Horizontal Scale of Roughness Variations for Realistic Landscapes

HORIZONTAL SCALE OF ROUGHNESS VARIATIONS FORREALISTIC LANDSCAPES

KOEN DE RIDDER�

Flemish Institute for Technological Research (Vito), Mol, Belgium

(Received in final form 15 November 2002)

Abstract. A method is described to infer the horizontal scale of roughness variations from land-usemaps, for use in flux aggregation schemes designed to account for sub-grid scale terrain heterogeneityin numerical models of the atmosphere. Based on simple statistical-geometrical considerations, itis shown that this horizontal scale is inversely proportional to edge density, which can be readilyderived from high-resolution land-use maps. The method is demonstrated for a highly heterogeneouslandscape in northern Belgium, the resulting length scale values approximately ranging between 200and 1500 m. Finally, the extension towards satellite imagery and coarse land-use data is discussed.

Keywords: Effective roughness length, Terrain heterogeneity.

1. Introduction

The continental land surface exerts a significant influence on the atmospherethrough its control on the vertical fluxes of momentum, energy, and matter. Tocalculate these fluxes, numerical models of the atmosphere rely on the frameworkof similarity theory. A key assumption made in the latter is horizontal homogeneity,which is rarely observed since many natural and man-made landscapes exhibitsignificant spatial variability. Several methods have been proposed to account forsub-grid scale land surface heterogeneity while keeping within the frameworkof similarity theory. Most are based on an effective roughness length, which isobtained by averaging drag coefficients at a blending height (e.g., Mason, 1988;Claussen, 1990; Schmid and Bünzli, 1995; Goode and Belcher, 1999).

Without exception, the cited studies rely on the specification of a horizontalscale of roughness variations, which will further be denoted Lx . However, most orall of these studies have been restricted to idealised terrain, with regular sequencesof smooth and rough patches. To our best knowledge, no method currently existsto produce maps of Lx for real terrain at scales of a few hundred kilometres, i.e.,a typical mesoscale model domain size. Therefore, to allow the application of theblending-height method to actual landscapes, the present study proposes a methodto infer Lx from high-resolution land-use maps.

� Corresponding address: Koen De Ridder, Vito-TAP, Boeretang 200, B-2400 Mol, Belgium, E-mail: [email protected]

Boundary-Layer Meteorology 109: 49–57, 2003.© 2003 Kluwer Academic Publishers. Printed in the Netherlands.

50 KOEN DE RIDDER

The remainder of this paper is organised as follows. Section 2 describes thetheory to calculate Lx from the edge density, defined as the edge length of roughsurface patches per unit ground area. An application is presented in Section 3, usinga detailed high-resolution land-use map of northern Belgium, and the extension ofthe methodology towards high-resolution satellite imagery and coarse-resolutionland-use maps is discussed.

2. Theory

From the outset, the assumption is made that a landscape can be described as con-sisting of rough patches separated by smoother areas. The rough patches representsurface types with a roughness length of the order of metres, such as forests andcities. The smoother areas correspond to low-roughness surface types, includinggrassland and crops with roughness lengths of a few tens of millimetres. Relegatinglandscape elements to only two classes may seem an oversimplification, but manynatural and manmade landscapes appear to exhibit a bimodal distribution.

In such a landscape, the spatial scale of horizontal roughness change Lx

corresponds to the average distance between rough patches. Whereas in a one-dimensional landscape this quantity can be determined unambiguously, the situ-ation is far more complex for real, two-dimensional landscapes. We adopt as adefinition of Lx the average distance between rough patches following a straightline. In principle, an ensemble average has to be taken, using all possible line posi-tions and orientations. However, in order to retain a workable theory, the landscapeunder investigation is assumed isotropic and statistically homogenous at the scaleof suitably sized sub-areas (typically 100 km2), so that a single straight line canbe used to derive Lx for each of those areas. Statistical homogeneity, which is notincompatible with landscape heterogeneity, requires that the overall properties ofthe smooth-rough landscape do not vary in a statistical sense, so that the densityand average distance between rough patches do not change within the consideredarea.

