Catchment Relief Characteristics for Hydro Logical Purposes

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Catchment relief characteristics for hydrological purposesSjur Kolberg

To cite this Article Kolberg, Sjur(1997) 'Catchment relief characteristics for hydrological purposes', Norsk GeografiskTidsskrift - Norwegian Journal of Geography, 51: 1, 15 — 22To link to this Article: DOI: 10.1080/00291959708552359URL: http://dx.doi.org/10.1080/00291959708552359

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Norsk geogr. Tidsskr. Vol. 51, 15-22. Oslo. ISSN 0029-1951

Catchment relief characteristics for hydrological purposes

S. KOLBERG

Kolberg, S. 1997. Catchment relief characteristics for hydrological purposes. Norsk geogr. Tidsskr.Vol. 51, 15-22. Oslo. ISSN 0029-1951.

It is argued that the use of topography based hydrological indices emphasizes differences betweenhillslope and channel hydrology, and that the dominant drainage process should be reflected in thechoice between a locally or globally based relief measure. A topographic wetness index is used toevaluate different relief measures for conceptualization of hillslope drainage. Locally based reliefmeasures show better correspondence with the index than globally based relief measures. Despite itssimple calculation, relative contour length sufficiently describes the average local slope. It is alsoshown that the local measures are linear with respect to area, while non-linearities are clearly evidentfor the global measures. The global relief measures tend to be insensitive for varying data resolution,whereas the local values are deeply dependent. For the wetness index, only a small part of theresolution dependency results from variations in the slope value.

Sjur Kolberg, SINTEF NHL, N-7034 Trondhcim, Norway.

Description of a catchment's response to precipi-tation input is vital for short-term forecasting,long-term water balance simulations and estima-tion of design floods. In ungauged catchments,where data are not available for calibrating amodel or estimating flood frequency parametres,hydrological characteristics must be estimatedfrom variables available from maps or othersources. An extensively mapped, easily obtain-able and hydrologically relevant property of abasin is its topography, increasingly available asdigital elevation models (DEMs).

Hydraulic attenuation, great heterogeneity andinteractions among processes strongly limit thenumber of characteristics which can be signifi-cantly related to catchment response. The neces-sity of a simple model structure encourages thesearch for a parsimonious and carefully selectedset of variables, as is emphasized by Anderson &Howes (1986). The predictable influence of to-pography on the hydrologic characteristics of anungauged catchment is not strong enough tojustify the large number of parameters providedby the DEM. This article thus focuses on thehydrological relevance of topographic measuresconsisting of a single number.

Traditional measures of relief are based on theratio of a characteristic elevation difference to acharacteristic length of the catchment. The eleva-tion information is provided by a hypsographiccurve, or simply read for key points in the catch-ment, i.e. the outlet and the upper end of the mainchannel. The characteristic length is typically thesquare root of area or the distance between the

chosen key points. The resulting measures tend toreflect the global relief of the catchment, whichcan be justified for routing flood waves in thechannel network. However, the global measuresare less influenced by hillslope topography, andfor the local gradient governing subsurfacedrainage, they can only serve as coarse indexes.

Hillslope hydrology and the wetnessindexThe importance of hillslope drainage is recog-nized in several hydrological models. The univer-sal soil loss equation (USLE) (Wischmeier &Smith 1978) uses slope length, gradient andprofile to estimate soil erosion caused by surfacerunoff. For a specific purpose, these parametersmay be' incorporated into one single variable,known as a wetness index. Beven & Kirkby(1979) and O'Loughlin (1986) have developedmodels for hydrologic response based on suchstatic indexes. The wetness index u-,, as definedby Beven & Kirkby (1979), is described by:

ir, = ln(/l/rtan/?),

where A is upslope area per unit contour length,T is transmissivity and /? is downslope gradient.This relationship results from Darcy's law, thecontinuity equation and an assumption of expo-nentially declining hydraulic conductivity withsoil depth. Where transmissivity data are notavailable, omitting T gives a simpler expression,which is used in this investigation.

