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HYDROLOGICAL PROCESSES  Hydrol . Process. 23, 301 7– 302 9 (200 9) Published online 26 August 2009 in Wiley InterScience (www.interscien ce.wiley.com) DOI: 10.1002/h yp.7413 Test of statistical means for the extrapolation of soil depth point information using overlays of spatial environmental data and bootstrapping techniques Helen E. Dahlke, 1 * Thorsten Behrens, 2 Jan Seibert 3,4 and Lotta Andersson 5 1  Biological and Environ mental Engineering, Cornell University, 165 Riley-Robb Hall, I thaca, New Y ork, 14853, USA 2 Physical Geography, Institute of Geography, University of Tuebingen, Ruemelinstrasse 19-23, 72070 T¨ ubingen, Germany 3  Department of Geography , University of Zurich, CH-8057 Zurich, Switzerland 4  Department of Physical Geography and Quaternary Geology, Stockholm University, SE-106 91, Stockholm, Sweden 5 Swedish Meteoro logical and Hydrological Institute, Department of Research and Development, SE-601 76 Norrk¨ oping, Sweden Abstract: Hydrological modelling depends highly on the accuracy and uncertainty of model input parameters such as soil properties. Since most of these data are eld surveyed, geostatistical techniques such as kriging, classication and regression trees or more sophisticated soil-landscape models need to be applied to interpolate point information to the area. Most of the existing inter polat ion technique s requir e a rando m or regul ar distributi on of points within the study area but are not adequ ate to satis fact orily interpol ate soil catena or trans ect data. The soil landsca pe model pres ente d in this study is predicting soil information from transect or catena point data using a statistical mean (arithmetic, geometric and harmonic mean) to calculate the soil information based on class means of merged spatial explanatory variables. A data set of 226 soil depth measurements cover ing a ran ge of 0– 6Ð5 m was used to test the model. The point data were sampled along four transects in the Stubbetorp catchment, SE-Sweden. We overlaid a geomorphology map (8 classes) with digital elevation model-derived topographic index maps (2–9 classes) to estimate the range of error the model produces with changing sample size and input maps. The accuracy of the soil depth predictions was estimated with the root mean square error (RMSE) based on a testing and training data set. RMSE ranged generally between 0Ð73 and 0Ð83 m š 0Ð013 m depending on the amount of classes the merged layers had, but were smallest for a map combination with a low number of classes predicted with the harmonic mean (RMSE  D 0Ð46 m). The results show that the prediction accuracy of this method depends on the number of point values in the sample, the value range of the measured attribute and the initial correlations between point values and explanatory variables, but suggests that the model appro ach is in gene ral scale invar iant. Copyrig ht  ©  2009 John Wiley & Sons, Ltd. KEY WORDS  soil-landscape mode lling; hydrological modelling; soil depth; bootstrapping; soil attributes; soil attribute prediction; statistical mean; root mean square error  Received 18 November 2008; Accepted 16 June 2009 INTRODUCTION Dig ita l hig h-r esolut ion soi l inf ormati on and new app- roache s to obtai n landsc ape hetero geneit ies face stil l a gro wing demand for imp rov emen ts of exi sting hyd ro- logical models and to capture the space–time variability of hydrological processes. Soil depth is seen as one of the essential input parameters for distributed hydrologi- cal and environmental modelling. Soil depth, or the depth fro m the groun d surface to the surface of the bedrock or an impermeab le layer, is see n as a maj or cont rol on soi l– wat er sto rage and availabil ity in many envi- ronments (Tromp-van Meerveld and McDonnell, 2006a). Soil depth signicantly affects spatial soil moisture pat- terns (Burt and Butche r, 1985; Freer et al., 2002; Tromp- van Meerveld and McDonnell, 2006b) as well as subsur- face and groundwater ow (Buttle and McDonald, 2002; Freer  et al., 2002 ; Stieg litz  et al., 2003). Soil depth or * Corr espo nden ce to: Hele n E. Dah lke, Biol ogic al and Envi ronmenta l Engineering, Cornell University, 165 Riley-Robb Hall, Ithaca, New York, 14853, USA. E-mail: [email protected] dept h to bed rock is thus a st andard vari able used in many hydrological models such as soil & water assess- ment tool (SWAT) (Arnold and Fohrer, 2005), distributed hydro logy soil vegeta tion model (DHSVM) (Wi gmost a et al., 1994), soil moisture distribution and routing model (SMDR) (Frank enber ger  et al., 199 9) or TOPMODEL (Beven  et al., 198 4). To face the growi ng demand for high-resolution spatial soil information, so-called quan- tit ati ve soi l-l andscape met hod s are applie d to ext end conventional soil survey point observations to the land- scape scale (Ryan  et al., 2000; McBratney  et al., 2003). Approaches applied to predict continuous soil attributes suc h as soil dep th compri se simple linear regres sio n, krigi ng and co-kr igin g (Odeh  et al., 1994, 199 5; Ryan et al., 2000), generalized linear models (McKenzie and Ryan, 1999), discriminant analysis (Sinowski and Auer- swald, 1999) and landform evolution models (Saco  et al., 2006). The development of these models has especially been faci lit ated by the achieved advance s in geogra phi cal inf ormati on sys tems (GIS), dig ita l ele vat ion mod els Copyright  © 2009 John Wiley & Sons, Ltd.

