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a Unité d'Agronomie Laon/Péronne, INRA, Laon, France
b Dep. of Physical Geography, Univ. of Utrecht, Utrecht, Netherlands
c MTT, Jokioinen, Finland
d Danish Institute of Agricultural Sciences, Foulum, Denmark
e Scottish Agricultural College, Penicuik, UK
f Faculty of Sciences, Univ. of Coruña, Coruña, Spain
eva{at}laon.inra.fr
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ABSTRACT |
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 TOP ABSTRACT INTRODUCTIONOverview of Roughness IndicesOverview of Predictive ModelsMaterials and methodsNOTESResults and discussionConclusionsREFERENCES |
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. Prediction of MDS in the field based on RR has an uncertainty of ± 3 mm (95% confidence interval). Variation was due to RR and its nonspatial nature. To improve predictions of MDS, the spatial configuration of the surface has to be taken into account.
Abbreviations: DEM, digital elevation model • LD, limiting elevation difference • LS, limiting slope • m, number of sub segments per transect • MDS, maximum depressional storage • MIF, microrelief index x peak frequency • MUD, mean upslope depression • RR, random roughness • T, tortuosity • TA, tortuosity index, Auerswald (1992) • TB, tortuosity index, Boiffin (1984) • TP, tortuosity index, Planchon et al. (1998) • TS, tortuosity index, Saleh (1993)
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INTRODUCTION |
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TOPABSTRACTINTRODUCTIONOverview of Roughness IndicesOverview of Predictive ModelsMaterials and methodsNOTESResults and discussionConclusionsREFERENCES |
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Water storage capacity, here referred to as MDS, is difficult to measure in the field because infiltration cannot be controlled. MDS consequently is estimated from DEMs of the surface. Under the assumption of zero infiltration, depressions from DEMs are virtually filling until the point of overflow (Ullah and Dickinson, 1979; Moore and Larson, 1979; Onstad, 1984; Huang and Bradford, 1990; Martz and Garbrecht, 1993). Digital elevation models need dense sampling in two lateral directions, which is laborious, especially in the field. Often the only available field roughness data are transects. Therefore, models have been developed to predict MDS from roughness indices and terrain slope (Onstad, 1984; Auerswald, 1992; Linden et al., 1988).
The objective of this study was to investigate the ability of selected roughness indices and models to describe MDS in view of improving MDS prediction. New techniques to treat DEMs, such as lasers and geographical information system (GIS) tools, allow better estimation of MDS, especially on surfaces with small roughness. Some roughness indices have never been related to MDS (Römkens and Wang, 1986; Huang and Bradford, 1992; Bertuzzi et al., 1990b). Until now, roughness indices have only been compared with respect to their ability to describe the change in roughness with rainfall (Bertuzzi et al., 1990b; Eltz and Norton, 1997), but have never been compared with respect to their ability to predict MDS. Furthermore, existing models predicting MDS have never been compared.
During 2 yr, roughness data were collected as part of a runoff research project on eight soils in five countries in northwestern Europe. Data analyses were performed to answer the following questions:
What range of roughness and MDS values can be expected for agricultural conditions in Western Europe?Can MDS be estimated from tillage?
Are roughness indices related to MDS?
Do the data obtained in this project validate existing predictive models based on these roughness indices?
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Overview of Roughness Indices |
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TOPABSTRACTINTRODUCTIONOverview of Roughness IndicesOverview of Predictive ModelsMaterials and methodsNOTESResults and discussionConclusionsREFERENCES |
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Tortuosity
Boiffin (1984) used the term tortuosity (T) for a roughness index. In this paper, T is extended to include any roughness index based on the ratio of surface profile length (L1) and the length of the straight line formed by its projection (L0), like TB, TP, TS, and TA (Table 1).
Tortuosity is often measured by the chain method (Saleh, 1993), where a roller chain of fixed length is laid over the soil surface and the lateral distance covered is measured, but it may also be determined from elevation measurements. Note that even though tortuosity indices of Auerswald (1992) and Saleh (1993) are the same, their limits are not. In Auerswald (1992), T is limited to a theoretical maximum of 29.3, based on the assumption that the maximum side slope of an aggregate is 45°.
