C S I R O L A N D a nd WAT E R

Topographic Scaling for the NLWRA

Sediment Project

NLWRA Sediment Project (CLW 12)

John Gallant

CSIRO Land and Water, Canberra

Technical Report 27/01, September 2001

Topographic Scaling for the NLWRA Sediment

Project

NLWRA Sediment Project (CLW 12)

John Gallant

Copyright

© 2001 CSIRO Land and Water.

To the extent permitted by law, all rights are reserved and no part of this publication covered by copyright may be reproduced or

copied in any form or by any means except with the written permission of CSIRO Land and Water.

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CSIRO Land and Water, Canberra

Technical Report Number: 27/01

2

Contents

ABSTRACT ........................................................................................................................3

1. INTRODUCTION ........................................................................................................4

2. MEASUREMENTS OF SLOPE AND HILLSLOPE LENGTH..................................4

2.1. Acquisition of high resolution DEMs ....................................................................4

2.2. Calculation of hillslope length ...............................................................................5

2.3. Calculation of slope................................................................................................7

3. SAMPLING AND MODELLING METHODS ...........................................................8

3.1. Model of hillslope length .......................................................................................9

3.2. Model of slope......................................................................................................11

3.3. Calculation of L and S factors..............................................................................12

4. APPLICATION OF MODELS...................................................................................13

5. ASSESSMENT OF RESULTS...................................................................................13

6. ACKNOWLEDGEMENTS........................................................................................15

7. REFERENCES ...........................................................................................................15

FIGURES 1 - 5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 - 21

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ABSTRACT

Prediction of hillslope soil erosion at the continental scale is complicated by the scale

mismatch between erosion processes and the scale of available data. The best

topographic data available across the Australian continent is the AUSLIG 9 second

DEM with a resolution of 250 m, whereas topographic control of soil erosion is at a

considerably finer scale. This report describes the methods used to derive the slope

length (L) and slope steepness (S) factors in the revised universal soil loss equation

(RUSLE) using statistical models based on measurements from high resolution

DEMs. Predictive variables in the statistical models included measures of relief and

topographic position from the 9 second DEM, several indicators of geology and soil

type, and various measures of climatic conditions. Hillslope length varies by an order

of magnitude, with a mean value of about 200 m, smaller than the resolution of the 9

second DEM. Slope varies over nearly two orders of magnitude, with a mean value of

about 13%, and for slopes less than 10% is approximately twice as large as slope

measured directly from the 9 second DEM.

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1. INTRODUCTION

The Water-Borne Erosion and Sediment Transport project (sub-project 4A in theme 5

of the National Land and Water Resources Audit) provides predictions of hillslope

erosion as one of the sources of sediment in river systems. The processes being

modelled – routing of overland flow with detachment and deposition of sediment –

are known to operate at a fine spatial scale, considerably finer than the resolution of

the 9 second DEM (Hutchinson et al., 2001) which is currently the best available

Australia-wide digital elevation data set. This scale mismatch is a well-known

problem and frequently occurs in hydrological modelling. While there is an extensive

literature describing the problem and suggesting possible approaches (Blöschl and

Sivapalan, 1995), there is as yet no accepted operational procedure for dealing with it.

In this project, erosion was modelled using the revised universal soil loss equation

(RUSLE) (Renard et al., 1997) which includes topographic effects via a length (L)

and slope (S) factor. The spatial variation in these two factors combined strongly

influences erosion intensity; the range of variation in the LS product is comparable to

the range of rainfall and soil erodibility (Rustomji and Prosser, 2001). Deriving these

factors directly from the 9 second DEM would be indefensible, as slopes would be

underestimated and hillslope length overestimated in most areas.

The approach used for this project was to calculate slope and hillslope lengths from

selected high resolution DEMs and then build statistical models using predictive

variables that are known everywhere.

