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WATER RESOURCES RESEARCH, VOL. 44, W12421, doi:10.1029/2007WR006667, 2008

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ArticleScaling of slope, upslope area, and soil water deficit:
Implications for transferability and regionalization
in topographic index modeling
Nawa Raj Pradhan,1 Fred L. Ogden,1 Yasuto Tachikawa,2 and Kaoru Takara3
Received 15 November 2007; revised 27 May 2008; accepted 15 September 2008; published 16 December 2008.

[1] Development of a generally applicable rainfall-runoff model and identification of

associated model parameters require understanding of connections between physicalprocesses at disparate scales and hydrological similarities between catchments. In thisstudy, we test the hypothesis that understanding of geomorphometric scaling relations can
reduce uncertainty when transferring model parameters between catchments when
applying the TOPMODEL concept. Scaling relations on contributing area, slope, andcontour length were successfully used to scale the topographic index distribution inwatersheds located in vastly different regions of the world: Japan, Nepal, Panama, and theUnited States. Model parameters were identified through calibration of TOPMODEL in
the 210-km2 Kamishiiba catchment in Japan. These parameters were transferred to two
Sun Koshi River subcatchments in Nepal, namely, the 850-km2 Likhu and 620-km2
Balephi catchments, the 414-km2 Upper Rio Chagres catchment in Panama, and the
37-km2 Town Brook catchment in the United States. Results show how a priori estimatesof the most sensitive model parameters can be used to make predictions in poorly gauged
or ungauged basins with some degree of confidence provided that scale effects are
considered. This result hints at the potential universality of the topographic index
distribution scaling relations in catchments where runoff is dominated by subsurface flow.
Citation: Pradhan, N. R., F. L. Ogden, Y. Tachikawa, and K. Takara (2008), Scaling of slope, upslope area, and soil water deficit:
Implications for transferability and regionalization in topographic index modeling, Water Resour. Res., 44, W12421,
doi:10.1029/2007WR006667.

1. Introduction[2] Many river basins in the world are either ungauged or
poorly gauged. The prediction of flow in those rivers is an
important task for the management of water resources. In
most regionalization studies, a relationship between theparameters of the model and the catchment descriptors isdeveloped [Parajka et al., 2005], so that the parameters aretransferable to similar regions [Blöschl, 2005]. A lack ofreliable methods to translate the spatial, and also temporal,
scale dependence relations into effective hydrological model parameters poses a serious problem [Kavetski et al., 2003;
Pradhan et al., 2006] that hampers predictions in ungauged
basins of developing countries, where the information
gained from one scale is to be used in making predictionsat other (usually coarser) scales [Pradhan et al., 2006]. Thisis due to the fact that coarsening of the resolution of spatial
data sets results in the loss of information at higher spatial
frequencies [Kuo et al., 1999], leading to significant errors
in simulation results, owing to strong nonlinearity in thescale-dependence effect on many hydrological processes.
1
Department of Civil and Architectural Engineering, University of
Wyoming, Laramie, Wyoming, USA.
2
Hydrology and Water Resources Engineering Laboratory, Department
of Urban and Environmental Engineering, Kyoto University, Kyoto, Japan.
3
Disaster Prevention Research Institute, Kyoto University, Kyoto, Japan.

[3] One issue that hampers generalized topographicallydriven saturation-excess modeling is that the topographic
index is dimensional, having units of ln(L). This dimensionality dictates that topographic indices developed for thesame catchment using DEMs with different resolution will
not be identical. Hence, the topographic index distribution is
not scale invariant to underlying data resolution. Overcoming this problem requires development of scaling relations.
[4] Thus, scale transformation and scale invariance appear
as fundamental steps for developing a process-based method
to identify similarities between catchments and for thedevelopment of an effective regionalization technique. The
problem of transferring information gained at one scale formaking predictions at a different hydrological scale is ascaling problem [Beven, 1995].
[5] In this study, we test the hypothesis that geomorphometric scaling relations may be used to transfer model
parameters between catchments where saturation-excess
runoff is dominant. Testing of this hypothesis could assist
in the development of an approach for a transferable
rainfall-runoff model and transferable model parameters toassist predictions in ungauged basins using a freely available global data set. Development of a transferable rainfallrunoff model and the identification of transferable model
parameters are key steps in establishing similarities andconnections among physical processes at disparate scales
and in the determination of possible linkage of hydrological

Copyright 2008 by the American Geophysical Union.
0043-1397/08/2007WR006667$09.00

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Table 1. Effective Parameter Values for the Saturation Excess
Runoff Generation Mechanism of TOPMODEL Identified at 50-m
DEM Resolution in the 210-km2 Kamishiiba Catchment
Parameter

Value

Lateral transmissivity of soil at saturation
condition, To (m2 h1)
Decay factor of lateral transmissivity withrespect to saturation deficit, m (m)Maximum root zone storage, Rzmax (m)

9.8
0.07
0.001

similarities between catchments making scale transformation and scale invariance the fundamental requirement.
[6] Section 2 presents the study objectives, methodology,
watersheds and data. In section 3 we discuss the scaledependence of the TOPMODEL state variables. In sections
4 – 6, we discuss the scale dependencies of dominating
geomorphometric parameters and development of methods
to accurately account for the effect of scale in upslope
catchment area, slope, unit contour length, and topographic
index. The effectiveness of our approach is demonstrated in
section 7 by testing whether the modeling outcome isreliable at a scale that is significantly different from thescale where the model was calibrated, without recalibration.
Discussion and conclusions follow in section 8.

