SCOPE 35 - Scales and Global Change

ABSTRACT

This chapter examines the influence of topography on the spatial distribution of soil properties and processes at different spatial scales. Three data sets are discussed to illustrate a variety of scaling problems. Topography is characterized by elevation, gradient, and vertical and horizontal curvature. Scale relationships between variables are examined using linear stochastic theory and the theory of regionalized variables. The soil pedon is described as the basic representative elementary volume of soil horizon properties. The data sets, however, indicate different properties have different spatial variance relationships. Even horizon thickness has different representative spatial scales depending on surface or subsurface influences. The nature of the spatial variance relationships are highly dependent on the spatial distribution of topographical parameters operating at specific scales which control the transfer of mass and energy in the landscape. Integration of small scale observation to large scale units should be based on the flow of mass and energy. Thus the drainage basin becomes an obvious level of integration; The development of methods for describing the joint distribution of soil and topography should continue to be a high priority for research.

INTRODUCTION

Soils are viewed as natural bodies formed on the land surface, occupying space and having unique morphology (Simonson, 1968). Soil as an isotropic or anisotropic body is also a continuum rather than particulate (Knox, 1965). Thus in the study, classification, and mapping of soils, there is an inherent dependence on scale of observation and scale linkages. Finkl (1982) has presented a collection of benchmark papers which present a historical development of the present day concept of soil. Included in these papers is the concept of a pedon (Soil Survey Staff, 1960) which is defined as the smallest three-dimensional spatial unit of the surface of the earth that is considered as a soil. A pedon has lateral dimensions such that it contains 'representative variations' of the soil being studied. The lateral dimensions are 1m if 'ordered variation' in genetic horizons is present. If horizons are cyclical or intermittent and are repeated every 2 to 7 m, then the lateral dimensions are one-half the cycle. The concept of a pedon has been the traditional method of dealing with short range spatial variations. Mean values of soil properties for the pedon are used to place that individual spatial soil unit into a classification system.

The concept of a pedon is still a topic of considerable debate. For non-repetitive variation no formal definition of 'representative variation' is given and the term 'ordered variation' is confusing. However, a more formal treatment of the problem using continuum theory, the concept of a representative elementary volume (Bouma, 1984; Waganet, 1984), and analysis of spatial variance structure using geostatistics can lead to a similar definition for a basic soil unit.

Soil classification systems are concerned with organizing information and ideas about soils in a logical and useful fashion; it is not generally concerned with spatial ordering. The purpose of soil survey is to produce a soils map. It is a land mapping system which breaks the soil continuum into spatial units that have less variance for selected soil properties than the continuum (Wilding, 1984). Spatial mapping units are theoretically formed by grouping similar and contiguous pedons into polypedons and grouping polypedons into larger and larger ensembles depending on the scale of the map. In general because of the close relationship between topography and soils, spatial units correspond with landform features. The description of the mapping unit usually involves classification of the included soils, thereby tying the soil classification system to the mapping system at different scales. In the process of mapping, surveyors subjectively decide on the variation allowed in the mapping unit (Bridges, 1982), both in the variation allowed within soil classes, and the level of inclusion of unspecified soils within map unit areas. In many cases however, at least half of the variability of soil properties exists on a scale of < 1 m ² (Webster and Beckett, 1968) which is less than the dimensions of the pedon (as defined).

The condensed summary above is somewhat superficial, but it does define the basic concept of a soils map and how soil variability is considered. Soil surveyors have been concerned with soil variability and scale on a day to day basis, but have generally dealt with it by constructing conceptual models of systematic variability that have ignored short range variability characteristics of most soil landscapes. This variability is important because soils maps are curently being used to extrapolate point information about behaviour of soils to larger scales (Bridges and Davidson, 1982; Jarvis, 1982). Mean values of soil properties for arbitrarily selected, typical pedons are used as input to models for prediction over a given spatial unit. The predictions from each unit are further averaged (weighted according to actual area of each unit) over larger areas until the region of interest has been covered. In some cases estimates of the probability density functions (PDF) of the soil variables within the spatial unit are used for prediction at a given level of probability.