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16 S. Kolberg NORSK GEOGRAFISK TIDSSKRIFT 51 (1997)

For raster data, n-,- is calculated cell by cellusing the multiple direction drainage algorithmof Quinn et al. (1991). Iteration proceeds down-wards, so, when focused, a cell has already re-ceived drainage from its upslope area. Theaccumulation is completed by adding the area ofthe cell itself. The accumulated upslope area isthen divided among downslope neighbours pro-portional to gradient and contour length.Downslope gradient is the weighted mean ofthese proportions:

tan/?=

Here /? denotes gradient angles, j iterates overdownslope neighbours, and Lj is a contour lengthweighting factor, 0.5 for cardinal directions, 0.35for diagonal. Calculation of ir, relies heavily onGIS functions accessing each individual cell andits neighbours, which even in an advanced GISinvolves extensive macro programming. Manyapplications must rely on simpler measures.

Under steady-state conditions, tr, is assumed todescribe the spatial pattern of depth to the watertable, with high values corresponding to moistareas and rapid response. Barling et al. (1994)have extended the concept to account for variabledrainage times since prior rainfall, relaxing thesteady-state assumption. The wetness index ap-proach has proved the significance of hillslopecharacteristics in both catchment hydrographpredictions (Beven et al. 1995) and flood fre-quency predictions (Beven 1986, 1987). In thisinvestigation it1,- is used to evaluate other reliefmeasures.

It follows from the definition and interpreta-tion of ir, that hillslope gradient has a negativeeffect on hydrologic response. This is opposite tochannel flow, where high gradients means lessreduction of flood peaks. The physical rationalebehind this concept is that steep terrain tends todrain effectively between rainstorms, keeping alow groundwater table and a high moisturedeficit which stores the next storm's water yieldwithin the soil matrix. A flat area keeps a highgroundwater table, where saturation is morelikely to cause overland flow and high floodpeaks. The different effects of gradients callsfor a separation of hillslope and channel pro-cesses in runoff and flood modelling if slope isto be used as a determinant of hydrologic re-sponse.

Data preparation and description ofrelief measuresThis project is based on 60 small river catchmentsfrom all parts of Norway, among them Sagelvaresearch catchment outside Trondheim (Fig. 1).Areas range from 1 to 20 km2. The data source isthe 100 m point elevation database of the Norwe-gian Mapping Authority, which is interpolatedfrom 20 m contours on 1:50 000 maps. Digitalcontours are generally not available, and werere-interpolated from the DEM. The DEM hassome artefacts, i.e. very high frequencies of thecontour values (20, 40,...), as well as clearbands appearing at shaded relief plots. Owing tothe first artefact, the DEM was slightly smoothedbefore interpolation. Sinks were removed to en-sure connected drainage of the entire area, smallsinks simply by raising the elevation a maximumof 20 m. Standard filling procedures were avoidedfor deep or large sinks, where elevation values areoften more correct for the sink itself than for anartificial threshold damming its outlet. In suchcases, corrections were done interactively andsubjectively.

For Sagelva catchment, another data set isobtained by scanning orienteering map printingfolies at approximately 1:5000 scale with 5 mcontour interval (Fig. 2), By containing morespatially detailed information, this data set is lessgeneralized than the 100 m database, and ahigher resolution is needed in the resulting DEM.For this data set, the original contours are used.

The first relief variable in the comparison is theaverage local slope. Its calculation involves func-tions which are standard in most GIS pro-grammes, and requires Iocational information;that is, that (x,y) positions are assigned to allelevation values z. In this case, values are takenfrom the wetness index computation; however,various techniques exist (see Skidmore (1989) fora comparison). Average local slope is dimension-less and linear with respect to scale, but is sensi-tive to the generalization level.

The other locally based relief measure is therelative contour length, R, defined as:

where / is the contour interval, A is the area, jiterates over all contours, and L, is the arc lengthof contour j . Relative contour length is dimen-sionless and linear with respect to area, but sensi-tive to generalization. Expressing the expected


NORSK GEOGRAFISK. TIDSSKRIFT 51 (1997) Catchment relief characteristics 17

Fig. I. Sagelva research catchment near Trondheim, with subcatchments Hestsjobekken [1], Svarttjonnbekken [2], the rest ofHokfossen [3] and the rest of Sagelva [4]. Data from Nonvegian Mapping Authority with 1:50 000 scale origin, cell size 100 m.Grayshades show DEM elevations, contour interval is 20 m, sink perimeters are marked with bold lines.

elevation shift per unit distance, relative contourlength is theoretically equal to average localslope; in this case, however, somewhat smallerdue to the data transformations. Locational in-formation is absent in the calculation, which issimple within a vector based GIS or with aplanimeter and a printed topographic map.

