Geomorphometric processing of digital elevation models

Computers & Geo.wtences Vol. 13. No. 6. pp. 603~:~09, 1987 Printed tn Great Britain. All rights reserved

0098-3004 87 $3.00 + 0.00 Copyright ~ 1987 Pergamon Journals Lid

GEOMORPHOMETRIC PROCESSING OF DIGITAL ELEVATION MODELS STEVEN E, FRANKLIN Department of Geography. Memorial University of Newfoundland. St. John's. Newfoundland. Canada AIB 3X9 (Received l January 1987: accepted 30 May 1987)

Abstract--The general system of geomorphometry is composed of elevation, derivatives of elevation at a point, and moments of the distribution of elevation over some area. All of the point measures in this system can be obtained by computer processing of a digital elevation model (DEM). and they can be used as input to the analysis and classification of terrain. A suite of FORTRAN programs implementing this system for dense grid DEMs has been designed and used in various operating environments. Attention has been given to the methods used in the approximation of terrain concepts such as slope and relief. An area of high relief in subarctic Canada is used to illustrate the discussion. Key lt'ordx: Geomorphology. Relief. Slope, Terrain analysis.

1981; Oswald and Raetzseh, 1984). However. the translation of complex geographic concepts to operaInformation extracted from dense grid digital eleva- tional methods is not straightforward always, as tion models (DEMs) can bc used in terrain classifica- shown by the several ways measures such as slope tion and applied gcomorphological studies (e.g. have bccn derived. In this paper, the literature is Craig, 1983; [:vans, 1980; Mark, 1979: Collins, 1975; sunuuarized briefly in that regard with the aim of 110il and Brych, 1981), and can bc combined with providing the initiate with some background on the other qttantitative data such as that derived from digital use of this relatively new and extremely powerremote sensing systems (e.g. tlutchinson, 1982; ful data source in physical geography. A number of Franklin, 1987; Strahlcr, Logan, and Bryant, 1978). operational definitions are clarified and one applicaTypically, the information is contained in measures tion of general geomorphometric processing in is highwhich are either highly site specific or which are a part relief environment in Canada's Yukon is illustrated. of :t general system. They :ire obtained by computer processing rather than manual techniques. Specific gcomorphometry, by definition, is applied GENERAL GEOMORPIlOMETRY in an analysis where specific landforms and terrain features are separated a priori from the adjacent par- Elevation is essentially an instantaneous point value. cels of land using clear and recognized criteria (Evans, In a dense grid format DEM (shown in Fig. I), each 1972, 1985). Examples can be cited in the analysis of elevation value represents the basic phenomcnocirques (Trenhaile, 1976), open rock basins (Sauchyn logical unit of analysis and therefore, is analogous to and Gardner, 1983), and in conventional geomorpho- pixel size in remote-sensor imagery, or photographic logical investigations of stream channels and drainage resolution in photographs. The terrain is viewed as a basins (see Mitchell, 1973; Townshend, 1981). two-dimensional signal where the pixel size or denseGeneral gcomorphometry is the field of measurement ness of the grid is analogous to the sampling rate of and analysis of those characteristics of landform that elevation. The general system of geomorphometry is are applicable to any continuous rough land surface based on this definition in the following discussions. INTRODUCFION

(Evans. 1972; Franklin, 1987). That analysis involves metric parameters of elevation, slope, aspect, relief, and convexity, which have been used in parametric landscape analysis alone (e.g. Speight, 1968; Collins, 1975), in integrated terrain mapping (e.g. Franklin, 1987), forest type mapping (e.g. Fleming and Hoffer, 1979) and the correction of radiance data (e.g. Holben and Justice, 1980; Teillet, Guindon, and Goodenough. 1982). Computer programs designed to process dense grid DEM data and extract measures of geomorphometry are plentiful (e.g. Collins, 1975; Grender, 1976: Franklin and Peddle, 1987: Collins and Moon,