LANDFORM ANALYSIS

A number of quantitative methods have been developed for describing landscape form, but their application for studying the relationship between soils and topography is limited. Description of landscape form is the basis of geomorphology. Specific geomorphometry is the measurement and analysis of specific land surface features which are defined and separated from adjacent land areas according to clearly defined criteria (Evans, 1981).

With an elevation grid (matrix) landform shape parameters at every point in the grid can be calculated (Young and Evans, 1978). Landform shape can be characterized in part by five parameters: elevation, gradient, aspect, vertical curvature, and horizontal curvature. Gradient and aspect describe a plane tangential to the surface at the specified point and together define the concept of slope. Gradient is the maximum rate of change of elevation while aspect is the direction of the gradient usually given in degree units using North as a reference. Vertical curvature is the rate of change of gradient or the second vertical derivative of elevation. By convention, concave and convex curvature have negative or positive values, respectively. Horizontal curvature is the rate of change of aspect (i.e. direction of flow) along a contour line with negative and positive values indicating convergence or divergence of flow lines respectively. The parameters are estimated by fitting a least square quadratic surface to a subgrid of elevation readings around each point of interest, and calculating the derivatives of the surface.

For basins, after a value for aspect (surface flow direction) has been obtained for each grid point, a sixth parameter, catchment area index (CAI) can be calculated. CAI is the total number of unit cell areas (area associated with one grid point) that contributes to flow at a given grid point. The CAI is calculated as a running summation of all flow lines converging at a specific location.

The mathematical definition of slope and curvature parameters is unambiguous and, ideally, an instantaneous point measurement. The measurements are, however, made over a specific area and equations using least squares calculation will give a smoothed surface form. A discussion of scale effects on estimation of topographic parameters has been given by Evans (1972). He defines the estimates on the smallest scale as 'local' values whose scale is pre-determined by the sampling interval. Estimates based on increasing scales are regional values, and are based on statistical moments of elevation rather than derivatives. Any area can be characterized by a probability distribution function of elevation, which can in turn be summarized by its statistical moments. For example, the third moment of elevation is skewness which is an inverse measure of the region curvature. Positive skew indicates proportionally more higher (than the mean) than lower elevations which is an indication of concave surfaces (negative curvature). Thus regional curvature is measured by the third moment of the original elevation while local curvature is measured by the first moment of the second derivative of elevation.

LANDFORM AND SOIL

Although relationships between topography and soils are well documented (Gerrard, 1981; Huggett, 1982), limited success in quantitatively or statistically describing the relationship has been achieved. With a significant proportion of soil variability occurring over very short distances, there appears to be as much variability within slope class designations as between classes (Joel, 1933; Ball and Williams, 1968; King et al., 1983). In addition there are fundamental differences in soil relationships on convex, concave and straight slopes (Gerrard,1981).

In a detailed examination of soil-landscape relationships in Saskatchewan, Canada, from a soil survey perspective, King et al. (1983) concluded that the only realistic division for map units is by convex upper slopes (shallow soils), concave lower slopes (deep soils) and depressional areas (gleyed soils). Short range variability was so great that it was not possible to create meaningful map-units on a smaller scale or by using more refined soil groupings such as soil series. As the authors stated, the divisions suggested by the study are significant in that they correspond with landscape elements recognized as meaningful in terms of crop and vegetative productivity.