Finally, two variants of global gradient mea-sures based on the hypsographic curve are in-cluded. Full basin gradient uses the wholeelevation range, while the highest 15% and thelowest 10% of the area are excluded from Reduced basin gradient. The characteristic length isthe square root of the catchment area. The first


18 S. Kolbcrg NORSK GEOGRAFISK TIDSSKRIFT 51 (1997)

Fig. 2. Sagelva research catchment, with subcatchments Hestsjobekken [1], Svarttjonnbekken [2], the rest of Hokfossen [3] and therest of Sagelva [4]. The database is in 5 m contours scanned from approximately 1:5000 scale orienteering map printing folies.Grayshades show DEM elevations, cell size is 5 m, contour interval shown is 20 m.

measure requires only two values easily readfrom printed maps, but is somewhat sensitive todistribution tails compared to the second, whichneeds a frequency distribution. Both global mea-sures are non-linear, with low sensivity to datageneralization.

Results and discussionTable 1 shows the correlation matrix with indica-tive scatterplots for the five terrain variables plusarea. Very high correlation is observed betweenthe two local measures average local slope andrelative contour length (r = 0.99). Relative con-


1NORSK GEOGRAFISK. TIDSSKRIFT 51 (1997) Catchment relief characteristics 19

tour lengths are somewhat lower than average localslope values, with a regression line gradient of 0.92.This reflects that some detail in the elevationinformation is lost by interpolating contours fromthe DTM. The two global basin gradients are alsowell correlated (r = 0.95), although slightly lessthan for the local measures. Relief measures fromdifferent groups show in comparison considerablyweaker correspondence (r = 0.77 to 0.80), confirm-ing the major difference between local and globalgradients. The presence or exclusion of elevationdistribution tails has no major influence on thisconclusion.

Table 1 shows that wetness index is more closelyrelated to the local measures ( r = —0.89) than to

the basin gradients ( r = —0.72 to —0.74), in allcases with a non-linear connection. This resultshows the importance of a local basis when reliefmeasures are used to reflect hillslope drainage. Thelocal relief measures are also more closely relatedto u',- than to the global basin gradients. In otherwords, extending the local relief variable with anupslope area term has less effect on the differencesbetween catchments than changing the gradientscope from local to global.

Table 1 shows a significant positive correlationbetween area and the local relief variables. Thelinear nature of these variables gives no reason forinterpreting this as a generally valid relationship,and it is assumed to reflect a biased sample

Table I. Correlation matrix for the five relief measures and catchment area, with Pearson's r and level of significance p, and n = 60observations. Corresponding scatterplots illustrate the value range of each variable and the shape of the connection. The>' axes inthe scatterplots correspond to the matrix columns. Several plots indicate a non-constant variance, whereas plots involving thewetness index show a non-linear connection.

Mean WetnessIndex

r = - 0,7158p = 0,000

r = - 0,7361p = 0,000

r = - 0,8909p = 0,000

r = - 0,8948p = 0,000

r = - 0,3929p = 0,002

Reduced BasinGradient10-85 %

r = 0,9516p = 0,000

r = 0,8049p = 0,000

r = 0,7775p = 0,000

r = 0,0457p = 0,731

Full BasinGradient0-100 %

r = 0,7927p = 0,000

r = 0,7709p = 0,000

r = 0.0679p = 0,610

* • - • .

o.oki;

RelativeContourLength

r = 0,9934p = 0,000

r = 0,3537p = 0,006

Average LocalSlope

r = 0,3897p = 0,002

.70i

.35

0,00

CatchmentArea


20 S. Kolberg NORSK GEOGRAFISK. TIDSSKRIFT 51 (1997)

Fig. 3. Scatterplot of catchment area versus the residuals from a regression model between reduced basin gradient and averageslope. The results show that the locally computed slope is increasingly underestimated by the globally based basin gradient ascatchment size increases.

of catchments. A negative correlation was ex-pected between area and the global relief mea-sures, but is not confirmed in Table 1. This maybe due to a biased sample, and, to test thisassumption, residuals from a regression betweenaverage local slope and reduced basin gradientare plotted against area (Fig. 3). The negativedependency (r= —0.56) shows that local relief isincreasingly underestimated by global relief mea-sures as area increases, and confirms the scaledependence of global relief measures comparedto local.

