PEDOMETRIC STUDIES OF SOIL VARIATION Alex. McBratney A summary of pedometric work by Alex. McBratney  and collaborators (1997 - 2003) In the following numeric superscripts correspond to the numbers given in the List of Papers at the end.  Introduction I present here a summary of  novel written contributions my co-authors and I have made to the study of soil variation in six areas. (a) Numerical soil classificationThe development of new approaches to soil classification and the mapping of soil classes, particularly based on multivariate statistics, numerical taxonomy and fuzzy sets. (b) Fine-scale (<1 m) soil variation and soil structureThe development of novel methods for analysing and modelling soil pore structure, based on image analysis, mathematical morphology and fractal geometry. (c) Soil variability and geostatisticsThe development of methods of spatial analysis for describing and predicting field soil attributes, principally based on regionalised-variable theory (geostatistics). (d) Precision Agriculture (e) Pedogenetc modeling (f) Pedotransfer functions Together, these may be considered under the general heading of Pedometrics, a neologism which may be explained by the following quotation, 'the term `pedometrics' was first proposed to me by Professor McBratney, and I know that he had in mind the whole quantitative approach to the study and description of soil, especially soil in the field.' Pedometrics has been defined^§ as 'that area of science concerned with the description, classification, formation and distribution of soil by quantitative mathematical and statistical techniques.' After a general background statement the pedometric developments are classified and discussed briefly under the three headings given above. Some of the work was done in Scotland, England, the Netherlands, and Wisconsin, but by far the majority pertains to eastern Australia, particularly southern Queensland and New South Wales (NSW). General^1-4 Soil exhibits continuous change in space and time. This variation is usually considered to be problematic in relation to sampling effort, quality of information and for optimal soil management. Pedometrics arose from a general need to quantify many of the conventional approaches to soil description, classification and mapping. This was required to evaluate precision and accuracy of statements about soil attributes and classes, to make procedures more reproducible, and results more comparable. Although pedometrics may appeal to a more objective approach, that might not be upheld philosophically and practically - there are still subjective decisions in which attributes are described, how they are described or measured and how they are analysed. It is hoped at least the approach may better allow the falsification of hypotheses. Pedometrics attempts to understand the soil quantitatively and to provide and purvey accurate and precise information concerning it. The first paper^1 summarises what pedometrics attempts to do in relation to environmental soil management; especially the way the quality and quantity of soil information may be modified by soil variation and various kinds of uncertainty. As such the paper acts as a proem to the scope of pedometrics. It reviews developments and concepts, suggesting a way forward by synthesising new meta-tools from basic ones.

	(a). Numerical soil classification ^5-8 The work has focused on the development of new approaches to soil classification and the mapping of soil classes, particularly based on multivariate statistics, numerical taxonomy and fuzzy sets. 2(a) Theoretical and methodological Soil classification has long been problematic. The traditional problems have been; (i) the definition of an individual soil, (ii) the creation and definition of soil classes using several attributes simultaneously, (iii) dealing with soil as a continuum and, (iv) a quantitative genetic basis for classification. The work presented here deals mainly with (ii) and (iii), and (i) and (iv), to a much lesser extent. 2a(i) The definition of an individual soil Soil horizons are the results of pedological processes and classes of soil horizon descriptions are seen as fundamental individuals for soil classification. The pedological (FitzPatrick^+) and the soil-physical (Bouma^~) approaches to soil horizon classes have close parallels and are useful in applications because of the added flexibility they bring.^12 2a(ii) The creation and definition of soil classes using several attributes simultaneously  Expert systems and numerical classification in general were reviewed and their relative ability to improve soil classification systems were discussed. The inductive and deductive elements of expert systems are seen as corresponding to the class establishment and class assignment phases of classification. To date, numerical classification has been useful in the analysis and organisation of local low-level soil data. It has been largely untried at the higher global levels of soil classification because of a lack of suitable data and scientific commitment. Numerical classification has a potentially useful part to play in establishing soil classes and generating rules for assignment in expert systems.