Proc. of the 2nd Asia Pacific Association of Hydrology and Water Resources (APHW) Conference,July 5-8, 2004, Suntec Singapore International Convention Exhibition Centre, Singapore,vol. 2, pp. 641-648, 2004

SCALE INVARIANT TOPMODEL FOR THE SOLUTION OF DEM
RESOLUTION EFFECTS AND PARAMETER INCONSISTENCYIN HYDROLOGICAL MODELING
Nawa Raj PRADHAN
Graduate School of Civil Engineering, Kyoto University
Yoshida Honmachi, Sakyo-ku, Kyoto, 606-8501, Japan
Yasuto TACHIKAWA and Kaoru TAKARA
Disaster Prevention Research Institute, Kyoto University
Gokasho, Uji, 611-0011, JapanThe topographic index of TOPMODEL is scale dependent which leads identified
parameter values to be dependent on a DEM resolution. This makes difficult to use
model parameter values identified with a different resolution TOPMODEL. To overcome
this problem, this research has developed a concept of resolution factor to account for thescale effect in up-slope contributing area per unit contour length in the topographic index
and a fractal method for scaled steepest slope as an approach to account for the scaleeffect on slopes. The method has been applied to Kamishiiba catchment (210 km2) inJapan and it is shown that the downscaled topographic index distribution is similar to atarget resolution DEM topographic index distribution. The method to downscale thetopographic index distribution is then coupled with the TOPMODEL to develop theScale Invariant TOPMODEL and is applied in Kamishiiba catchment (210 km2). It isshown that the simulated runoff from the Scale Invariant TOPMODEL applied at 1000m
grid resolution DEM, with the same set of effective parameter values derived from 50m
grid resolution DEM, have matched with the simulated runoff of the 50m DEM
resolution TOPMODEL. It is also shown that the simulated runoff from the Scale
Invariant TOPMODEL applied at 1000m grid resolution DEM, with the same set of
effective parameter values derived from 50m grid resolution DEM, have matched withthe observed runoff with high efficiency without recalibration.
INTRODUCTION
In the field of hydrology, since the introduction of the first blueprint of a distributedhydrological model (Freeze and Harlan [1] ) the desire to develop more physicallyrealistic distributed models has been motivated for forecasting changes in hydrologicalbehavior due to a variety of land use and climate changes and for hydrologic predictionsin ungauged basins.
The fact that a model may be physically based in theory but not consistent withobservations results primarily from the mismatch in scales between the scale ofobservable state variables and the scale of applications. Conceptualizations of the12
homogeneity of the hydrological quantities inside the grid of a DEM result in different
performances in the models themselves with variations in the assumed scale.
Several researches (Wolock and Price [2]; Zhang and Montgomery [3] ) havediscussed the effects of digital elevation model map scale and data resolution on thedistribution of the topographic index, concluding that there is interdependence between
DEM scale and topographic index distribution. Band et al. [4] point out that higher
frequency topographic information is lost as the larger sampling dimensions of the grids
act as filter. Lack of a translation method of the scale dependence relation into effectivehydrological models have posed a serious problem for the ungauged basins of developing
countries where only coarse resolution DEM data is available (Pradhan et al. [5] ).
In this study we focus on the influence of DEM resolution on slope angle, upslope
contributing area, and develop a method to downscale the topographic index ofTOPMODEL by incorporating scaling laws. By using the method, the topographic index
distribution of fine resolution DEM is successfully derived by using only coarse
resolution DEM (Pradhan et al. [6]). Then we develop a Scale Invariant TOPMODEL bycoupling the method to down scale the topographic index distribution with TOPMODEL
and it is shown that the Scale Invariant TOPMODEL is consistent with observation whenapplied at coarse resolution DEM with the parameter identified at fine resolution DEM.
DEM RESOLUTION EFFECTS ON TOPOGRAPHIC INDEX DISTRIBUTION
Topographic index of TOPMODEL is defined asTI
a= ln  (tan β ) (1)where a is the local up-slope catchment area per unit contour length and β is the slope
angle of the ground surface. TOPMODEL allows for spatial heterogeneity by making
calculations on the basis of the topographic index distribution. Topographic index being
scale dependent, leads identified parameter values to be dependent on a DEM resolution.
In Figure 1, topographic index distribution is shifted towards higher value, as DEM
resolution gets coarser in the Kamishiiba catchment (210 km2) in Japan. Table 1 shows
DEM resolution effect on a spatial mean value of the topographic index λ in Eq. (2).