Even though considerable variability of landforms and soils exist, it is generally conceded that the variance is not necessarily random. That is, measurements of a property taken close together are more alike than values taken farther apart. This connectiveness is called spatial variance structure. Methods for describing the spatial variance structure are usually derived from the theory of regionalized variables using the semi-variogram or covariogram (Matheron, 1971) or from linear stochastic theory using the autocorrelogram, cross-correlogram or spectral analysis (Jenkins and Watts, 1968). Spectral analysis is a one-way analysis of variance as a function of frequency or scale. Inter-relationships between variables can also be examined in the frequency domain. All of the methods are part of space-time systems analysis (e.g. Bennett, 1979). Recent reviews of spatial statistical analysis of soil properties have been given by Peck (1983), Warrick et al. (1986) and Nielsen and Bouma (1984). A historical review or spectral analysis of landforms has been given by Pike and Rozema (1975). Problems associated with the scale of observation in the application of autoregressive theory to topographic data have been discussed by Thornes (1973).

The purpose of this paper is to present results from studies where the spatial distribution of soil properties and the influence of topography at different scales are the focal point. Three data sets will be given which illustrate a variety of scaling problems. The relationship of the soil properties to soil processes occurring over long and short time scales will be discussed.

Description of data sets

The Delhi data set will be used to illustrate the problem of soil variability at different scales. The Weyburn data set will be used to examine the relationship between topographic parameters and soil variability on a scale of < 50 m. With the Floral data we examine the relationship between soil electrical conductivity (a function of the water and salt content of soil and soil texture) and landform across a small agricultural drainage basin. Spatial autocorrelations were calculated using equations given by Davis (1973). Power spectra, co-spectra, and coherency estimates were calculated using smoothed Fourier methods (Brillinger, 1981; Otnes and Enochson, 1978). A summary of the equations used has been given by Kachanoski et al. (1985a).

Delhi data set

The site is located on the Delhi Agricultural Research Station in Southern Ontario, Canada. The samples have been collected, analysed and tabulated by Protz et al. (1987) as part of an ongoing study on soil variability. Three transects, 1000 m, 100 m, and 10 m long, were sampled in both virgin forest and cultivated fields. The site was selected for its general uniformity in topography over the study area ( < 1% slope). Sampling intervals were 10 m, 1 m, and 0.1 m in the 1000 m, 100 m and 10 m long transects respectively. There were 100 points, termed profiles, where soil horizons were measured from the soil surface to a depth of about 2 m. For the 1000 m and 100 m long transects, soil pits were excavated at each location and the soil horizon thicknesses recorded and soil cores taken. The 10 m transect data were obtained from a 10 m trench, sampling the face of the trench every 0.1 m. Approximately 3600 soil horizon samples were taken.

Horizon thicknesses for profiles along the forest transects are illustrated in Figure 9.1. The statistical moments of the major soil horizons (Table 9.1) indicate that a significant amount of variability exists on a scale of < 10 m. However, the 10 m scale also shows a certain amount of 'smoothing out' compared with the other transects (Figure 9.1). Similar variations were present in the cultivated transects. The 1000 m transect has a significantly greater horizon and solum thickness than the other transects, which is due to the first 38 profiles. The last 62 profiles of the 1000 m transect have the same moments as those of the 10 m and 100 m transect. 

Figure 9.1 Horizon thickness for three scales of observation for the Delhi forest transects

Table 9.1 Means and coefficients of variation (%) for the thickness of soil horizons in the Delhi data set

Soil Horizon	Transect
	10 m	100 m	1000 m
LFH	0.026(46)*	0.030(45)	0.034(39)
Ah	0.119(19)	0.118(30)	0.128(19)
B	0.754(50)	0.720(52)	0.840(43)
SOLUM	0.899(42)	0.868(44)	0.100(35)
*mean (% coeff. of variation)