^4 2a(iii) Dealing with soil as a continuum  The major contributions here have been the development of multivariate fuzzy classification ^2,3,5,7,8,9,10,11and allocation^4,13 techniques for soil studies. The theory of fuzzy sets was first introduced as a possible way of dealing with the continuity of climatic data.^3 The method of fuzzy c- or k- means was used to create fuzzy groups for two sets of climatic data, one from Australia and the other from China. The resulting groups showed the inherent continuity of the data and a reasonable geographical contiguity; this showed promise for soil variation. The problem of choosing c or k (the number of groups) and m (the degree of fuzziness) simultaneously was discussed.^3 A subsequent paper showed that classifications resulting from the method of fuzzy k-means are not necessarily suitable for prediction of properties from class memberships. A new method was developed to incorporate the concept of extragrades in order to enhance its predictive power.^5 The first illustrations of the new method were for artificial data sets.^5 Applications to soil were then illustrated^7,8 and software called MacFUZZY developed.^11A method for allocating new individuals to continuous soil classes was developed and detailed examples given. This method has advantages over more conventional methods of soil identification.^13Finally, a method of mapping fuzzy classes by optimally predicting k+1 continuous classes onto a fine grid. The resulting raster maps can be manipulated in various ways to produce isogram, choropleth or chorochromatic maps.^10,9 2a(iv) A quantitative genetic basis for classification Multivariate soil-environment interactions have been explored using multivariate techniques.^6,12 The robustness of various multivariate ordination methods were applied to elucidate the relationship between a hillslope soil and its environment in a South Australian subcatchment. Canonical correspondence analyses were found to be less attractive than the linear methods (i.e., principal-components and redundancy analyses) because interrelations among soil variables and between soil and the landform attributes are more linear than unimodal.^6 2(b) Applied (Papers 14-19) 2b(i) Fuzzy classes in alluvial landscapes The applications have dealt mainly with the use of fuzzy sets for the creation of soil classes. The work has mainly been used in river valleys in southern Queensland and northern New South Wales where soil stratigraphic relationships are complex. The use of layer classes is particularly apposite in those areas. In the alluvial plain of the Lockyer Valley, southeast Queensland, 133 soil profiles on a 25-metre interval transect were used to generate fuzzy groups centroids and memberships for both profiles and horizons.^14,15 Distribution of fuzzy group membership on the transect reflected changes in landscape position, soil profile classes, soil stratigraphic units and soil factors related to internal drainage. The distribution of horizon groups generated by fuzzy classification showed the difficulty of creating a fixed number of profile classes based on homologous relationships between horizons. A subsequent study^16,17,18 of detailed soil physical and chemical data from 210 soil profile sites arranged on an equilateral triangular grid with approximately 2.8 km spacing between sites was carried out in the lower Namoi valley of northern NSW. Two major separate classifications were devised.^16,17,18 The first used soil sample layers and fuzzy k-means algorithms. A soil layer classification was defined largely in terms of quantitative chemical and textural attributes. These layer classes showed reasonable contiguity and could be identified and related to pedological, geological and geomorphological entities. The stratigraphic nature of the landscape was also better represented by the fuzzy k-means soil layer classification, as evidenced by the soil layer sequence profiles identified and defined.^16,17 The second classification used point information at each soil profile site in a fuzzy land suitability evaluation. The use of fuzzy-set operators provided definitions and quantification of concepts such as land versatility and rotation suitability.^16,18 2b(ii) Allocation to fuzzy classes The novel method of soil allocation^13 was used on the Australian Great Soil Groups (GSG) system of classification, which is implicitly a fuzzy classification. Central concepts had not been defined explicitly. Centroids were generated, together with fuzzy group membership, using data from the 147 soil profile descriptions in the Handbook of Australian Soils.^*,19 Some of the GSG, such as Siliceous Sands and Red and Brown Hardpan Soils, were divided into their component parts for better and easier quantification and allocation. The centroids of GSG were examined, and the method of fuzzy k-means with extragrades was then used to allocate unknown profiles to the GSG. The results showed that the system is intuitively reasonable.

