-merly the Division of National Mapping).This section describes the three major methods of structuring a DEM, the sources of elevation data inAustralia and the United States, how D EM s are comm only produ ced a nd gives som e indication of theirquality. The section concludes with a discussion of some simple techniques for analysing elevation d ata forthe estimation of topograph ic attributes such as slope and aspect and gives examples of methods that can beapplied to each of the three DEM structures.Dutu networks ,for Digital Elevation Models

When discussing the use of DEMs it is impo rtan t to consider the way in which the surface representation isto be used. The ideal structure for a DEM may be different if it is used as a structure for a dynamic.hydrologic model than if it is used to determine the topographic attributes of the landscape.There are three p rincipal ways of structuring a network of elevation da ta for its acquisition an d analysis, asillustrated in Figure 1 . Triangulated Irregular N etwork s (T IN S) usually sam ple surface specific-points, such

Figure I . Methods of structuring an elevation data network: (a) square-grid network showin g a moving 3 x 3 submatrix ccntrcd onnode 5; (b ) triangular irregular netw ork- TIN; and ( c) contour-bascd network

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DIGITAL TERRAIN MODELLING I : REVIEW OF APPLICATIONS 5as peaks, ridges, and breaks in slope, and form an irregular network of points stored as a set of x, y, an d zcoordinates together with pointers to their neighbours in the net (Peucker et al., 1978; Mark, 1975). Th eelemental area is the plane joining three adjacent points in the network and is known as a facet. Grid-basedmethods may use a regularly-spaced triangular, square, or rectangular mesh or a regular angular grid, suchas the 3 arc-second spacing used by the U S . Defence M appin g Agency. The choice of grid-based m ethod isrelated primarily to the scale of the area t o be examined. The da ta can be stored in a variety of ways, but themost efficient is as z coordinates corresponding to sequential points along a profile with the starting pointand grid spacing also specified. Th e elemental area is the cell bounde d by three or four adjacent grid-pointsfor regular triangular and rectangular grid-networks, respectively. Contour-based methods consist ofdigitized contour lines and are stored as Digital Line Gr ap hs (DL Gs ) in the form of x, y coordinate pairsalong each conto ur line of specified elevation. These can be used to subd ivide an are a into irregular polygonsbounded by adjacent contour lines an d adjacent streamlines (Moo re, 1988; M oor e and Grayson , 1989, 1990)and ar e based on the stream path analogy first proposed by On stad an d Brakensiek ( 1 968).The most widely used data structures consist of square-grid networks because of their ease of computerimplementation an d com putatio nal efficiency (Collins and M oo n, 1981). However, they d o have severaldisadvantages, including: (1) they cannot easily handle a bru pt changes in elevation, although Fra nke an dNielson (1983) discuss possible ways of modelling these d iscontinuities; (2) the size of grid m esh affects theresults obtained an d the com putatio nal efficiency (P an us ka et al., 1990);(3) the com puted upslope flow pathsused in hydrologic analyses tend to zig-zag and therefore are somewhat unrealistic; and (4) precision islacking in the definition of specific catchment areas. Since regular grids must be adjusted to the roughestterrain, redundancy can be significant in sections with smooth terrain (Peucker et al., 1978), whereastriangulated irregular networks are more efficient and flexible in such circumstances. Olender (1980)compares triangulated irregular networks with regular grid structures.Topographic attributes such as slope, specific catchment area, aspect, plan, and profile curvature can bederived from all three types of DE Ms . M ethods of doing this an d potential uses of these and o ther c om pou ndattribu tes are discussed in detail later. However, the most efficient D EM structu re for the estimation of theseattributes is generally the grid-based method. Contour-based methods require an order of magnitude m oredata storage and do not provide any computational advantages. With TIN structures, there can bedifficulties in determining the upslope connection of a facet, although these can be overcome by visualinspection and manual manip ulation of the TI N netw ork (P alacio s and C uevas, 1989). The irregularity of theTIN makes the computation of attributes more difficult than for the grid-based methods.For dynamic hydrologic modelling, there are quite different considerations. Mark (1978) noted that gridstructures for spatially p artitioning topo graphic da ta are not appropriate for many geom orphological andhydrological applications. He stated that 'the chief source of this structure should be the phenomena inquestion, and not problems, data, or machine considerations, as is often the case'. Hydrologic modelssimulate the flow of water across a surface so the elemental areas of the DEMs should reflect thisrequirement. Contour-based m ethods have important advantages in this regard (M oore, 1988; Moo re andGra yson , 1989, 1990) because the structu re of their elemental areas is based o n the way in which w ater flowson the land surface. Orth ogo nals to the contours are streamlines so the equa tions de scribing the flow of waterca n be reduced to a series of coupled one-dimensional equations.Digital elevation data sources

Digital elevation dat a are available for selected areas in the United States from the Natio nal C artog raph icInforma tion Center (NC IC ) in several forms (De pt. Interior-U.S.G.S., 1987). D E M s are available on a 30msquare grid for 7-5 minute quadrangle coverage, which is equivalent to the 1:24000-scale map seriesquadrangle. These data are produced by the U.S.G.S. for the NC IC from: ( I ) Gestalt Photo Mapper I1 (Kellyet al., 1977); (2) manual profiling from photogrammetric stereomodels; (3) stereomodel digitizing ofcontours; and/or (4) digital line graph data. DE M s based on a 3 arc-second spacing have been developed bythe Defense Mapping Agency (DM A) for 1 degree coverage, which is equivalent to the 1 :250000-scale m apseries quadrangle. Line ma p data in digital form, known as Digital Line Gr ap h (D LG ) dat a, are available for7.5 minute an d 15 minute topo grap hic qua drangles, the 1: 100000-scale quad rang le series, and 1 :2000000-

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6 1. D. MOORE, R . B.G R A YS O N A N D A . R . LADSONscale maps. A 30 arc-second DEM is now available for the conterminous United States from the NationalGeophysical Data Center (N G D C ) of the National Oceanic and Atmospheric Adm inistration (NOAA ).Mo ore and Simp son (1982) developed a coarse resolution D E M of Australia with a 6 minute (10 km) gridspacing and the Australian Survey and Land Information G ro up (A US LIG ) is currently developing a1 :1000000-scale grid DE M using a 18 arc-second grid-spacing (i.e. approximately 500 m) (Trezise an dHutch inson, 1986). Recently, Hutch inson and Dow ling (1991) developed a 1.5 minu te grid D E M ofAustralia. Both the 1.5 minute an d 18 arc-second D EM s were developed using the procedures described byHutchinson (1989a) which use both spot height data and stream line data. At the global scale, a 5 minuteglobal DEM is available from NGDC-NOAA.As noted above, direct photogrammetric techniques are available to produce DEMs from stereophotopairs. With the Gestalt Photomapping system, photo coordinates of control points are measured and theirlocations and elevations input (Kelly et al., 1977). A least squares solution is used to produce a stereo modelthat can be processed off-line to generate a D E M . Stereo images available from the French earth observationsatellite, SPO T (Le Systeme Po ur IObservation de la Terre), can be used to produce orthop hotos and D EM sin much the same way as conventional air photography. Satellite data has the advantage that it can bepurchased in digital form and directly accessed by computers.

I f digital elevation data for a particular study area are not available they can be derived by digitizingexisting topographic maps. Contour lines can be digitized automatically using the processes of rasterscanning and vectorization (Leberl and Olson, 1982)or by manually using a flat-bed digitizer and softwarepackages. A variety of low-cost flat-bed digitizers and softw are packages ar e now av ailable for most personalcomputers. Individual contours can be digitized and retained in con tou r form as DL G s or interpolated ontoa regular grid or TIN. Similarly, spot heights can be digitized and analysed as an irregular network orinterpolated onto a regular grid.Hutc hinson (1988, 1989a) describes a new efficient finite difference metho d of interpo latin g grid D E M sfrom co ntour line data or scattered surface-specific point elevation dat a. Th e method is innovative in that ithas a drainage enforcement algorithm that automatically removes spurious sinks or pits and calculatesstream lines and ridge lines from points of locally maximum curvature on contour lines. The most commonmethod of performing a c ontou r to TIN transformation is via Delaun ay triangulation (McL ain, 1976), butthis method can produce poorly configured trian gular facets where a facet edge crosses a cont our segment orall three vertices are on the same contour. C hristensen (1987) developed a parallel pairs pro cedure as part ofa medial axis transformation method that overcomes these problems.Digital elevution model quality