2. MEASUREMENTS OF SLOPE AND HILLSLOPE LENGTH

2.1. Acquisition of high resolution DEMs

High resolution DEMs were obtained to represent most of the combinations of

landform, climate and geology in Australia. An initial set of readily available DEMs

was acquired, then gaps in the coverage were identified and additional DEMs

obtained to fill those gaps. The final set of DEMs covered areas from all states, and

included both coastal and inland areas. Resolutions were mostly 20 to 50 m and were

derived either from 1:25 000 scale source data (1:50 000 in some cases) or from radar

altimetry collected during airborne geophysical surveys. One DEM covering the wet

5

tropics of northern Queensland was at 80 m resolution and was based on 1:100 000

scale source data.

Measurements of terrain properties such as slope and hillslope length are, to a degree,

sensitive to DEM resolution (Wilson et al., 2000). The minimum hillslope length

obtainable from a DEM is somewhat larger than DEM resolution, so 50 m resolution

can produce larger minimum hillslope lengths than 20 m resolution. However, there

were very few locations where measured hillslope length was shorter than 80 m. The

80 m resolution of the wet tropics DEM is considerably coarser than the other DEMs

and it is likely that this has resulted in reduced slopes and a lack of short hillslope

lengths. However this region was considered to be sufficiently different from other

areas that omission of this data set would significantly degrade the predictive

accuracy of the model. In the absence of any information on which to base a

correction, the results in this area were used without modification. Figure 1 shows the

location and extent of the high resolution DEMs used for model building.

2.2. Calculation of hillslope length

The algorithm used to calculate hillslope length is based on the classification of DEM

cells into one of four classes: top, bottom, hillslope and indeterminate. The class is

assigned by examining the set of cells within a circular context of a given size and

marking the cells that are the maximum and minimum value in the circle. The circular

context is constructed at each grid cell, so each cell is included within the context a

number of times. After all positions have been analysed, cells may have been a

minimum and/or a maximum one or more times. Any cell that is never a maximum is

classed as a bottom (valley) cell for that circle size; any cell that is never a minimum

is classed as a top (ridge) cell; and cells that have been both maximum and minimum

are classed as hillslopes. Those cells that were neither maximum nor minimum do not

have a clear interpretation, although they include saddle cells, and are termed

indeterminate cells. This analysis is performed using a range of different sized circular

contexts, from one pixel in radius (25 – 80 m) up to a user-defined maximum (1 km in

this application).

The property that makes this classification useful for determining hillslope length is

that most cells are classified as hillslope points at small radii but indeterminate at

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large radii. The radius at which the number of indeterminate points equals the number

of hillslope points is the measure of the hillslope length. This radius cannot be

determined for individual cells, since it is the frequency of different classes amongst a

group of cells that determines the hillslope length at a site. The method therefore

requires a user-defined length scale over which the frequency of cell classes is

determined (1 km in this case). Producing results at the DEM resolution is wasteful

because of the degree of smoothing implied by this length scale, so the resolution of

the output grid is also user-specifiable (250 m in this case) but is rounded to the

nearest multiple of the DEM resolution. Note that the classes are determined at the

DEM resolution even though the output is reported at coarser resolution.

This algorithm is implemented in the program HillLength2. For the sake of

efficiency, the program analyses all the radii at each cell before moving to the next

cell, progressively adding cells to the boundary of the circular context, locating the

minimum and maximum values and setting indicators for minimum or maximum in

the appropriate cells for each radius.

Following calculation of the hillslope length at 250 m intervals, a mean length over a

1.5 km radius is computed. This step was performed to smooth out fine-scale

variations in hillslope length that could not realistically be predicted from the coarser

scale predictive variables. The 1.5 km radius was selected based on the observation

that the terrain in 9 second DEM becomes reliable over distances of approximately

1 km.

A further processing step was required to exclude areas with low relief, as hillslope

length in very flat landscapes is unreliable both conceptually (what is a hillslope when

there are no hills?) and practically (noise or subtle variations in DEM produce small

hillslope length values). Low relief areas for this purpose were defined as areas

having a standard deviation of elevation over a 2 km radius that was less than 5 m.

The 2 km resolution was chosen to be consistent with the scale of analysis in other

parts of the project. The smoothed and masked result was then converted to

geographic projection at 9 second resolution.