Table 2 shows topographic variables from theSagelva area, computed from both data sets.Again, important differences occur between thelocal and the global relief measures. Averagelocal slope and relative contour length varystrongly with generalization; higher values fromthe most detailed data set. For contour lines, theeffect corresponds to the shoreline length para-dox known from fractal geometry, and is clearalso in Figs. 1 and 2. The difference in theaverage local slope illustrates the part of eleva-tion variance occurring at scales between 5 and100 m.

For the 50 000 data set, Table 2 confirms thedifference between relative contour length and

average local slope indicated in Table 1. Thishigher level of detail in the raster DEM than inthe interpolated contours may be due to theinterpolation process itself, to the slight smooth-ing of the DEM, or to the equation used tocalculate slope from the DEM. The similar effectis almost absent in the orienteering map data set,where the DEM is interpolated from the con-tours, hence the slope calculating procedure isnot suggested as the primary reason for the dif-ference.

All the local measures appear linear with re-spect to scale, in the sense that the value for anarea may be found by averaging the subareavalues. This is not the case for the global mea-sures, which clearly decrease with catchmentarea. The global measures are much less depen-dent on data generalization than the local. Again,this suggests the linkage of local slope measuresto hillslope properties and global gradients tochannel slope.

The average local slope does not explain thedifference in wetness index between the two datasets. A 100% increase in average local slope cor-responds analytically to a 0.69 unit reduction inthe index value, while the observed index reduc-tion of 3 units corresponds to a 20-fold increase


NORSK GEOGRAFISK TIDSSKRIFT 51 (1997) Catchment relief characteristics 21

Table 2. Relief measures and area for different parts of Sagelva research catchment originating from the two data sets. The dataset at 1:50 000 scale is based on a 100 m point elevation grid (originally interpolated from 20 m contours), while the data set at1:5000 scale orieinates from scanned 5 m contours.

Catchment[map reference]

Hestjobekken [1]

Svarttjonnbekken [2]

Rest area, Hokfosscn [3]

Hokfossen [1, 2, 3]

Rest area, Sagelva [4]

Sagelva [1, 2, 3, 4]

Originalmap scale

50 0005000

50 0005000

500005000

50 0005000

50 0005000

50 0005000

Fullbasin

gradient

0.1570.149

0.1290.127

0.1050.114

0.0940.093

0.1830.201

0.1150.119

Reducedbasin

gradient

0.0480.049

0.0550.049

0.0450.048

0.0360.035

0.1040.112

0.0380.037

Averagelocalslope

0.1140.181

0.1470.259

0.1470.244

0.1480.236

0.1750.281

0.1530.242

Relativecontourlength

0.1040.177

0.1330.261

0.1250.237

0.1240.232

0.1540.275

0.1280.237

Meanwetnessindex

9.056.33

8.995.65

9.285.74

8.895.85

9.045.66

8.835.83

Area(km2)

1.701.9

3.733.42

2.952.93

8.388.31

1.201.16

9.589.47

in the Ajtznfi ratio. So while the slope termaccounted for 80% of the index variance betweencatchments (Table 1), the specific upslope area Adescribing land surface shape clearly dominatesthe scale dependency of the index. This finding isin agreement with Wolock & Price (1994).

The DTM resolution necessary to describe hill-slope drainage patterns varies with terrain com-plexity and with the connection between the landsurface and the water table. Quinn et al. (1991)showed substantial differences between 12.5 and50 m grids in an African catchment. Zhang &Montgomery (1994) compared different DEMsranging from 2 to 90 m scales, and suggested thatsignificant improvements follow increasing detaildown to about 10 m. On this basis, the Norwe-gian 100 m database is probably not sufficient todescribe the flow paths in hillslopes.