Terrain shape and aspect In physical geography, slope is considered a point valuc (Young, 1974). Slope is the first derivative of elevation and can be estimated in a number of ways. For example, surface fitting has been used by Letts and Rochon (1980) with digitized topographic map data, although with dense grid models there may be some difficulty in fitting a smooth plane to highly irregular elevation values. The slope (and aspect) angles are calculated using the partial derivatives of the elevation values with respect to the X (East/West) and Y (North/South) directions. The partial deriva-

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F i g u r e I. I s o m e t r i c v i e w o f d e n s e g r i d d i g i t a l e l e v a t i o n m o d e l

of Aishihik Lake area. southwestern Yukon• Canada. iAI view from southeast (azimulh 135 ) at elevation of 23 ; JR) view from southwest (azimuth 225 ) at elevation of 23 . rives can be determined using two 3 x 3 first-order filters (Sobel operators). The slope and aspect then are calculated its follows: slope = aspect

=

tan i{[(?,:/&~.): + (~:/?y),.],.,} tan t i( _ i~:/,~.r)/( _ ?.7/?.~.)].

The calculation esscntialiy fits a second-order surface to the neighborhood based on a minimum meansqu:trcd error criterion. But, because smoothing does occur, there is a danger of attributing characteristics of the mathematical surface to terrain characteristics. This may be acceptable with topographic map data, ~hich are generalized already. The best-fit surface slopes are the slopes of the plane, and not, in the situation of stcreocorrelation or survey models, of the elevation surface characterized by the data-acquisition process• In physical terms, continuity between all points in the window must be assumed for the derived slope to have meaning; this assumption is based on the fact that the procedure involves derivafives of the surface at the center of the plane which may or may not be representative of slope over the selected window. Another approach to slope is to reject areal continuity except between two points, where the points represent current elevation and its lowest (or arbitrarily, its highest} nearest neighbor (Collins, 1975; Oswald and Raetzsch, 1984). There is an assumption that the slope between any two adjacent points is linear. If this assumption is not justified, mathematical interpolation will not improve the situation from a physical metric point of view (Yoeli, 1983)• Slope is simply in the direction water flows to (or from) the current point--the difference in elevation divided by

Figure 2. Isometric view of derivative slope surthcc I\~r Yukon DEM. Slope values are derived using nearest netghbor algorithm to compute direction water would flow to o~er 22-m grid spacing from any elevation point. Result is replicate grid containing magnitude of slope. View is from southeast (azimuth 135 ) at elevation of 35 . the cell resolution (distance between the two points). The application of an algorithm based on this concept results in a slope surface depicted in Figure 2 Ik~r the Yukon grid. Such an approach eliminates undesirable surface smoothing and is a consistent calculation f o r all areas regardless of the dcgrcc of discontinuity (Collins, 1975). Moreover, the calculation closely emulates licld measurements of slope (reviewed by Young. 1974) and replaces the elevation grid with a point by point replicate containing slope values. A higher susceptibility to noise in the elevation grid may bc apparent because the approach uses only two points. In stcreocorrelation models this may not b c a limiting lactor because patch (neighbor) values arc determined sire ultancously. Surlace aspect is the directional component ol" slope and can be calculated (as noted) as the direction that slope faces. However, it is specified traditionally as the direction on the compass, usually in categories (e.g. Fleming and Hoffer, 1979). Compass azimuth is not a metric; for example, 10 ' is closer to 360' than to 30 . Thus, terrain aspect specified in this way must bc analyzed using circular statistics, or at least, discussed separately from other metrics extracted from the model. Alternatively, transformations of aspect such as those which occur in discussions of geometric relationships between inclined surfaces and solar illumination (Iqbal, 1983) could be used. One practical method was used by Justice (1978) and further discussed by Townshend (1981) for use with high sun elevation satellite images. Incidence values are calculated for slope as a function of aspect, solar elevation and solar azimuth: Incidence =

cos(~) + sin (:t) . cot (/~) . cos (0)

where u is slope, f/ is sun elevation, and 0 is the difference between terrain aspect on a compass and the solar azimuth. Each term is expressed in degrees. The result represents an estimate of the ratio of direct solar illumina-

Geomorphomctric NW.