Many published DEMs are derived from topographic maps so their accuracy can never be greater tha n th eoriginal source of the da ta. For example, the most accurate D E M s produced by the U.S.G.S. are generated bylinear interpolation of digitized contou r maps and have a maximum root mean square error (R M SE ) of one-half contour interval and an absolute error no greater than two contour intervals in magnitude (Dept.Interior U.S.G.S., 1987). Care must be exercised when fitting mathem atical surfaces to D E M s as theresulting models give the appearance of generating data, but the accuracy of the data is unknown. TheU.S.G.S. DE M s are referenced to true elevations from published map s that include points on co nto ur lines.bench marks, or spot elevations (De pt. Interior-U.S.G.S., 1987). However, these tru e elevations alsocontain errors. I t is therefore apparent that all DEMs have inherent inaccuracies not only in their ability torepresent a surface but also in their constituent d ata a nd i t behoves users to become aw are of the natu re andthe types of errors (Carter, 1989).Th e quality of the U.S.G.S. DE M dat a a re classified as Level I , 2, or 3 (D ept . Interior-U.S.G.S., 1987).Level 1 is the standard format and has a maximum absolute vertical error of 50 m and a maximum relativeerror of 21 m. Most 7.5 minute quadrangle coverage is classified as Level 1 . Level 2 da ta have been sm oothedand edited for errors. DEMs derived by contour digitizing are classified as Level 2. Th ese da!a have amaximum error of two contour intervals and a maximum root mean squ are error (RM SE ) of one half of acon tou r interval. Level 3 data have a maximum error of one contour interval and a maximum RM SE of onethird of a contour interval-not to exceed 7 m. Th e U.S.G.S. does not currently prod uce Level 3 elevation

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DIGITAL TERRAIN MODELLING I : REVIEW OF APPLICATIONS 7data. The vertical accuracy of the Gestalt Photomapping system is 0.022 per cent to 0-03 per cent of theaircraft flying height and is therefore capable of producing a DEM of level 3 accuracy (Kelly et al., 1977).Konecny et a / . (1987) showed th at SP O T stereoscopic data may be used for topographic mapping up to ascale of I :25000 and is capable of producing a DEM with a grid spacing of 500m and a root mean squareerror of 7.5 m.The errors contained in the U.S.G.S. grid-based D EM s were classified by C arte r (1989) as either global orrelative. Global errors are systematic errors in the DEM that can usually be corrected by applyingtransformations including linear or nonlinear translation, rotation, and scaling to the whole DEM. LikeCarter, the authors have found mismatching elevations along the boundaries of adjacent 7.5 minute DEMsto be the most pervasive global errors. Relative errors occur when a few elevations are in obvious errorrelative to the neighbouring elevations (Carter, 1989). Th e errors in DL G d at a include artificial peaks an dblocks of terrain extending above the surro unding terrain, particularly alo ng ridge lines. These errors m ay bedu e to deficiencies in the interpolation a lgorithm s that produced the DL G da ta or to an excessive spacingbetween sampled coord inate pairs along the line data. Tha pa (1990) describes a method of optima lcompression of DLG data that may overcome some of these problems.Analysis of elevation data

The topographic attributes of a landscape can be calculated directly from a DEM using only the pointvalues without the assistance of surface fitting, smoothing operations or the assumptions of continuity(Collin s, 1975). Th is app roac h has limited usefulness, is restricted to grid-based DE Ms , and does not producephysically realistic results particularly in the calculation of flow directions in flat areas (Douglas, 1986).The most common method of estimating topographic attributes involves fitting a surface to the pointelevation data using either linear or nonlinear interpolation. The sequential steepest slope algorithmdeveloped by Leberl and O lson (1982) is an exam ple of a linear interpolation technique for generating gridelevation data. The interpolation is carried out firstly along ridge and drainage lines and then elevations areinterpolated for all other points in the grid using ridge, drainage, and contour lines as lines of knownelevation. Barnhill (1983) classifies nonlinear surface fitting methods in t w o ways: as local or global andpatch or point schemes. Glob al m ethods utilize all or most of the elevation da ta to characterize the surface ata point and have the advantage of preserving continuity. Their chief disadvan tage is their high compu tationalcost which is proportional to n3 , where n is the number of data points (Hutchinson, 1989a). For localmethods, the fitted surface (often a simple polynomial) at a point depends only on nearby data and thecomputational costs are proportional to n (Hutchinson, 1989a). Patch surfaces consist of small curvedpatches that are joined together smoothly, whereas point meth ods construct the surface using onlyinformation given at discrete points.A wide variety of methods are available for fitting elevation surfaces to point da ta a nd so me ar e describedby Hutchinson (1984). They include: kriging (Ga nd in, 1963; M ath ero n, 1973; Delfiner and Delhomm e, 1975;Ju pp and A domeit, 1981; and Dub rule, 1984), for which num erous com mercial packages a re available; localinterpolation m ethods (surveyed by B arnhill and Boehm, 1983); moving average m ethods; a nd splineinterpolation (Dub rule, 1984; Hutch inson, 1984). M ore recently, Hutchinson (1988,1989a) has developed aniterative finite difference interpolation method for use with irregularly distribu ted da ta t hat has the efficiencyof a local method without sacrificing the advantages of the global m ethods. Th is metho d uses a nested gridstrategy that calculates grid D E M s at successively finer resolutions.

Most digital terrain analysis meth ods ar e based o n grided d at a structures an d for these local interpolationmethods are the simplest and easiest to implement. This appr oac h has been used by Evans (1980), Zaslavskyand Sinai ( 1 98 1 ), Mark ( I 983), Jenson (1985,1987), Zevenbergen and Thorne (1987), Jenson and Dom ingue( 1 988), and Moore and Nieber (1989). One simple grid-based local interpolation method that appears verypromising is Snyder et al.s (1984) sliding polynomials. I t has the advantage over methods like Zevenbergenand Thornes (1987) in that the continuity of the fitted function and its first and second derivatives arepreserved between adjacent patches.One problem with the analysis of digital elevation data for hydrologic applicatiow is the definition ofdrainage paths when the DEM contains depressions or flat areas. Some depressions are da ta errors while

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8 1. D. MOORE, R . B. GRAYSON AND A. R . LADSONothers are natural features or excavations (Jenson an d Dom ingue, 1988; Hutchinson, 1989a). The hydrologicsignificance of depressions d epends o n the type of landscape represented by the DE M . In some areas, such asthe prairie pot-hole region in the upper Midwest of the United States, surface depressions dominate thehydrologic response of the landscape. In areas with co ordin ated drainag e, depressions are an artifact of thesampling and generation schemes used to produce the DEM. In landscapes with natural depressions, thenumerical filling of depressions in the DEM is used as a method of determining storage volumes and toassign flow directions that approximate those occurring in the natural landscape once the depressions arefilled by rainfall and runoff (e.g. M oore and Larson , 1979). M ark (1983) discusses the smo othing of da ta setsto remove depressions, but this has the disadvantag e of affecting all the elevations in the d at a set and will no tremove large depressions.O'Callaghan and Mark (1984) and Jenson (1987) proposed algorithms to produce depressionless D EM sfrom regularly spaced grid elevation d ata . I f the depressions are hydrologically significant then their volum ecan be calculated. Jenson a nd D om ingue (1988) used the depressionless DE M as a first step in assigning flowdirections. Their procedure is based o n the hydrologically realistic algorithm discussed by Ma rk ( 1 983) andO'Callaghan and Mark (1984), but is capable of determin ing flow paths iteratively w here there is more th anone possible receiving cell and where flow must be routed across flat areas. Hutchinson's (1988, 1989a)method, which was briefly described earlier, includes an automatic drainage enforcement algorithm thatremoves spurious sinks or pits. This can greatly simplify the task of obtaining a depressionless DEM.

If a surface defined by the function F ( x , y , z) is fitted to the D EM , then a number of hydrologicallyimportant topographic attributes can be derived from this function at the point (.yo, yo, zo). As an example.we will now dem onstra te how two of these attribu tes, aspect an d the max imum slope can be determined fromthe function F(x. y, z ) using elementary geom etry. We will then briefly describe three elementary loc al-pointmethods of representing surfaces using triangulated irregular networks, square-grid networks, and contour-based networks, respectively.Thefitted s u t j i x ~ ~he equation of the tangent plane to the point (xo. yo, zo) is

Now, if we let

then the equation of the plane tangent at the point (x0, yo, zo) is

ora b cx + - y + i + - = Od d d (4 )

where c = - x , - yo - zo =cons tant. The partial derivatives defined by Equa tion 2 may be estimateddirectly from the analytical form of F ( x , y, 2 ) or , i f the D EM uses a squa re grid. as finite differences. T hemaximum slope angle, /I, is defined as the intersecting angle of the tangent plane with the horizontal plane(i.e. z =0) and is given by

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DIGITAL TERRAIN MODELLING I: REVIEW OF APPLICATIONS 9The aspect is the orthogonal t o the tangent to ( x o ,y o , zo) in the horizonta l plan e (i.e. with z =zo =constant).Simplifying Equations 1 to 4 with z =z , =constant, the equation of the tangent to ( x o , y o , z,) in thehorizontal plane is

a cY = - - X - -b bwhere c = - x , - y , =constant. This is the equation of a line with slope - a / b , so the slope of itsorthogonal is b/a. Therefore, the aspect, $, measured in degrees clockwise from north is