Zhang & Montgomery (1994) found the maineffect of larger grid size on the u1,- distribution tobe a positive shift; the mean value increases whilethe distribution shape is less altered. Using theindex in hydrograph forecasting, model predic-tions have been shown to be mainly dependenton the mean value (Wolock & Price 1994). Ifcalibration data are available, the scale depen-dency may be compensated. However, differentgeneralization levels in the original data seriouslyalter both the statistical and the spatial distribu-tion of values, and affect the description of slopeprocesses or spatial variation of soil moisture or

saturation (Wolock & Price 1994). Reducing thecell size does not improve the generalization ofthe original data.

ConclusionsThe results confirm that there are substantialdifferences between locally and globally basedmeasures of relief. Recognizing that gravitydriven hillslope processes depend on local slope,it is suggested that global relief measures areavoided in terrain-based conceptualizations ofhillslope runoff generation.

Relative contour length is shown to correspondvery well to a complete GIS-based calculation ofaverage local slope. However, it requires far lesseffort in its calculation, and represents a simpleand efficient estimate of mean hillslope gradient.

The results also indicate that the variation inthe wetness index In(a/tanfi) among catchmentsis well described by locally based slope measures.For simple regression relationships, where therelative differences between catchments are moreinteresting than the absolute values, the complexcalculation of the index may be substituted by asimpler local slope measure.

The spatial generalization in the elevation data-base strongly affects the computed wetness index,mainly through variation in the specific upslopearea, which reflects the shape of the terrain fea-


22 S. Kolberg NORSK GEOGRAFISK TIDSSKRIFT 51 (1997)

tures. In a physically based analysis of hillslopeprocesses, the 100 m database cannot be expectedto reflect the hillslope flow paths or distributionof saturated areas.

Acknowledgements. - This project is partly funded by theNorwegian Research Council. The support is greatly appreci-ated.

Manuscript accepted March 1996

ReferencesAnderson, M. G. & Howes, S. 1986. Hillslope hydrology models

for forecasting in ungauged watersheds. In Abrahams, A. D.(ed.) Hillslope Processes. Allen & Unwin, Winchester.

Barling, R. D., Moore, I. D. & Grayson, R. B. 1994. Aquasi-dynamic wetness index for characterizing the spatialdistribution of zones of surface saturation and soil watercontent. Water Resources Research 30 (4), 1029-1044.

Beven, K. J. 1986. Runoff production and flood frequency incatchments of order n: an alternative approach. In Gupta, V.K. et al. (eds) Scale Problems in Hydrology. Reidel, Dor-drecht.

Beven, K. J. 1987. Towards the use of catchment geomorphol-ogy in flood frequency predictions. Earth Surface Processes12, 69-82.

Beven, K. J. & Kirkby, M. J. 1979. A physically based, variablecontributing area model of basin hydrology. HydrologicalSciences Journal 24 (1), 43-69.

Beven, K. J., Lamb, R., Quinn, P. F., Romanowicz, R. & Freer,J. 1995. TOPMODEL. In Singh, V. P. (ed) Computer Modelsof Watershed. Hydrology, Water Resources Publications,Highlands Ranch.

O'Loughlin, E. M. 1986. Prediction of surface saturation zonesin natural catchments by topographic analysis. Water Re-sources Research 22 (5), 794-804.

Quinn, P., Beven, K. J., Chevalier, P. & Planchon, O. 1991. Theprediction of hillslope flow paths for distributed hydrologicalmodelling using digital terrain models. Hydrological Processes5, 59-79.

Skidmore, A. K. 1989. A comparison of techniques for calculat-ing gradient and aspect from a gridded digital elevationmodel. International Journal of Geographical Information Sys-tems 3 (4), 323-334.

Wischmeier, W. H. & Smith, D. D. 1978. Predicting rainfallerosion losses - a guide to conservation planning. Agricultur-al Handbook no. 537. US Dept of Agriculture, WashingtonDC.

Wolock, D. M. & Price, C. V. 1994. Effects of digital elevationmodel map scale and data resolution on a topography-basedwatershed model. Water Resources Research 30 (II), 3041-3052.

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Catchment Relief Characteristics for Hydro Logical Purposes