605

processing of digital clevatron models

315.

N.0. va1u*

heldwlea

NE ’ 45’

-1.34

I

slop’

-0.41

5.1 le

L.”

8.

I\

-0.48

I

I\

-n,n

020 0.13 0.07 a28

I.34

0.49a66

Il.64

bO/

/

3.72

o,e5

\ .71

--I/

.61 1.47 1.33 s*ieo* Azimuth

9 151’

Elavatlon

= 33’

f?gurc 3. lncidcncc value schematic for Yukon DEM and solar conditions during LANDSAT image acquisition on 29 Augusl 1978. Incidcncc values arc computed as function of tcrmin slope and aspect and solar illumination geometry. Ncgativc values in this schematic correspond with brain in shadows.

tion and a dilruse component. in shadow

Terrain

aspects totally

rcccivc only ditTuse radiation,

and may

have a ncgativc incidcncc value. No consideration backscattcr,

adjacent

slope flux or horizon

ing is given.

The variations

of

brighten-

in slope and aspect that

lure features Calculations

angular

second moment

rely on cooccurrcncc

and entropy.

malriccs scl up in

moving windows. The number of times pairs of clevulion values

occur together provides an indication

surface roughness.

Summary

ot

measures of homoge-

result in varying incidence values for the solar illumi-

neity or energy can produce a stable estimate of tcr-

nation conditions

rain relief which can be used to characterize

MSS

image

Yukon

that prevailed during a LANDSAT

acquisition

in August

1978

over

the

test site arc depicted in Figure 3.

faces depicted facilitates

in the model

the interpretation

at various

of frequency

tions and the relation between landform

the sur-

scales. This distribu-

and gcomor-

Terroin rrlie/ and curvumre Relief is a well known and well understood

topo-

graphic concept used to convey an appreciation vertical

extent

without

rcfercnce to absolute slope or elevation.

example.

or dispersion

Gardner,

Smith,

the range in clcvation

to characterize

the standard

rcliefcstimatc;

features

and Deslogcs (1983)

large area in the Canadian prcferrcd

of landscape

of the

Rockies.

deviation

For used

rclicf over a Evans

(1972)

of elevation

as a

it is less sensitive to the arca of calcula-

tion and does not rely on few point samples. In digital imapcs. (Haralick.

the variance

is used as a textural

Shanmupam.

1978) which

and

Dinstein,

can be interpreted

spatial distribution

of elevation

More sophisticated ted by Franklin

feature

1973;

Hsu,

as roughness-the variability

(Fig. 4).

measures of relief w&e presen-

and Peddle (1987) based on the lex-

Figure 4. Isometric view of derivative relief surface for Yukon DEM. Relief values are generalizations of variance in elevation over I2Om square. This view is from southwest (azimuth 225”) at elevation of 35’.

606

S.E. FgA.~gLX.~

Figure 5. ()rthophotogr:tph of Yukon study site showing tcrr:fin features of interest. Area has been classilied into nine I,mdseapc classes: w:tlcr. I\~rcst,woodland, upland shrub, :dpinc shrub, alpine tundra, alpine barrens, marshhmd, and exposed slo~,s. Within each class, samples are generated to characterize gcomorphometric conditions of terrain. phological process in applied geomorphometric studies. Local surface convexity also generalizes terrain characteristics from point v:tlues; it is defined its the rate of change of slope or the second derivative of elevation (Evans, 1972; Papo and Gelbman, 1984). A function:tl interpretation of convexity lacks unified treatment in the physical geography literature. In essence, convexity takes on high values when the rate of elevation change increases between points that are close together. Papo and Gelbm:m (1984) suggested the intuitive interpretation of slopes and their curvature made "topographic sense" in terrain analysis. They referred to applications in military and engineering studies. Unfortunately this treatment is ditlicult to quantify although an intuitive definition of curvature as second-order information in the processing of slopes is helpful. Convexity can take on values that are cross-slope or downslope in plan or profile. The calculations are straightforward using best-fit techniques or nearest neighbors. The same arguments used in ration:dizing slope calculations apply here. If neighbors are used the interpretation can be simple; if the four neighbors are at greater elevation than the center, the value will be negative and the surface concave: if the neighbors are at lower elevation, the center may represent a peak