I) = 180- arctan - +90 -(3 ( (7 )when x is positive east and y is positive north.Triangulated irregular network Tajchman (I98 1) divided the landscape into a triangulated irregular networkand represented the surface of each triangular segment or patch by a plane. The equation of a planedetermined by three points P , ( x , , y I , l ) , P z ( x z , y 2 , z 2 ) , an d P , ( x , , y , , 2 3 ) is z =A x +B y +C,where theconstants A , B, and C are determined by simultaneous solution of the equation at P , , P 2 , and P , . Followingfrom Equations 1 t o 7, the slope, B, of this plane in the aspect direction is given by

/3 =arctan[(A2 +B2)2], (8 )and the aspect of the plane, Ijl, is given by

Ijl= 180 - rctan - +90 -(3 (:) (9 )where y is positive north and x is positive east, an d the area of the triang ular segment in the horizontal plane,A, is given by

Tajchman (1981) used modifications of these relationships to examine the distribution of slope andazimu th classes an d their interrelationships over the Little Run catchment in the App alachian Mo untains.Because the surface is assumed to be planar this method does not provide point estimates of slope or aspect,but rather estimates average values over each triangular segment. For similar reasons, the method cannotbe used to calculate topographic attribu tes such as profile and plan c urva ture because the second derivativesof the surface functions do not exist. More complex techniques that use piece-wise continuous curvedpatches, such as those described by Barnhill an d Boehm (1983), d o have these capabilities.Regular gridnetwork Evans (1980) fitted a 5-term qua dratic polynom ial to the interior grid p oint of a moving3 x 3 square-grid network such as that show n in Figure l a. Heerdegen and Beran (1982) used this scheme tocalculate a variety of distributed catchment topographic attributes including slope, aspect and plan, andprofile curvature. Zevenbergen and Thorne (1987) modified this method by fitting the following quadraticpolynomial to this moving grid.

z =A x 2 y 2 +BXY +C x y 2 +D x 2 +E y 2 +F x y +G x +H y +I ( 1 1 )This 9-term polynomial exactly fits all 9 points in the 3 x 3 moving grid, whereas Evans (1980) 5-termpolynomial doe s not. D etermining the values of the coefficients in this equatio n is simplified by having a

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10 1. D. MOORE. R . B. GRAYSON AND A. R . LADSONuniform grid, which allows the coefficients to be expressed solely as functions of the grid-point elevations andthe grid-spacing, i. he slope,/I,spect, $, profile curvature , 4, plan curvature, ( 1 1 , and curvature, 1.which isthe L aplacian-V2 of the function defining the surface (Zaslavsky and Sinai, 1981). of the m id-point in themoving grid can then be calculated using the following relationships.

11=arctan[(G2 +H* ) ] (12)( Z ) +90(, :-,)I/ = 180 - arctan

DG2 +E H 2 +FG HG 2 +H 2 .~4 = - 2

D H 2 +EG2 - F G H(IJ =2 G 2 +H 2The plan area in the horizontal plane characterized by each node or grid-point is A , = I.*. Jcnson andDom ingue (1988) describe a computationally efficient algorithm for estimating flow directions and hencecatchment areas and drainage path lengths for cach node in a regular grid DEM based on the concept of adepressionless D EM which was described earlier. They assume that water flows from a given node t o one ofeight possible neighbouring nodes, based on the direction of steepest descent. Moore and Nieber (1989)combined this algorithm with Zevenbergen and Thornes ( I 987) approach to estimate a wide variety ofhydrologically significant topog raphic attributes.Coiitoirr-h(~r,se(~~wrnwrks. Moore et a/. (l988a), Moore (1988) and Moore and Grayson (1989, 1990) havedeveloped 21 method of computing contour-based networks for topographic analysis. Their scheme. calledTA PE S-C , Topographical Analysis Program for the Environmental Sciences-Contour, partitions iicatchment into irregularly shaped elements or patches such as is shown in Figure Ic. These elements arebounded by equipotcntial lines o n two sides (contour lines) and streamlines, that are orthogonal to theequipotential lines on the other two sides (n o flow boundaries). Both the conto urs and the streamlines or flowtrajectories between contours are approximated by short straight line segments. Streamlines arc computedusing two criteria: ( I ) minimum distan ce between adjacent con tou rs, that is applied in ridge areas; and ( 2 )orthogonals to the downslope contour, t ha t is applied in valley areas. Application of the two criteria isdctermincd by the plan curvature. The two criteria are used in an attempt to overcome the error caused byusing straight line segments to define streamlines. Bell (1983) has developed a program , S LO PR O FI L. 2, forconstructing contour orthogonals from user specified points on a square-grid network. I t has considerablepotential for simplifying Moo re an d G raysons (1990) scheme. Hutchinsons (1988) scheme can also be usedto simplify this task by interpolating intervening conto ur lines with respect to the strea m line and ridge linestructures.TAPES-C computes the average slope of each element as

where h- is the contour interval, L , an d L , are widths of the element along the upslope and downslopecontours, respectively. and A , is the area of the element in the horizontal plane. Aspect, $ is calculated at themidpoint of the lower conto ur segment of each element a s the downslop e direction o rth ogo nal to the co nto urat that point. The plan curvature, w. is com puted using a finite difference approxim ation t o successive x, ycoordinate pairs along each contour. The area A , is calculated using trapezoidal numerical integralion.TAPES-C also determines the connectivity between upslope and downslope elements in the network andthis is used to com pute the specific catchment are a and drainage path length for each element.

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DIGITAL TERRAIN MODELLING I : REVIEW OF APPLICATIONS 1 1HYDROLOGI CALLY, GEOMO RP HOLOGI CALLY, AND BI OLOGICALLY S I GNI F I CANTTOP OGRAP HI C ATTRI BUTES

Topographic attributes can be divided into primary and secondary (or com poun d) attr ibutes. Primaryattributes are directly calculated from elevation data and include variables such as elevation and slope.Compound attributes involve combinations of the primary attributes and are indices that describe orcharacterize the spatial oariability o j specific pro cesses occurring in the landscape such as soil water contentdistribution or the potential for sheet erosion. These attribu tes can be derived empirically but it is preferableto develop them through the application and simplification of the underlying physics of the processes. In thepast we have been able to m odel specific point processes in considerable detail but have not been able torepresent the spatial variability of these processes very successfully. With the index approach we sacrificesome physical sophistication to allow improved estimates of spatial patterns in the landscape.In this section we describe a range of topographic at tributes and indices, their underlying assum ptions andlimitations and the hydrological, geomorphological, and biological processes they attempt to characterize.We also outline their physical basis, but do not attem pt to give detailed derivations-the interested readershould refer to the cited literature for these.We envisage that the topographic indices described below could be used in a variety of ways. The primaryindices can be used directly in processes modelling ( e g slope and aspect c an be used in the estimation ofenergy budgets) or the compo und indices can be used a s surrogates of very complex hydrological,geomorphological, and biological processes (e.g. use of the wetness and radiation indices to predict thespatial distribution of different plant species). In many cases it is not possible to carry out directmeasurements of these environmental processes because of physical, time, or economic constraints. Elevationdat a are usually accessible, and thro ugh the terrain analysis meth ods described above, topographic attr ibu tescan be readily calculated without these constraints.In many developing countries there is a general lack of environmental data for planning projects, but thefirst information obtained is usually a topographic map. Fr om this elevation da ta, topographic attributes canbe calculated and used in the planning of data collection netw orks (i.e. hydrological m onitoring, soil surveys,biological surveys), and in the initial planning of agricultural and forestry developments (as an example, seeSrivastava and M oore, 1989). As additional environm ental da ta become available, these da ta can be used toprovide improved estimates of the terrain-based indices. For exam ple, the susceptibility of the landscape tosheet and rill erosion can be initially estimated using only topographic data. As hydrological and soilsinformation become available, this information can be integrated into the predictions and finally, asinformation on vegetation and cover is developed, the full Universal Soil Loss Equation (Wischmeier andSmith, 1978) or a complete physically-based erosion model can be used. Hence, in such situations we seedifferent layers or levels of data being developed over time with elevation data and the related topographicattributes constituting a minimum data set. Adoption of this approach by international developmentagencies has the potential for achieving major cost savings, particularly during the critical initial planningand implementation stages of a project.