2. Objectives, Methods, Watersheds, and Data
[7] In this study we examine the utility of scaling
relations for transferring geomorphometric model parameters between scales and basins for the saturation-excess
TOPMODEL, and is motivated by the need to makepredictions in ungauged basins. The objective of this study
was to test the hypothesis that detailed consideration of the
impact of scale on model parameters will also improvetransferability of topographic-index-derived parameters between different catchments. This hypothesis is tested bytransferring TOPMODEL parameters identified by calibration in the Kamishiiba catchment in Japan, shown in Table 1,
to two subcatchments of the Sun Koshi river in Nepal, theUpper Rio Chagres catchment in Panama, and the Town
Brook catchment in the northeastern United States.
2.1. TOPMODEL Formulation Used in This Study[8] The TOPMODEL formulation used in this study isessentially the same version of TOPMODEL in wide
distribution today. Section 3, represents the overland flow
and base flow in the TOPMODEL formulation used in thisstudy. Calibration of TOPMODEL was first performed on
runoff volume. Muskingum-Cunge channel routing wasadded to the simulation of the Kamishiiba catchment only
after the TOPMODEL parameter set in Table 1 was identified. Rainfall on a saturated grid is the overland flow to berouted. Routing improved simulation results at the catchment by increasing the Nash-Sutcliffe coefficient in comparison with observed hydrographs. The addition of channelrouting had no other effect on the simulation results, as there
is no feedback in this formulation between the channel andrunoff generating components of TOPMODEL.
[9] Given the high-quality data set from Kamishiiba
catchment, we selected to use it for parameter estimationin this regionalization study. Pradhan et al. [2006] identified the parameters in the Kamishiiba catchment adopting

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trial and error approach in the calibration. Pradhan et al.
[2006] derived an estimate of the parameter m in the
Kamishiiba catchment from the first-order hyperbolic function that fitted the recession curve obtained from theobserved discharge of rainfall [Ambroise et al., 1996]. Prior
estimation of the m parameter also helped to identify the setof calibrated parameters that is rather less affected byequifinality [Beven, 2006] in calibration.
2.2. Study Watersheds[10] Our study uses calibration data from the Kamishiiba
catchment in Japan, and tested the utility of scaling relations
for improved regionalization using data from two tributariesof the Sun Koshi River in Nepal, the Upper Rio Chagres inPanama, and Town Brook in New York, United States.
These watersheds are discussed in the following sections.
2.2.1. Kamishiiba Catchment, Japan[11] This study uses a research-quality data set from the
210-km2 Kamishiiba catchment, which is located in Japan.
This same data set was used by Pradhan et al. [2006] tovalidate the scaling relations employed in the present study.
The Kamishiiba catchment lies in Kyushu Island of Japan.
Kyushu, the southernmost of the four major islands ofJapan, has a mountainous interior, is subtropical andremains warmer throughout the year than other parts ofJapan. The summer months are hot and humid with highrainfall intensity. The mean annual rainfall in the Kamishiiba catchment is approximately 2000 mm. Since the simulations of the Kamishiiba catchment were done in thewarmest season, snow processes were not simulated.
2.2.2. Likhu and Balaphi Catchments, Nepal[12] The two basins used in Nepal are the 850-km2 Likhu
and 620-km2 Balaphi catchments. Nepal, with an area of
147,181 km2, has more than 6000 rivers stretching from theHimalayas to the southern plain. This dense river network
flows through steep topography. Approximately 80% of the2000 – 3500 mm total annual rainfall is concentrated in therelatively short monsoon season [Gyawali, 1989], which lasts
approximately 4 months [Japan International Cooperation
Agency, 1985]. These peculiar characteristics lead to a setof hazards, including frequent floods and debris flows.
Although there are some gauging stations installed in somerivers, many river basins in Nepal are either ungauged or
poorly gauged. The lack of understanding of the scaledependence relations into effective hydrological models has
up to now posed a serious problem for making predictionsin ungauged basins in developing countries. Often in thiscase, either only coarse-resolution DEM data are available
or the information gained at fine scale is to be used in
making predictions at coarser scales.
[13] The Likhu and Balaphi catchments are tributaries of
the Sun Koshi River in Nepal. The Sun Koshi basin has
steep terrain and monthly average relative humidity rising
up to 80% during the rainy season. Humidity in the dry
season, on the other hand, varies widely during a 1-day
period and is typically much lower than of the rainy season.
The rainfall data used in Likhu and Balaphi catchments are
the daily rainfall data from the nearest available gauging
stations. Precipitation gauging stations number 1224 and1103 were used for estimating rainfall over the Likhu
catchment. In the Balaphi catchment, data were used fromprecipitation gauging stations number 1006, 1008, 1009,
and 1103. Rainfall-runoff data were obtained from the2 of 12