anti a high positive value (convex surface) will occur at that point. Of course, more complex interpretations are needed in variable terrain. TERRAIN ANAI,YSIS Terrain analysis is the "'set of activities which leads to the compilation of terrain characteristics or terrain qualities" (Townshend, 1981, p. 10). In this paper, one obvious set of terrain characteristics of importance in such activity, geomorphometry, has been discussed. From these general measurements we can describe and discriminate parcels of land from adjacent terrain; for example, the Yukon grid was described by Franklin (1987)in the separation of nine landscape classes in conjunction with spectral remotesensing imagery. The approach was biophysical and the relative improvement over a spectral classification alone was 29% when compared to the results of an aerial photography interpretation (Fig. 5). Sirnilar classification results have been noted by Strahler, Logan, and Bryant (1978) and Fleming and Hoffer (1979) in using spectral and geomorphometric data sets in mountain terrain. The general measures have been subjected to additional simple processing steps and used in planning and engineering studies. lleil and Brych (1981) and Collins (1975), among

607

Geomorphometric processing of digital elevation models Table I G e o m o r p h o m e t n c correlations by landscape class in Yukon study area Relief (a) Forest n = Ele'.atmn Relief

Convexity

Slope

Aspect

Mean

Variance

IO0 • "

* 065 0.53

0.41 " " -0.22

989.80 6.58 0.33 13.39 0.95

53.47 2.76 5.93 7.13 0.27

0.45

•

034 0.84 0.51

047 • -0.24 •

1200.47 6.67 0.18 12.42 0.94

86.43 2.39 4.37 6.04 0.21

0.32

0.33

0.39 0.59 0.71

" - 0.35 - 0.36 -0.68

1243.40 8.05 0.24 15.13 0.84

23.14 2.36 6.98 846 022

0.02 0.85 0.43

0.40 0.50 " 036

1074.77 I 1.63 0.05 20.35 109

100.03 7.18 827 12.08 0.37

- 0.22 0.80 050

° - 066 - 0 31

1368.16 6.30 0.85 12.62

18.24 3.95 6.4.4 7.78

0.87

0.24

Con',elltV Slope (b) Woodland n =

I00

Elevation Rehef Conve ~tlU'. Slope
Slope (d) Alp,he shrub n = I00 Ele'~atton Rehef

0.72

*

(.'O l i v e "Ki t %

Slope (e) Alpine tundra n -

75

Ele'.atton Rebel

0.47

- O. 36

{~'OII v e x l l %

Slop,= 11+) Alpine barrens n -

511

I!l¢'+allon

0.37 -0.32

Rchcf ( +Oll','e~ll%

* 0,~2 *

Slol~ (g) M.tr'~hktnd n +- 50 I']cwttlon Rchcf

- 0.511

0.21

{ '¢)[|VC~.LI Y

"V'I

0...

('Oil', exit y

Slope

1244.86 1845 O. I0 29.98 0.34

79.71 4.34 6.(R+ 6 57

0.23

- 0.28 057 0.61

* - 0 21 * "

963.00 3.00 - I.I)2 6.63 0.99

16.53 1.91 7.01 7.(H) 0.17

• 0.84 0.40

* - I).48 " -0.57

935.(H 8.29 - 1.12 14.81 0.77

211.85 5.1)1 6.78 9.~ 0.31

Slol~e (h) Exp,,,,ed sh,l'~es n ~ 511 I!levatton Reh¢f

* - 039 * - (1.45

*Correlatmn not signilicant at 0 O I .