Priniciry topographic uttrihute.~Speight (1974) described over 20 topographic attribu tes that can be used to describe landform. A series ofhydrologically related primary topographic attributes, adapted from Speight (1974, 1980), is presented in

Table 1. The attributes identified with an asterisk in this table can only be appraised along drainage lineswhile the others are spatially distributed properties. These can theoretically be evaluated at every point in th ecatchment, althoug h the values of many become unstable (i.e. have increasing erro r) close to the drainagelines. The primary topographic attributes that can be easily estimated using the computer-based methodsdescribed above include slope, /I;spect or azimuth, $; specific catchment area, A , , that is defined as theupslope area draining across a unit width of con tou r (Speight, 1980);flow pat h length, 7, profile curvature , 4;and plan curvature, w .Slope or slope steepness has always been an im portan t and widely used top ograp hic attribute. M any landcapability classification systems, some of which are described by Mo rgan (1986), utilize slope as the primary

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12 1. D. MOORE, R.B. GRAYSON AND A. R.LADSONTable I. Primary topographic attributes (adapted from Speight, 1974, 1980)Attribute Definition Hydrologic significanceAltitudeUpslope heightAspectSlopeUpslope slopeDispersal slopeCatchment slope*Upslope areaDispersal areaCatchment area*Specific catchmentFlow path lengthUpslope lengthDispersal lengthCatchment length*Profile curvaturePlan curvature

area

ElevationMean height of upslope areaSlope azimuthGradientMean slope of upslope areaMean slope of dispersal areaAverage slope over the catchmentCatchment area above a short lengthArea downslope from a shortArea draining to catchment outletUpslope area per unit width ofcontourMaximum distance of water flowto a point in the catchmentMean length of flow paths to a pointin the catchmentDistance from a point in thecatchment to the outletDistance from highest point tooutletSlope profile curvatureContour curvature

of contourlength of contour

Climate, vegetation type,Potential energySolar irradiationOverland and subsurface flowRunoff velocityRate of soil drainageTime of concentrationRunoff volume, steady-state runoff rateSoil drainage rate

potential energy

velocity and runoff rate

Runoff volumeRunoff volume, steady-staterunoff rateErosion rates, sediment yield,time of concentrationFlow acceleration, erosion ratesImpedance of soil drainageOverland flow attenuationFlow acceleration, erosion/Converging/diverging flow,deposition ratesoil water content

*All attributes except these ar e defined at points within the catchment

means of ascribing class, together with other factors such as soil depth, soil drainability, and soil fertility(Moore and Nieber, 1989). Dimensionless slope and flow path (i.e. slope steepness and length) are included asparameters in the Universal Soil Loss Equation-USLE, which was the basis for quantifying sheet and tillerosion by water in the recent assessment of the United States National Resources Inventory (NationalResearch Council, 1986). These two primary topographic attributes are being commonly used to assesserosion hazard. Moore and Thornes (1976) developed LEAP-Land Erosion Analysis Programs to examinethe spatial distribution of slope length, slope steepness, and the plan curvature for assessing 'topographicerosion potential '. More recently, Bork and Rohdenburg (1986) have developed a 'Digital Relief Model' forestimating and displaying the distributed morphographic parameters mentioned above.Hydrologically, the specific catchment area, A,, is a measure of surface or shallow subsurface runoff at agiven point on the landscape, and it integrates the effects of upslope contributing area and catchmentconvergence and divergence on runoff. For the ideal case of temporally and spatially uniform rainfull excess,the steady-state discharge per unit width, q. is directly proportional to A,. Even though this ideal conditionrarely exists in the natural environment this assumed relationship is used extensively in hydrology. Examplesinclude the application of the Rational formula, many applications of the USDA-Soil Conservation Servicecurve number technique, and most spatially lumped hydrologic models.Soil water content has been related to curvature (R2 =0.81) by Zaslavsky and Sinai (1981), slope, aspect,and specific catchment area by Moore et al . (1988b) and slope and plan curvature by Burt and Butcher(1986). Zaslavsky and Sinai (1981) observed that in their experimental catchment soil water contents werehigher in hollows even though there was no water table, perched water, or impermeable layer. Moore and

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DIGITAL TERRAIN MODELL ING I : REVIEW OF APPLICATIONS 13Burch (1986a) and others also dem onstrated t hat profile curva ture is an imp ortant determinant of erosionand deposition processes at the hillslope scale.Since the pioneering work of Whittaker (1967), there has been a concerted effort to describe the vegetationcontinuum in terms of environmental gradients. The term phytogeomorphology was coined by Howardand Mitchell (1985) in recognition of the interdependence of plants and landforms. They state thatphytogeomorphology reflects those sensitive landform -vegetation attributes tha t are visibly dom ina nt in thelandscape. Vegetation characteristics such as species com position, total basal area, and grow th habitat, havebeen correlated with gradient position defined by environmental scalars. These environmental scalars mayinclude primary topographic attributes such as altitude, slope and aspect, or derived attributes such asradiation indices (Austin et al., 1983, 1984) which are discussed later.In many cases altitude is used as an indirect means of accoun ting for spatial variation s in temperature and/or precipitation. Im proved estimates of the spatial distribution of climatic variables, including m ean seasonaland annual rainfall (isohyetal maps) and monthly mean daily maximum and minimum temperature, can beobtained from thin plate smoothing spline surfaces that f it the historical climatic dat a in terms of irregularlyspaced elevation da ta (Hutch inson, 1989b; Hutch inson an d Bischof, 1983). Hutchinsons method isparticularly attractive because it allows the variance of the dependent param eter to be spatially varied. Forexample, Hutchinson and Bischof (1983) assumed that mean rainfall was distributed independently withvariance a 2 / n ,where a s the local standard deviation estimate derived from the historical data and n is thenumber of years of record.Elevation an d a series of temperature a nd precipitation attr ibu tes derived from these elevation-basedclimate gradients have been used to develop a num erical classification scheme for examining the environm en-tal domain of rainforest structural types in the wet tropics World Heritage Property of northeastQueensland, Australia (Mackey et al., 1989). Mackey and No rton (1989) argue that physical environmentaldat a such as elevation, slope, aspect, and catchm ent a rea can be used as indices of species diversity a nd foreststructure and should be included in the recently proposed Australian National Forest Inventory.Dispersal area, which Speight (1974) indicated was related t o the soil drainage rate, cann ot be calculated ina m anner th at is consistent with th e estimation of specific catchm ent are a with existing grid-based m ethods ofanalysis. These methods have no provision for drainage from one node in the grid to multiple nodesdownslope in areas where flow is diverging. However, contour-b ased meth ods can directly calculate dispersalarea because they partition a catchment using streamlines.Analytically derived compound topographic indicesSoil water content and surface saturation zones Th e most comm only used hydrologically-based com poundtopographic attribute or index is

w r = l n ( T t2n 8 ) Or = In(&)where T is the transmissivity when the soil profile is saturated and w r and w are both often referred to aswetness indices. T he form represented by w assumes uniform soil properties; i.e. that the transmissivity isconstant throughout the landscape and equal to unity. Beven and Kirkby (1979) and OLoughlin (1986)derived slightly different forms of the wetness index, but both relate i t to the spatial distribution and size ofzones of saturation or variable source areas for runoff generation. Runoff from saturation zones is athreshold process and areas producing satu rati on overland flow can be identified using a threshold wetnessindex. Because significant percolation of soil water to groun d water occurs only w hen the soil water conten texceeds field capacity, a threshold concept could possibly be used to also relate the potential for groundwater recharge and non-point source pollution to the wetness index (Moore and Nieber, 1989).Moore et a / . (1988b) found a strong correlation between the distribution of w and the distribution ofsurface soil water content in a small fallow catchment. Burt and Butcher (1986) found that the product ofwetness index and plan c urvature ( w o ) produced the best correlations, whereas in the study of Moore et al.

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14 I . D. MOORE. R . B. GRAYSON AND A. R . LADSON(1988b) a linear relationship involving the wetness index, w , and aspect, $, performed best. Jones (1986, 1987)discusses the usefulness and limitations of the wetness index as an indicator of the spatial distribution of soilwater content and soil water drainage. Care must be exercised in applying a static index like w or wT topredict the distribution of a dynamic process like soil water content that shows hysteretic effects, especiallywhen threshold changes occur in the area of saturation (Burt and Butcher, 1986).Examining the surface water chemistry of the Llyn Brianne catchments in Wales, Wolock and Musgrove(1988) found that the mean value of the distribution of w T ,

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DIGITAL T E R R A I N M O D ELLIN G I : REVIEW OF APPLICATIONS 15cultivated fields with ridges (Shar on et al., 1988). Sh aro n (1980) also found t ha t the variations of hydrologicalrainfall between windward a nd lee slopes are 'reduced near wind-exposed ridges, owing to systematic small-scale variations of meteorological rainfall and of rain inclination'.Solar rudiurion dependent processes The surface energy budget is a driving force for evaporation andtranspiration processes occurring a t the land surface and is highly depende nt on top ography . E vapotransp ir-ation accounts for a major p art of the total water budget of a catchment. The potential solar radiation ratio,F =Ro/Roh,which is the ratio of the potential solar radiation on a sloping surface to that on a horizontalsurface, has been used widely in hydrological and ecological contexts as an approximate method ofexamining the spatial distribution of radiation across a catchment (Lee, 1978: Moore et al., 1988a). Th epotential solar radiation is the radiation received at a sloping site in the absence of the atmosphere and canbe expressed as:

241nr, =- os C#I cos 6 (sin q - q cos q )

where I is the solar con stant, S s the solar declination, r is the ratio of the earth -su n d istance to its mean, andC#I and are functions of the terrestrial latitude an d the topograp hic attr ibu tes of slope, /?,and aspect, $ (Lee,1978). Equation 21 can be numerically integrated over any time period rangin g from on e day to one year toestimate the seasonal potential solar radiation on an inclined surface. Hence, on a given catchment, thevariation in potential solar radiation over the catchment is a function of only slope, aspect, and the time ofyear.The major shortcoming of the above approach is that i t ignores all atmospheric effects which tend toreduce the contrast between sloping and horizontal sites (Fleming, 1987). Th e total shortw ave radia tionreceived on an inclined surface, R,, actually consists of three com ponents: direct-R, diffuse--Rd, an dreflected-R, radiation (Ga tes, 1980; Bristow an d Campbell, 1985; Flint and Childs, 1987).