The solum thickness is the depth from the surface to the C_K horizon boundary and is an indication of long term leaching and weathering intensity where soils are freely drained. The autocorrelation values of solum thickness of the native forest (Figure 9.2) for the 10 m and 1000 m transects do not approach 1.0 as the lag approaches zero. As indicated by the autocorrelation functions of the 100 m and 1000 m transects, a discontinuity at the origin exists which is called the nugget effect. The nugget effect can be attributed to either sampling error or variance microstructure (Rendu, 1978). Because the sampling error for horizon thickness would be small, the nugget must be due to the small scale spatial dependence which is illustrated by the autocorrelogram of the 10 m transect. The 1 m and 10 m sampling intervals are too wide to pick up the short range ( < 2 m) variance structure indicated by the 0.1 m sampling interval. The strong autocorrelation for the 10 m transect at small lags ( < 0.5 m) and the convergence of the autocorrelation to 1.0 as the lag approaches zero (Figure 9.2) indicate the 0.1 m sampling interval is sufficiently small to characterize the spatial structure of the horizon continuum. In addition, the strong auto correlation peak at the 2 m lag indicates that sudden increases in solum thickness occur, on average, every 2 m. Other smaller fluctuations in the autocorrelogram are present and would not normally be considered significant. However, the solum thickness autocorrelograms (Figure 9.3) for the 10 m transect from both the forested and cultivated site (the transects were about 1 km apart) are remarkably similar.

The stability of the spatial variance structure over space (in this case over 1 kg) is encouraging since it suggests a regularity or stationarity of the soil forming processes which can be used for spatial prediction using linear stochastic models (e.g. kriging or auto regressive equations). Examples of using spatial variance structures of soil properties for constructing linear prediction models such as Kriging and autoregressive equations have been given by Webster and Burgess (1980, 1980b), Sisson and Wierenga (1981), Vieira et al., (1981), Vauclin et al., (1983), and others. However the non-stationarity of the mean such as in profiles 1 to 38 of the 1000 m transect, (Figure 9.1) may limit the transferability of these variance structures.

Figure 9.2 Autocorrelation for solum thickness for 3 scales of observation for the Delhi forest data

Figure 9.3 Comparison of solum thickness autocorrelation for the cultivated and forest 10 m transects (Delhi data, dotted line is the cultivated site)

Figure 9.4 Autocorrelation of the LFH horizon thickness for the Delhi 10 m forest transect

The average 2 m repetitive nature of the solum thickness is consistent with the concept of a pedon. In this case the pedon would have average lateral dimensions of 1 m (half the average cycling frequency). The 30-60 m section of the l00 m transect (Figure 9.1) indicates that the A horizon increases in thickness over a short distance (44-46 m) while the lower horizons are affected at successively wider and wider distances with soil depth. This is consistent with a point source of water at the surface and subsequent radial flow affecting the weathering intensity over wider and wider areas with depth. The above scenario is not entirely consistent with the concept of a pedon since the scale of observation for horizons is changing with depth. Solum thickness is the cumulative effect of soil forming conditions over thousands of years. The surface litter layer (LFH horizon) is a more dynamic horizon and is an expression of more recent soil forming conditions. The autocorrelation of the LFH thickness (Figure 9.4) indicates a spatial pattern which is different from the mineral horizon.

Weyburn data set 

The study area is situated approximately 30 km east of Weyburn, Saskatchewan, Canada. The parent material of the soil is glacial till underlain by a second till formation of earlier age. The uppermost till is approximately 1-2 m thick and was deposited about 20000 years ago B.P. The second till lies below the first till, has significantly different physical and chemical characteristics than the overlying till, and was deposited approximately 38 000 years ago B.P . The stratigraphic surface between the two tills were characterized by a sand-gravel layer of variable thickness.

Soil cores (76 mm diameter, 2 m long) were taken every 1 m in a number of transects in both a native (never cultivated) grassland and a portion of the field which had been cultivated for 30 yrs (Figure 9.5). The soil cores were separated into horizons, horizon thickness recorded, and a variety of soil properties measured including bulk density, soil water, and organic carbon. Around each sampling point, elevation readings were taken every 1 m in a 3 m x 3 m grid. Microtopographic parameters including gradient and vertical curvature were calculated at each point using the methods of Young and Evans (1978). The study area is situated on a relatively smooth (overall slope < 0.5% ) upper slope portion. Thus, the soil variability would be classified as 'within slope variability' according to soil survey. The soil type is similar as that studied by King et al. (1983) who concluded that within slope variability was so great that soil survey could not practicably delineate meaningful units other than upper (shallow soils) and lower (deep soils) slope areas. A more detailed analysis and description of the Weyburn data set is contained in recent papers (Kachanoski et al., 1985a, 1985b, 1985c; Singh et al., 1985; Selles et al., 1985).