Figure 2 shows measured hillslope length (after smoothing) for the Warragul

1:250 000 map sheet in south-eastern Victoria including Wilson’s Promontory

(145°30′ E to 147° E, 39°15′ S to 38° S, approximately 130 × 130 km). Hillslope

7

lengths in this landscape range from 60 to 400 m; in some other landscapes hillslope

lengths up to 1 km are obtained

2.3. Calculation of slope

Slope was calculated from the high resolution DEMs using conventional methods

(the SLOPE function in Arc/Info Grid, with the percent option). For DEMs in

geographic coordinates, corrections were applied to account for the different units of

measure for horizontal distance (degrees) and elevation (metres) and for the difference

in spacing in the x and y directions due to the spherical coordinate system. The

direction of slope was first determined using the ASPECT command, and the slope

was calculated using:

( )( )( )[ ]2)2cos(1)cos(11111190%

aspectlat

slopeslopegeo −−−

=

where slopegeo is the corrected slope, slope% is the percent slope computed by the

conventional finite difference method, lat is latitude and aspect is the aspect with

north being 0. The 111190 factor is the number of metres in one degree at the mean

radius of the earth. The latitude term accounts for the reducing spacing in the x

direction as latitude increases away from the equator, while the aspect term modifies

the latitude correction according to the direction of the slope. When the slope is facing

east or west the full correction applies, while for slopes facing north or south no

correction is used because the y spacing remains constant. This correction for aspect is

only approximate; more accurate slope values could be calculated using finite

differences accounting for the x and y spacing explicitly. However this is much more

difficult within the GIS and the error introduced by this approximation is not expected

to be a large factor in the total error budget, including the effects of sampling and

interpolation.

From the raw slope, a mean slope over a circle of 250 m radius was calculated and the

results converted to geographic projection at 9 second resolution.

Figure 3 shows measured mean slope for the Warragul map sheet. Mean slope ranges

from 0 to 50 % in this area which is the typical range for the steepest areas of

Australia.

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3. SAMPLING AND MODELLING METHODS

The statistical models were constructed using the Cubist data mining tool (version

1.08; Rulequest Research, 2001). This software takes a set of samples, each with a

target value and a collection of potential predictive variables, and constructs a list of

rules each comprising a set of conditions and a linear model that provides the

predicted value when the conditions are met. Cubist can also use an independent set of

samples to test the model, and reports both summary statistics and the predictions for

each validation datum. For this application 30% of the points in the 9 second

resolution data were used for model building and a further 10% of points for model

testing. The sample locations were generated using a locally written program called

sampimg, which allows the distribution of samples to be constrained by a class map,

resulting in approximately equal numbers of samples in each class. Relief was divided

into 3 classes and combined with hillslope length in 8 classes (slope in 7 classes) to

produce 24 (21) classes. This ensured relatively even sampling of the full range of

hillslope lengths (slopes) in each of the low, moderate and high relief parts of the

landscape.

The resulting sample set, combined from all the high resolution results, contained

approximately 200 000 sample points, each with a calculated value and the values of

all predictive variables at that location.

The predictive variables were selected to represent the major factors presumed to

control landscape form:

• Material (geology and soil)

• Climate (temperature, rainfall and seasonality)

• Geomorphology (relief, slope, slope position)

Sixteen variables were used for prediction:

• Two aggregated geology classifications derived from the 1:2.5M scale

geology map of Australia (geol_agec and geol_lith)

• A more detailed lithology surface provided by BRS (lithology)

• The Australian Soil Classification derived from the Atlas of Australian Soils

(soil_asc)

9

• Mean annual rainfall, rainfall seasonality index and annual moisture index

(clim12, clim15, clim28)

• Mean annual temperature, temperature seasonality and diurnal temperature

range (clim1, clim4, clim7)

• Relief, relative elevation and slope position within land units defined by ridge

and stream networks from the 9 second DEM (relief, relev, slopos)

• Standard deviation of elevation and elevation percentile (an indicator of slope

position) within 2 km radius circular regions from the 9 second DEM (sd2km,

pctl2km)

• Slope from the 9 second DEM (slopepercent)

One question that arises in statistical modelling of this type is which variables are

actually important, or alternatively how many variables can be discarded without

reducing the power of the model. Examining all combinations of variables is

prohibitively time consuming, so a stepwise approach was adopted. For the first step,

each variable was used independently and the best variable identified using statistical

diagnostics from the modelling package (correlation and relative error). This one

variable was then used with each other variable, and the best second variable

identified. This procedure was repeated until all variables were included. The program

combinations and shell script stepbystep implement this procedure.