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HYDROLOGICAL PROCESSES Hydrol. Process. 23, 3017– 3029 (2009)Published online 26 August 2009 in Wiley InterScience(www.interscience.wiley.com) DOI: 10.1002/hyp.7413

Test of statistical means for the extrapolation of soil depthpoint information using overlays of spatial

environmental data and bootstrappingtechniques

Helen E. Dahlke,1* Thorsten Behrens,2 Jan Seibert3,4 and Lotta Andersson5

1  Biological and Environmental Engineering, Cornell University, 165 Riley-Robb Hall, I thaca, New York, 14853, USA2 Physical Geography, Institute of Geography, University of Tuebingen, Ruemelinstrasse 19-23, 72070 T¨ ubingen, Germany

3  Department of Geography, University of Zurich, CH-8057 Zurich, Switzerland 4  Department of Physical Geography and Quaternary Geology, Stockholm University, SE-106 91, Stockholm, Sweden

5 Swedish Meteorological and Hydrological Institute, Department of Research and Development, SE-601 76 Norrk¨ oping, Sweden

Abstract:

Hydrological modelling depends highly on the accuracy and uncertainty of model input parameters such as soil properties.Since most of these data are field surveyed, geostatistical techniques such as kriging, classification and regression trees ormore sophisticated soil-landscape models need to be applied to interpolate point information to the area. Most of the existinginterpolation techniques require a random or regular distribution of points within the study area but are not adequate tosatisfactorily interpolate soil catena or transect data. The soil landscape model presented in this study is predicting soilinformation from transect or catena point data using a statistical mean (arithmetic, geometric and harmonic mean) to calculatethe soil information based on class means of merged spatial explanatory variables. A data set of 226 soil depth measurementscovering a range of 0– 6Ð5 m was used to test the model. The point data were sampled along four transects in the Stubbetorpcatchment, SE-Sweden. We overlaid a geomorphology map (8 classes) with digital elevation model-derived topographic indexmaps (2–9 classes) to estimate the range of error the model produces with changing sample size and input maps. The accuracyof the soil depth predictions was estimated with the root mean square error (RMSE) based on a testing and training data set.RMSE ranged generally between 0Ð73 and 0Ð83 m š 0Ð013 m depending on the amount of classes the merged layers had, butwere smallest for a map combination with a low number of classes predicted with the harmonic mean (RMSE  D 0Ð46 m).The results show that the prediction accuracy of this method depends on the number of point values in the sample, the value

range of the measured attribute and the initial correlations between point values and explanatory variables, but suggests thatthe model approach is in general scale invariant. Copyright  ©  2009 John Wiley & Sons, Ltd.

KEY WORDS   soil-landscape modelling; hydrological modelling; soil depth; bootstrapping; soil attributes; soil attributeprediction; statistical mean; root mean square error

 Received 18 November 2008; Accepted 16 June 2009

INTRODUCTION

Digital high-resolution soil information and new app-

roaches to obtain landscape heterogeneities face still a

growing demand for improvements of existing hydro-

logical models and to capture the space–time variability

of hydrological processes. Soil depth is seen as one of the essential input parameters for distributed hydrologi-

cal and environmental modelling. Soil depth, or the depth

from the ground surface to the surface of the bedrock 

or an impermeable layer, is seen as a major control

on soil– water storage and availability in many envi-

ronments (Tromp-van Meerveld and McDonnell, 2006a).

Soil depth significantly affects spatial soil moisture pat-

terns (Burt and Butcher, 1985; Freer et al., 2002; Tromp-

van Meerveld and McDonnell, 2006b) as well as subsur-

face and groundwater flow (Buttle and McDonald, 2002;Freer   et al., 2002; Stieglitz   et al., 2003). Soil depth or

* Correspondence to: Helen E. Dahlke, Biological and EnvironmentalEngineering, Cornell University, 165 Riley-Robb Hall, Ithaca, New York,14853, USA. E-mail: [email protected]

depth to bedrock is thus a standard variable used in

many hydrological models such as soil & water assess-

ment tool (SWAT) (Arnold and Fohrer, 2005), distributed

hydrology soil vegetation model (DHSVM) (Wigmosta

et al., 1994), soil moisture distribution and routing model

(SMDR) (Frankenberger   et al., 1999) or TOPMODEL

(Beven   et al., 1984). To face the growing demand for

high-resolution spatial soil information, so-called quan-

titative soil-landscape methods are applied to extend

conventional soil survey point observations to the land-

scape scale (Ryan  et al., 2000; McBratney  et al., 2003).

Approaches applied to predict continuous soil attributes

such as soil depth comprise simple linear regression,

kriging and co-kriging (Odeh   et al., 1994, 1995; Ryan

et al., 2000), generalized linear models (McKenzie and

Ryan, 1999), discriminant analysis (Sinowski and Auer-

swald, 1999) and landform evolution models (Saco  et al.,

2006).

The development of these models has especially been

facilitated by the achieved advances in geographical

information systems (GIS), digital elevation models

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3018   H. E. DAHLKE  ET AL.

(DEM), terrain analysis, statistical analysis and the

increasing computing capacity during the last decade.