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Department of Hydrology and Meteorology, Government of
Nepal. For both the Likhu and Balaphi catchments, evaporation data used were obtained from climatological records
of Nepal (see http://www.dhm.gov.np/index.php?p =publication). Four climatological seasons can be identified
in Nepal [Nayava, 1980]. The pre – monsoon season(March – May) is characterized by hot, dry weather with
scattered rainfall; toward the end of this period, it becomes
more humid with thundershowers. The summer monsoon
season (June – September) is governed by the southeasterly
flow of moisture-laden air from the Bay of Bengal. The
post – monsoon season (October –November) has a typicalonset date coincident with the autumnal equinox. Rainfall
activity is substantially reduced with November typicallythe driest month. The winter season (December – February)is generally dry, although westerly weather systems may
bring cold air and winter precipitation (in the form of snow)that contributes to low-flow regimes of these rivers.
Groundwater is primarily recharged by the monsoon rainfall, and is the most significant contribution to low-flow
regimes of these rivers in Nepal. Some portions of thesecatchments lie above the snow line; however, it is very
difficult to clearly distinguish meltwater contributions dueto the overwhelming influence of the summer monsoon rain
[Hannah et al., 2005]. For these reasons, snowmelt was notconsidered an important process in these catchments.
2.2.3. Upper Rio Chagres Catchment, Panama[14] The Greater Panama Canal Watershed is a hydrologically complex and ecologically diverse. The watershedconsists of a mix of pristine old-growth forest managed asnational park, and partly deforested grazed areas in 13
subbasins. There are two significant reservoirs in the greater
Panama Canal watershed [Harmon, 2005]. The Rio Chagres, notable as the only river in the world to flow into two
oceans, is the largest river by annual flow volume entering
into the Panama Canal and provides approximately half thewater required for operating the canal while comprising
only a third the area of the greater Panama Canal watershed.
The 414-km2 Upper Rio Chagres is a protected National
Park. As such it is predominantly covered by old-growthforest, less than 10% of the watershed disturbed by humanactivities.
[15] Panama is located in the seasonal tropics, withpronounced wet and dry seasons. Typically, precipitation
peaks during June– July and October – November when theintertropical convergence zone passes across the region,
with very little rain falling during the dry season between
January and March. Rainy season storms generated in the
Caribbean tend to track from north to south, so that there isa pronounced N – S precipitation gradient across the Panamanian isthmus. The Atlantic coastal region receives anannual average of >3000 mm of precipitation as comparedto <2000 mm for the Pacific coastal region on the leeward
side of the continental divide although the year-to-year
variations in total precipitation can be significant. For
example, annual precipitation in the Upper Rio Chagres
watershed is quite variable, with annual values ranging from1,800 to 3,200 mm between 1972 and 2005. It never snowsin the Rio Chagres catchment.
2.2.4. Town Brook Catchment, New York[16] The 37-km2 Town Brook catchment lies in Catskill
mountains in southern New York State. In humid, well-

W12421vegetated areas, such as in the northeastern United States,
most direct runoff is generated by the saturation-excess
mechanism, that is, via precipitation on or exfiltration fromsaturated areas in the landscape [Ward, 1984]. Runoff
generation in Town Brook and adjacent watersheds was
examined by Lyon et al. [2004] and Steenhuis et al. [1995].
The Town Brook catchment is smaller and significantly
different from the other catchments used in this study. Whilethe catchments in Japan, Nepal, and Panama are relatively
young as a result of geologically recent tectonic or volcanic
activity, the Town Brook catchment is in the northern
Appalachian province, which is geologically much older
than the other study catchments.
2.3. DEM Data Sources[17] DEM data at 1000- and 90-m resolutions for thecatchments in Nepal and Panama were downloaded from
Global 30 Arc-Second Elevation (GTOPO30) and Shuttle
Radar Topography Mission (SRTM) data sets from theUnited States Geological and Survey (USGS) EROS data
center web site, respectively. The 50-m resolution DEM forthe catchments in Japan was obtained from Geological
Survey of Japan (GSJ). The DEM at 10- and 30-m resolution for Town Brook catchment in the United States wereobtained from National Elevation Data set (NED), UnitedStates Geological and Survey (USGS) Web site.

3. Effect of DEM Resolution on Saturated Area
and Base Flow Predictions From TOPMODEL[18] The topographic index distribution for the 210-km2
Kamishiiba catchment was calculated using a 50-m resolution DEM. TOPMODEL was calibrated using rainfall-runoff
data from this catchment, which resulted in Nash-Sutcliffe
efficiency, hereafter referred to as N– S efficiency, of 94%.
The corresponding identified value of lateral soil transmissivity parameter was 9.8 m2/h (see Table 1). Figure 1ashows the model predicted that 7% of the catchment area
was saturated at the initial time step.
[19] When a 1000-m DEM was used to develop the
topographic index distribution and the model parameters
identified through the original calibration were applied, theinitial percentage of saturated area predicted by TOPMODEL increased to 59% as shown in Figure 1b. This is quite
an unreasonable prediction for a conceptual state variableowing to the change in DEM resolution, and highlights the
problem with the dimensionality of the topographic index.
This very large error in the TOPMODEL-predicted discharge results in a negative N– S efficiency as a consequence of the overestimation of the saturated area.
Furthermore, the base flow predicted by the TOPMODEL
using the topographic index distribution derived from the1000-m DEM is very small compared to 50-m DEM
resolution TOPMODEL as shown in Figure 2. This is
attributable to the fact that too much water is stored in thesoil as demonstrated by the 52% overprediction of thesaturated area.
[20] This overprediction of saturated area is primarily
caused by the effect of DEM resolution on the topographic
index distribution [Zhang and Montgomery, 1994]. Figure 3
shows the scale effect in the cumulative topographic index
distribution using topographic data from catchments indifferent parts of the world. In the TOPMODEL framework,

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W12421[21] Equation (2) is the relation between S(t) [L], thespatial mean storage deficit and S (i, t) [L], local saturation
deficit at each location i in a catchment as [Beven andKirkby, 1979]
 


S ði; tÞ ¼ S ðt Þ þ m TI  TI þ ln To  ln To ;Figure 1. Analysis of distributed saturated area output of
TOPMODEL at different digital elevation model (DEM)
resolutions in the 210-km2 Kamishiiba catchment in Japan
using model parameters identified using 50-m DEM
resolution in the Kamishiiba catchment. (a) The 7%
saturated area is predicted by TOPMODEL at the initialtime step, when 50-m DEM resolution is used. (b) The 59%
saturated area is the output of TOPMODEL at the initialtime step, when 1000-m DEM resolution is used.
the topographic index is used to distribute the local saturation deficits, given knowledge of the mean storage deficit[Beven and Kirkby, 1979]. The topographic index [Kirkby,
1975] of TOPMODEL is defined asa;
TI ¼ lnðtan bÞ

ð1Þ

where a is the local upslope contributing area per unit
contour length [L] and b is the slope angle of the groundsurface [L L1].