As more variables are introduced into the model and the rule conditions derived by

Cubist become more selective, the number of points that do not match any condition

increases, leaving undefined cells in the predicted result. This is an undesirable effect

that was minimised as part of the selection of model complexity.

Final selection of the model was based on the statistical diagnostics, visual

comparisons of predicted and measured maps and the relative rates of unpredicted

points.

3.1. Model of hillslope length

Models for hillslope length with a small number of variables performed poorly, but

the performance improved as more variables were included. There were few problems

with unpredicted values, so the selected model included all 16 variables and contained

56 rules.

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Lithology was the most predictive single variable (correlation = 0.35), with three

climate variables (mean annual temperature, diurnal temperature range and mean

annual rainfall) producing slightly lower correlations. The order in which variables

were selected was:

• lithology

• mean annual temperature

• temperature seasonality

• standard deviation of elevation

• mean annual rainfall

• diurnal temperature range

• slope position

• rainfall seasonality

• elevation percentile

• annual moisture index

• Australian soil classification

• slope from 9” DEM

• relief

• relative elevation

• geology (geollith)

• geology (geol_agec)

The last two variables actually decreased the model performance slightly so could

have been omitted, but this effect was not noticed during the model selection and they

were included in the model. The final model comprised 55 rules and has a correlation

coefficient of 0.71 on the validation data.

The importance of lithology in this model is not surprising, but the degree to which

climatic variables explain hillslope length is surprising. Climatic variables make up 5

of the 8 most predictive variables. Standard deviation of elevation, essentially a

measure of relief, is the only geomorphic variable appearing high on the list. Mean

annual temperature, the second variable included, could be acting as a surrogate for

elevation above a base level since temperature decreases systematically with

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elevation. Temperature seasonality, the third variable included, may be a direct

measure of climatic regime distinguishing alpine, arid and coastal environments.

The lithology predictive variable was not available for some areas in Western

Australia and the Northern Territory. For these areas, a separate model without

lithology was developed and used for prediction. This model contains 54 rules and has

a correlation coefficient of 0.69.

Figure 4 shows the predicted hillslope length for the Warragul map sheet. Compared

with the measured values of Figure 2, the predicted map shows similar overall

patterns with differences in detail, and a reduced range: both short and long hillslopes

are under-represented in the predictions. The spatial variation of hillslope length at the

regional scale is the desired result, and this prediction is considered to be a

satisfactory result for the purposes of this project.

3.2. Model of slope

While slope from the 9 second DEM was considered unreliable compared to slope

from the higher resolution DEMs, it was expected to be well correlated with mean

slope. This was indeed the case, and slope was by far the best single predictor, but

additional variables improved the result both in terms of statistics and spatial patterns

(as assessed by the visual comparison of the predictions and measurements). The

results ceased to improve after 10 variables, and the frequency of unpredicted values

increased with increasing numbers of variables, so the final model used only the first

10 variables:

• slope from 9” DEM

• standard deviation of elevation

• lithology

• rainfall seasonality

• annual moisture index

• annual mean temperature

• elevation percentile

• relative elevation

• diurnal temperature range

• slope position

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This model comprised 53 rules and had a correlation coefficient of 0.87.

The list of included variables contains a mixture of geomorphic, geologic and climatic

variables. As expected, slope was the single variable with most predictive power. The

early inclusion of standard deviation of elevation and lithology are unsurprising. What

is surprising is the lack of predictive power from annual mean rainfall; instead rainfall

seasonality and moisture index appear as relatively important predictors. It may be

that the effect of mean annual rainfall is already captured by the 9” slope, with the

seasonality and moisture index accounting for subtler effects of rainfall-related

climate.

Two additional models were derived for special cases. A second model with 58 rules

was developed and used for the areas where lithology was unavailable (parts of WA

and NT). An additional model using only 9” slope as the predictive variable was

developed for use in the interior of the continent where many of the predictive

variables were not available and the extrapolation of the model is not appropriate.