λ =1ln( a i tan β i ) dAA ∫A(2)Table 1 DEM resolution effect on topographic constant, λ, in Kamishiiba catchment.
DEM Resolution (m)

50

150

450

600

1000

Topographic constant λ [ln(m2)]

6.076

7.423

9.222

9.622

10.353

335

33.4

31.7

30.1

28.4

18.5

16.9

15.2

13.6

11.9

8.6

10.3

5.3

6.95

23.65026.8

0.1
0.05

25.1

0.2
0.15

23.5

Density

0.25

21.8

0.3

20.2

Topographic index from 50m DEM
resolution
Topographic index from 150m DEM
resolution
Topographic index from 450m DEM
resolution
Topographic index from 1000m DEM
resolution

0.35

Topographic index value [ln(m2)]

Figure 1. Effect of DEM resolution on density distribution of topographic index.
Considering lateral transitivity to be constant in a subcatchment or catchment, then
the key role for hydrological similar condition is played by the distribution function oftopographic index. Higher frequency topographic information contained in topographicindex is lost as the larger sampling dimensions of the grids act as filter. This makes thehydrological similarity condition accounting combined soil-topographic index, to varywith the variation in DEM resolution used. To overcome this problem, this research hasdeveloped a method to downscale topographic index.
THE METHOD TO DOWNSCALE THE TOPOGRAPHIC INDEX
Resolution factor in topographic index
The smallest contributing area derived from a DEM resolution is a single grid of theDEM at that resolution. Thus area smaller than this grid resolution is completely lostwhich makes the topographic index distribution from coarse resolution DEM to swift
towards higher values (see Figure 1). But as we use finer resolution DEM, the smaller
contributing area that is the area of finer grid resolution is achieved. From this point ofview we introduced Resolution Factor Rf concept in topographic index as shown in Eq.
(3), for detail explanation refer Pradhan et al. [6].
TI

(3)


Ci= ln  (W i R f tan β i )

where Ci and Wi is the upslope contributing area and unit contour length at a point i of thecoarse resolution DEM. Taking W*i as the unit contour length of the target resolution
DEM, Resolution Factor Rf is defined as
Rf=Coarse DEM
T arg et DEM

Re solution
Re solution=WiW∗i(4)Fractal method for scaled steepest slope
Slope derived from coarse resolution DEM is underestimated. To scale the local slope,
we followed the fractal theory in topography and slope proposed by Klinkenberg and4Goodchild [9] and Zhang et al. [10]. Considering the significant role of the steepest
descend slope in hydrological modeling, we developed a modified fractal method toaccount for the DEM resolution effects on the steepest slope, which is shown in Eq. (5).

S scaled = α steepest d scaled(1 − D )(5)

where Sscaled is the scaled steepest slope from a caorse resolution DEM to the target
resolution DEM. dscaled is the scaled steepest slope distance of the target resolution DEM.
The direction of the scaled steepest slope in the target resolution DEM is taken as thesame direction of the steepest slope in the coarse resolution DEM. It is found thatstandard deviation of elevation in the same sub area is quite stable for change in DEM
resolution. Thus fractal dimension D in Eq. (5) is related to the standard deviation ofelevation σ in 3 x 3 moving window pixels as per Zhang et al. [10] as shown by Eq. (6).

D = 1 . 13589+ 0 . 08452 ln σ(6)The parameter α is a coefficient in the fractal method for slope proposed by Zhang et al.
[10] and its values is found to fluctuate very high from one local place to another incomparison to D value. Unlike the method by Zhang et al. [10] we developed a new
method in which α values are derived directly from the steepest slope of the available
coarse resolution DEM, αsteepest in Eq. (6), keeping the fact that steepest slope itself
represents the extreme fluctuation, for details refer Pradhan et al. [6].
SCALE INVARIANT TOPMODEL
By combining Eq. (3) and Eq. (5), the method to downscale topographic index which
includes resolution factor to account for the effect of scale in up slope contributing area
per unit contour length and a fractal method for scaled steepest slope as an approach toaccount for the effect of scale on slope is given by Eq. (7).

Ci {W i R f (tan β i )F }

(TI )scaled = ln (7)where, (TI)scaled is the scaled topographic index and (tanβi)F is Sscaled of Eq. (5) which isthe scaled steepest slope by fractal method. The method to downscale the topographic
index is combined with TOPMODEL to develop the Scale Invariant TOPMODEL.
RESULTS AND DISCUSSION
The method to downscale the topographic index and the Scale Invariant TOPMODEL isapplied to Kamishiiba catchment (210 km2), Japan.