Based on differences in the quality and type of field

measurements of soil properties and the availability of 

additional spatial environmental explanatory variables,

the available methods can be categorized into continu-

ous and discrete approaches (Burrough, 1993). Commoncontinuous approaches analyze the spatial continuity of 

a specific soil variable based on the variance of their

distribution using geostatistical methods (e.g. kriging)

or they include known environmental information (e.g.

topographic, land use and substrate information) for the

spatial distribution of the soil variable based on a regres-

sion model (Mertens   et al., 2002). Discrete approaches

such as Bayesian expert systems model categorical (nom-

inal, ordinal or interval) soil attributes or soil classes

through the integration of soil and landscape information

into a semantic net and/or the definition of logical rules

(Skidmore  et al., 1996). Other methods that predict con-ventionally mapped soil-landscape units are fuzzy logic

approaches (Zhu, 2000) and neural networks (Lehmann

et al., 1999; Behrens  et al., 2005) that use learning algo-

rithms to train a network that predicts the desired output

units based on mapped soil units.

Despite the great variety and advances that have been

made in the development of continuous and discrete soil-

landscape models, the approaches have limitations in

their applicability to provide input parameters for dis-

tributed hydrological models. Discrete approaches pro-

vide soil information for spatial entities and provide

hence the data structure required in most of the distributed

or hydrological response units (HRU)-based hydrologicalmodels. HRUs describe areas of homogeneous hydro-

logical response based on similar topographical, pedo-

logical and geomorphological characteristics, which are

extracted from an overlay of topographic, soil and land

use data. The concept is based on the assumption that

hydrological processes within a delineated hydrological

response unit show a certain degree of homogeneity and

therefore less variability as compared with surrounding

area units. In comparison to raster-based hydrological

models, it aims to reduce parameterization complexity

and computing time, especially at regional and catchment

scale applications (Flugel, 1995; Leavesley and Stannard,1995). Following the HRU concept, discrete soil model

approaches effectively facilitate the reduction of the spa-

tial variability of hydrological processes in the landscape

and reduce the time and effort to collect necessary soil

attribute data in a study area (Park and van de Giesen,

2004). However, they bear the risk that the hydrologi-

cal model application is bound to the scale of the pre-

existing conventional soil surveys, which exist mostly in

the range of 1 : 50 000 to 1 : 1 000 000 (e.g. 1 : 1 000 000

in Sweden) and are rather inflexible to scaling of the soil

information (Olsson, 1999; Behrens and Scholten, 2007).

Moreover, the development of soil unit-based quantita-

tive soil models reached a degree of complexity in user

expertise and user knowledge, both on the soil survey

and on the model side that challenges their short-term

applicability as simple tools to generate soil input data

for hydrological models and modeller.

Continuous approaches have the advantage that they

are easy applicable, have little demands in computation

software (e.g. implemented in common GIS) and user

expertise. However, most of the geostatistical methods

require a large number of samples or frequent sam-pling for accurate predictions and bear the problem that

even with established model functions, the capabilities to

extrapolate the results outside the study area or catch-

ment remain limited (Kravchenko, 2003). Geostatistical

methods also assume a certain data structure such as a

regular grid or uniform distribution (Odeh  et al., 1994,

1995; Lane, 2002; Kravchenko, 2003; Lyon  et al., 2006).

Methods such as kriging and inverse distance weighting

(IDW) and regression trees require a regular or random

distribution of the point data that are scattered over the

observation area. However, transect or catena data are

usually not object of interpolation techniques, becausetheir spatial representation for a defined area of inter-

est is limited to the proximate surrounding of the catena

and the incremental distance of the points along the

catena. The application of common interpolation tech-

niques (e.g. kriging and IDW) to catena point data results

in a decrease of the predictive capacity the farther a

point/cell needs to be predicted from the field-measured

points. Typical artefacts such as stripes or facets are pro-

duced in the prediction maps showing the decreasing

ability of the interpolation algorithm to predict in areas,

which lack point observations.

The interpolation of soil information sampled with the

catena approach remains therefore a challenge for geosta-tistical methods and soil-landscape modelling techniques.

Most studies that use catena soil information are, thus,

limited to small-scale applications such as single hill-

slopes and avoid predictions of larger landscape areas.

Most of the interpolation of catena-sampled soil infor-

mation is facilitated through the integration of digital

terrain analysis into the interpolation process (Moore

et al., 1993; Sommer and Schlichting, 1997; Gessler

et al., 2000; Chamran   et al., 2002). Statistical correla-

tions among soil properties such as soil moisture, net

primary productivity, soil organic carbon, soil texture

classes and especially soil depths and terrain attributesgenerated from a DEM have been investigated since

the end-1970s and have greatly enhanced the quanti-

tative investigation of hydrological processes in soils

(Beven and Kirkby, 1979; O’Loughlin, 1986; Moore

et al., 1991). These studies contribute to the under-

standing of relations between topography, water move-

ment and ecosystem processes and support quantitative

and dynamic modelling of eco-hydrological processes

through the integration of GIS-based terrain analysis and

field observations (Chamran  et al., 2002).

This study presents a soil-modelling technique to

extrapolate soil-depth information from four transects

(soil depth as understood as depth to bedrock) to a

small catchment in Sweden based on different maps of 

explanatory variables. Three statistical means (arithmetic,

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THE EXTRAPOLATION OF TRANSECT SOIL DEPTH POINT INFORMATION   3019

geometric and harmonic) are tested to predict soil depths

based on class means derived from an overlay of the

point observations with each class of a geomorphology

and different terrain maps. Using bootstrapping, the

capability of the statistical means to predict soil depths

and the model uncertainty is estimated for different

spatial disaggregation.