This model contained 17 rules (correlation = 0.80), and the composite effect of those

rules was manually modelled with a single non-linear model:

9

99

001.01

014.1

1.21

s

sss pred

++

+=

where spred is predicted slope (%) and s9 is slope (%) from the 9” DEM. This model

predicts a minimum slope of 1%, increasing with 9” slope at a rate that decreases as

9” slope increases.

Figure 5 shows predicted slope for the Warragul map sheet. As for hillslope length,

the predictions are similar in overall pattern with considerable differences in detail.

The predicted slope has fewer areas of high slope and less detail, but for the purposes

of this project the prediction is once again considered to be satisfactory.

3.3. Calculation of L and S factors

The RUSLE L and S factors are calculated from mean hillslope length and mean slope

using the standard RUSLE equations (Renard, et al. 1997). In some land use classes

where surface runoff does not appear to increase with flow path length, the L factor is

set to 1.

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Some applications of hillslope length for erosion prediction require an L factor for the

interior of the continent. In this non-agricultural area no high resolution DEMs were

obtained to provide samples for rule building. Extrapolation of the rules into the

different geomorphic and climatic conditions of this area is inappropriate, so a unit L

factor value was used. For low relief areas this is reasonable as the L factor is close to

1 when slope is low. High relief areas would in reality have a higher L factor.

Further details of these calculations are presented in Lu et al (2001).

4. APPLICATION OF MODELS

In order to apply the rules over the entire agricultural region of Australia, the rules

need to be converted to a form that can be applied efficiently over large areas. The

program CubistToC++ was written to accomplish this; it reads the model output

file produced by Cubist and converts this to C++ source code and compiles it; a

complementary Arc/Info script (AML) file is also generated that extracts the required

data as raw binary files, runs the C++ program and imports the prediction back into

Arc/Info.

Small areas (one or two cells) that have no prediction because they do not meet any of

the conditions in the model are filled using averages of surrounding cells.

For hillslope length, which has large areas unpredicted due to low relief, additional

processing was required to eliminate small holes and small areas of predicted values

within larger unpredicted areas. The guiding principle here was that single 9 second

pixels are irrelevant at the continental scale so there should be a smallest region size

for both predicted and unpredicted regions. This size was set to 200 cells or 12.5 km2

based on visual assessment of the resulting maps.

5. ASSESSMENT OF RESULTS

The hillslope length algorithm has not been validated, but manual measurements from

contours in a small number of locations agreed with the results of HillLength2.

The error from calculating hillslope length is deemed to be small. The error in

calculating mean slope is also expected to be small.

The accuracy of the predictions is assessed both by the statistical diagnostics of the

Cubist program and visual comparisons of patterns. The correlations reported by

14

Cubist are 0.71 for hillslope length, which indicates a reasonably good prediction but

with substantial variation unaccounted for, and 0.87 for slope indicating quite a good

model accounting for most of the variation.

Root-mean-square (RMS) error for each prediction was analysed using the

approximately 67000 pairs of measured and predicted values. For hillslope length the

RMS error was 85 m, compared to a mean hillslope length of about 200 m. Error

generally increases with the magnitude of the predicted hillslope length, and the mean

relative error is about 30% for predicted lengths between 100 and 500 m. The model

performs poorly for low hillslope lengths; for predicted lengths less than 70 m, the

average error is over 100 m. It also under-predicts large hillslope lengths: there are

few predictions greater than 400 m in spite of the measured values reaching 1000 m.

For slope the RMS error was 7%, compared to a mean slope of 13%. Error generally

increases with slope value, with under-prediction increasingly prevalent at higher

slopes. The mean relative error is about 0.6 for slopes less than 5% decreasing to 0.1

for slopes above 60%. The error is approximately 0.4 times the slope value up to 20%

slope, above which the error is 8% slope. The minimum predicted slope is about

0.5%, indicating that very low slopes are not correctly represented.

The rules produced by Cubist have not yet been analysed to ascertain their

acceptability from a geomorphic perspective. It is possible that some of the variables

are acting as surrogates for other effects, in which case more physically realistic

models could be constructed by using a predictive variable more closely related to the

effect. One example of this could be temperature, which is a strong predictor of

hillslope length but may be reflecting some effect of elevation on hillslope length,

rather than temperature per se. A close analysis of the rules may suggest areas for

research into the physical processes underlying the development of slope forms, but

this has not been attempted at this stage.