This relationship can be approximated by:

where R,, an d Rd, are the total and diffuse radiation on a horizontal surface, respectively, f s the fraction ofthe sky which can be seen by the sloping surface [=cos '(p/2)], p is the albedo of the su rroun ding area, and pis the slope angle (Paltridge and Platt, 1976; Bristow and C amp bell, 1985). T he total and diffuse radiation ona horizontal surface are often expressed as functions of the total an d diffuse transmittances of the atmosphe reand the potential solar radiation on a horizontal surface, Rob. These transmittances are functions of thethickness and composition of the atmosphere, particularly the water vapour (i.e. cloudiness), dust, andaerosol content (Lee, 1978; Ho una m , 1963). Several models for predicting total shortwave radiation inmountainous terrain have recently been evaluated by Isard (1986). The reader is referred to the abovecitations and to Kondratyev (1977) for an in de pth discussion of these relationships.Equation 23 ignores the effect on computed radiation input t o a surface of shading from direct sunlight bysurrounding terrain at enclaved sites. Failure to account for this effect can have a significant impact on theability to predict snowmelt processes and snowpack conditions in alpine regions and on the ability tocorrelate vegetation diversity a nd distribu tion to radiation. Solutions of the horizon problem are computa-tionally intensive because they cannot be calculated from elevation data restricted to the immediateneighbourhood of a point (Dozier and Bruno, 1981). However, Dozier and Bruno (1981) present a solutionscheme with an efficiency proportion al to n, where n is the number of data points, for use with grid-basedDEMs. Dozier (personal communication) has recently extended this method to provide estimates of direct,diffuse, and reflected radiation and Duguay (1989) has adapted it for the estimation of net radiation

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16 I. D. MOORE. R . B. GRAYSON AND A. R . LADSON(shortwave and longwave) in mountainous terrain. An approximate radiation index that accounts for diffuseand direct radiation and t o a limited extent, the horizon problem , is also reported by Fleming (1987) and hasbeen used by Austin p r a / . (1983, 1984) and Mackey ef al. (1989 ). An excellent review of radiatio n mo delling isprovided by Duguay (1989).Vegetation diversity and biomass production have been shown to be related to radiation input in theUnited States (Hutchins er al., 1976; Tajchman and Lacey, 1986). Hutchins ef ill. (1976) found that slopesreceiving greater amounts of solar radiation had higher temperature s and greater evap orative dem and whichwas reflected in less dense tree stands and less well developed vegetation. The distributions of vegetation andsolar radiation in small mountainous catchments near H okka ido, Japa n are highly corrclatcd (Takahashi,1987) and Austin er al . (1983, 1984) have shown th at the distribution and zonation of eucalypt species insoutheastern Australia is related to altitude, rainfall, and annual radiation index. The radiation index incombination with the wetness index also has potential for characterizing biological distribution and speciesdiversity (Moore et al., 1988a).Snowmelt processes also respond to radiation inputs and are sensitive to topographic position (i.e.elevation, slope, and aspect). Slope and aspect m aps derived from D EM da ta can be used in conjunction withmultispectral scanner (M SS ) da ta and map s of the areal extent of snow cover, obtained from A dvanced VeryHigh Resolution Radiom eter (A VH RR ) satellite dat a, to estimate areal snow w ater equivalent a t a resolutionof approximately 500m (Carroll and Yates, 1989). Leavesley and Stannard (1989) are developing adistributed-parameter, energy-budget, snowmelt-runoff model that uses digital terrain analysis and geo-graphic information systems to provide objective basin characterization procedures. They are specificallyusing slope and aspect in their energy budget calculations and use elevation data in the computation of lapserates for air-temperature and precipitation extrapolation.Erosion processes Stream power, f2 =py q tan l j , where p g is the unit weight of water and y is the dischargeper un i t width, is the time rate of energy expenditure and has been used extensively in studies of erosion,sediment transport and geom orphology as a measure of the erosive power of flowing water. Th e compoun dtopographic index A , ta n/ ? is, therefore, a measure of stream power since 4 is often assumed to beproportional to A,. Moore et al. (1988b) found that the indices A , tan /I and w (i.e. AJtan ,4), when usedtogether, were good predictors of the location of ephemeral gullies in a sm all fallow agricultural c atchm ent.Fo r example, on a semiarid catchment in Australia ephemeral gullies formed w here MI>6-8 and A , tan /3 >18 (M oor e et al., 1988b) and on a small catchm ent in the island of Antigun in the Caribbean they formedwhere w >8.3 and A, tan f i >18 (Srivasta va and Mo ore, 1989). Th e threshold values of the indices differbecause of differences in soil properties. Thorne et al . (1986) developed the index A p an /i, where (1) is theplan curvature, to identify the location of ephemeral gullies. They also proposed the following equation forpredicting the cross-sectional area of an ephemeral gully, X , , after one year of development:

X , =0 . 2 ( A S o an / ? ) 0 ' 2 5 (24)The stream power index, or its derivatives, could be used to identify places where soil conserva tion measuresthat reduce the erosive effects of concentrated surface runoff, such as grassed waterways, should be installed(M oo re and Nieber, 1989).The physical potential for sheet and rill erosion in upland catchments can be evaluated by the productR K L S , a comp onent of the Universal Soil Loss Equation, where R is a rainfall and runoff erosivity factor, Kis a soil erodibility factor, and L S is the length-slope factor that accounts for the effects of topography onerosion (Wischmeier and Sm ith, 1978). T o predict erosion at a point in the landscape the LS factor can bewritten as:

LS =(n + ])(- ".>y Cn22.13 0.0896where n =0.4 an d m = 1.3. This equation was derived from unit stream power theory by M oor e and Burch(1986b) and is more amenable to landscapes with complex topo graphies tha n the original empirical equation

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DIGITAL T E R R A I N M O D E L L I N G I : R E V I E W OF APPLICATIONS 17because it explicitly accounts for flow convergence and divergence through the A , term in the equation(M oo re and Nieber. 1989). The erosion po tential can be normalized using the concept of an allowable soilloss, E , , to give the erodibility index, R K L S I E , , which can be mapped on the basis of the topographicattributes of slope and specific catchm ent area, an d soil and climatic att ribu tes via K , E , , and R (Moore andNieber, 1989).Soilpropert ies There have been numero us att em pts to relate soil properties, soil erosion class, and to a lesserextent, productivity to landscape position in the pedology and soil survey literature (e.g. Walker et al., 1968;Daniels et al., 1985; Stone et af., 985; Kreznor e t af . ,1989). For example, organic matter content and Ahorizon thickness, B horizon thickness and degree of development, soil mottling, pH and depth tocarbonates, and water storage have all been correlated to landscape position (Kreznor et al., 1989). Mo st ofthese studies use general mapping units such as head slopes, linear slopes, and footslopes to characterize soilposition, rather th an the specific topograph ic attributes described above. However, W alker et al. (1968) didattempt to correlate a range of depth characteristics, such as thickness of the A horizon, to slope, aspect,curvature, elevation, and flow path length (distance to hillslope summit), although there was muchunexplained variance which he attributed to relic features such as the prior streams postulated by Butler(1950).

A 1989 USDA-Soil Conservation Service task force on The Utility of Soil Landscape Units concludedthat future soil surveys must include more information on the shape of the land surface and that theselandform parameters should reflect the combined effects of both the hydrological and erosional processestaking place at different locations in the landscape, as proposed by Moore et al . (1986a). Digital elevationmodels were seen as the basic data for providing this information that would be delivered throughgeographic information systems.Information s ystem s and the display of topographic attributes and indices

Information systems and data bases for addressing resource and environmental problems must, bynecessity, be capable of integrating spatial information to representative and environmentally diverselandscapes and must be capable of being interfaced with m odels and o ther analytical tools used for analysis.Geographic Information Systems (GISs) are an exciting primary tool for manipulating, storing, andaccessing this data because they: integrate spatial and other information within a single system; offer aconsistent framework for analysing the spatial variation across landscapes; allow geographic knowledge tobe manipulated and displayed in a variety of forms including maps; and allow connections to be madebetween entities based on geographic proximity and characteristics that are vital to understanding andmanaging activities and resources (Unwin and Wilson, 1990).In Australia and elsewhere there has been a rush by many organizations responsible for resourcedevelopment, conservation, and the environment to develop GIS s with a view tha t they will be a panacea forproviding solutions to their assessment a nd m anagement problems. H owever, in many cases half digestedsecondary data are being input into them and data obtained at quite different scales are being used with noconsideration of scale effects. Prima ry d ata that mod ulate la ndscape processes and biological responses, suchas the primary topographic attributes described above, should be used as far as possible to construct thebasic data sets that a CIS needs to access. As ideas and concepts change and evolve subjective da ta a re oftenrendered useless. Wha t is also often lacking within a G IS a re knowledge-based (i.e., physically-based)approaches for analysing and interpreting data. Such approaches allow realistic answers to questions thatmight be asked, such as: Where are high erosion hazard zones located in a landscape? The compoundtopographic indices described above or even the more complex hydrological, geomorphological, andbiological models that are described later do provide this knowledge-based approach and can be imbeddedwithin the data analysis subsystems that any GIS must have. Most GISs are based on a pixel or cellularstructure so that the grid-based methods of terrain analysis described here can provide the primarygeographic data for them and can be integrated within these analysis subsystems.One of the reasons that CIS technology has been readily ado pte d is because i t allows spa tial informationto be displayed in integrative ways that are readily comprehensible and visual. The spatial distribution of