Note that it is assumed that landscapes can be characterised by a single value ofLx . In the context of deriving the blending height from Lx (using either of the meth-ods cited in the Introduction), the question arises as to whether this is correct whenconsidering landscapes containing a whole range of different patch sizes rather thana single dominant one. Stated otherwise, do landscapes with equal Lx but differentpatch size distributions exhibit equal blending heights? Fortunately, this appearsto be the case, at least up to a certain extent. Indeed, Goode and Belcher (1999),on addressing this issue with their detailed numerical model, concluded that theapproximation of using average patch length is not too severe.

In the theory developed here, Lx is obtained by first counting the number ofrough-patch edges intersecting with a straight line per unit length of the latter(Nε/L, see below). This count is performed by means of an infinitesimal rectangle

HORIZONTAL SCALE OF ROUGHNESS VARIATIONS FOR REALISTIC LANDSCAPES 51

R with area L dx (Figure 1); L is taken sufficiently large so as to intersect with asignificant sample of rough patches. The quantity dx, on the other hand, is infinites-imally small (dx → 0), so that edge segments of irregularly shaped rough patchesintersecting with R can be considered straight. The determination of average inter-patch distance along the rectangle’s long axis is done by counting the number ofedges traversing R, associating any pair of edges with one intersection of a roughpatch.

Defining edge density ε as the total edge length of rough patches per unit surfacearea (units of m−1), the total edge length contained within R is given by εLdx. Inorder to obtain the number of edge intersections within R, it suffices to divide thetotal edge length by the average edge intersection length 〈dl〉, viz.

Nε = εLdx

〈dl〉 . (1)

Formally, the average edge intersection length is given by

〈dl〉 =∫ π/2

0dl(θ)f (θ)dθ, (2)

where dl(θ) is the infinitesimal length of an individual edge intersection as a func-tion of orientation θ with respect to the long axis of R (see Figure 1), and f (θ) is anormalised edge direction distribution function.

It is crucial to realise that, even though the isotropy requirement mentionedabove implies that edge segment directions are uniformly distributed, this is not thecase for the edge intersections within R. Indeed, edge intersections nearly alignedwith the rectangle’s long axis appear in a lesser number than intersections that areperpendicular to the long axis. This is caused by the fact that nearly-aligned edgeintersections present a smaller cross-section to R, hence have a reduced probabilityof intersecting with the latter. As f (θ) must be proportional to the projection of anedge segment onto the vertical of R, one finds that f (θ) = sin θ . Based on Figure1, one also easily finds that dl(θ) = dx/ sin θ . Inserting this information into (2),it follows that

〈dl〉 = π

2dx. (3)

As a consequence, the number of edges occurring over the length L of the rectangleis given by

Nε =(

2

π

)εL. (4)

Using Figure 1 as an aid, it can be seen that any pair of edge intersections is asso-ciated with one intersection of a rough patch with R, so that putting Nε = 2 in (4)

52 KOEN DE RIDDER

Figure 1. Geometry used in the theoretical model for Lx . The upper part of the graph shows atwo-dimensional view of an idealised landscape containing rough patches (grey) in a smooth land-scape, with a superimposed rectangle R of dimensions Ldx. The black boxes in the lower part ofthe graph are the one-dimensional projection of the intersections of the rough patches with R. Theintersections of the edges with R are shown as thick lines, and the rough-patch intersections ashatched areas. The angle between the rectangle’s long axis and the edge intersections is denotedθ .

implies L = Lx . Stated otherwise: If a number Nε of edge intersections occurs overa distance L, then half that number, i.e., Nε/2 rough-patch intersections occur overthat distance, since any intersection of a rough patch with rectangle R contributestwo edge intersections. The average distance between rough patch intersectionswith R is then given by Lx = L/(Nε/2), thus finally leading to

Lx = π

ε. (5)

As will be shown in the next section the quantity ε, and therefore also Lx , can becalculated in a straightforward manner from high-resolution land-use maps.

3. Application and Discussion

As a demonstration of the above theory, (5) was applied to the Flanders-Brussels(FB) region, an area of approximately 250 × 100 km2 located in the northern partof Belgium. A very detailed land-use map is available for this region at a spatialresolution of 20 m, containing 29 different land-use types. A resolution of a fewtens of metres is taken to be acceptable at this point; the topic of resolution andresolution-scale invariance of the method will be addressed in more detail later.