Overall, the results are considered adequate for the purpose of this project. The errors

form a significant but not overwhelming contribution to the error budget for hillslope

soil erosion.

The value in this modelling approach based on high resolution data is demonstrated

by two observations. Firstly, the average hillslope length (200 m) is less than the

resolution of the 9 second DEM. Secondly, the model of slope based solely on the 9

15

second DEM slope demonstrates that the mean slope is more than twice the value

measured directly from the 9 second DEM, although this factor reduces as slope

increases above 10%. Determining hillslope lengths and slopes directly from the 9

second DEM would have caused gross errors in erosion estimates.

6. ACKNOWLEDGEMENTS

This work was supported by the National Land and Water Resources Audit and the

Cooperative Research Centre for Catchment Hydrology.

Much of the high resolution DEM data was provided free of charge or at reduced cost

for the purpose of this project. The Queensland Department of Natural Resources

provided the Injune DEM. The Western Australian Department of Land

Administration and CSIRO Mathematics and Information Sciences provided the upper

Blackwood River DEM. The South Australian Department of Environment and

Heritage provided the data for the Fleurieu Peninsula area. Tasmania’s Department of

Primary Industry, Water and Environment provided the data for the Tasmanian DEMs

at a reduced cost. The Victorian Department of Natural Resources and Environment

provided the Gippsland DEM. The Australian Geological Survey Organisation

provided the Clarence DEM. Sydney and Murrumbidgee data was supplied by Land

and Property Information NSW. The Northern Territory Geological Survey provided

the Rum Jungle and Pigeon DEMs. The South Australian Office of Minerals and

Energy provided the Chowilla DEM. The Geological Survey of Victoria provided the

Colac and Mildura DEMs. The northern Queensland DEM was provided by the Wet

Tropics Management Authority. Data for the Dotswood DEM was supplied by (to be

completed). The south-east Queensland DEM was supplied by (to be completed).

Thanks to Graeme Priestley and Alan Marks who organised the acquisition and

processing of the high resolution topographic data.

7. REFERENCES

Blöschl, G. and Sivapalan, M. (1995) Scale issues in hydrological modelling: A

review. Hydrological Processes 9(3/4): 251-290.

Hutchinson, M.F., Stein, J.A. and Stein, J.L. (2001) Upgrade of the 9 second

Australian digital elevation model. World Wide Web http://cres.anu.edu.au/dem.

16

Lu, H., Gallant, J.C., Prosser, I.P., Moran, C., and Priestley, G. (2001) Prediction of

sheet and rill erosion over the Australian continent, incorporating monthly soil loss

distribution. Technical Report 13/01. CSIRO Land and Water Canberra, Australia.

Renard, K.G., Foster, G.A., Weesies, D.K., McCool, D.K., and Yoder, D.C. (1997)

Predicting soil erosion by water: A guide to conservation planning with the revised

universal soil loss equation. Agriculture Handbook 703. United States Department of

Agriculture Washington DC.

Rulequest Research (2001) Rulequest Research Data Mining Tools. World Wide Web

http://rulequest.com.

Rustomji, P.J. and Prosser, I.P. (2001) Spatial patterns of sediment delivery to valley

networks: Sensitivity to sediment transport capacity and hillslope hydrology relations.

(In preparation).

Wilson, J.P., Repetto, P.L., Snyder, R.D. (2000) Effect of data source, grid resolution,

and flow-routing method on computed topographic attributes. In Wilson, J.P. and

Gallant, J.C. (eds) Terrain Analysis: Principles and Applications. John Wiley and

Sons, New York.

17

Figure 1. Location of high resolution DEMs used in the project.

Colours indicate the DEM resolutions.

18

Figure 2. Measured hillslope length for the Warragul map sheet.

19

Figure 3. Measured mean slope for the Warragul map sheet.

20

Figure 4. Predicted hillslope length for the Warragul sheet.

21

Figure 5. Predicted slope for the Warragul map sheet.