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18 I. D. MOORE, R . B. GRAYSON AND A. R . LADSON

Slope (YO)Figure 1. Areally wcightcd cumulative frequency distribution of slope. in per cent. of a 210ha subcatchment of the Cottonwood Creekin Montana

topographic attributes and indices can be displayed as isoline or coloured planimetric maps, colouredisometric projections or numerically in the form of cumulative, areally weighted, frequency distributions.Examples of the distribution of slope classes on a 210 ha subcatchment of the north fork of CottonwoodCreek in southwestern Montana (45"32'N, 1 1 I"39'E)are presented as a cumulative frequency distributionand as a grey-scale superimposed on an isometric projection of the 30 m grid-based D EM of the catchment inFigures 2 an d 3 , respectively. Thes e types of presentation allow the re ader to quickly get a 'feel' for the spatialvariability of a wide variety of processes occurring in a catchment.

BASIN DELINEATION AND STREAM NETWORKSDepending on the scale, drainage basins, catchments, or subcatchments are the fundamental unit for themanagement of land and water resources. Many distributed parameter hydrologic models, such as theStanford Watershed Model -SWM (Crawford and Linsley, 1966) and the Finite Element Storm HydrographModel-FESHM (Ross cr nl., 1979),use the partitioning of catchments into subcatchments or landunits, eachhaving a single soil mapping un i t and landuse type, as a means of crudely representing spatially varyingcatchment characteristics within the structures of the models. In FESHM these are called hydrologicalresponse units. The subdivision of the landscape into catchments and subcatchm ents is performed aroun dthe stream network and these areas are defined on the basis of the drainage network above the catchmentoutlet and above interior nodes or junctions where upstream channels join, respectively. The 'blue lines'drawn on topographic maps are a conservative representation of the stream network as they only representpermanently fowing or major intermittent or ephemeral streams (Mark . 1983). In the last ten years there hasbeen an increasing effort directed towards the development of automatic, computer-based channel networkmodels that use digital elevation d ata as their prim ary inp ut. Hydrological, geomorpho logical, an d biologicalanalyses using geographic information systems and remote sensing technology and computer-basedcartographic systems can be more easily integrated with these methods than traditional ma nual m ethods,with a high degree of objectivity. Excellent reviews of channel network analysis are presented by Jarvis(1977), Smart (1978), A b r a h a m (1984), Mark (1988) and Tarboton er al . (1989).

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DIGITAL TERRAIN MODELLING I : REVIEW OF APPLICATIONS 19

Slop. (%)ABOVE 40

30 - 4020 - 30BELOW 10

rntxl 10 - 20Figure 3. Spatial distribution of slope classes superimposed on an isometric projection of the 30 m D EM of a 210 ha subcatchment ofthe Cottonwood Creek in Montana

Mark (1988) identified several algorithm s for determ ining total accu mu lated dra inag e at a point forapplication to grid-based DEMs. However, some are computationally impractical to implement from anoperational and c om putatio nal efficiency point of view. On e of the earliest algorithms prop osed, C ollins'(1975) method, requires the elevation data to be sorted into ascending order and although a num ber ofefficient sorting algorithms are now available, the method is not practical for large elevation matricescompared to the alternative techniques described below. A second, and probably the most commonly used,algorithm for determining drainage or contributing areas and stream networks was proposed by O'Cal-laghan and Mark (1984). After producing a depressionless D E M they use the resulting drainage directionmatrix and a weighting matrix to iteratively determine a drainag e accum ulation m atrix th at represents thesum of all the weights of all elements draining to that element. Th e element w eights range from 0.0(no runofffrom the element) to 1-0 entire element contributes to runof f )and the drainage direction of each individualelement is the f low direction from the element to one of its eight nearest neighb ours based on th e direction of

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20 1. D . MOORE, . B. GRAYSON AND A. R. LADSONsteepest descent. If all the weights equal one, then the drainage accumulation matrix gives the totalcontributing area, i n number of elements, for each element in the matrix. Streams are then defined for allelements with an accumulated drainage area abov e some specified threshold. A somew hat sim ilar, but fasterand more operationally viable method of defining stream channels and drainage basins has been developedby Jenson (1985) and Jenson and Domingue (1988) and is being widely used. Another variation ofOCallaghan and Marks (1984) algorithm is described by Band (1989).

Another approach that uses Puecker and Douglas (1975) algorithm for marking convex and concaveupward points as ridge and stream points, respectively, is described by Band (1986). For each element thehighest of the neighbouring eight elements is marked and after one sweep through the elevation matrix theunmarked elements identify the drainage courses. Similarly, identifying the lowest of the eight neighbouringelements defines ridge lines which are ap proxim ations to the drainage divides. The drainage and ridge linesmay not be continuous and may consist of multiple adjacent lines that must be interpolated and thinned toone-element wide continuous lines. A connected channel network is developed using a steepest decentmethod that includes an elementary pit removal algorithm that tends to break down in flat terrain. I n thefinal step each stream junction element is defined as a divide, thu s anchorin g the divide grap h to the streamnetwork, and the divide network is refined. Th us subca tchm ents ar e defined for each stream link. Th e mcthodrequires initial smoothing of the DE M but does have the advantage th at no arb itra ry threshold area need bespecified (Tarboton et ul.,1989).None of these methods accurately predict where stream chan nels initiate. Chan nel heads may adv ance andretreat depending on discharge cycles (Dietrich et al., 1987)so their location is dynamic. Channel initiation iscontrolled by four processes: incision by Hortonian overland flow, incision by saturated overland flow.seepage erosion, and shallow landsliding (Dun ne, 1980; Mo ntgom ery a nd Dietrich, 1989). Mo ntgom ery an dDietrich (1989) found that local valley slope at channel heads is inversely related to source a rea a nd source-basin length as well as specific catchm ent area at th e channel head. In first-o rder catchments there is also aninverse relationship between the geometric properties of source-basin length and drainage density. On steepslopes channel head location is controlled by subsurface flow-induced instability; at abrup t channel heads o ngentle slopes it is controlled by seepage erosion; and gradual channel heads are controlled by saturationoverland flow (Montg ome ry an d Dietrich, 1989). Wilson et al . (1989) also indicate t hat soil failure, and hencechannel initiation, is more likely to occur in hollow areas with well coupled stormflow systems in both thebedrock and the overlying colluvium where high positive pore pressures exist. Channel initiation is a very

important process in the evolution of landscapes but is poorly represented in many geomorphologicalmodels of landscape evolution. Some dynamic geomorphological models are described below.S O M E T O PO G R A PH IC A L L Y -B A S ED H Y D R O L O G I C A N D G E O M O R P H O L O G I C M O D E L S

The period from about 1960 to 1975 can be viewed as the era of hydrologic modelling in which manydevelopments in the hydrological sciences occurred because of the growing awareness of the deficiencies ofthe old methods of analysis and the increasing availability of digital computers. This led to the use ofnumerical methods in hydrology and hydrologic modelling. The first version of the Stanford WatershedModel (S W M ) was released in 1962 and a subsequent version, the Mark IV SWM (Crawford and Linsley,1966). released in 1966, has probably become the most widely used hydrologic model over the last 25 years.Hydrologic models developed at this time w ere generally concerned with predicting water qu antities, such asrunoff volumes and discharges, at a catchment or subcatchment outlet. During this era there was a rapiddevelopment of mathematical descriptions of many individual hydrologic processes that were incorporatedinto many hydrologic models. Most hydrologic models were essentially lumped param eter m odels, or at bestonly crudely accounted for spatially variable processes an d catchm ent characteristics.Th e decade between ab out 1975 and 1985 can be viewed as the tran spo rt modelling era duri ng whichtime environmental problems became a top priority, including all forms of pollution. During this eratransport models were seen as the best way of predicting water pollution. Many of the hydrologic modelsdeveloped in the late 1960s an d early 1970s were used a s the flow com ponents in these tr ans por t models,often with little modification. These models poorly account for the effects of topography on catchment

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DIGITAL TERRAIN MODEL LING I : REVIEW OF APPLICATIONS 21hydrology where flow convergence and divergence has a major impact on observed and predicted flowdepths and velocities and are generally incapable of providing realistic spatially distributed estimates of thesehydrologic characteristics. Spatially variable flow characteristics such as flow depth and velocity are thedriving mechanisms for sediment and nutrient transport in landscapes and unless they can be predictedreasonably well water quality modelling cannot be successful and the results from these models may beinappropriate for making land-use planning and management decisions.