ð2Þ

where To [L2T1] is the local value of lateral (horizontal)transmissivity when the soil has zero storage deficit; ln To isthe catchment average of ln To; m is a decay factor ofsaturated transmissivity of soil with respect to saturation
deficit with dimensions of length [L]; and TI is thecatchment average of the topographic index.
[22] Areas with S (i, t)  0 in equation (2) represent
contributing areas for saturation excess overland flow asshown in Figure 1. Considering an effective To value, thelocal saturation deficit is totally controlled by local topographic index and the catchment average of the topographic
index. The effect of DEM resolution on the cumulative
topographic index distribution as shown in Figure 3 is thereason for the different predictions of saturation shown in
Figure 1 as given by equation (2).
[23] In the TOPMODEL framework, subsurface contributions to streamflow, Qb(t) [LT1] can be derived fromequation (2) asS ðt ÞQb ðt Þ ¼ To eTI e

=m

;

ð3Þ

which is also controlled by the catchment average of the
topographic index. The DEM resolution effect on the
topographic index shown in Figure 3a is the reason forthe difference in subsurface flow shown in Figure 2 as given
by equation (3).
[24] The scale dependence of the topographic index withdecreasing resolution is caused by the loss of high spatial

Figure 2. Comparison of subsurface flow obtained from 50-m DEM resolution TOPMODEL, 1000-m
DEM resolution TOPMODEL, and TOPMODEL applied at 1000-m DEM resolution with scaled
topographic index to 50-m DEM resolution. The applied effective parameters are identified by 50-m
DEM resolution TOPMODEL. A rainfall event 6 days in duration (22 September 1999 to 27 September1999) in the 210-km2 Kamishiiba catchment in Japan was used to analyze the subsurface flow.
4 of 12

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W12421Figure 3. DEM resolution effect on density distribution of topographic index and applicability of the
topographic index distribution scaling method in different topographic regions of the world: (a) 210-km2
Kamishiiba catchment in Japan, (b) 850-km2 Likhu catchment in Nepal, (c) 414-km2 Upper Rio Chagres
catchment in Panama, and (d) the 37-km2 Town Brook catchment in the northeastern United States.
frequency components in the upslope catchment area, andthe underestimation of the slope when using a coarseresolution DEM. The error in the predicted saturated statevariable and base flow response can be removed by scaling
the topographic index distribution from the available scale
to another scale of interest [Pradhan et al., 2006]. Understanding the scaling of the topographic index can only beobtained by deriving a scaling formulation of the dominantgeomorphological parameters like upslope catchment area,
unit contour length and slope.

4. Scaling Solution to the Effect of DEM
Resolution on Geomorphometric Information[25] Scaling relations for upslope contributing area andslope developed by Pradhan et al. [2006] are showed in thissection. This section also introduces a new method to
produce an unbiased measure of unit contour length thatis independent from DEM resolution.
4.1. Scaling of Local Upslope Contributing Area
and Slope Angle[26] Many physically based models of hydrological andgeomorphic processes rely on spatially distributed characterization of drainage area and slope angle. Pradhan et al.
[2006] showed when upslope catchment area is calculated atfine resolution, information is lost when the calculation isrepeated using coarse-resolution DEMs. Pradhan et al.
[2006] also showed that slopes are underestimated whenusing coarse-resolution DEMs. To avoid this scaling effect,
which can seriously affect the accuracy of hydrological andgeomorphological models, Pradhan et al. [2006] success-fully derived the scaling formulation for the upslope catchment area and slope – angle. The scaling method for upslope
contributing area derived by Pradhan et al. [2006] isdefined as

Ai;scaled ¼

8
<
:

Ai

9
=

ð1Ni ÞH  ;N ;N s e No

ð4Þ

where the suffix i is a location in a catchment, Ai is the
upslope catchment area obtained from a coarse-resolution
DEM, Ai,scaled is the scaled upslope catchment area at apoint i, Ns is the total number of desired target fineresolution grids included in an available course-resolution
grid, Ni is the number of the coarse-resolution grids
contained in the upslope contributing area at a location iin the catchment, No is the number of the coarse-resolution
grids contained in the upslope catchment area at the outletof the catchment, and HN is a factor [Pradhan et al., 2006].
[27] The method for scaling local land-surface slope
developed by Pradhan et al. [2006] is defined asqi;scaled ¼ ai;steepest di;scaled ð1Di Þ ;ð5Þwhere i is the central grid location in 3  3 moving window
pixels, qi,scaled is the downscaled steepest slope in 3  3moving window pixels, di,scaled is the steepest slope distanceof the target-resolution DEM, Di is the fractal dimension,and ai,steepest is a coefficient.
[28] Pradhan et al. [2006] found that equation (4) works
for any range of scaling provided that there are enough