Figure 2. The upper panel shows the rough cover fraction of the Flanders-Brussels area, definedas the proportion of the surface covered by rough patches, interpolated to a 1-km grid. Light anddark tones correspond to low and high values, respectively. As an example, the areas determined bythe black and white square windows in the upper panel are enlarged to show detailed geographicaldistributions of rough-smooth land use in the left and right lower panels, black areas correspondingto rough patches.

As required by the theory developed in the previous section, land-use types arefirst assigned to two different roughness classes, that is, smooth and rough. Theformer contains all types with roughness lengths ranging from a few to tens ofmillimetres (grassland, crops, water), and the latter contains those types with aroughness length of the order of a metre (forest, built-up areas). A field ri,j is thenassociated with the land-use map, having a value of 1 if cell (i, j ) belongs to therough class, and 0 otherwise. An example of the resulting binary roughness-classmap is shown in Figure 2. Rough patches are shown in black, and the interspersedsmooth areas in white.

54 KOEN DE RIDDER

As a next step, the perimeter length contributed by each grid cell is calculatedas

pi,j = �x[4 − (ri+1,j + ri−1,j + ri,j+1 + ri,j−1)]ri,j , (6)

where �x is the resolution of the land use map (20 m). Practically, (6) impliesthat for every black grid cell, a count of the number of white ‘4-neighbours’ isperformed, and the result multiplied by the grid cell’s length. For instance, anisolated black grid cell contributes a perimeter length of 80 m, and a black cellsituated at the edge of a straight forest edge contributes 20 m. Black grid cellstotally surrounded by other black pixels contribute nought, as does any white gridcell.

The final step consists of the spatial smoothing (boxcar average) of pi,j over asquare neighbourhood, and dividing the result by (�x)2 to yield edge density ε.The smoothing width is chosen as a trade-off between the following arguments:(i) Taking it too small would result in certain areas containing too few edges to

constitute a statistically significant sample required by the theoretical model ofthe previous section;

(ii) taking it too large would wipe out any regional contrasts.For the case of the FB land-use map, it was found that using a window width of10 km preserves spatial gradients, while containing a sufficient number of edgeseverywhere. The resulting Lx map, calculated from ε using (5), is displayed inFigure 3. It shows mostly low Lx values, with a range between 200 and 1500 m.

Even though the feasibility of deriving Lx from a detailed land-use map hasbeen demonstrated, there remains the practical problem of having at one’s dis-posal high-resolution land-use information. Indeed, land-use data such as the FBmap employed here are not available for most areas. High-resolution (order oftens of metres) satellite imagery constitutes a very valuable alternative: multi-spectral imagery from optical sensors onboard, e.g., Landsat or SPOT platforms,are routinely being used to derive land use, and the latter is readily linked withroughness length values (Hasager and Jensen, 1999). These instruments also offerpanchromatic sensing, in which case use can be made of the property that roughsurfaces stand out as dark patches in a landscape (Stanhill, 1970). Satellite radar isalso a possibility, having the advantage that the backscatter signal is physicallyrelated to surface roughness characteristics, and that it is much less hamperedby atmospheric perturbation of the signal (de Vries, 1999). Finally, combininghigh-resolution spectral reflectances in the red and near-infrared wavebands allowsdiscrimination between areas of green vegetation and bare soil, which could beused to map an Lx-equivalent for heat fluxes.

A possible drawback of high-resolution imagery is that the domain coveredis generally limited to a few hundred kilometres at most, which may be a seri-ous restriction in larger-scale applications. Therefore, the ultimate data source forthe methodology outlined above would consist of coarse-resolution (few hundred


Figure 3. The upper panel contains the geographical distribution of Lx , in metres. Contour lines aregiven every 250 m, and the lighter grey tones indicate higher values. The dashed line corresponds tothe location of the transect shown in the lower panel, which displays Lx as a function of east-westdistance along this transect.

metres) land-use or satellite imagery, which generally cover large areas. An ex-ample of such information is the CORINE land-use map (European Commission,1993), covering most of Europe at a spatial resolution of 250 m. This is not suffi-cient for our purpose since the roughness elements, i.e., individual trees, are farfrom resolved. To investigate the potential of medium-resolution imagery, con-cepts from fractal theory and self-similar structures (appearing similar whateverthe scale) are used. It can be found from, e.g., Soille (1999; his Equation (3.17))that, in the case where a landscape exhibits statistical self-similarity, Lx varies withresolution �x (‘scale’) as