The last five years has seen an increasing emphasis on the need to predict spatially variable hydrologicprocesses at quite fine resolutions. We are now in the era of spatial modelling. Digital elevation data andremotely sensed catchment characteristics such as vegetation cover are now viewed as essential dat a inp uts tothe new generation of hydrologic and water quality models. Furthermore, we are beginning to see majorchanges in the basic structures of hydrologic models that allow them to more readily use this spatial inputdata and predict spatially distributed landscape processes. This section identifies and discusses a series ofhydrologic and geomorphological models that have been built around grid-, TIN- and contour-based DEMnet works.

Mod els using grid -base d structuresGrid or cellular approaches to subdividing the landscape provide the most common structures for

dynamic, process-based hydrologic models. Some examples of models based on this structure include theAreal Non point S ource Watershed E nvironment Response Simulation model- AN SW ER S (Beasley andHuggins, 1982), the AGricultural Non-Point Source pollution model-AGNPS (Young et a / . , 1989) and thevery detailed Systeme Hydrologique Europeen model-SHE (Abbot et al., 1986). AGNPS is a semiempirical,peak flow surface water quality model and does no t route the flow as such. SH E employs a two-dimensionalform of the diffusion wave equations to partition flow. In ANSWERS, a weighting factor is derived bypartitioning a cell on the basis of the line of max imu m slope. A problem with these app roach es is that the flowpaths a re not necessarily down the line of greatest slope so the resulting estimates of flow characteristics (e.g.flow depth and velocity) are difficult to interpret physically. ANSWERS uses a preprocessing program,ELEVAA, to generate slope steepness and aspect from grid-based DEMs for input into the model. Panuskaet al., (1990) interfaced a grid-based terrain analysis method a nd grid-based D EM s with the A GN PS m odelto generate the terrain-based model parameters which represent about one-third of the input parameters forthe model. Use of grid-structures fo r these models has a majo r benefit in that i t allows pixel-based remotelysensed data , such as vegetation types and cover (i.e. canopy ) characteristics, to be used to estimate the modelparameters in each element or cell because of the inherent compatibility of the two structures.TheGeom orphic Instantaneous U nit Hydro graph -GIU H, first proposed by Rodriguez-Iturbe and Valdez(1979) and adapted by Gupta et a / . (1980, 1986), Rosso (1984) and others, is becoming one of the mostcomm only used meth ods of determining the rainfall-runoff response of catchments. Th e method assum esrun off is generated uniformly throughout a catchmen t, but has more recently been adapted to include bothHortonian (Diaz-Granados et al., 1984; Gupta et al., 1986, and others) and partial area (Sivapalan andWood, 1990) runoff generation. The GI U H parameterizes the Instantaneous Unit H ydrog raph- IUH , whichrelates catchment response to rainfall excess, via the characteristics of the channel network expressed byHortons ( 1 945) laws through the Strahler (1952) ordering scheme of channel networks. On the basis of theassumptions of linearity, time invariance and a lumped system, the catchment discharge, Q ( r ) , can beexpressed in terms of the rainfall excess rate, i ( t ) , and the IUH, u ( t ) , as

Q ( t )= Jd ~ ( t ) i ( r )d sFollowing from Rosso (1984), a useful analytical form of the IUH is the two-parameter gamma probabilitydensity function:

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22 1. D. MOORE, R . B. GRAYSON AN D A. R . LADSONwhere r( is the gamma function and (x an d k are shape and scale parameters, respectively, that can beestimated empirically from the geomorphic structure of the catchment where:

and r is the mean streamflow velocity, L is the mean length of higher order streams and R,, R, and R , areHortons area. bifurcation, and length ratios. respectively. The product of the peak discharge and time topeak of the GIUH, U * , can be shown to be a function of catchment geomorphology only, such that:

U * =038(R,/R,)O RoO5 (29)In Equations 29 and 30 the three geomorphic ratios are defined as:

R, =N i / N i + ,R , =Li/Li- , R, =A i / A i - , (30)where N , , L i . and A i are the number of streams, the mean length of the streams, and the mean basin area,respectively, of orde r i (Horton. 1945).These quantities can be easily determined from the structure of thechannel network determined from a DEM of the catchment using the methods briefly described in theprevious section.Beven and Kirkby (1979) have developed a physically-based, topographically driven flood forecastingmodel, TOPMODEL. that is beginning to be widely used. The model is based on the variable contributingtheory of runoff generation and at the core of the model is the relationship:

Si =rnt - mln[AJ(T, tan j?)Ii+S (31)where S i is the soil storage deficit at point i , m is a recession parameter, i s the mean value of theln[A,/(Ti tan /I)] distribution for the catchment (the flow path partitioning index), T is the transmissivity,and S is the mean value of S i or the entire catchmen t. This equation permits the prediction of the soil waterdeficit pattern and the saturated source area (when Si I ) from a knowledge of topography and soilcharacteristics (Beven, 1986). Recent versions of TOPMODEL also consider Hortonian overland flow(Beven, 1986; Sivapalan e l al., 1987). Applications and developments of the model are described by Bevenand Wood ( 1 983). Beven et al . (1984),Hornberger r t al . (1985), Kirkby ( 1 986),and Beven (1986). Sivapalan etal. (1987)used the model t o examine hy drologic similarity a nd developed a series of dimensionless similarityparameters, one of which included a scaled soil-topographic parameter which was recently used to accountfor partial area runoff with th e GIUH (Sivapalan and Wood, 1990).The TOP M OD EL approach is mos tcommonly driven by a grid-based method of analysis but can easily be adap ted t o contour-based methods.Several geomorphological models have been developed t o simu late landscape evolution on geologic timescales. Landscapes are modelled as open dissipative systems where sediment transport is the dissipativeprocess and the state equation for elevation is the sediment mass continuity equation, written in the form(Willgoose et 01.. 1989):

&(x, y, t ) =sources - inks +spatial coupling? I (32)where z(x, v , t ) represents the land surface which is twice continuously dimerentiable in space (x , y ) and oncecontinuously differentiable in time, I (Smith and Bretherton, 1972). The spatial coupling is achieved viasediment transport which is usually expressed as a function of slope, /?,and discharge, Q,which in turn isspatially linked via the water continuity equation (usually the overland flow equations). Examples of theprocesses represented in these models include: tectonic uplift (sources), bedrock weathering, rockfall, slowmass movement (creep), rain splash and overland flow erosion an d sediment t ran spo rt (Kirkby, 1971;Smith

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23IGITAL TERRAIN MODELLING I : REVIEW OF APPLICATIONSand Bretherton, 1972; Ahnert, 1 987; and Willgoose et al., 1989). Th e evolution of both one-dimensionalhillslopes (e.g. Kirkby, 1971; Smith and Bretherton, 1972; Ahnert, 1976) an d two-dimensional landscapes(Smith and B retherton, 1972; Ahn ert, 1976,1987; and W illgoose et al. , 1989,1990) have shown that the shapeof the final hillslope or landscape is dependent on the form of the governing sediment transport equation.Most models, such as Ahnerts (1976, 1987) SLOP 3D model, which simulates a wide range of landscapeprocesses, do not specifically account for the linkages between the land system and the channel network.However, Willgoose et a / . 1 989,1990) recently developed a channel network and catchment evolution modelthat does account for these interactions. All th e two-dimensional landscape evolution models developed todate are driven by grid-based DEMs.Models using TIN-based structures

TINs have also been used for dynamic hydrologic modelling (e.g. Silfer et a / . , 1987; Vieux et al., 1988;Maidment et al., 1989; Go odric h a nd Woolhiser, 1989; Palacios an d C uevas, 1989). Vieux et al. (1988)minimized the problem s of using TIN s for dynam ic modelling by orienting the facets so tha t one of the threeedges of a facet formed a strea mline and there was only on e outflowing edge. However, this required a prioriknowledge of the topography, which was obtained from a contour map, thereby detracting from theadvantages of the TIN method. For the more common case where the facets are not oriented so that thejunction between facets forms a stream line, the mod elling of the flow on the facets is not trivial b ecause of thevariable boundary conditions (i.e. either one or two inflowing and outflowing edges). A method ofdistributing the inflow and /or outflow for each facet to downslop e facets und er these co nditio ns is describedby Silfer et al. ( 1 987). Go odric h a nd Woolhiser (1 989) divided each facet int o 15 subelements and applied atwo-dimensional finite element solution of the kinematic equations. Maidment et al. (1989) solved the tw o-dimensional St. Venant equations for flow on the facets.Models using conlour-bused strucrures