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As contour length is varied from the target fine-resolution
grid at the catchment divide region to the available coarseresolution grid at the catchment outlet, the influence of Rf on
Wi must gradually decrease in equation (6) from value 1 at
the catchment divide portion to R1f as Wi becomes larger. For
this reason, in equation (6) the influence factor If isintroduced having the value 1 at the catchment divide
portion so that the unit contour length given by a target fineresolution grid is obtained, having value R1f at the outlet sothat unit contour length is defined by the available coarseresolution DEM. Using exponential decay, the influence
factor is equal to 1 at the catchment divide and transitions toa value of R1f at the catchment outlet as the upslope
catchment area increases. Thus If is defined as
Figure 4. Scale effect on the conceptual derivation ofcontour length. (a) Unit contour length representation of acoarse-resolution DEM. (b) Unit contour length representation of a fine-resolution DEM.
pixels of a grid size to represent the distribution in acatchment. In equation (5), slope is a function of themeasurement scale by assuming that topography is unifractal in a specified range of measurement scale. This unifractal concept can break down at very fine scales limiting
the scaling range as shown by Pradhan et al. [2006].
4.2. Development of a Method for Scaling Contour
Length in the Topographic Index[29] With coarse DEM grids, the simple use of either the
single-flow direction method or multiple-flow direction
method overestimates the contour length near the catchment
divide compared with estimates developed from fine DEM
grids. With fine DEM grids, the simple use of either the
single-flow direction method or multiple-flow direction
method underestimates the contour length in the vicinityof the catchment outlet when fine-resolution DEM is usedcompared with that from coarse DEM grids. For example, atthe outlet, the upslope drainage area is the same (Figures 4aand 4b), but the contour length representation of a coarseresolution DEM, W, in Figure 4a is larger than the contour
length representation of a fine-resolution DEM, W*, inFigure 4b.
[30] To solve the scale problem in defining the contour
length using the single-flow or multiple-flow direction
method, contour length is varied from the target fineresolution grid at the catchment divide region to theavailable coarse-resolution grid at the catchment outlet.
The downscaled contour length is defined as
Wi;scaled ¼

Wi
;
Rf If

Coarse DEM
Target DEM

Resolution:
Resolution

ð8Þ

where Ni is the number of the coarse-resolution grids
contained in the upslope contributing area at a location i in
the catchment and No is the number of the coarse-resolution
grids contained in the upslope catchment area at the outletof the catchment. Considering that the influence of Rf on Wi
in equation (6) is almost negligible at the outlet of thecatchment, where Ni = No and If = R1f , the value of HR
introduced as a factor in equation (8) can be obtained fromequation (8) by substituting Ni = No and If = R1f as
HR ¼

No
ln Rf :
No  1

ð9Þ

[31] Equations (6) and (8) give a scale-independent
method of representing unit contour length as
8
<

9
=
Wi
Wi;scaled ¼
ð1Ni ÞH  :R ;:Rf e Noð10Þ

The differences in the mathematical formulation for upslope
catchment area in equation (4) and unit contour length inequation (10) is that equation (4) uses Ns and HN whereas
equation (10) uses Rf and HR.

5. Development of a Scaling Approach in
Hydrogeomorphology[32] Substituting the value of a and b in equation (1)Afrom equations (4) and (10) and qi,scaled fromby Wi;scaledi;scaled
equation (5), the downscaled topographic index, TIscaled isdefined as0ð6Þ

where Wi, scaled is the scaled contour length from theavailable coarse-resolution DEM to the target fine-resolution DEM at location i, Wi is the unit contour length of theavailable coarse-resolution DEM at location i, and If is theinfluence factor per unit contour length. Rf is resolution
factor defined as
Rf ¼

ð1Ni ÞH 
R
If ¼ e N o ;B
TIscaled ¼ lnB
@

Ai
ð1N

ÞH

1

C
Ns e
C:
W
i
ð1N ÞH  qi;scaled A
i R
No
Rf e
i
No

N

ð11Þ

Substituting Ns = R2f [Pradhan et al., 2006] in equation (11)and simplifying, the downscaled topographic index, TIscaled
is defined as

ð7Þ
6 of 120 ð1N ÞðH H Þ 1
i
R
N
No
Ai e
A:
TIscaled ¼ ln@
Wi qi;scaled Rf

ð12Þ

PRADHAN ET AL.: TOPOGRAPHIC INDEX MODELING

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W12421Figure 5. Upscaling slope and catchment area in the 210-km2 Kamishiiba catchment. (a) Distributionfunctions of steepest slope obtained from different DEM resolutions and upscaled slope. (b) Distributionfunctions of upslope catchment area obtained from different DEM resolutions and upscaled upslope
catchment area.
Substituting the value of TI in equation (1) with the value of
TIscaled from equation (12) in equation (2), the scaled local
saturation deficit is defined asapplying equation (14), it is shown in Figure 2 that thesimilar base flow response as that of 50-m DEM TOPMODEL is obtained from the 1000-m DEM resolution
TOPMODEL without recalibrating.
S ði; t Þ ¼ S ðtÞ þ m[36] In this paper we applied the upscaling relations for82
90 ð1N ÞðH H Þ 13
i
R
N
<
=
slope
and upslope contributing area published by Pradhan
N
o


Ai e
A5 þ ln To  ln To ; et al. [2006] for the first time to make predictions in 4TI scaled  ln@:;
Wi qi;scaled Rf
diverse catchments. To upscale, we simply replaced theð13Þ fine-resolution DEM with a coarse-resolution DEM towhich the geomorphometric information is to be upscaled.
where TI scaled is the catchment average of TIscaled in Thus while upscaling, Rf in equation (7) is defined as
equation (12).
[33] Substituting the value of TI by TI scaled in equation
Available Fine DEM Resolution
:
ð15Þ
Rf ¼
(3) the scaled base flow is defined asTarget Coarse DEM Resolution
Q b ðt Þ ¼ T oeTI scaled eS ðtÞ

=m

:

ð14Þ

Equations (13) and (14) are used to produce scale-independent
saturated area and base flow response in TOPMODEL
framework.