others, reported the use of the hypsometric integral and area by slope histograms in classifications. Advanced terrain amtlyses producing hydrological maps and land use study inputs also were discussed. Applied geomorphological studies have been supported in several important ways. in general, a preliminary geomorphometric analysis would involve the calculation of indices (e.g. hypsometric integral) to summarize conditions, the fitting of equations to characterize surfaces (e.g. gradient trend surface) or the interpretation of frequency distributions in bounded areas (e.g. landscape units or geological zones). The analysis then would be related to landform and process. For example, Evans (19801 interpreted four descriptors calculated from the distributions of each of five derivatives from elevation models obtained

from many diverse locations. That analysis facilitated geomorphological comparison of areas subject to different processes and having dissimilar morphology. EXAMPLE

APPLICATION

In Table I the geomorphometric relationships sampled from particular land units representative of the nine landscape classes noted in Figure 5 for the Yukon study area are described. In terms of precision, this tabulation by class represents a significant improvement over the relatively subjective interpretation available through photointerpretation. Of particular interest are the relationships between elevation and relief and between convexity and slope. The correlation coefficient in the former situation ranges from

S. E. FgANKLI.~

608

insignificant in the forest areas to - 0.47 in the alpine tundra and +0.72 in the alpine shrub regions. The negative relationship occurs as a result of the plateau morphology of tundra in this geographic region; as elevation increases between about 1150 and 1400m a.s.I, the terrain becomes level and less variable. In the alpine shrub zone. much lower in absolute elevation. the variance in both relief and elevation is high because the slopes are much more steep and variable. The relationship between convexity and slope is not straightforward, partly as a result of the difficulty inherent in physical interpretations of average surface curvature over an entire landscape class. However, although convexity is extremely variable in this environment, there are trends worthy of note. For exampie, the correlation coefficient is not significant in the barrens class, but is consistently about 0.40 to 0.55 except for the upland shrub class where it is 0.71. There is a trend in units having weaker slope/relief relationships to have stronger slope/convexity relationships, and convexity and relief are not related except in the barrens class. The interpretation here might focus on the steep nature o f the ghtciated valleys and the presence of broken terrain in the form of talus slopes and scrce occurring adjacent to the barrens class along the elevation gradient. CON('I,USIONS

I)igital methods of terrain analysis and classification provide a method of reducing analog decisions, such as those used in landscape mapping from aerial photographs, to a repeatable, consistent process in v,hich the subjective elements and assumptions are made explicit. The derivation o f quantitative terrain data such :ts gcomorphometry from digital elevation grids is an essential step in successful application of these methods as well as in applied geomorphological study. This paper has summarized some of the general processing methods and used a mountain environment in Canada's Yukon to illustrate geomorphometric applications. One main objective in this paper was to focus an introductory discussion around the translation of geographic concepts such as slope and relief to operational techniques. Interested readers are invited to contact the author for copies of the programs or comments and suggestions on ideas expressed here. .4ckmm'h'dgments--This research was funded by the Natural Sciences and Engineering Research Council of Canada; Nordco Ltd.. of SI. John's provided computer support. Professor Ellsworth LeDrew (University of Waterloo) assisted in acquisition of the DEM and Mr Derek Peddle (Memorial University) provided programming help. I am indebted to Mr Bill Johnstone for his lucid approach to slope calculations.

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Collins, S. H., and Moon, G. C., 1981, Algorithms for dense grid digital terrain models: Photogrammetric Engineering and Remote Sensing, v. 47. no. I. p. 71-76. Craig. R G.. 1983. Criteria for constructing optimal digital terrain models, m Craig. R. G., and Croft, J. L., eds., Applied geomorphology: The Binghamton Symposia Series in Geomorphology No. 11: George Allen and Unwin. London. p. 108-130. Evans, i. S., 1972, General geomorphometry, derivatives of altitude and descriptive statistics, in Chorley. R. J.. ed., Spatial analysis in geomorphology: Methuen, London. p. 17-90.