Figure 9.5 Sampling pattern for Weyburn data

Statistical moments (Table 9.2) and the correlation matrix (Table 9.3) for selected soil properties of the native grassland are given. Although the A horizon thickness and mass were significantly correlated to vertical surface curvature, only 14% of the variability could be explained. Thickness and mass of the B horizon were not significantly correlated (5% level) to any microtopographic parameter. Multiple correlation between microtopographic parameters and horizon thickness accounted for 25% and 9% of the variability of the A horizon thickness and B horizon thickness respectively. In contrast, the depth to the intertill sand-gravel layer was significantly correlated (1% probability level) to thickness and mass of the B horizon, but not the A horizon.

Table 9.2 Means and coefficients of variation (%) for the Weyburn data set

Soil Property	A horizon	B horizon	Solum (A + B)
Thickness (m)	0.172(28)*	0.215(33)	0.384(34)
Bulk density (Mg m -3)	1.14(8)	1.47(6)	1.32(5)
Mass (Mg m -2)	0.197(29)	0.315(34)	0.512(22)
Total Carbon (Kg m -2)	7.2(24)	4.0(41)	11.2(22)
*mean (% coeff. of variation

The power spectra of curvature and A horizon mass (native grassland) are remarkably similar (Figure 9.6) with a strong increase in variance power centered around 0.14 cycles m ^-1 (7 m period) and secondary increases around 0.32 cycles m ^-1 (3 m period). The spectrum of A horizon mass does not show a significant increase in variance at low frequencies (large scale) which indicates that the variability is mainly on a scale of  £ 10 m. Coherency estimates indicated significant correlation (5% level) between curvature and A horizon mass at the 7 m cycling frequency.

The power spectra of B horizon thickness and depth to the intertill sand lens are given in Figure 9.7. Both spectra have a significant peak at 0.23 cycles m ^-1 (4.5 m period) and significant coherency (correlation) at that frequency. The concentration of variance at 0.23 cycles m ^-1 is not found in the A horizon spectrum.

Table 9.3 Correlation between topography, depth to sand layer and soil properties for the Weyburn data set

Soil Property		Microtopography Properties			Sand lens depth
		Elevation	Gradient	Curvature
A horizon	Thickness	0.14	0.17	0.38**	0.15
	Bulk density	-0.18	0.34*	0.12	-0.22
	Mass	0.07	0.27	0.37**	0.08
	Total Carbon	0.01	0.08	0.14	0.12
B horizon	Thickness	-0.08	-0.03	-0.27	-0.41 **
	Bulk density	-0.41**	0.46**	-0.01	-0.03
	Mass	-0.16	0.06	-0.28	-0.41
	Total carbon	-0.14	-0.06	-0.10	-0.38
Solum	Thickness	0.01	0.08	-0.03	-0.30*
	Bulk density	-0.50**	0.50**	-0.17	-0.35*
	Mass	-0.12	0.19	-0.07	-0.35*
	Total Carbon	-0.10	0.02	-0.03	-0.17
* sign. at 0.05 probability
** sign. at 0.01 probability

Figure 9.6 Comparison of the A horizon mass and surface curvature spectra, native Weyburn site

The data suggest that processes controlling A horizon formation do not have the same spatial variance relationships as those controlling the formation of B horizons. The A horizon variability is affected by the variability of surface curvature which would influence the redistribution of rainfall and the moisture in the rootzone of the vegetation and, therefore, both biomass production and leaching potential. At greater depths, the factor affecting the redistribution of soil water appears to be depth to the intertill sand layer. For unsaturated flow, which is the normal condition at this site, the sand interlayer would act as an impedance resulting in higher moisture conditions and thus increased weathering potential in the layers above it. The net result is the significant negative correlation between depth to the stratigraphic sand layer and B horizon thickness and mass.