6. Upscaling and Downscaling[34] Equations (7), (8), (10), (12), (13), and (14) are
explained from the downscaling point of view. Thus inexplaining the derivation of these equations, a coarseresolution DEM is taken as an available DEM and a fineresolution DEM is taken as the target-resolution DEM atwhich the geomorphometric information is to be obtained.
Applying equation (12), to scale the cumulative topographic
index distribution developed from a coarse-resolution DEM
shows a close fit with the topographic index at a fineresolution DEM in Figure 3. Figure 3 shows the applicabilityof the scaling method of topographic index distribution inwatersheds located in vastly different regions of the world.
[35] Applying equation (13), the saturated area obtained
dropped down from 59% in Figure 1b to 10%, very close tothat obtained using 50-m DEM resolution in Figure 1a.
Thus by scaling the topographic index distribution andThus, Rf in equation (15) is less than 1 while upscaling.
When upscaling, Rf used in all the scaling relations isdefined by equation (15) and Ns is the square of the Rf
defined by equation (15) and Ni and No are defined at a fineresolution DEM. Furthermore, while upscaling, qi,scaled inequation (5) is the upscaled steepest slope in 3 x 3 moving
window pixels and di,scaled is the steepest slope distance of atarget coarse-resolution DEM. While upscaling the slope,values of ai,steepest in equation (5) are derived directly fromthe steepest slope of the available fine-resolution DEM.
[37] Figure 5a shows the upscaled slope distributionfunction from 150- and 1000-m DEM resolutions. The
RMS error between the cumulative distribution of slope
from 1000- and 150-m DEM resolutions in Figure 5a is0.342. The RMS error between the upscaled cumulative
distribution of slope from 150-m DEM resolution to 1000-m
DEM resolution and the slope at 1000-m DEM resolution in
Figure 5a is 0.015. Thus by the use of the slope upscaling
relation, the RMS error was reduced by 95%.
[38] Figure 5b shows that the frequency distribution of
upslope contributing area from 1000-m DEM resolutionagrees quite well with the upscaled upslope catchment area
frequency distribution from 150-m DEM resolution to the

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target 1000-m DEM resolution. Without the use of the
upscaling relations, the RMS error in dimensionless (0 – 1)
upslope contributing area was 0.36%. The RMS error of the
upscaled dimensionless contributing area was 0.04%. The
use of the upscaling relation reduced the RMS in contributing area by 89%.

7. Tests of Improved Model ParameterTransferability by Accounting for Scale Effects
on Geomorphometric Parameters[39] This section describes our test of the hypothesis thatgeomorphometric scaling relations can be used to improve
the transfer of rainfall-runoff model parameters between
catchments where saturation-excess runoff is dominant.
These tests involved calibrating TOPMODEL on the
Kamishiiba catchment in Japan, and using the derived
scaling relations to transfer those parameters to catchments
in Nepal, Panama, and the United States. In the following
sections, downscaled TOPMODEL means downscaling
equations (12), (13), and (14) to a target fine-resolution
scale using an available coarse-resolution DEM and upscaled
TOPMODEL means upscaling equations (12), (13), and (14)to a target coarse scale using an available fine-resolution
DEM.
7.1. Transferring the Model Parameters From
Kamishiiba Catchment in Japan to Sun Koshi Basin
Subcatchments in Nepal[40] Similar to the Kamishiiba catchment, the Likhu andthe Balephi catchments experience high humidity in summer and comparatively lower humidity in winter. The
defined catchments in Japan and Nepal have tectonic
activity and are geologically young mountains. The mountainous setting in both the regions in Japan and Nepal
creates rivers that have steep channel slopes. Thus, there
are possibilities that the parameter values obtained at theKamishiiba catchment in Japan are applicable to the subcatchments of the Sun Koshi basin in Nepal.
7.1.1. Tests on the 850-km2 Likhu Catchment, Nepal[41] The Kamishiiba-calibrated parameters were appliedin the Likhu catchment represented by a digital terrain
model of resolutions of 1000 m and 90 m. To match thescales of model application, equations (12), (13), and (14)are applied to scale the topographic index, soil water deficit
variable, and base flow.
[42] Figure 6a shows the simulated discharge when the
Kamishiiba-calibrated parameters were applied at 1000-m
DEM resolution in Likhu catchment. The N– S efficiency isnegative. Figure 6c shows the simulated discharge ofdownscaled TOPMODEL from 1000-m DEM resolution
to 50-m DEM resolution in the Likhu catchment with the
Kamishiiba-calibrated parameters. The N – S efficiency
increased from a negative value in Figure 6a to 81% in
Figure 6c.
[43] Figure 6b shows the simulated discharge when the
Kamishiiba-calibrated parameters were applied at 90-m
DEM resolution in Likhu catchment. The N –S efficiency
is 72%. Figure 6d shows the simulated discharge of downscaled TOPMODEL from 90-m DEM resolution to 50-m
DEM resolution in the Likhu catchment with the Kamishiiba-calibrated parameters. The N – S efficiency increased
from 72% in Figure 6b to 82% in Figure 6d.