SITE DESCRIPTION

The Stubbetorp catchment (58°440N, 16°210E) is located

about 120 km southwest of Stockholm in the eastern part

of central South Sweden (Figure 1). The hilly catchment

belongs to the upper part of the Kolmarden mountain

ridge, a region dominated by low-weathering gneissic

granites that bounds the northern shore of the deeply

incised bay Braviken of the Baltic Sea (Wikstrom,

1979). The main valley and the two side valleys of Stubbetorp catchment, which covers an area of 0Ð94 km2,

are northwest–southeast orientated following the major

fault line in this region. Altitude in the catchment ranges

from 80 m above sea level (asl) at the gauge to 130 m asl.

The Stubbetorp catchment was completely covered with

water after the last deglaciation period (Persson, 1982).

Both glacial ice movements and the action of ocean

waves, which left the top of the hills with little soil cover,

influenced the present geomorphology and topography.

In large parts of the catchment (46%), the bedrock is

covered with till on which usually rather conductive, very

stony and in fine materials depleted soils are developed.

The eroded gravel and fine sediments have accumulated

in depressions and in the main valley where ombrotrophic

peatlands and swamp forests (in total 10Ð5%) with a

maximum peat depth of 6Ð5 m occur. The catchment

is largely dominated by podzolic forest soils, whereas

lithosols with rocky outcrops are especially occurring in

the southeast part of the catchment. The mean slope of thecatchment is 5Ð9° with a maximum slope of 26° in the area

of the catchment outlet. Most of the catchment is forested

(83%) with  Pinus sylvestris   and  Picea abies   of different

age, deciduous tree species are less important and occur

only in the wetland areas. The climate in the catchment is

characterized by a mean annual precipitation of 666 mm

and an annual potential evaporation of 432 mm (period

1985–1994). Mean annual runoff measured for the same

time period was 230 mm (Pettersson, 1995).

MATERIALS

Soil depth measurements

Soil depth measurements (depth to bedrock) were

available for two longer transects (485 m length) crossing

the main valley in the upper part of the catchment

and in two shorter transects in the central (210 m

length) and lower part (120 m length) of the catchment

(Figure 1). These soil depth measurements were obtained

in 1994 using Georadar (Olofsson and Fleetwood, 1994).

The derived data set consists of 226 points with an

incremental distance of 5 m with soil depths varying

between zero and 6Ð5 m (Figure 2).

Figure 1. Study area: Stubbetorp catchment, central-southeast Sweden. Dots indicate locations of soil depth measurements used in this study. Greyareas indicate wetland areas, mapped in July 2005 in the catchment

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THE EXTRAPOLATION OF TRANSECT SOIL DEPTH POINT INFORMATION   3021

Table II. Terrain parameters calculated for Stubbetorp catchmentand Pearson product– moment correlation coefficients (r ) esti-

mated between soil depths and terrain attributes, respectively

Terrain Attributes   r 

Vertical distance to channel network (Olaya,

2004)

0Ð58

Elevation above channel (McGuire  et al., 2005)   0Ð54Relative profile curvature (Behrens, 2003)   0Ð52Relative hillslope position (Hatfield, 1999)   0Ð42Minimum curvature (Wood, 1996)   0Ð41Waxing/waning slopes (Huber, 1994)   0Ð36Longitudinal curvature (Wood, 1996)   0Ð35Mean curvature (Shary  et al., 2002)   0Ð33Mean curvature (Zevenbergen and Thorne, 1987)   0Ð33Mean curvature (Bolstad  et al., 1998)   0Ð33Mean curvature ‘high pass filter’ (Behrens,

2003)0Ð33

Mean curvature (Mc Nab, 1989)   0Ð32True surface distance from streams (Behrens,

2003)0Ð31

Relative aspect curvature (Lehmeier and Kothe,1992)

0Ð28

Profile curvature (Shary  et al., 2002)   0Ð27Minimum curvature (Shary  et al., 2002)   0Ð25Height above channel (Behrens, 2003)   0Ð23Maximum curvature (Shary  et al., 2002)   0Ð23Maximum curvature (Wood, 1996)   0Ð16Horizontal curvature (Shary  et al., 2002)   0Ð16Plan curvature (Zevenbergen and Thorne, 1987)   0Ð14Difference curvature (Shary  et al., 2002)   0Ð12Solar insolation (Shary  et al., 2002)   0Ð12Vertical excess curvature (Shary  et al., 2002)   0Ð12Plan curvature (Shary  et al., 2002)   0Ð09Surface volume above minimum elevation

(Nogami, 1995)0Ð06

Topographic roughness (Behrens, 2003)   0Ð06Surface area (Jenness, 2004)   0Ð03Unsphericity (Shary  et al., 2002)   0Ð02Ring-curvature (Shary  et al., 2002)   0Ð02Aspect (Moore  et al., 1993)   0Ð01Gaussian curvature (Shary  et al., 2002) 0Ð02Slope (Horn, 1981) 0Ð04Surface runoff velocity (Moore  et al., 1991) 0Ð04Gradient Factor (Shary  et al., 2002) 0Ð04Gradient Factor (Behrens, 2003) 0Ð04Total accumulation curvature (Shary et al., 2002) 0Ð04Horizontal excess curvature (Shary  et al., 2002) 0Ð07Cross-curvature (Wood, 1996) 0Ð09Rotor curvature (Shary  et al., 2002) 0Ð11Reflectance map (Florinsky, 1998) 0Ð12

Topographic index (Beven and Kirkby, 1979) 0Ð13Slope-length-factor (Moore  et al., 1991) 0Ð16Relative height curvature (Behrens, 2003) 0Ð17Cross-curvature (Moore  et al., 1991) 0Ð24Hemispherical dispersion (Hodgson and Gaile,