5Table 2 Topographic constant λ value for scaled DEM from 1000 m grid resolution tofiner grid resolutions in Kamishiiba catchment.
Topographic constant, λ [ln(m2)], value for scaled DEM from 1000 m grid resolution to50 m target grid
150 m target grid
450 m target grid
600 m target grid
resolution
resolution
resolution
resolution
6.474
7.573
9.11
9.604

Application of the method to downscale the topographic index distribution
The down Scaled values of λ in Eq. (2) from 1000m grid resolution DEM to finer DEM
resolutions in Table 2 are almost equal to the λ values in Table 1 from the fine grid
resolution DEMs.
Figure 2 shows the perfect fit of density function of scaled topographic index
distribution from 1000m grid resolution DEM to various grid resolution DEMs by using
scale invariant model. It is found that between 50m-grid resolution DEM and 150m-grid
resolution DEM where the slope obtained is more precise and does not vary significantly,
resolution factor Rf alone played the dominant role in the scale invariant model.
Figure 3a is the topographic index distribution using 1000m DEM. Figures 3 b, c, dand e are the scaled topographic index distribution obtained by using the scale invariantmodel with the same 1000m grid resolution DEM to 600m, 450m, 150m and 50m grid
resolution DEM respectively. Figure 3f is the topographic index distribution using 50m
DEM. Distinct difference can be seen between spatial distribution of the topographic
index in Figure 3a and Figure 3f that are from 1000m-grid resolution DEM and 50m-grid
0.3

0.2
0.15
0.1

35

32.8

30.6

28.4

24

26.2

21.8

19.6

17.4

1315.28.6

10.8

Topographic index from 450m grid resolution
DEM235

32.8

30.6

28.4

24

26.2

21.8

19.6

17.4

1315.28.6

010.806.40.05
20.05

6.4

Topographic index from 50m grid resolution DEM

0.1

Scaled topographic index from 1000m gridresolution DEM to 450m grid resolution DEM

0.25

4.20.15

4.2

Density

Scaled topographic index from 1000m gridresolution DEM to 50m grid resolution DEM

0.2

Density

0.3

0.25

2

Topographic index value [ln(m )]2Topographic index value [ln(m )]

0.35

0.3
0.2
0.15

Topographic index from 150m grid resolution
DEM

0.1

Scaled topographic index from 1000m gridresolution DEM to 600m grid resolution DEM

0.25
Density

0.2
Topographic index from 600m grid resolution
DEM

0.15
0.1

0.05

Topographic index value [ln(m )]

35

32.8

30.6

28.4

26.2

24

21.8

19.6

17.4

1315.2

8.6

6.4

235

32.3

010.8

2

29.5

26.8

24

21.3

18.5

13

15.8

10.3

7.5

2

4.75

0.05

0

4.2

Density

0.3

Scaled topographic index from 1000m gridresolution DEM to 150m grid resolution DEM

0.25

2

Topographic index value [ln(m )]Figure 2. Comparison of scaled topographic index distribution from 1000m DEM
resolution to finer grid resolution DEM and at the fine scale in Kamishiiba catchment.

6Figure 3. Spatial distribution of scaled topographic index applied to Kamishiiba
catchment (210 km2). (a) topographic index distribution using 1000m DEM resolution,(b) scaled topographic index distribution obtained from 1000m DEM resolution to 600m
DEM resolution, (c) scaled topographic index distribution from 1000m DEM resolution
to 450m DEM resolution, (d) scaled topographic index distribution from 1000m DEM
resolution to 150m DEM resolution, (e) scaled topographic index distribution from1000m DEM resolution to 50m DEM resolution, (f) topographic index distribution using50m DEM resolution.
resolution DEM respectively. The spatial distribution of topographic index displayed byFigure 3e has matched the existing reality displayed by Figure 3f.
Application of Scale Invariant TOPMODEL
Figure 4a and 4b shows simulation results by TOPMODEL (with and without coupling
the downscaling method of topographic index) in Kamishiiba catchment (210 km2) for
the same rainfall event. For all the simulation results indicated in Figure 4, the usedeffective parameter of the TOPMODEL are identified by 50m DEM resolution
TOPMODEL (see Table 3) which gave the Nash efficiency of 94%. When applying thesame parameter values to 1000m DEM resolution TOPMODEL, Nash efficiency
tremendously dropped down to negative value, -45% (see Figure 4a). Figure 4a showscompletely different and erroneous performance of 1000m DEM resolution TOPMODEL

Precipitation(m/hour)

Precipitation(m/hour)