W12421[44] The identical simulation results are shown in Figures
9c and 9d, demonstrating that the derived scaling relations
are effective in transferring the parameters identified at afine scale to different coarse scales at which the model isapplied. At this point the utility of the downscaling method
presented in this research is quite clear. Reduction of thegrid size also means a considerably increase in the work
involved in data collection and processing [Vázquez et al.,2002].
7.1.2. Tests on 620-km2 Balephi Catchment, Nepal[45] Figure 7 shows a long-term simulation result, from1991 to 1996 on the Balephi catchment in Nepal. This
simulation result was obtained by scaling the TOPMODEL
parameters from the 1000-m original DEM to 50-m DEM
resolution and applying the Kamishiiba-calibrated parameters. As catchments in Nepal are ungauged or poorly
gauged, Balephi catchment discharge data for the years
1993 and 1995 are missing. Thus the simulation results
for the years 1993 and 1995 in Figure 7 are blind predictions. The saturation information simulated at the end of
previous years, that is year 1992 and year 1993, was given
as initial state variables in those blind prediction years 1993
and 1995. The N – S efficiencies for 1991, 1992, 1994, and1996 are 78%, 88%, 67%, and 70%, respectively, with anaverage of 75%.
7.2. Tests on 414-km2 Upper Rio Chagres Catchment,
Panama[46] TOPMODEL is used in the humid tropics, andstudies have shown that it performs well when the catchments are in high-moisture states [Campling et al., 2002;
Molicova et al., 1997]. Using data provided by the Panama
Canal Authority, we calibrated TOPMODEL at 90-m DEM
resolution in the Upper Rio Chagres catchment, 414 km2, inPanama, which is located at 9°N in the seasonal tropics.
Figure 8 shows the TOPMODEL simulation output fromJanuary to September 1986. In Figure 8a, the calibrated topranked parameter set gave a N – S efficiency of 50%.
Reasons for not achieving a higher efficiency could be theinabilities of TOPMODEL to take account the variety ofrunoff generation processes and the importance of deep
groundwater in producing base flow in this tropical catchment [Niedzialek and Ogden, 2005] as well as the lack ofdetailed spatial description of rainfall data. Regional generalizations strictly require that the various processes ofrunoff generation acting in these regions should be carefully
examined [Pilgrim, 1983]. We tested the transferability of
the TOPMODEL parameters from the Kamishiiba catchment to the Upper Rio Chagres catchment. Given that direct
calibration of TOPMODEL yielded a N– S efficiency of50%, we would not expect the best performance of thetransferred parameters to exceed this value.
[47] Figure 8b shows the simulated discharge when the
Kamishiiba-calibrated parameters were applied at 1000-m
DEM resolution in the Rio Chagres catchment. In this case
the N– S efficiency was negative, 20%. Downscaling of
the TOPMODEL parameters from 1000- to 50-m DEM
resolution in the Upper Rio Chagres catchment with the
Kamishiiba-calibrated parameters gave the simulation
results shown in Figure 8c, where the N – S efficiency
increased to 45%. Figure 8c nearly matches the simulation
results in Figure 8a, which were achieved by direct calibration. Despite the fact that the TOPMODEL formulation is

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W12421Figure 6. Comparison of TOPMODEL predictions at different DEM resolutions, with and without scalecorrection in 850-km2 Likhu catchment in Nepal. All simulation results are for the same period of rainfall
using basin-specific topographic index distributions and the Kamishiiba-calibrated parameters: (a) 1000-m
DEM resolution, (b) 90-m DEM resolution, (c) 90-m DEM resolution scaled to 50-m DEM resolution,and (d) 1000-m DEM resolution scaled to 50-m DEM resolution.
incomplete in this watershed, as indicated by the relatively
low N – S efficiency, the scaling relations presented in thisstudy result in model performance using the scaled andtransferred Kamishiiba catchment parameters that is nearly
equal to that seen with direct calibration.
7.3. Tests on the 37-km2 Town Brook Catchment,
New York[48] Figure 9 shows the TOPMODEL simulation result
for the year 2003 using daily rainfall-runoff data from the
97-day period beginning in September to the first week ofDecember, in the Town Brook, catchment in New YorkState, United States. Figure 9a shows the simulated discharge when the Kamishiiba-calibrated parameters wereapplied without scaling at 10-m DEM resolution in the
Town Brook catchment resulting in a N – S efficiency of
31%. Using the scaling relations to transfer the Kamishiiba

parameters from 50- to 10-m DEM resolution increased the
N – S efficiency to 50% in Figure 9b. Although, the scaling
relations increased the N – S efficiency from 31% to 50%,
we found that TOPMODEL performed better in this catchment with recalibration at 10-m DEM itself. So in this case,given the significant difference in catchment size, thescaling relations do not explain all of the variation.
[49] To further explore this, the TOPMODEL was calibrated on the Town Brook at 10-m DEM resolution, resulting in To = 3.0 m2/h and m of 0.02, and a N– S efficiency of
86%. The prior estimate of the m parameter from the firstorder hyperbolic function that fitted the recession curve
obtained from the observed discharge of rainfall in the
Town Brook catchment was found to be 0.015 which isvery close to the calibrated m value of 0.02 in the catchment. Using the Kamishiiba To value, and upscaling the

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W12421

W12421Figure 7. Long-term simulation in Balephi (Nepal) catchment at 1000-m DEM resolution andtopographic index distribution scaled to 50-m DEM resolution with the Kamishiiba-calibrated
parameters. Simulations for years 1993 and 1995 are blind predictions as the corresponding years
have no available observed data.
TOPMODEL from 10- to 50-m DEM resolution, with thecalibrated Town Brook m parameter, the N– S efficiency
was 84% in Figure 9c.

age timescale or ‘‘storage half life’’ of 45±15 days. The
quest for fundamental hydrological variables as a result of
ecological, geological or other controls is an area of active
research.

8. Discussion[50] The use of routing increased the N – S efficiency in
the Kamishiiba catchment, as it did when the Kamishiiba
calibration parameters were transferred to the Nepal catchments. However, the incorporation of routing but did not
improve the simulation performance in the Upper Rio
Chagres and Town Brook catchments. The most likely
reason for this is that the response time of the 414-km2
Upper Rio Chagres and the 37-km2 Town Brook catchment
is less than the 1-day model time step employed in thisstudy. This paper does not consider the effect of model time
step on parameter identification. This is a limitation of ourstudy. However, the scaling relations tested in our study did
improve model performance using the Kamishiiba parameters to almost equal those obtained using direct calibration.
[51] We demonstrate that accounting for grid-size scaleeffects improves model performance in regionalization tests.
We did not seek to answer the question ‘‘is there an optimal
grid size.’’ As found by Andrle and Abrahams [1989] andKlinkenberg and Goodchild [1992], unifractal scaling ofslope breaks down at very fine scales owing to different
processes that control microtopography. This prevents application of equation (5) for downscaling to scales at whichmicrotopography is important. This limits the applicabilityof our approach to scales on the order of a few meters to a
few tens of meters.
[52] In this study, the value of the lateral transmissivity
T = 9.8 m2h1 identified in calibration on the Kamishiiba,
Japan, catchment was found to be applicable to some degree
in all four test catchments. We cannot draw any conclusionfrom this result. However, Nimmo [2007] analyzed 64 macropore flow data sets and found that flow velocities inmacropores and fractures are well bounded, with a meanvalue of 13 md1. An argument posed by Brutsaert [2008]
from analysis of river recession data proposed that there is afundamental geomorphological control wherein stream
drainage density develops to produce a characteristic drain-