The amount of accumulated organic carbon and the formation of A and B horizons in prairie soils (Mollisols) are intimately related. In native conditions, for a given slope position, vegetative growth will ultimately determine soil organic carbon levels because it will control the type and amount of organic matter added to the soil and will vary mainly with moisture conditions. The similarity of the spectra for total solum carbon and depth to intertill sand layer (Figure 9.8) confirms the influence this layer is having on the long term moisture regime.

Figure 9.7 Comparison of the spectra for B horizon mass, and depth to sand layer spectra (Weyburn data)

The interrelationships among horizon mass, curvature, and depth to the sand layer produce an interesting relationship between surface curvature and total solum carbon. The coherency and cospectra for these variables (Figure 9.9) indicate significant (1-5% level) negative correlation for scales greater than 4.0 m (frequency <0.25 cycles m ^-1) and significant (1-5% level) positive correlation on a scale less than 4 m (frequency > 0.25 cycles m ^-1 ). The positive correlation at one scale cancels the negative correlation at the other scale so the overall (standard) correlation is essentially zero (i.e. r = 0.03, Table 9.3).

Figure 9.8 Comparison of the spectra of Total Solum Carbon and depth to sand layer (Weyburn data)

Because the native grassland was sampled at the end of the growing season approximately 25 days after the last rainfall, the measured soil water content values would represent the permanent wilting point (PWP) of the profile. The semi-variograms of soil water (Figure 9.10) indicate significantly different spatial correlation ranges for the A horizon and B horizon PWP.

The relationships discussed so far have been for the native grassland. The spectrum of total solum carbon for the cultivated (30 yrs) portion of the field (Figure 9.11) also indicates increases in variance at the 7 m and 3 m periods which are the same as those of the curvature spectrum in the native field. The spectrum of surface curvature in the cultivated field showed no spectral peaks at these scales due to smoothing and infilling from the tillage operations (Kachanoski, 1985c). Coherency analysis (Figure 9.11) indicated the native surface curvature was correlated to solum properties in the cultivated field, while present day local curvature was not correlated.

Figure 9.9 Coherency and Co-spectrum between total solum carbon and surface curvature. (Dashed line indicates 5% probability level.)

Figure 9.10 Semi variograms for A horizon and B horizon soil water (Weyburn data)

Figure 9.11 Comparison of the total solum carbon spectra from the native and cultivated fields including the coherency to curvature (Weyburn data, dashed line = 5% probability)

Floral data set
The previous data sets have dealt with soil properties that take medium to long periods of time to change perceptibly. In this data set, the spatial distribution of soil electrical conductivity (EC) is examined in relation to topographic parameters measured over a small drainage basin. The EC is influenced by soil moisture and dissolved salts, dynamic properties strongly affected by recent hydrological processes.

The study area is a 57 ha agricultural drainage basin approximately 30 kg east of Saskatoon, Saskatchewan, Canada. The soils are silt loam to sandy loam textured Mollisols. The basin was part of the Saskatchewan Research Council basin study (1960-1969). Contour lines (0.3 m interval) had been determined for the basin. The contour map of the basin was digitized and interpolated to produce an elevation grid (6 m interval). Topographic parameters were calculated in a manner similar to the Weyburn data set. EC readings (0-1.0 m depth) were taken on a 6 m grid in eight 540 m long transects which ran perpendicular to the main drainage pattern, using a non-contacting electromagnetic probe (Geonics EM-38). A diagram of the drainage basin showing the 1.5 m contour lines is given in Figure 9.12.

