1
Appendix S1
Topographic Controls on the Distribution of Tree Islands in the High Andes of South-western
Ecuador
Coblentz, D. and Keating, P.L.
Topographic Analysis and the Eigenvalue Ratio Method
Information contained in the topographic character of the landscape can provide valuable
information about the geodynamic, tectonic, and climatic history of a region. Geomorphometry,
or the use of digital information to aid the study of landscape geometry, began with Chorley et
al. (1957) and provides the framework for the quantitative description of the land surface (also
see Zakrzewska, 1963; Turner & Miles, 1967; Hormann, 1969; Mark, 1975; Zevenbergen &
Thorne, 1987). Computer-based methods of geomorphometric analysis were first developed by
Zakrzewska (1963), Turner & Miles (1967), Hormann (1969) and Evans (1972), among others,
who evaluated the relationship between classical statistical parameters and computed means of
terrain “character.” Many subsequent investigations of geomorphometric parameters have
helped establish the information important for basin analysis, slope stability and other
applications (for a review see Mark, 1975). Collectively these techniques provide an ideal
approach for extracting the information about the topographic character of the landscape that is
needed to evaluate the relationship between the terrain and vegetative land cover.
There are several measures of the topographic character that are important in the context
of understanding the relationship between topography and vegetative landcover. These quanities
include: 1) topographic roughness (defined as a measure of variability in the landscape, or the
irregularity of a topographic surface – first investigated digitally by Stone & Dugundji, 1965); 2)
topographic fabric or grain, defined the tendency to form linear ridges (for a review of the
definition of the term "grain" in the literature see Pike et al., 1989 and discussions in Guth, 1999,
2
2003); 3) Topographic organization (where large values indicate a dominant linear fabric to the
terrain, low values isotropic topography, see discussion in Guth (2003)), and 4) the topographic
gradient.
Quantification of the topographic fabric is accomplished using the Eigenvalue Ratio
Method for evaluating the topographic roughness and organization (Watson, 1966; Woodcock,
1977). Certain ratios of the three principal eigenvalues, derived from the “orientation tensor”
constructed from the collection of surface normal vectors, yield information about the
topographic roughness and organization, and provide a useful method for analyzing the
topographic fabric of a DEM. Chapman (1952) and Woodcock (1977) explored the use of the
eigenvalue method for fabric shapes in structural geology, and more recently, the two methods
have been combined for automatic characterization of terrain organization (Guth, 1999). An
excellent review of the application of this technique for the evaluation of landslide surfaces can
be found in McKean & Roering (2003).
The surface normals (constructed by taking the unit vectors perpendicular to each cell in
the DEM) are defined in polar coordinates by the direction cosines: xi=sini cosi, yi=sinisini,
and zi=cos i, where i is the colatitude (90o - latitude) and i is the longitude of a unit orientation
vector. The surface normals for rough topography show considerably more scatter compared to
those of flat surfaces (Fig. S1). The normals to the Earth's surface can be viewed as a cloud of
vectors in space, and the three eigenvectors associated with the vector cloud define a three
dimensional ellipsoid that reflects their distribution (Woodcock, 1977). Local variability in the
orientation data is evaluated statistically, and certain ratios of the eigenvalues provide
quantitative information about the topographic character (roughness, organization and grain
orientation). If (x1, y1, z1) ... (xn, yn, zn) represent a collection of n unit vectors perpendicular to
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the n topographic surfaces described by a DEM, then the orientation matrix T (Equation 1) can
be formed by the sums of the cross products of the direction cosines (Fara & Scheidegger, 1963;
Scheidegger, 1965; Watson, 1966; Fisher et al., 1987).
  2 y 