Figure 8. Transferability of parameters with scaled
topographic index in the upper Rio Chagres catchment inPanama. All simulation results are for the same rainfall
period (January to September 1986). (a) TOPMODELcalibrated discharge at Shuttle Radar Topography Mission
90-m DEM resolution. (b) TOPMODEL-simulated discharge with Global 30 Arc-Second Elevation 1000-m
resolution DEM using the Kamishiiba-calibrated parameters. (c) Simulated discharge with 1000-m topographic
index scaled to 50-m DEM resolution using the Kamishiibacalibrated parameters.

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W12421Figure 9. Transferability of parameters with scaled topographic index in the 37.1-km2 Town Brook
catchment, in the northeastern United States. (a) TOPMODEL-simulated discharge with 10-m DEM
resolution using the Kamishiiba-calibrated parameters, (b) simulated discharge with topographic index
upscaled from 10- to 50-m DEM resolution using the Kamishiiba-calibrated parameters, and(c) TOPMODEL-simulated discharge with upscaled topographic index from 10-to 50-m DEM resolution
with the Kamishiiba-calibrated To value and the Town Brook – calibrated m value.
[53] In hydrologic modeling, fine-resolution DEM offers
advantages over course-resolution DEM for identifying
grid-resolution depended effective parameter values likethe lateral soil transmissivity T [Pradhan et al., 2006].
Finer-resolution DEMs allow for better description ofsmall-scale topographic features such as low-order channels
and hillslopes. Identifying a grid size for a faithful landscape representation is a future research work.

9. Conclusions[54] The topographic index is dimensional. At a point, it
is derived from local upslope contributing area, unit contour
length and slope. Available methods to quantify the topographic parameters drainage area, contour length per unit
area, and slope, are influenced by DEM grid resolution.
This makes the topographic index at a point dependent on
grid resolution. Subsequently TOPMODEL parameters thatwork in one watershed at one grid resolution will not work inthe same basin at a different grid resolution. Knowledge of
how to correct for this effect allows transfer of TOPMODEL
parameters not only from one grid size to another, but also
from one basin to another. Scaling relations on contributing
area, slope, and contour length were successfully used toscale the topographic index distribution in watersheds
located in vastly different regions of the world; Japan,
Nepal, Panama, and the United States. Accounting for scaleeffects using the geomorphometric scaling relations

increases transferability of TOPMODEL parameters form
one catchment to another.
[55] The use of geomorphometric scaling of topographically derived parameters from a catchment in Japan to twocatchments in Nepal, one catchment in Panama, and one
catchment in the United States, demonstrates that the gridresolution-dependent effective parametric value of lateralsoil transmissivity T is quite stable from one place toanother in the regionalization of TOPMODEL in humid
catchments. Results indicate that the stability of the lateraltransmissivity of soil requires that there should not be ascale mismatch between the parameter identification scale
and model application scale, or topographic scaling relations should be applied to match the scales of model
application and parameter identification. From the analysis
made in the Town Brook catchment we observed that thetransmissivity decay parameter m value is not dependent on
the grid resolution. The values of m given by Beven [1997]
do not seem to have any significant correlation with theother corresponding TOPMODEL parameter values. While
there is a theoretical link between the exponential transmissivity profile and the recession curve, effectively m is mostimportant in controlling the decay of the recession andhence values can be derived from recession analysis[Ambroise et al., 1996]. This means that there will be alink between soil moisture storage capacity and catchment
size rather than grid size. Larger catchments are more likely

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to have longer-term storage. A prior estimate of the parameter m from the first-order hyperbolic function that fits therecession curve obtained from the observed discharge could
be a good indication for the transferability of the mparameter value if data are available. The prior estimateof m obtained using the first-order hyperbolic function
method for the two Nepalese catchments and the Upper
Rio Chagres Panama, resulted in m = 0.1 and 0.06 respectively, which is close to the Kamishiiba-calibrated m valueof 0.07. We recommend use of a prior estimate of the mparameter when the prior estimate significantly differs fromthe Kamishiiba value of 0.07, as was the case in our analysisof the Town Brook watershed data set.
[56] Acknowledgments. The authors thank Tammo Steenhuis and theCity of New York, Department of Environmental Protection, for providing
the Town Brook data. The Upper Rio Chagres Panama data were provided
by Carlos Vargas, Jorge Espinosa, and Mike Hart of the Panama Canal
Authority. The Balephi and Likhu data were provided by the Department ofHydrology and Meteorology, Government of Nepal. This research was
supported in part by grant W911NF-07-1-0389 from the U. S. Army
Research Office to the second author.

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F. L. Ogden and N. R. Pradhan, Department of Civil and ArchitecturalEngineering, University of Wyoming, Laramie, WY 82071, USA.
(fogden@uwyo.edu; nawa_water@yahoo.com)
Y. Tachikawa, Hydrology and Water Resources Engineering Laboratory,
Department of Urban and Environmental Engineering, Kyoto University,
Kyoto 615-8540, Japan. (tachikawa@mbox.kudpc.kyoto-u.ac.jp)
K. Takara, Disaster Prevention Research Institute, Kyoto University,
Kyoto 611-0011, Japan. (takara@mbox.kudpc.kyoto-u.ac.jp)

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