Evans, I. S., 1980. An integrated system of terrain analysis and slope mapping: Zeitschrift fiir Geomorphologie. N.F. Supplementband, v. 36. p. 274--295. Evans. I. S.. 1985, The morphometry of specific landforms: Paper presented to the First International Geomorphology Conference. Manchester. 20 p. Fleming. M. D.. and Hoffer. R. M., 1979. Machine processing of LANDSAT MSS and DMA topographic data for forest cover type mapping, in Proc. Fifth Symposium on Machine Processing of Remotely Sensed Data. LARS, Purdue University, p. 377 390. Franklin, S. E., 1987. Terrain amdysis from digital patterns in geomorphometry and LANDSAT spectral response: Photogrammetric Engineering and Remote Sensing. v. 53, no. I. p. 59 65. Franklin. S. E., and Peddle, D. R.. 1987, Texture analysis of digital image data using spatial cooccurrence: Computers &Geosciences, v. 13. no. 3, p. 293 311. Gardner, J. S., Smith. D. J., and Deslogcs, J. R.. 1983. Dynanlic geomorphology of the Mr. Rae area: high mountain region in southwestern Alberta: Dept. Geography Publ. Set. No. 19. Univ. Waterloo. Waterloo. Ontario. 237 p. Grender, G. C., 1976. Topolll: a FORTRAN program for terrain :malysis: Computers & Gcosciences. v. 2, no. 2, p. 195 2119. Ilaralick, R. M., Shanmugam, K., and Dinstcin, I., 1973, Textural features for image classitication: I I-El" Transaclions on Systems. M;m and Cybernetics. v. 3, no. 6, p. 610 621. I leil, R. J., and Brych, S. M., 198 I, The digital terrain model: a tool for quantifying terrain characteristics, in Proc. Annual Meeting of the American Congress on Surveying and Mapping, Washington. D. C., p. 156 171. |[olben, B. N., and Justice, C. O., 1980. The topographic effect on spectral response from nadir pointing sensors: Photogrammetric Engineering and Remote Sensing, v. 46, no. 9. p. 1191-1200. Hsu, S. Y., 1978. A texture tone analysis for automated land use mapping with panchromatic images, m Proc. Annual Meeting of American Society for Photogrammetry, Washington. D. C. p. 203 215. Hutchinson. C. F., 1982. Techniques for combining LANDSAT and ancillary data for digital classilication improvemerit: Photogrammetric Engineering and Remote Sensing, v. 48, no. I, p. 123-130. Iqbal, M., 1983. An introduction to solar radiation: Academic Press (Canada), Don Mills, Ontario, 390p. Justice. C. O., 1978. An examination of the relationships between selected ground properties and LANDSAT MSS data in an area of complex terrain in southern Italy, in Proc. Fall Technical Meeting of American Society for Photogrammetry. Albuquerque, New Mexico, p. 303328. Letts, P., and Rochon, G.. 1980, Generation and use of digital elevation data for large areas, m Proe. Sixth Canadian Symposium on Remote Sensing, Victoria, B. C., p. 61-71. Mark. D. M., 1979, Phenomenon-based data structuring and digital terrain modelling: GeoProcessing. v. 1, no. I, p. 27-36.

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609

SAT by incorporating topographic information, in Proc. Twelfth International Symposium on Remote Sensing of Environment, Ann Arbor, Michigan, p. 927-q42. Teillet. P. M.. Guindon, B., and Goodenough, D G . 1982. On the slope aspect correction of MSS data: Canadian Jour. Remote Sensing. v. 8, no. 2, p. 84--106. Townshend. J. R. G., ed., [98l, Terrain analysis and remote sensing: George Allen and Unwin. London, 232 p. Trenhai[e. A. S.. 1976. Cirque morphometry in the Canadian Cordillera: Annals Am. Assoc. Geographers, v. 66. no. 3, p. 451-462. Yoeli. P.. 1983. Digital terrain models and their cartographic and cartometric utilization: Cartographic Jour.. v. 20. no. [. p. 17-22. Young. A., 1974, Slope profile survey: British Geomorphological Research Group. Technical Bull. No. II. GeoAbstracts. Norwich, 52 p.