	(b). Fine-scale variation and soil structure ^9-14 This has concerned the development of methods for analysing fine-scale (<1m) soil geometry and pore structure using image analysis based on stereology, mathematical morphology and fractal geometry.  3(a) Theoretical and methodological (Papers 20-32) 3a(i) Soil micromorphometry A pedometric method for the volumetric estimation of weathering of an in-situ soil profile was devised. The technique, based on stereology and the statistics of spatial point processes, uses the distances between, and the size of, zircon grains in undisturbed, horizontal, thin sections.^20 3a(ii) Rapid analysis of soil pore structure (RAPS) The utility of soil pore structure measurements has been restricted by the lack of an appropriate spatial model and the tardiness of specimen preparation and data generation. A completely new methodology was devised.  Specimen preparation A technique for the production of an undisturbed planar face through soil in the field condition was developed. Field impregnation with epoxy resin containing a UV-fluoresecent dye, that cures in moist soil, is followed by sawing, laboratory impregnation of the exposed face, and grinding back beyond the original surface to a smooth finish.^21 Subsequent modifications included making large undisturbed vertical soil pore-structure monoliths field-impregnated with epoxy resin,^25,26 and the use of two differently coloured fluoresecent dyes to separate surface-connected pore space from the rest.^27 For comparative purposes, an ancillary laboratory method using continuous flow and dehydration of acetone or 1,4 dioxane followed by slow impregnation was developed.^34 Image analysis Grinding is followed by digitisation, and digital grey-level image segmentation to produce a binary^21 or ternary^27,28 image. The method was modified to show roots contrasted from the soil matrix by enhancing autofluorescence using UV light and appropriate filters.^24,26 Mathematical morphology and stereology Using digital binary images of vertical planes in the soil and assuming a vertically non-stationary, horizontally isotropic model for the pore geometry, estimates of several pore-structure attributes were made using contiguous parallel linear probes. The attributes are porosity, surface area, and the pore star length and solid star length.^22 A C program, STRUCTURA, was developed for estimating these attributes.^23Depth functions of these attributes are modelled statistically using Laplacian smoothing splines. These depth functions provide a means of comparing pore geometry across soil types or between treatments.^22 The method was modified for measuring three-component images; namely, field-impregnated pore space, laboratory-impregnated pore space, and soil solid. Eleven pore-structure attributes were generated from the three-component image.^27,28 3a(iii) Soil fractal geometry Fractal geometry has proven to be a plausible model for soil structure.^29,30,31,32 Soil aggregates have a fractal mass and have scale-dependent bulk density, i.e., larger soil aggregates have a smaller bulk density.^29,31 The mass fractal dimension of soil (Dm) may be calculated from the bulk density-aggregate size relationship. There are several fractal dimensions that may be measured on an undisturbed soil structure, e.g., the mass-fractal (Dm), the surface-fractal (D), and the spectral-fractal dimensions (d). These dimensions were estimated on images of six soil thin sections and Dm (1.65-1.85) and d (1.24-1.67) in particular, were shown to discriminate the different structures.^32 Fractal theory showed how fractal geometry mediates physical processes such as diffusion within the soil.^32 The work on soil pore structure was integrated by building a two-dimensional geometric model using fuzzy random fractal sets. The model produces realisations of the fuzzy, random set 'porosity' using a kind of simulated annealing.^30 3(b) Applied (Papers 33-42) The rapid image analysis (RAPS) method described above was used to compare the effect of management practices on soil structure of Red-Brown Earths and Grey Clays - the two most economically important soil groups for agriculture in eastern Australia. Additionally a field method for measuring soil structural stability was devised. The effect of soil surface cover was also modelled. Tillage trials on Red-Brown Earths at Cowra and Forbes, NSW were investigated to assess the long-term effect of direct drilling with more conventional tillage practices.^33,35 Infiltration, bulk density and image analysis data lead to similar conclusions about changes in pore structure. Under direct drilling there was greater macroporosity (>0.175 mm diameter in section) and greater root and faunal activity.^33In a subsequent study on Red-Brown Earths at West Wyalong, NSW, adjacent 90-year cultivated and never-cultivated areas of land were compared for a variety of chemical and physical properties. The properties that changed most as a result of clearing and cultivation were pH, electrical conductivity, organic carbon, nitrogen and profile macroporosity. These changes are due to various processes including soil mixing, fertilising, crop-growing and an altered water balance. The cultivated soil was found to be much less diverse than the adjoining virgin soil.