1999)0Ð26

Longitudinal curvature (Moore  et al., 1991) 0Ð27Steepest downslope (Tarboton, 1997) 0Ð28Profile curvature (Zevenbergen and Thorne,

1987)0Ð35

For the final selection of the terrain attributes as

input data sets for the soil model, both a clustering

of the four single terrain parameters in a number of 2

to 9 classes and parameter combinations of two, three

and all four terrain parameters were tested, resulting in

104 data sets. Parameter combinations were tested in

the sense to artificially generate terrain maps with a

varying number of classes whose spatial disaggregation

could explain best the spatial variability of the measured

soil depths. Since one of the aims of this study is to

test the model’s applicability to predict soil depth for

various spatially disaggregated input data sets, the lack of sufficient environmental data sets as input data in

the model was substituted by terrain maps of variable

number of classes generated through the combination

of different terrain parameters. To extract the terrain

parameters or parameter combination that showed the

highest class dissimilarity, the   F-value of a one-way

analysis of variance was calculated for each terrain data

set. The  F-value is a measure for how representative the

spatial variance of the fractioned terrain maps for the

distribution of soil depth in the catchment is and whether

the terrain map can be selected as input data set in the

conceptual soil model or not (Table III).

Soil model approach

The soil model approach is aimed to allow generating

spatial maps of soil characteristics (in this study: soil

depth) based on catena point information. The approach

is applicable to generate either user-defined discrete

landscape units like entities used in HRUs or semi-

continuous raster maps. The general approach is based

on class means resulting from an overlay of the soil-

depth measurements with each class of any nominal

data set (e.g. geomorphology and terrain layer). The

approach assumes that each environmental data set usedin the model represents actual differences in the soil

characteristic to be modelled in an area of interest. The

class means are calculated as arithmetic means over

all points located in spatial units with the same class-

id. Assuming that the catena of soil depths points is

crossing several spatial units in each spatial data layer,

the information of each class can be spread over the study

site, if overlaid with other spatial data sets and their

class means. Analogue to the regionalization concept

(Diekkrueger   et al., 1999), the overlay of two or more

spatial data sets results, thus, in the disaggregation of 

the study site into smaller discrete units whose ‘real’ soil

depth will be approached, the more data sets are used in

the model, the higher the explanatory variables correlate

with the measured soil attribute.

In this study, we tested three statistical means (arith-

metic, geometric and harmonic mean) to predict the soil

depth for Stubbetorp catchment from class means of the

generated terrain maps and the geomorphology map.

 Model fitting and validation

The set of 226 soil depth points was split into training

and testing data sets of pre-defined size to evaluate the

spatial soil depths predictions and the model error of the

different soil models. To estimate the model performance,

we applied a bootstrapping technique. Bootstrapping   is a

statistical method to estimate standard errors by sampling,

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3022   H. E. DAHLKE  ET AL.

Table III. Variability of  F-values as measure of class dissimilarity in mean soil depth tested for all possible terrain parameters andparameter combinations of the cluster analysis

Terrain Parameter Combinations Number of Classes

2 3 4 5 6 7 8 9

1 vd 11Ð27 9Ð86 8Ð18 8Ð10 6Ð84 8Ð11 7Ð30 7Ð51rhp 37Ð28 27Ð54 17Ð14 17Ð66 13Ð10 16Ð46 10Ð72 9Ð61eac   46Ð85   24Ð13 19Ð33 17Ð07 17Ð41 16Ð86 19Ð29 17Ð12rpc 38Ð81 55Ð78 24Ð28 20Ð36 23Ð75 18Ð90 15Ð68 13Ð37

2 eac/rhp 46Ð85 24Ð13 21Ð94 17Ð07 21Ð66 25Ð45 27Ð62 14Ð87eac/rpc 40Ð10 52Ð91   30Ð56 28Ð46   25Ð82 25Ð45   27Ð62 25Ð64

eac/vd 46Ð18 25Ð74 21Ð76 16Ð85 17Ð82 15Ð89 15Ð33 13Ð59rhp/rpc 39Ð53   59Ð21   25Ð98 23Ð64 24Ð81 19Ð45 15Ð14 16Ð30vd/rhp 22Ð18 11Ð41 19Ð61 19Ð92 20Ð72 11Ð14 10Ð16 9Ð45vd/rpc 39Ð53 53Ð20 26Ð80 21Ð81 25Ð13 19Ð46 16Ð23 15Ð04

3 eac/vd/rpc 40Ð10 53Ð15 30Ð56 25Ð99 25Ð89   26Ð29   23Ð18 24Ð02eac/vd/rhp 46Ð18 25Ð74 22Ð51 19Ð99 19Ð99 17Ð73 17Ð78 14Ð99eac/rhp/rpc 39Ð74 52Ð91 30Ð56 25Ð99 25Ð89 25Ð45 25Ð58 21Ð68

4 eac/rhp/rpc/vd 8Ð95 52Ð91 30Ð56 26Ð94   28Ð44   21Ð66 23Ð18 21Ð69

Note: Vd, vertical distance to channel network; eac, elevation above channel; rpc, relative profile curvature; rhp, relative hillslope position. The higherthe  F-value, the better is the class separation of the arithmetic class means. The highest  F-value reached for each group of classes is highlighted inbold.