Tortuosity has a disadvantage when describing surface storage or soil roughness. Tortuosity depends on soil roughness and scale (Skidmore, 1997). If one looks at the same profile but at a bigger scale, both with a chain with smaller links or with a profile meter with a higher lateral resolution, one sees more detail and T inevitably increases. This also means that two profiles with roughness of a different order of magnitude, measured at a different scale, may have exactly the same T value. One could imagine a degraded surface of a plough layer measured every 5 cm and a seedbed measured every 1 mm having the same T, but different depressional storages. Tortuosity values obtained from measurements at different sample spacings cannot be compared, whereas RR values can be compared.
Limiting Elevation Difference and Slope
Limiting elevation difference (LD) and slope (LS) were developed to take the spatial aspect of roughness into account. Limiting elevation difference and LS are based on the first-order variance or mean absolute elevation difference (
Zh) as a function of lag (h) (Table 1). A linear regression is fit to the transformed variogram, so the first-order variogram is described by a hyperbolic function:
 | (1) |
Variance rises with increasing lag and reaches a plateau or sill at a certain lag distance or range. The LD (1/b) is the asymptote value of the first-order variance (i.e., the sill); LD is related to RR (Linden and Van Doren, 1986). Indeed, RR corresponds with the square root of the sill of the second-order variogram (Isaaks and Srivastava, 1989). Linden and Van Doren (1986) stated that LS is surface slope at small intervals because Z/X (considering
), where Z is height, would approach LS when X (or h) approaches zero. But in fact, the derivative of the hyperbolic function equals LS when X is zero:
 | (2) |
Therefore, LS is not surface slope, but variogram slope at the origin. Together with LD, it determines the form of the variogram, which is assumed to be hyperbolic. Slope and random roughness are independent (Linden and Van Doren, 1986). Bertuzzi et al. (1990b) stated that LS and T are related and give information on small intervals.
Fractal Indices
Fractal indices have been developed based on variance, like fractal dimension (D) and crossover length (l), and based on tortuosity, like FB and FM (Table 1). Use of a fractal index assumes implicitly that there is self-similarity in the surface profile across a range of scales. Burrough (1983) showed that the soil surface does not behave as a fractal but merely as a pseudo fractal; fractal behavior holds only over a certain range of scales, until the semivariogram reaches its sill. Furthermore, a fractal index assumes a power function relation between semivariance and lag, or between T and sample spacing. Fractal indices are regression parameters. They are not a measure of roughness or scale itself, but describe the form of the variogram, like LS. Even though a fractal index is an indicator of morphological roughness response to rainfall, it cannot be used to quantify MDS. Thus, the fractal index will not be considered further in this study.
Microrelief Index and Peak Frequency
Microrelief index and peak frequency (MIF) (Table 1) is a simplified, nonunique expression of surface roughness (Römkens and Wang, 1986). This means that surfaces with a different topography may have the same MIF value.
Again, MI is similar to RR; MI is the mean absolute difference between elevations, while RR is the root of the mean square difference. F, the number of peaks per unit length, is not clearly defined because it depends on scale. If a profile is described with a smaller sample spacing or link length, the number of peaks will increase. So in practice, MIF is not a useful parameter and it will be excluded from this study.
Mean Upslope Depression
Mean upslope depression (MUD) (Table 1) was the first roughness index developed to predict depressional storage (Hansen et al., 1999). On a transect a subsegment is chosen. The most downslope point is the reference point. Mean height difference between the reference point and all other points is calculated. The subsegment is displaced by one point and the procedure is repeated. The MUD is the mean value of all subsegments. Hansen et al. (1999) did not give any guidelines to decide on subsegment length, but concluded that a length of 30 cm was best for their study conditions.
Mean upslope depression is dependent on roughness and slope. Thus, a model to predict MDS will need only MUD as input and not a combination of a roughness index and a slope.
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Overview of Predictive Models |
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TOPABSTRACTINTRODUCTIONOverview of Roughness IndicesOverview of Predictive ModelsMaterials and methodsNOTESResults and discussionConclusionsREFERENCES |
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Relating Maximum Depressional Storage to a Combination of Limiting Elevation Difference and Limiting Slope
Linden and Van Doren (1986) combined LD and LS to predict MDS (Model 3, Table 2), with the aim of reaching a higher accuracy than with a single roughness index. The model explained 79% of the data variation. The database consisted of 159 DEMs measured by a pin meter on field plots with varying tillage. Grid spacing was 2 or 5 cm in both directions, and total plot surface was 2.0 or 0.9 m in both directions.