^40 In a later study in the Goulburn Valley of NE Victoria, physical and chemical soil properties were compared in adjacent biodynamic and conventionally managed Red-Brown Earths under improved, summer-irrigated dairy pastures. The biodynamic soil had greater macroporosity to a depth of at least 420 mm and larger organic matter content in the upper 50 mm.^41 The structural degradation of Grey Clays as a result of growing cotton under furrow irrigation has resulted in declining cotton yields. This degradation may be ameliorated with suitable management. A field experiment at Warren, NSW, assessed the long-term effects of repeated deep ripping plus gypsum, as compared with the more traditional shallow cultivation, on the physical condition of a sodic Grey Clay. The soil was sampled seven years after the experiment began. Results from both image analysis and conventional soil measurements indicated there was a structurally degraded zone from 25 to 60 cm in the shallow cultivated treatment. This was in contrast to a more porous and weaker soil in the deep ripped plus gypsum treatment that produced slightly greater cotton yields.^36 A further study compared an adjacent cotton ridge and furrow. Examination of vertical and horizontal faces showed the high variability of aggregate size to a depth of 250 mm under the ridge and the presence of fine cracks connecting the compacted furrow with relatively loose soil under the ridge.^38Image analysis methods have difficulty measuring the short-term dynamics of soil structure, e.g., stability to wetting. An instrument was devised for measuring the soil structural stability in situ, to be used in conjunction with RAPS. Stability is assessed by measuring the change in intrinsic permeability that may occur when the soil is permeated by air and then by water. A ratio of the intrinsic permeability of the soil to these two fluids is then used as a structural stability index, with unity representing complete stability. Baseline data using a relatively stable sand medium, and in-situ site data from several soil types in NSW, showed that the index ranged from 6 to 574 and varied significantly between soil types and management practices.^42 Some tentative work attempted to measure roots along with soil structure. The techniques of sample preparation, image acquisition and root measurement are outlined. Consideration was given to orientation of soil sections, and the consequences for root-distribution measurements.^39 Finally, the effectiveness of crop-residue cover was measured and modelled.Change in the spatial distribution of cover through the fallow season was shown for both images from photographs and cover simulations.^37

	(c). Soil variability and geostatistics ^15-39 Perhaps the most significant work has been done on the development of methods of spatial analysis for describing and predicting field soil attributes. This has been principally based on regionalised-variable theory (geostatistics), but more recently other statistical methods have been used.  4(a) Theoretical and methodological The developments can be considered under; (i) spatial dependence and the variogram, (ii) spatial prediction, and (iii) soil sampling strategies.  4a(i) Spatial dependence and the variogram The variogram is central to geostatistics and is the single most important tool in geostatistical applications to soil. One of the first published applications was to soil measured along a transect on the University of Aberdeen's Tillycorthie Estate.^43 Mathematical functions for variograms must be conditional negative semi-definite, and there are only a few families of simple function that meet this demand. These include the transitive spherical and exponential models. If more complex models are needed they can be formed by combining two or more simple models.^49 The Akaike Information Criterion is recommended for selecting the best model from several plausible ones to describe the observed variation in soil, though for kriging it may be desirable to validate the chosen model.^51 For routine analysis, fitting models to sample variograms by weighted least-squares approximation is preferred to the more demanding statistical procedure of maximum likelihood.^49 The cross-variogram was introduced to soil science, to describe spatial variation between soil variables.^48A particular problem is the efficient estimation of the spatial variogram when no information on magnitude or scale of variation of a variable is available. Practical, spatial and parsimonious considerations lead to a design-model-based approach of staggered designs on linear transects in three orientations. Estimation of the parameters of the accompanying variance-components model was done by restricted maximum likelihood (REML).