Table IV. Number of soil depth points in each class of the raster maps

Number of Classes Raster Maps Used Class id Totalin the Overlay

1 2 3 4 5 6 7 8 9

2 eac 33 193 2263 rhp rpc 24 55 147 2264 eac rpc 101 80 11 34 2265 eac

 rpc 71 94 34 7 20 2266 eac vd rhp rpc 5 17 69 45 12 80 2267 eac vd rpc 86 30 15 68 5 7 17 2268 eac rpc 49 47 0 20 67 22 14 7 2269 eac rpc 64 14 11 15 45 53 4 0 20 2268 geomorphology 12 3 40 83 37 17 30 4 226

Note: Vd, vertical distance to channel network; eac, elevation above channel; rpc, relative profile curvature; rhp, relative hillslope position.

where the samples are repeatedly replaced (Efron, 1981).

In this study, we used bootstrapping to estimate the root

mean square error (RMSE) between predicted soil depths

calculated of the training set and the soil depths of the

testing data set, used as expected values. Although the

original data set was split into equally sized training and

testing data sets (113/113 points), we expected the RMSE

to be largely influenced by the sample size of some of 

the raster map classes. Some of the terrain maps with

a high number of classes contain a low number of soil

depth points or even no soil depth points (empty classes)

(Table IV). Due to the large data range of measured soil

depths, the sample mean of these classes and the RMSE

are greatly influenced by the values picked during the

bootstrapping.

We calculated the RMSE for different scenarios to

estimate the quality of the predicted soil depth mapsusing bootstrapping and 5000 iterations for each test. In

detail we tested three different scenarios for validation

and calculated the RMSE as follows:

 RMSE D

 

1

niD1

x i y i2

where   x i   is the estimated soil depth calculated of the

arithmetic class means of two classes when combining

two input maps using one of the statistical means and

y i   is a soil depth point of the testing data set. The three

measures for validation were the following:

1. The RMSE was calculated between the estimated

soil depth ( x i) of a certain class combination of the

training data set and each of the respectively soil

depth points of the testing set (y i) of exactly the same

class combination, in the following referred to as the

RMSEsingle value.

2. The RMSE was calculated based on the estimated soil

depth ( x i) of a class combination of the training data set

and the class average of the soil depth points of either

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THE EXTRAPOLATION OF TRANSECT SOIL DEPTH POINT INFORMATION   3023

the geomorphology or the terrain map class (y i) that

the class combination is consisting of, in the following

referred to as the RMSEclass value.

3. The RMSE calculated on class level averaged to

a total RMSE of a given map combination (e.g.

Geomorphology and 2-classes terrain map) to compare

the quality of the different dissolved soil depth maps,for the remainder of this article defined as RMSEtotal.

RESULTS

Terrain layer selection and classification

F-values were calculated for all combinations of ter-

rain attributes. For each class category (number of 

classes), the highest  F-value was estimated and the ter-

rain map among all raster maps selected that showed the

best class separation. Table III shows  F-values obtained

for all single terrain parameters and terrain parameter

combinations. The best  F-value reached for each class

category is highlighted in bold. In case that more than

one terrain map reached the best  F-value, we chose the

terrain map with the lowest number of combined ter-

rain parameters on the basis of Ockham’s razor (Wolpert,

1990).

Variability of soil depth measurements and class

combinations

Table V summarizes the available number of soil depth

points for each class combination, when the geomorphol-

ogy map is merged with a terrain map assuming all 226

soil depth points in the model. However, with respectto the three validation scenarios stated in section model

fitting and validation, the best validation method of the

estimated soil depths is to compare the estimated soil

depth of an area to soil depth points that are exactly

located in the same area. Since the soil depth points in our

study show a non-uniform distribution over the catchment

(see Figure 1), the estimated soil depth can only directly

be verified for a few class combinations with soil depth

points located in exactly the same area. Table III sum-

marizes the number of maximal available points for each

class combination to estimate the soil depths that can be

directly or indirectly validated with soil depths from thetesting data set. The 8-classes and the 9-classes terrain

maps both have ‘empty’ classes (class 3 of eac rpc8; class

8 of eac rpc9) and contain no representative soil depth

points for the calculation of a class mean (Table IV).

Soil model test using bootstrapping

Results of the total RMSE averaged over 5000 boot-

strapping iterations using the harmonic mean are shown

in Figure 3. Tests of the arithmetic and geometric mean

to predict soil depths for each geomorphology and ter-

rain map class combination were also performed. How-

ever, the results of the total RMSE, RMSEclass value   and

RMSEsingle value   indicated a poorer performance of the

statistical means as predictors, compared with the har-

monic mean. Both statistical means showed in general

higher RMSE in all validation scenarios and predicted

lower soil depth ranges in the output maps compared

with the original measurements and the predictions made

with the harmonic mean. The 226 point observations of 

soil depth ranged from 0 to 6Ð5 m. The use of the arith-

metic mean to calculate class means would have resulted

in non-zero values and would have caused a bias of pre-dicted soil depth in areas (e.g. bare soil areas) where the

majority of soil depth points is zero. Initial test comput-

ing the coefficient of determination between the predicted

soil depth maps and class means of the original soil depth

measurements resulted in lowest coefficients for the maps

predicted with the arithmetic mean (max.   R2D 0Ð60)

and highest coefficients for the maps predicted with the

harmonic mean (max.   R2D 0Ð73). Consequently, only

assessments based on the harmonic means were selected

for further analyses.

The different map combinations shown in Figure 3

resulted in similar mean RMSE values for the comparedstatistical means with slightly decreasing RMSE values

with increasing number of classes. The means of the cal-

culated total RMSE values decrease from approximately

0Ð82 m (12Ð6% of the total data range) for the 2-classes

terrain map combination to about 0Ð73 m (11Ð1% of total

data range) for the 9-classes terrain map combination.