Relating Maximum Depressional Storage to Mean Upslope Depression for True Random Roughness
Using MUD as roughness index, Hansen et al. (1999) developed a model explaining 80% of the data variation (Model 4, Table 2). The database consisted of DEMs measured by an automated pin meter (Schjønning, 1993, unpublished data) on plowed or direct drilled field plots. Grid spacing was 6 by 3 cm, and total plot surface was 2.6 m by 1.3 m.
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Materials and methods |
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TOPABSTRACTINTRODUCTIONOverview of Roughness IndicesOverview of Predictive ModelsMaterials and methodsNOTESResults and discussionConclusionsREFERENCES |
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Elevation Measurements
Roughness conditions were created in field (ploughed, harrowed, and seedbeds) and lab (reconstructed seedbeds) plots. Degraded situations were obtained by natural rain (field) and simulated rain (lab). We created raster-based maps (DEMs) from roughness measurements on regular grids. Elevations were measured with pin meters when laser relief meters were not available (Burwell et al., 1963; Schjønning, 1993; Mäkelä, 1998; Bertuzzi et al., 1990a). Soil roughness variability was ensured by establishing tilled plots on selected fields across Europe. We evaluated 221 soil surfaces including eight soil types. Sample spacing, or distance between points on a transect and between transects, varied from 2 to 30 mm, and total plot area varied from 0.2 to 3.5 m2 (Table 3)
. The DEMs with a particular sample spacing are from now on called a sub-data set.
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Boundary conditions are important for MDS calculations. If closed boundaries of fixed height were supposed around the plot, MDS would be overestimated, but if open boundaries were supposed around the plot, MDS would be underestimated. The best solution would be to use open boundaries, then disregard the watersheds that touch the border and estimate MDS over the rest of the surface, but our plots would probably have become too small to be representative. Therefore, we chose to surround the plot by copies of itself (Fig. 1) and fill this enlarged plot using open boundaries. MDS is then calculated over the original plot surface. This arrangement of boundary conditions accounted for both random roughness and oriented roughness. Random roughness is filled since depressions at an edge are filled to the level of the lowest outflow point inside the map. Oriented roughness is not filled because furrows that extend across the plot allow outflow. One edge of the plot must be chosen parallel to the tillage direction to account for oriented roughness. This was the case for all plots.
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Calculation of Roughness Indices
To calculate roughness indices, we developed a computer program that uses maps in PCRaster format as input. The program first sampled 30 equidistant transects sampled across each DEM, regardless of plot size or sample spacing. We selected transects parallel to the tillage direction, if the direction was evident. The program then calculated the indices according to their equations as cited in literature for each transect. It finally calculated the statistical mean of thirty to obtain the final roughness value. Final roughness value is a mean value for random roughness on top of ridges and inside furrows (in case of oriented roughness).
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Results and discussion |
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TOPABSTRACTINTRODUCTIONOverview of Roughness IndicesOverview of Predictive ModelsMaterials and methodsNOTESResults and discussionConclusionsREFERENCES |
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(ANOVA, Tukey test, with STATISTICA 98, Statsoft). Evaluation of MDS values per tillage shows variation within each class and overlap between classes. Because of this variation, MDS predictions from the mean MDS per tillage type would not be useful. Variations of MDS within a tillage type can be explained by factors affecting the effect of tillage on roughness and factors changing roughness after tillage. The effect of tillage on roughness depends on the number of passes of the tillage tool, tractor speed, clay content, and soil water content (Zobeck and Onstad, 1987). Rainfall and runoff after tillage reduce roughness at a rate that depends on aggregate stability; aggregate stability is influenced by soil type, soil water content, and biological activity related to how crop residue is managed (Zobeck and Onstad, 1987; Le Bissonnais, 1990, 1996). Freezing and thawing also can reduce roughness (Pardini et al., 1996). Plant growth may increase roughness (Martinez-Turanzas et al., 1997).