Lx,�x ∝ (�x)D−1, (7)

where D ∈ [1, 2] is the fractal dimension of the considered landscape. Assum-ing (7) is indeed observed, it could be used to derive ‘true’ Lx values (in thesense of yielding similar values as those obtained from high-resolution maps) frommedium-resolution maps. In order to test this assumption, the spatial resolution ofthe FB land-use map was degraded tenfold (to mimic medium-resolution data),assigning the dominant roughness class to the resulting 200-m grid cells. The

56 KOEN DE RIDDER

parameter Lx was then calculated as before, though based on this degraded map,for sixty different 10 × 10 km2 areas within the FB domain. A relatively constantratio Lx,200/Lx,20 ≈ 5 was found for most of these areas, indicating D ≈ 1.7.Unfortunately, it was also found that (7) fails for areas with a low (< 20%) orhigh (> 80%) rough cover fraction, implying that self-similarity breaks down underthese conditions, and so limiting the applicability of medium-resolution maps. Adetailed investigation regarding ways of circumventing this problem, perhaps usinginformation from the medium-resolution maps themselves, would be more thanworthwhile.

4. Summary

A methodology has been presented to produce maps containing the horizontalscale of roughness variation Lx for real landscapes, for use in flux aggregationschemes that account for sub-grid scale landscape heterogeneity in atmosphericmodels. Based on a statistical-geometrical approach, it was shown that Lx is in-versely proportional to the edge density of the rough patches occurring in thelandscape. Subsequently, it was demonstrated that edge density can be calculated ina straightforward manner from high-resolution land-use maps, using the Flanders-Brussels area as an example. In this highly fragmented area, values obtained forLx range between 200 and 1500 m. It was argued that the methodology couldbenefit from high-resolution satellite imagery, which has the advantage of beingavailable for most land areas on the globe. An analysis was made of the usefulnessof coarse-resolution data sources as input for the methodology, using conceptsof fractal theory. Some problems were pointed out, related to the breakdown ofself-similarity.

Acknowledgements

The work presented here was performed as part of the project ‘Determination ofacidification and eutrophication sensitivity of Flemish forests with modelled de-position fluxes’ (VLINA-9801), and funding from the Flemish Impulse Fund forNature Development is acknowledged. OC-GIS Flanders is thanked for graciouslyproviding the high-resolution land-use map for Flanders and Brussels.

References

Claussen, M.: 1990, ‘Area-Averaging of Surface Fluxes in a Neutrally Stratified, HorizontallyInhomogeneous Atmospheric Boundary Layer’, Atmos. Environ. 24A, 1349–1360.

de Vries, A.: 1999, Land Surface Roughness and Remote Sensing, Ph.D. Thesis, RijksuniversiteitGroningen, The Netherlands, 130 pp.


European Commission: 1993, CORINE Landcover Technical Guide, EC Directorate-General Envir-onment, Nuclear Safety and Civil Protection, Luxemburg, 136 pp.

Goode, K. and Belcher, S. E.: 1999, ‘On the Parameterisation of the Effective Roughness Length forMomentum Transfer over Heterogeneous Terrain’, Boundary-Layer Meteorol. 93, 133–154.

Hasager, C. B. and Jensen, N. O.: 1999, ‘Surface-Flux Aggregation in Heterogeneous Terrain’,Quart. J. Roy. Meteorol. Soc. 125, 2075–2102.

Mason, P. J.: 1988, ‘The Formation of Areally-Averaged Roughness Lengths’, Quart. J. Roy.Meteorol. Soc. 114, 399–420.

Schmid, H. P. and Bünzli, B.: 1995, ‘The Influence of Surface Texture on the Effective RoughnessLength’, Quart. J. Roy. Meteorol. Soc. 121, 1–21.

Soille, P.: 1999, Morphological Image Analysis: Principles and Applications, Springer-Verlag,Berlin, 316 pp.

Stanhill, G.: 1970, ‘Some Results of Helicopter Measurements of the Albedo of Different LandSurfaces’, Solar Energ. 13, 59–66.

Horizontal Scale of Roughness Variations for Realistic Landscapes

Documents

Transcript of Horizontal Scale of Roughness Variations for Realistic Landscapes