Figure 9.12 Floral drainage basin with location of first and last EC transects

A summary of the average topographical parameters of the transects is given in Table 9.4. At the basin scale there is a significant correlation ( < 0.001 probability level) between EC and surface elevation (Figure 9.13). Multiple correlation indicated that the deviations from the regression line (Figure 9.13) were significantly correlated to local surface curvature (i.e. measured on an areal scale = 0.014 ha). Elevation and curvature together accounted for approximately 60% of the variability of measured EC values. The negative relationship between EC and elevation is probably a reflection of the regional (basin) groundwater flow system which would influence the overall soil moisture and salt accumulation regimes. Spatial variations in local curvature cause fluctuations away from this regional relationship. This can be seen in transect seven (Figure 9.14) where there is a general increase in EC with decrease in elevation, with local fluctuation superimposed. The spectra of local curvature and EC for this transect (Figure 9.15) indicate that for fluctuations on a scale < 25 m ( > 0.24 cycles per 6 m) the variance decompositions are almost identical.

Table 9.4 Mean elevation and soil conductivity for the Floral Basin Transects

Basin Transect	Relative Elevation (m)	Soil Conductivity  (0-0.5 m, dSm -1 x 100)
1	23.0	14.1(46)*
2	19.3	14.2(51)
3	17.2	29.0(54)
4	16.9	28.4(46)
5	16.4	27.0(54)
6	15.1	48.4(47)
7	13.8	52.5(37)
8	11.3	62.1(27)
*Mean (% Coeff. of variability)

Figure 9.13 Regression relationship between EC and surface elevation, Flora drainage basin

Figure 9.14 Relative elevation and EC measurements for the seventh transect (Floral data)

Figure 9.15 Comparison of the EC and surface curvature spectra for the seventh transect (Floral data)

The increase in EC variance at low frequencies (large scale) is due to the general increase (trend) in EC over the length of the transect (Figure 9.14) which, as mentioned earlier, can be related to the general decrease in elevation. Coherency analysis indicated significant correlation between elevation and EC at low frequencies, and curvature and EC at high frequencies. Strictly speaking, a power spectrum is not defined if a trend is present, however, in practice this is not a serious problem because the spectrum still isolates the variance contribution at different scales and the presence of a trend will only result in increased variance at low frequencies (Jenkins and Watts, 1968). The spatial variance distribution of soil EC is clearly a combination of large scale (low frequency) variations in elevation and smaller scale (high frequency) variations in surface curvature. In other transects, (e.g. transect eight in Figure 9.16), the relationship between local changes in surface curvature and EC are so apparent that spectral analysis is not necessary. 

Figure 9.16 Comparison of EC and surface curvature  for the eighth transect (Floral data)

SUMMARY

 The purpose of any spatial model is to simplify, organize, and extrapolate information about the soil system. Soil systems are particularly complex because of the large number of interacting variables. The utility of a simplified description of a space-time system will depend on how well the linkages between spatial units are described (Bennett, 1979).

Two frequently occurring spatial linkages are (1) hierarchical processes and (2) contiguity (lag) processes (Bennett, 1979). Anderson et al. (1983) merged the concept of hierarchical land classification systems and the level of integration concept of ecology (Rowe, 1961) and proposed a hierarchical classification of agro-ecosystems for the basis of compiling and managing data, and extrapolation into the future. Ecosystem levels include the pedon, polypedon, soil catena, and higher levels. Contiguity (lag) processes are governed according to lag, or the co-spatial dependency of adjacent spatial  units. The behaviour of larger regions is dependent on lagged diffusion terms between subregions. These lagged diffusion terms are frequently described by the partial differential equations of flow, flux, and potential in air and water movement (Bennett, 1978). Contiguity (lag) processes describe the law that everything is related but near things are related more than distant things. This is represented by the autocorrelograms, spectra, cospectra, etc. which have been presented for the data sets. The co-dependency of spatial processes due to spatial positioning and distribution has led to the suggestion that the erosional drainage basin, which accommodates the three dimensional characteristics of soil and soil related processes, should be adopted as the basic functional study unit (Vreeken, 1973; Huggett, 1973, 1982).