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Maximum Depressional Storage vs. Grid Spacing
Before addressing the question whether roughness indices are related to MDS, we need to evaluate whether MDS data from DEMs with different resolutions are comparable. Dependence of MDS on varying grid spacing was studied in four plots (Fig. 3)
. Plots initially had a grid spacing of 2 or 4 mm. Then, sample density was progressively diminished in both x and y directions by leaving points out. So 2-mm grids became 4-, 6-, 8-, and 10-mm grids; 4-mm grids became 8-, 12-, 16-, and 20-mm grids. At each step, all possible surface configurations were tried and their MDS calculated (Fig. 3). Contrary to the conclusion of Huang and Bradford (1990), MDS did not structurally decrease with increasing grid spacing. Variance analysis on MDS for each surface at different resolutions (n 
9) showed that we can admit equal means for all resolutions (P 
0.05). Comparison of the general mean MDS at all resolutions (n 9) and MDS at the smallest sample spacing
with a bilateral Student's t test proved a significant difference for all surfaces (P 0.01). After an initial increase, MDS stabilized in the range of sample spacings as used in this study (10–30 mm), which implies we are allowed to compare values from different pin meters. Lab surfaces show a significant but yet negligible increase in MDS (<0.2 mm). The reason why MDS does not decrease may be that even though resolution in the x and y directions diminishes, the resolution in z direction stays the same; since the cells are larger, elevation differences between cells enhance storage across a larger cell area, so MDS may be maintained, even though the degraded DEM is a less precise representation of the soil surface.
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(Fig. 4)
. No single relationship could be found between MDS and TA, TP, or LS because data segregate to groups of DEMs having the same sample spacing (Fig. 5)
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Random roughness had the highest correlation with MDS of all indices; it explained 80% of the variance. Limiting elevation difference explained 75% of the variance. Limiting elevation difference was somewhat more variable than RR, possibly because experimental variograms did not always correspond with the theoretical form on which LD is based. In theory, the variogram must be rising and stabilizing at a sill. In field plots, a nonrandom component of roughness existed, altering the form of the semivariogram.
The irregularity of experimental variograms also made it difficult to decide on the maximum lag at which the sill is attained (range). Linden and Van Doren (1986) chose 200 mm as a mean value for the whole data set after primary investigation of variograms. Likewise, we chose an overall value per sub-data set, knowing that an overall value may be inaccurate for certain individual semivariograms. With an overall value, the linear regression from which LD and LS are calculated may be performed on a range of lag values that is too big or too small. Thus, difficulties in determining the range or differences between transects may have altered regression parameters, explaining part of the variation in LD values.
Mean upslope depression had the same coefficient of variation as
. A disadvantage in the calculation of MUD is that no rules are given to determine the length of the subsegment. By repeated regressions of MUD against MDS, Hansen et al. (1999) determined that five intervals of 120-cm, seven of 90-cm, or 11 of 60-cm length were required to attain 70% of explained variance in MDS. Since a sensitivity analysis of MUD for subsegment length is beyond the scope of this paper, we have arbitrarily chosen a fixed subsegment length per sub-data set, so that there are always 20 to 30 height readings per subsegment. Since sample spacing and number of height readings per transect were different for each sub-data set, inevitably subsegment length and number of subsegments per transect were different as well (Table 4)
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Three reasons for our contradictory conclusion are: 1. Our criterion is the relation of indices to MDS, whereas all studies above used the criterion of the indices' response to roughness decline during natural or artificial rainfall.
2. We analyzed the effect of varying sample spacing on T and LS, while others did not.
3. Generally LD and LS are thought of as parameters with a physical meaning. We believe they are statistical parameters, describing the form of the semivariogram.
Evaluation of Existing Models to Predict Maximum Depressional Storage
Performances of four existing models were relatively poor, except for the model MUD–MDS (Fig. 7)
. The model of Onstad (1984) underestimated MDS for the range of roughness values. The model was biased, but the chosen roughness index seemed justified.
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The model of Auerswald (Morgan et al., 1998; Auerswald, 1992) did not predict MDS. The problem was the choice of roughness index, T being dependent on sample spacing (Fig. 6). Also, we found TA values superior to the theoretical maximum of 29.3%. Auerswald's assumption of maximum slope of 45° did not hold. We observed slopes of up to 90°. This explains the predicted MDS values greater than the theoretical maximum of 3.5 mm.