As noted earlier, contour-based methods of partitioning catchm ents provide a natural way of structuringhydrologic and water quality models. This metho d was first proposed by On sta d and Brakensiek (1968), whotermed i t the stream path analogy. With this form of partitioning only one-dimensional flow occurs withineach element, allowing water movement in a catchment to be represented by a series of coupled one-dimensional equations. The equations can be solved using a one-dimensional finite-differencing scheme.This method has been used by Kozak (1968) and Onstad (1973) to examine the distributed runoffbehaviour of small catchments. In each case the catchment partitioning was carried out by hand. Theapplications of Onsta d an d B rakensiek (1968) an d Ko zak (1968) involved small catch me nts with very simpletopographies, while that of Onstad (1973) was for a simple hypothetical symmetric catchment. Recently,Tisdale et al. (1986) used this technique together w ith an implicit one-dimensio nal kinem atic wave equa tionto examine distributed overland flow and achieved good accuracy when co mpared to steady-state solutionsfor flow depth and discharge over a hypothetical cone-shaped catchment. They found the method to becomputationally faster than two-dimensional finite element modelling methodologies, because it solvessimpler model equations, while retaining an equivalent physical realism. Mo ore an d Gra yson (1990), Mo oreet al. (1990), and G rayso n (1990) present two simple process-oriented hydrolog ic models that simulatesaturation overland flow and Hortonian overland flow using a contour-based network of elements and thekinematic wave equ ations for rou ting subsurface and overland flow between elements. In the solutions of thekinematic equations, the models use the slope and area of each element, the widths of the element on theupslope and downslope contours bounding the element, the flow path length across the element, and theconnectivity of the elements calculated by the TAP ES -C terrain analysis suite of programs described earlier.Adopting an approach similar to TOPMODEL, Moore et al. (1986b) used OLoughlins (1986) steady-state, contour-bas ed m ethod of comp uting the In(AJtan p ) versus per cent saturated a rea relationship (an d inthe process fitted the mean catchment transmissivity) to simulate the saturated source area expansion a ndcontraction as a function of catchmen t wetness. This relationship , together with the assu mp ion of successivesteady-states in the time domain, provided the basic structure of a simple lumped-parameter model of

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24 I . D. MOORE, . B. GRAYSON AN D A. R. LADSONforested catchment hydrologic response. In the m odel runoff was assumed to occur via subsurface flow a ndsaturation overland flow.

FRACTAL DIMENSION OF LANDS CAP ESTh e idea that different processes dom ina te hydrologic and geom orphic response at different scales is implicitin the literature describing the modelling of these systems. In the preceeding sections of this paper we haveattemp ted to show how these processes relate to land form and th e characteristics of the stream network. Th elast decade has seen an explosion in the application of fractal theory to the characterization of topographicsurfaces (Goodchild, 1980; Mark and Aronson, 1984; Clarke, 1986, Culling and Datko, 1987; Roy et al..1987; Huang and Turcotte, 1989 and others) and river networks (Tarboton et al., 1988; Hjelmfelt. 1988; LaBarbera and Rosso. 1989 and others). Na tural topographic surfaces show remarkable similarity to syntheticfractal surfaces generated by fractional Brownian processes with fractal dimensions, D, ranging from 2. I to2.3 (Mandelbrot. 1975, 1982; Goodchild, 1982). Most landscapes can only be considered to be fractal in astatistical sense. T he application of fractal theory in hydrology, geomorphology, and biology is in its infancyand there is great potential for major advances in this area of research in the coming years.The topological dimension, D,, familiar from Euclidean geometry is 1.0 for a curve and 2.0 for a surface,whereas the corresponding fractal or Hausdorff-Besicovitch dimension, D, can range from 1.0 -2 .0 and2.0-3.0, respectively. Fractals em body the concept of space-filling. River netwo rks and landscapes, whichcan b e visualized as a series of bran chin g lines and folded surfaces, respectively, ar e said to be space-filling iftheir fractal dimensions are 2 and 3 , respectively. Fractal dimensions have been defined by spectral analysisusing Fourier transforms (e.g. Frederiksen, 1981; Voss, 1985; Huang and Turcotte, 1989) and by variograms(e.g. Goodc hild, 1980; Mark and Aronson, 1984; Roy et al., 1987) which can be written as

E[ ( z , - ~ + ~ ) ] d2 ( 3 3 )where d is the distance between points i an d i +d, z is the observed value at the tw o p oints (e.g. the elevationof the land surface), and Culling and Da tko (1987) called H the Hurst scaling parameter which ranges from0.0.-1.0.The fractal dimension is then

D = ( D , + I ) - H (34)Goodchild (1980) presents examples of simulated surfaces with fractal dimensions ranging from 2.1 (smoothsurfaces) to 2-7 (rough surfaces).In natural landscapes D appears to be constant for only limited areas over limited ranges of scales(Goodchild, 1980). Mark and Aronson (1984) computed the surface variograms for 17 7.5 minute U.S.G.S.DE M s from Pennsylvania, Oregon, and Co lorado in the United States and found at least two different valuesof D. At small scales, for distances between 0.4 and 1.8 km , D ranged from -=2-2.48; at intermediate scales,for distances up to 4-7 km , D ranged from 2.39-2.8; while in some cases at larger scales the surfacevariograms displayed periodicities. They suggest that the breaks represent characteristic horizo ntal scales atwhich different surface processes operate. Similar results were obtained by Culling and Datko (1987) whosuggested that D ranges from 2.0 to 2.3 a t the hillslope scale an d from 2.4 to 2.6 at scales associated with thedrainage network. Huang and T urcot te (1989) found th e mean fractal dimension of the state of Arizona to be2.09. Roy et al. (1987) calculated fractal dimensions of fluvial, summit, and glacial areas in eastern NorthAmerica of 2.13, 2-10,and 2.21, respectively, but found that the values of D varied spatially within the DEMand w ith elevation. Roy et al . also attribute the higher fractal dimensions obtained by Mark and Aronson tothe sampling plan used which creates biases towards the longer scales.

I f a surface is self-similar then the fractal dimensions of the surface and of profiles along the surface shoulddiffer by unity. However, Huang and Turcotte (1989), using spectral analysis techniques, found the meantwo-dimensional fractal dimension to be 2.09, and the one-dimensional fractal dimension to be 1.52(H =0.48). The process is Brownian for H =0.5.

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25IGITAL TERRAIN MODELLING I : REVIEW O F APPLICATIONSBoth T arboton et al . (1988) and La Barbera and Rosso (1989) have examined the fractal dimensions ofriver networks and compared them with those derived from Hortons (1945) laws. Th e fractal dimension ofindividual rivers range from 1.1-1.2 and that of the network as a whole is about 2.0, implying that thenetwork is space filling which is consistent with classical fluvial geomorphology and the popular randomtopology m odel (Tarb oton et al., 1988). Hjelmfelt (1988) and La Barbera and Rosso (1989) demonstratedthat if channel netw orks ar e fractal, then a length or drainage density measurement, L , , at scale d , is related

to that ( L 2 )measured at scale d 2 by:

They refer to Pilgrims (1986) work on design flood procedures in Australia which incorporates a series ofparameters that are related to the topology of the river network and hence dependent on scale. Equation 35can be equated to the empirical map scale adjustment relationship proposed by McDermott and Pilgrim(1982), producing a value of D of 1.09, which is consistent with the fractal dimension of an individual stream(Hjelmfelt, 1988).S U M M A R Y A N D C O N C L U S I O N S

The relative magnitudes of many hydrological, geomorphological, and biological processes operating innatural landscapes are sensitive to topographic position. Indices of these processes can be developed asfunctions of distributed soil, vegetation, and , particularly, topo graphic attributes. C om puterized meth ods ofterrain analysis for use on small inexpensive personal com puters have the capability of quickly a nd efficientlyderiving these topographic attributes. This paper attempts to introduce some of the basic concepts involvedin terrain analysis using digital elevation d ata and dem on stra te the usefulness of the technique forhydrological, geomorphological, and biological applications. These techniques have the potential forwidespread application to resource evaluation, planning, and management. Finally, these methods providethe opportunity for realistically representing the three-dimensional nature of natural landscapes inhydrologic and geomorphologic modelling under the constraints of maintainin g physical rigour, simplifyingthe governing equations that must be solved, and reducing the computational requirements.This review has been divided into five sections. The first section discusses the general characteristics ofdigital elevation models and describes the three major methods of structuring a DEM (grid-, TIN- andcontour-based methods), the sources of elevation data in Australia and the United States, how DEMs arecomm only produced, and gives some indication of their quality. The section concludes with a discussion ofsome simple techniques for analysing elevation data for the estimation of topographic attributes such asslope and aspect and gives examples of methods that can be applied to each of the three D E M structures.In the second section we describe a range of topographic attributes and indices, their underlyingassumptions and limitations and the hydrological, geomorphological, and biological processes they attemptto characterize. We also outline their physical basis, but d o not atte m pt to give detailed derivations. Th e useof grid-based DEMs for this form of analysis is recommended because grid-based DEMs are the mostcomm only available form of digital elevation da ta, the m ethods of analysis are com putationally efficient an dsimple and this structure is compatible with remotely sensing techniques and geographic informa