118

109

100

91

82

73

64

55

46

37

28

19000.005
0.01
0.015
0.02
0.025
0.03

0.005
0.01
0.015
0.02
0.025
0.03
0.035

0.035
1400

Simulated discharge of50m DEM resolution
TOPMODEL

600

Observed discharge

3400600
3800

Simulation discharge of
1000m DEM resolutionTOPMODEL

1000

Simulated discharge of
Scale Invariant
TOPMODEL applied at
1000m DEM resolution
Simulated discharge of50m DEM resolution
TOPMODEL

800

Discharge (m /s)

1200

Discharge (m /s)

101118

109

100

91

82

736455463728

19101

7400

Observed discharge
2002000(a)

118

10991

82

100Time (hour)

73645546372819110118

10991

100Time (hour)

82736455463728191100(b)Figure 4. Simulation results of TOPMODEL (with and without coupling the downscaling
method of topographic index) applied to Kamishiiba catchment (210 km2). (a)Completely different and erroneous performance of the 1000m DEM resolution
TOPMODEL from that of the 50m DEM resolution TOPMODEL when applying thesame set of effective parameter values calibrated at the 50m DEM resolution. (b) Perfectmatch of the simulated discharge of the Scale Invariant TOPMODEL with that of thesimulated discharge of 50m DEM resolution TOPMODEL and the observed discharge
under the same effective parameter values calibrated at the 50m DEM resolution.
from that of the 50m DEM resolution TOPMODEL when applying the same set ofeffective parameter values calibrated at the 50m DEM resolution. This blunder is theconsequence of the dependence of the topographic index distribution on a DEM
resolution, which is shown earlier in Figure 1 and Table 1. Figure 4b shows perfect matchof the simulated discharge of the Scale Invariant TOPMODEL with that of the simulated
discharge of 50m DEM resolution TOPMODEL and the observed discharge under thesame effective parameter values calibrated at the 50m DEM resolution, Nash Efficiency
obtained is 90%.
Table 3 Effective parameter values identified by 50m DEM resolution TOPMODEL.
Lateral transmissivity of soil atsaturation condition To (m2/hr)
9.8

Decay factor of lateral transmissivity
with respect to saturation deficit m (m)0.07

Maximum root zone
storage Rzmax (m)0.001

8CONCLUSION
This research has developed a Scale Invariant TOPMODEL, which is independent ofDEM resolution effects and consistent in parameter values with observations althoughscale of observable state variables and scale of application are mismatched. Analyzing
the scale laws this research has developed concept of resolution factor to account for theeffect of scale in up slope contributing area per unit contour length in topographic index
and a fractal method for scaled steepest slope as an approach to account for the effect ofscale on slopes, which are combined to develop the method to downscale topographic
index distribution. The method to downscale the topographic index is then coupled with
TOPMODEL to develop the Scale Invariant TOPMODEL. It is hoped that the findings ofthis research seeks its applicability as a tool to a wider range of boundary as per the scale
problems in hydrological processes and solution approach is concerned.
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[3] Zhang, W. and Montgomery, D. R., Digital elevation model grid size, landscape
representation, and hydrologic simulations, Wat. Resour. Res. 30(4), 1994, 10191028.
[4] Band, L. E. and Moore, I. D., Scale: Landscape attributes and geographicalinformation systems, Chapter 9 in Scale Issues in Hydrological Modelling, edited byJ. D. Kalma and M. Sivapalan, Wiley & Sons Ltd, Chichester, 1995, 159-179.
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Nepalese drainage basin, Weather radar information and distributed hydrologicalmodeling , IAHS publication, no. 282, 2003, 288-293.
[6] Pradhan N. R., Tachikawa Y., and Takara K., A scale invariance model for spatial
downscaling of topographic index in TOPMODEL, Annual Journal of HydraulicEngineering, JSCE, vol. 48, 2004, 109-114.
[7] Beven, K. J. and Kirkby, M. J., A physically based, variable contributing area modelof basin hydrology, Hydrol. Sci Bull. 24, 1979, 43-69.
[8] Beven. K.J., Rainfall-Runoff Modelling, John Wiley & Sons Ltd., New York, 2000.
[9] Klinkenberg, B. and Goodchild, M. F., The fractal properties of topography: acomparison of methods, Earth Surface processes and Landforms, 17(3), 1992, 217234.
[10] Zhang, X., Drake, N. A., Wainwright, J. and Mulligan, M., Comparison of slope
estimates from low resolution DEMs: Scaling issues and a fractal method for theirsolution, Earth Surface Processes and Landforms, 24, 1999, 763-779.