The model of Hansen et al. (1999) has an appropriate linear form and described all data. Point variation around predicted values was around 3 mm.
Of these four models, the one of Hansen et al. (1999) clearly was best suited, but the one of Onstad (1984) could perform at least as well after reestimation of its parameters.
Model to Predict Maximum Depressional Storage at Zero Slope
We developed a model to predict MDS from RR at zero slope. By regression analysis, a model was fitted and upper and lower limits of the confidence interval on the prediction of a new value were determined (Tomassone et al., 1992). Variation in data increased with increasing RR and MDS (Fig. 4). In other words, residual error is dependent on RR. This is the case of a lognormal model. After log transformation of RR and MDS, a linear model was fitted. Then transformed variables were used to express MDS as a function of RR (Fig. 8a and 8b)
. Predictions of MDS and 95% confidence interval limits for prediction of a new value are shown. Confidence limits ranged from 3 ± 2 to 10 ± 7 mm MDS. The proposed model is biased for low MDS values. Also, it seems to overestimate the error on the prediction for high MDS values. Low values corresponded with surfaces constructed in the lab; therefore, separate models for lab and field measurements were developed. (Fig. 8c and 8d). For lab data, MDS ranged from 0 to 2 mm, and a lognormal model was fitted. For field data, MDS ranged from 1 to 13 mm, and a linear model was fitted. Prediction error was between 0.2 and 0.5 mm for lab data and 3 mm for field data. These two submodels are better suited. The lab model eliminates the bias for the prediction on low MDS. The field model normalizes the residual errors and describes them correctly.
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Maximum Depressional Storage Prediction
A predictive model for MDS can be made from RR, if slope is included as a second independent variable. Random roughness in itself is independent of slope, even though it is recognized that the presence of a slope may affect the roughness changes during rainfall and runoff. But our results show that such a model would predict MDS with an error of at least 3 mm. This error is due to the nonspatial character of RR. Random roughness is merely an indicator of the vertical component of soil surface roughness, and surfaces with the same RR may have a different MDS.
The only way to make a better prediction of MDS is to work in three dimensions instead of using a roughness index. If we would be able to reconstruct the DEM itself from single transect measurements, then we could calculate MDS directly. GIS packages such as PCRaster become more and more available to researchers, which makes it possible to work on DEMs.
Studies have simulated DEMs on the basis of transect measurements (Chadoeuf et al., 1989, 1996; Bertuzzi et al., 1995) and found that a three-dimensional soil surface can be simulated from roughness measurements on transects with the help of a boolean model that supposes that the surface is composed of half spheres. These surfaces look like the original experimental soil surface and have similar RR values. The following question remains: Do reconstructed soil surfaces have the same MDS as experimental data?
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Conclusions |
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TOPABSTRACTINTRODUCTIONOverview of Roughness IndicesOverview of Predictive ModelsMaterials and methodsNOTESResults and discussionConclusionsREFERENCES |
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3 mm. If we want to improve MDS prediction based on a roughness parameter, the best suited index would be RR. However, a predictive model based on RR would contain an error of 3 mm, due to variability in MDS values for a given RR. This error arises because RR does not have spatial sensitivity, whereas spatial configuration of the surface is essential for depressional storage. Further improvement of MDS prediction cannot be reached with a method using a transect-based roughness index, but requires a three-dimensional soil surface sufficient to calculate MDS from the digital elevation model. Height measurements on soil surfaces, often obtained by laser meters or photogrammetry, are still expensive, thus mathematical modeling of soil surfaces based on single transect measurements is needed.
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ACKNOWLEDGMENTS |
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NOTES |
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TOPABSTRACTINTRODUCTIONOverview of Roughness IndicesOverview of Predictive ModelsMaterials and methodsNOTESResults and discussionConclusionsREFERENCES |
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Received for publication April 29, 1999.
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REFERENCES |
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TOPABSTRACTINTRODUCTIONOverview of Roughness IndicesOverview of Predictive ModelsMaterials and methodsNOTESResults and discussionConclusionsREFERENCES |
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