^56 4a(ii) Spatial prediction The principle of optimal estimation using regionalised-variable theory (kriging) is only one of a variety of two-dimensional soil prediction methods available. The methods were classified as global or local, interpolating or non-interpolating, and smooth or non-smooth, predictors.^50 Two carefully-designed topsoil pH surveys compared the prediction performance of these methods.^50,53 Interpolating methods were generally very poor predictors. Of the non-interpolating methods, Laplacian smoothing splines and kriging generally performed best. Estimates of variance derived from models which assume independent errors were greater than estimates of variance derived from neighbouring pairs of data sites, suggesting that short range correlations are real.^50 In the second study, outliers seriously affected the performance of all prediction method. Smooth interpolators, Laplacian smoothing splines, and intrinsic random functions all behaved problematically. Universal kriging using parameter estimates obtained by REML was consistently best.^53 The principle of optimal estimation using regionalised-variable theory was extended from that of a single soil property to situations where there are two or more spatially interdependent ones. The technique of co-kriging was illustrated by a case study of soil particle-size distribution at Woburn Experimental Farm, Bedfordshire, England. There was a strong co-regionalisation between soil textural variables. This allowed topsoil silt to be estimated and mapped by co-kriging more precisely than by kriging topsoil silt alone.^48 Another study showed that splitting the region into two geomorphic zones resulted in a 65% reduction in mean absolute deviation and a 14% reduction in mean squared deviation of predicted clay contents compared with a global model.^55 Ancillary information provides a way of incorporating soil knowledge into the spatial prediction process. For example, landform attributes may be derived from digital elevation models (DEMs). From a pedological point of view these should be useful for soil prediction. Ancillary information may be incorporated in the prediction process through co-kriging or combinations of Multiple Linear Regression (MLR), Generalised Linear Model (GLM) and Generalised Additive Model (GAM). The latter two , generally termed regression-kriging methods, solve the problems of non-linearity, multiple exogenous variables and discontinuous scales of measurement of ancillary variables. There was a clear advantage in using the regression-kriging methods, over ordinary- and co- kriging, on those variables which had a small correlation with the landform attributes.^57 In subsequent studies, GAM and GAM-kriging proved to be the most superior methods when corroborated independently using test sites, and based on mean error (ME) and root mean square error (RMSE) of prediction.^58,59 Geostatistical approaches have the advantage of giving estimates of uncertainty of the predicted soil surface. A Dutch study calculated the average ratio of actual square errors of prediction to the estimated kriging variances at 75 test locations. These were used to adjust the kriging variance estimates on the regular grid to get more realistic estimates. These empirically-derived kriging variances were then used to construct an isolinear map, with three sets of contours, allowing the user to obtain realistic confidence limits, as well as the predicted value, for any point on the map.^54 4a(iii) Soil sampling strategies A model-based sampling strategy was developed for soil survey in which an individual soil property is of interest and can be measured. It depends on first determining accurately the variogram for the property.^44,46,56From the variogram, estimation variances can be found for any combination of block size and sampling density by the methods of kriging. An equilateral triangular configuration of sampling points is best where variation is isotropic, but a square grid at the same density will usually be preferred for convenience.^44,46 A FORTRAN IV program, OSSFIM, was written to carry out this procedure automatically.^45 The method was modified for estimating the regional mean^47 and for local estimation of a variable with spatially dense observations of ancillary variables.^48

	4(b) Applied (Papers 60-65) 4b(i) Soil-landscape models The strong relationships between landscape attributes and hydrological and pedogeomorphic processes have enabled terrain-analytical techniques for spatial prediction of soil properties. Two examples from southeastern Australia showed the potential usefulness of explicit and repeatable statistical methods that exploit the correlation between the soil and environmental factors.^65This lead to the conclusion that integration of models describing processes continuous between the soil system and the surrounding environment, linking pedological, earth surface and ecological processes is essential to an understanding of soil genesis. Prediction and management of current and future environmental change at global, regional, and local scales is also possible.