For convenience, the number of classes in the respec-

tive terrain maps is used in the remaining sections to

distinguish the tested map combinations in further inter-

pretations.

Validation results of the single-RMSE

(RMSEsingle values

), class-RMSE (RMSEclass values

) and a

comparison of estimated and predicted soil depths are

shown for the harmonic mean in Figure 4. For the

majority of the estimated soil depths, the single and

Figure 3. Box-and-whisker plot of total RMSE reached for the harmonicmean and different map combinations. The RMSE are sorted according

to the number of classes of the terrain map used in the overlay withthe geomorphology map. The diagram shows for each map combinationthe median, the upper and lower quartile and the smallest and largest

observed RMSE during the 5000 bootstrapping iterations

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3024   H. E. DAHLKE  ET AL.

Table V. Maximum number and number of exactly located soil depth points available for the prediction of soil depths for eachgeomorphology-terrain map combination based on all 226 points

Maximal Available Number of Points Exactly Located Points

Number of TerrainClasses Maps Geomorphology Geomorphology

Class id 1 2 3 4 5 6 7 8 1 2 3 4 5 6 7 8

  12 3 40 83 37 17 30 4

2 eac1   33 45 36 73 116 70 50 63 37 3 7 19 4

2   193 205 196 233 276 230 210 223 197 12 3 37 76 18 17 3

3 rhp rpc1   24 36 27 64 107 61 41 54 28 16 82   55 67 58 95 138 92 72 85 59 12 5 4 2 6 263   147 159 150 187 230 184 164 177 151 3 35 63 27 11 4 4

4 eac rpc

1   101 113 104 141 184 138 118 131 105 4 1 23 4 3 13 16 12   80 92 83 120 163 117 97 110 84 2 13 31 31 33   11 23 14 51 94 48 28 41 15 9 24   34 46 37 74 117 71 51 64 38 8 4 3 1 4 14

5 eac rpc

1   71 83 74 111 154 108 88 101 75 13 28 28 22   94 106 97 134 177 131 111 124 98 4 3 21 37 13 163   34 46 37 74 117 71 51 64 38 8 4 3 1 4 144   7 19 10 47 90 44 24 37 11 5 25   20 32 23 60 103 57 37 50 24 2 1 6 2

6 eac vd rhp rpc

1   5 17 8 45 88 42 22 35 9 2 12   17 29 20 57 100 54 34 47 21 6 1 2 83   69 81 72 109 152 106 86 99 73 6 1 19 1 13 24   45 57 48 85 128 82 62 75 49 7 19 17 25   12 24 15 52 95 49 29 42 16 2 2 6 26   80 92 83 120 163 117 97 110 84 2 12 5 12 2 2

7 eac vd rpc

1   86 98 89 126 169 123 103 116 90 2 15 55 1 2 22   30 42 33 70 113 67 47 60 34 4 18 83   15 27 18 55 98 52 32 45 19 6 1 2 64   68 80 71 108 151 105 85 98 72 6 1 18 8 13 225   5 17 8 45 88 42 22 35 9 2 16   7 19 10 47 90 44 24 37 11 3 3 1

7   17 29 20 57 100 54 34 47 21 14 3

8 eac rpc

1   49 61 52 89 132 86 66 79 53 2 13 23 112   47 59 50 87 130 84 64 77 51 5 1 7 6 8 23   0 12 174   20 32 23 60 103 57 37 50 24 2 14 45   67 79 70 107 150 104 84 97 71 1 17 38 7 46   22 34 25 62 105 59 39 52 26 14 87   14 26 17 54 97 51 31 44 18 6 2 68   7 19 10 47 90 44 24 37 11 3 4

9 eac rpc

1   64 76 67 104 147 101 81 94 68 1 17 33 9 42   14 26 17 54 97 51 31 44 18 6 2 63   11 23 14 51 94 48 28 41 15 9 24   15 27 18 55 98 52 32 45 19 2 7 4 25   45 57 48 85 128 82 62 75 49 5 1 7 6 6 2

6   53 65 56 93 136 90 70 83 57 2 13 23 157   4 16 7 44 87 41 21 34 8 1 38   0 12 179   20 32 23 60 103 57 37 50 24 14 6

3 40 83 37 30 4

3 40 83 37 30 4

Note: N   is the maximum number of soil depths points located in each class of each map. Light grey highlighted cells show class combinations thoseestimated soil depths can directly be validated with soil depths points that are exactly located in the same class combination. Dark grey highlightedcells indicate class combinations that do not comprise direct validation points, but that can be compared with the class mean of the testing dataset. Black cells highlight class combinations that occur in the final prediction maps, but those soil depths cannot be calculated due to a lack of soildepth points located in one or both of the combined classes (empty classes). White cells highlight class combinations that do not occur in the finalprediction maps.

class RMSE stay in the range of the calculated total

RMSE and the data set’s standard deviation of 1Ð09 m.