The data sets given in this chapter have been concerned with representing and describing the continuous distribution of soil properties by examining the spatial variance structure, that is the contiguity (lag) relationships. The nature of the contiguity (lag) relationships has been shown to be highly dependent on the spatial distribution of topographic parameters (at different scales) which control the transfer of mass and energy in the landscape. This is as it should be, because soil properties are dependent variables, determined by the major soil forming factors.

A complex system such as the soil will have a variety of linkage processes occurring. The hierarchical agro-ecosystem classification (Anderson et al., 1983) requires that the integration levels (scales) be related to each other in a definable and quantitative sense. Integration levels are related in a definable way in that the objects at one level are the environment for those at the next lowest level. The levels, however, are defined in a descriptive manner and are thus morphological models (Dijkerman, 1974). More realistically, levels should be related in a functional and quantitative manner; the cascading system defined by the path followed by flows of energy and mass (Dijkerman, 1974) seems most appropriate. If the quantitative criteria are based on the physical laws determining the flow of mass and energy to higher and lower levels then the drainage basin becomes an obvious level of integration. Sub-basins, geomorphic elements within basins (valley sides, uplands, flood- plains), soil catenas, and convex (shallow soil) and concave (thick soil) slopes as described by King et al. (1983) are all successively lower levers of integration. Soil catenas rather than surface form or topography alone is an important element, in that the properties of the soil (texture, structure) become part of the system. Local changes in surface curvature can be used to determine even smaller levels of integration within a slope position as indicated by the relationship between EC and surface curvature on a scale < 25 m (Figure 9.15, Floral data set) and surface curvature and A horizon thickness (Figure 9.7, Weyburn data set). In many cases this level will coincide with the scale of a pedon. The interpolation can continue to smaller scales using the spatial variance (contiguity) structure of representative elementary volumes (REV) of  porous media (Wagenet, 1984; Bouma, 1984). The REV represents the smallest sample (soil) in a practical sense since it determines the stability of the phenomenological relationships used to describe the flux of energy or mass (e.g. Fick's Darcy's and Fourier's law) which have transfer coefficients with area (L^-2) units.

The pedon is a concept for describing the REV of soil horizon properties. Unfortunately, as the data sets indicate, different properties have different spatial variance relationships. Even horizontal thickness has different spatial scales depending on surface or subsurface influences. In the Weyburn data the native A horizon was influenced by the local curvature of the present surface curvature while the B horizon was influenced by the surface which existed 20000 B.P. (i.e. the depth to the sand layer which lies on the older till at the site). In addition because mass and energy transfer with depth is not uni-directional (vertical), surface properties influence what is happening both ahead and behind in the horizontal plane. These co-spatial interrelationships pose significant difficulties in interpreting and using mean values for soil horizon properties of a pedon for input into existing soil process models at a more general scale. The problems associated with sampling on a pedon basis have resulted in most spatial process models using point sampling techniques. However, similar problems exist at this sampling scale with respect to the size of the core used.

In many cases it will not be possible to identify the factors influencing the spatial variance structure of soil variables. For example, the local surface curvature which controlled the variance distribution of the native A horizon (Weyburn data) is no longer present because of the effects of cultivation. Aggregation to the next level of integration (convex upper, etc.) as done by King et al. (1983) on this soil type may be necessary. Prediction at smaller scales may require measurement of the microvariance structure which is time-consuming, but may have wider application if the variance structure remains stationary over larger areas as in the Delhi data (Figure 9.3). In some soil systems such as the Floral basin it may be possible to explain the local fluctuations of soil processes based solely on digital topographic analysis.

Almost all point soil process models will require water content as an input. On this basis alone, it is necessary that 'real' space-time modelling uses digital terrain analysis methods. In addition, because surface landform shape is a good indication of the spatial variance distribution of soil properties, it seems reasonable that finding methods of describing the joint distribution of soil and topography should continue to be a high priority.

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The electronic version of this publication has been prepared at the M S Swaminathan Research Foundation, Chennai, India.