 xi xi i xi z i 
2


T    y x i y  y z i 
i
i
i

2 
  z i x i z i y i z i 


(1)
Performing an eigenvalue analysis on the T matrix yields three eigenvalues and three
eigenvectors. The relative magnitudes of the three eigenvectors, and their orientation, define the
distribution. If the three eigenvalues (S1, S2, S3) are normalized with respect to n, then Si=i/n
and S1+ S2+ S3=1. For most landscape surfaces the relationships between the three eigenvalues
(and their associated eigenvectors) can be generalized as follows: 1) the eigenvalue S1 is much
greater than S2; 2) S2 is approximately equal to S3; 3) the eigenvector associated with S1 is
approximately vertical and those associated with S2 and S3 lie in the horizontal plane; and 4) the
eigenvector S3 points in the direction of the dominant topographic fabric or the topographic
grain. Various ratios of the eigenvalues yield useful geomorphometric parameters, including the
topographic roughness [R1 = 1/(ln(S1/S2 ))], the topographic organization [R2 = ln(S2/S3)],
strength [R3 = ln(S1/S3)], and a parameter K defined as the ratio of flatness to organization:
[ln(S1/S2 ) / ln(S2/S3)]. Because there are only two independently varying eigenvalues, the
flatness (the ratio ln(S1/S2 )) can be plotted against organization (ln(S2/S3)) to describe the pattern
of vector orientations, which range from clusters to girdles (Woodcock, 1977).
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Example distributions are shown in Fig. S2. The orientation data are considered
clustered when S1> S2~ S3, and form a girdle distribution when S1~ S2> S3. The strength
parameter increases with radial distance from the origin. Some examples of possible end
members for topographic distributions include: 1) uniformly scattered surface vectors (perfectly
rough topography) in which case S1= S2= S3=0.33 and the data would plot at the origin of the
graph; 2) uniformly flat topography that produces uniform surface normal vectors, which would
plot in the upper left corner (both extremely large K and strength values); and 3) perfectly
random surface which would plot along the K=0 axis.
References
Chapman, C.A. (1952) A new quantitative method of topographic analysis. American Journal of
Science, 250, 428-452.
Chorley, R.J., Malm, D.E.C. & Pogorzelski, H.A. (1957) A new standard for measuring
drainage basin shape. American Journal of Science, 255, 138-141.
Evans, I.S. (1972) General geomorphology, derivatives of altitude and description of statistics.
Spatial Analysis in Geomorphology (ed. by R.J. Chorley) pp. 17-90. Methuen & Co. Ltd.,
London.
Fara, H.D. & Scheidegger, A.E. (1963) An eigenvalue method for the statistical evaluation of
fault plane solutions of earthquakes. Seismological Society of America Bulletin, 53, 811-816.
Fisher, N.I., Lewis, T. & Embleton, B.J. (1987) Statistical Analysis of Spherical Data.
Cambridge University Press, New York.
Guth, P.L. (1999) Quantifying and Visualizing Terrain Fabric from Digital Elevation Models.
Geocomputation 99: Proceedings of the 4th International Conference on GeoComputation
5
(ed. by J. Diaz, R. Tynes, D. Caldwell, and J. Ehlen)
http://www.geovista.psu.edu/geocomp/geocomp99/index.htm.
Guth, P.L. (2003) Terrain Organization Calculated From Digital Elevation Models: in Evans,
I.S., Dikau, R., Tokunaga, E., Ohmori, H., and Hirano, M., eds., Concepts and Modelling in
Geomorphology: International Perspectives: Terrapub Publishers, Tokyo, p.199-220. Online
at http://www.terrapub.co.jp/e-library/ohmori/pdf/199.pdf.
Horman, K. (1969) Geomorphologische Kartenanalyse mit Hilfe elektronischer Rechenanlagen.
Zeitschrift für Geomorphologie, 13, 75-98.
Mark, D.M. (1975) Geomorphometric parameters: a review and evaluation. Geografiska
Annaler, Series A, Physical Geography, 57, 165-177.
McKean, J. & Roering, J. (2003) Objective landslide detection and surface morphology
mapping using high-resolution airborne laser altimetry. Geomorphology, 57, 331-351.
Pike, R.J., Acevedo, W. & Card, D.H. (1989) Topographic grain automated from digital
elevation models. Proceedings, 9th International Symposium on Computer Assisted
Cartography, Baltimore, April 2-7, 1989, ASPRS/ACSM, p.128-137.
Scheidegger, A.E. (1965) On the statistics of the orientation of bedding planes, grain axes, and
similar sedimentological data. US Geological Survey Professional Paper 525-C, C164-C167.
Stone, R.O. & Dugundji, J. (1965) A study of microrelief – its mapping, classification and
quantification by mean of a Fourier analysis. Engineering Geology, 1, 89-187.
Turner, A.K. & Miles, C.R. (1965) Terrain analysis by computer. Proceedings of the Indiana
Academy of Sciences, 77, 256-270.
Watson, G.S. (1966) The statistics of orientation data. Journal of Geology, 74, 786-797.
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Woodcock, N.H. (1977) Specification of fabric shapes using an eigenvalue method. Geological
Society of America Bulletin, 88, 1231-1236.
Zakrzewska, B. (1963) An analysis of landforms in a part of the central Great Plains. Annals of
the Association of American Geographers, 53, 536-568.
Zevenbergen, L.W. & Thorne, C.R. (1987) Quantitative analysis of land surface topography.
Earth Surface Processes and Landforms, 12, 47-56.
7
Figure S1. Topographic surface roughness in a hypothetical DEM as defined by the unit surface
normal vectors. For smooth topography (a), the vectors are coherent as indicated by the
clustering in the steronet of the vector orientations. Conversely, for rough topography (b), the
vectors have greater dispersion as reflected in the steronet mapping. After Hobson (1972).
Figure S2. The location in K-space (a plot of topographic flatness, ln[S1/S2] plotted vs. the
topographic organization, ln[S2/S3]) of several example distribution of surface normals
(Woodcock, 1977). Tightly clustered distributions fall towards the flatness axis (large K values),
while loosley clustered distributions fall below the K=1 line, which demarks the gridle-cluster
transition. Distance from the origin is governed by the strength parameter which is defined by
ln[S1/S3]. Thus coherent, flat topographic surfaces (which tend to be poorly organized) plot in
the upper-left corner of the K-space while relatively incoherent topographically rough
distributions will plot lower and to the right. At the furthest extreme a purely random
distribution will plot in the lower-right corner of K-space. After Woodcock (1977).