The single and class RMSE exceed the mean total

RMSE for estimated soil depth greater than 1 m. This

was expected considering the value range of mea-

sured (0–1 m) and predicted soil depths (0–0Ð54m)

(Figure 4b). RMSEclass values   are generally larger than

RMSEsingle values  because of the greater data range result-

ing from the comparison of an estimated soil depth

point to the mean soil depth of a layer class. The small

RMSEsingle values   indicate that the estimated soil depths

predicted with the harmonic mean differ only little from

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THE EXTRAPOLATION OF TRANSECT SOIL DEPTH POINT INFORMATION   3025

Figure 4. Comparison of RMSE and estimated soil depths calculated with the harmonic mean. Diagram (a) shows the estimated soil depths (black dots), results of the two validation scenarios RMSEsingle value   (black crosses) and RMSEclass value   (grey diamonds). The RMSEsingle value   result froma comparison of the estimated soil depth of a certain class combination to validation points located in areas with the same class combination andRMSEclass value  show the comparison of the estimated soil depth of a certain class combination to all validation points of either one of the combinedclasses in a map combination. Diagram (b) shows a comparison of minimum and maximum estimated soil depths predicted with the testing and

training data set

the soil depths actually measured in the area of a certain

class.

Predicted soil depths maps

Maps of estimated soil depth were generated with the

harmonic mean for each map combination (Figure 5).

The predicted soil depth maps show an increasing degreeof spatial disaggregation the more classes the spatial

data sets in the overlay process have. The number

of entities increases in the prediction maps from 128

to 1438 for the overlay of the geomorphology with

a terrain map consisting of minimum two classes to

maximum 9 classes. Similarly, the size of the largest

spatial entity in the predicted soil maps decreases from

maximum 160 800 m2 to 32300 m2. Soil depth maps

predicted with the 8- or 9-classes terrain layer exhibit

‘empty’ or ‘no-data’ areas, where the soil depth cannot

be modelled. Both terrain layer lack soil depth points

in one of the classes to calculate the class mean. Thesize of the ‘no data’ areas in the geomorphology/8-

terrain classes map covers 0Ð034 km2 and 0Ð031 km2

in the geomorphology/9-terrain classes map. The areas

equal 3Ð9% and 3Ð3% of the catchment area (0Ð942 km2),

respectively.

Soil depth maps with a higher degree of spatial

disaggregation show also a greater range of predicted

soil depths. Minimum, maximum and average soil depths

increased from 0Ð50 m to 0Ð31 m, 2Ð24 m to 3Ð04 m and

1Ð2 m to 1Ð68 m, respectively with increasing number of 

included terrain classes in the predicted soil depth map

(Figure 6).

RMSE were calculated between the soil catena points

and the cell values in the soil depth prediction maps

to estimate the most suitable soil depth prediction map

(Table VI). The map combinations of the geomorphology

map with the 2-terrain-classes map reached the best

coefficients among all map combinations and tested

statistical means. The lowest RMSE (RMSE  D 0Ð46 m)

was reached for the geomorphology/2-terrain classes map

predicted with the harmonic mean, which also showed

the highest   R2

. The second lowest RMSE (RMSE   D0Ð61 m) was reached for the geomorphology/5-terrain

classes map. The prediction error of these two map

combinations was less than 10% of the overall soil depth

range measured in the catchment.

DISCUSSION

The  R2 reached in the soil depth prediction maps agrees

well with accuracies achieved for most quantitative

spatial soil models (Beckett and Webster, 1971; Ryan

et al., 2000). According to Beckett and Webster (1971),

 R2

greater than 0Ð7 are unusual for most spatial modelsand   R2 of 0Ð5 or less are common. In this study, the

RMSE of the final soil depth prediction maps showed

an error smaller than 10% of the data range. This shows

that the presented soil model approach provides an easy

applicable method in terms of computation requirements

that predict spatial variability of soil depth more accurate

than a single explanatory variable.

The fact that the geomorphology/2-terrain classes map

reached the lowest RMSE among all tested statistical

means was unexpected, because both the value range of 

estimated soil depths and the degree of spatial disaggrega-

tion were smaller in the final prediction map than in map

combinations with more classes. However, this fact can

be explained with the clustering approach that has been

used to reclassify the terrain attributes to generate second

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3026   H. E. DAHLKE  ET AL.

Figure 5. Maps of estimated soil depths using the harmonic mean as prediction model. Each map shows a map combination of the geomorphology

layer consisting 8 classes and a terrain layer with varying number of classes (2–9 terrain classes). Grey areas indicate ‘empty’ classes, where soildepths could not be estimated due to lacking point data in the training data set

Figure 6. Comparison of minimum, maximum and mean soil depth foreach produced soil depth map using the harmonic mean. Statistics issorted according to the number of classes in the terrain map used in the

overlay with the geomorphology map

input layer for the overlay process. The  k -means cluster-

ing algorithm used in this study randomly generates   k 

clusters from the continuous terrain attribute maps. The

final location and size of the clusters are, however, statis-

tically determined by the convergence criterion that needs

to be met for each cluster (Hartigan and Wong, 1979).

The terrain classes resulting from the clustering depend

on statistical differences in topography, but might not

reflect the actual soil depth variability in the watershed.

An expert-based differentiation and reclassification of the

terrain attributes as input layer are therefore suggested for

future applications.Although the best RMSE suggests that the soil depth

map with the lowest disaggregation is the best choice for

further applications, if a higher spatial disaggregation is

desired, the user has to balance between the prediction

accuracy and the number of classes used in the overlay

process. The use of input layers with more classes may

lower the probability to calculate the layer class means

(e.g. soil depth). The overlay of several explanatory

variables with a low number of classes will likely

increase the probability to ensure complete coverage in

the prediction maps and higher prediction accuracies.

However, in case of the occurrence of unpredictable

areas, post-processing is needed to complete the soil

depth information. Several approaches can be applied

such as taking only the information from one of the

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