^60 4b(ii) Soil pH sampling A general model was devised for soil pH measurement that includes instrumental drift, random measurement error, and random and correlated spatial variation. Methods for estimating these four components are described in detail. For soil pH in water, instrumental drift, random measurement error and random spatial variation (nugget effect) were greater than the corresponding quantities for soil pH in CaCl2. Measurement error and nugget effect were of a similar size. The kriging method was modified to take into account the four-component model. For measuring soil chemical attributes, grid layouts should be supplemented by additional sites for the estimation of short-range variation, laboratory sampling designs should include controls, and field measurements should be adjusted for instrumental drift prior to being used for spatial prediction.^61 4b(iii) Soil contamination The total concentrations of lead, zinc, copper and cadmium in the topsoil of Glebe, an inner-city suburb of Sydney, were investigated. Stratified-random sampling was conducted within one-hectare square areas taking two samples at one-metre separation from each stratum as a means of investigating spatial variation. Fifty percent of total lead, zinc and copper concentrations and two and a half percent of cadmium concentrations were greater than the Australia & New Zealand Environmental Conservation Council guidelines of 300, 200, 20, and 3 mg/kg, respectively. Some spatial clustering was evident and a geostatistical analysis showed some large high-risk areas. Soil disturbance and distance from roads explained 24% of the variation in total lead concentration, 15% in total zinc and copper concentrations, and 13% in total cadmium concentration.^62 Two types of probability sampling schemes for assessing the degree and extent of soil contamination were considered.^63 First, schemes for areas that have received a constant and widespread input of contaminants, e.g., irrigation of waste water, spreading of sewage sludge, aerial fallout, contaminated fertiliser, are amenable to geostatistical analyses. Two-phase sampling schemes can be designed to minimise the uncertainty in the degree of contamination for a given effort. Secondly, for areas where contamination is expected to be localised to 'hotspots', e.g., around point sources, some kind of purposive sampling is to be preferred to a geostatistical one. Difficulties occur in cases where areas expected to have a larger probability of being contaminated are unknown a priori. A sequential sampling procedure seems appropriate and may be optimised to find local maxima using a simplex procedure. This approach is much more efficient if field-testing procedures are available.^63 4b(iv) Trace metals The topsoil of more than 3 500 fields in the Border Region of Scotland had been sampled and copper and cobalt measured to identify places where these metals are deficient for grazing animals. Classification at soil association level accounted for 22% of the variance in copper, representing the effect of parent material. Soil on the Old Red Sandstone, especially that of the Mountboy and Hobkirk associations, contained less than 1 mg/kg of copper, regarded as deficient, in many places.^64 The variograms for both copper and cobalt were isotropic and appeared to combine three components of variation; a field and farm component extending up to 3 km, a long-range or geological component extending to 15 km, and a non-spatial component which accounted for 69% of the variance in cobalt.^64 Isarithmic maps of the metal concentrations and their error variances were constructed. Several small areas of copper-deficient soil, associated with the Old Red Sandstone sediments and Fluvio-Glacial sands, were identified. There were also small patches with large concentrations of copper, some near towns probably resulting from pollution, or associated with volcanic rocks. Much of the region had soluble cobalt concentrations less than 0.25 mg/kg, the deficiency threshold.^64

	(d) Precision Agriculture ^40-55 (e) Pedogenetic modeling ^56-57 (f) Pedotransfer functions ^58-63 EPILOGUE The work presented here are a few tentative steps towards the goal of creating a mechanistic theory, rather than simply a quantitative description, of soil variation. This inevitably will entail building quantitative models of soil genesis at various space and time scales.

	LIST OF PEDOMETRIC PAPERS  used in review (up to 1997) General (1) A.B. M^cBratney (1992). On variation, uncertainty and informatics in environmental soil management. Australian Journal of Soil Research 30, 913_935. (a) Numerical soil classification (a) Theoretical and methodological (2) R. Webster and A.B. M^cBratney (1981). Soil segment overlap in character space and its implications for soil classification. Journal of Soil Science 32, 133_147.** (3) A.B. M^cBratney and A.W. Moore (1985). Application of fuzzy sets to climatic classification. Agricultural and Forest Meteorology 35, 165_185.*** (4) M.B. Dale, A.B. M^cBratney and J.S. Russell (1989). On the role of expert systems and numerical taxonomy in soil classification. Journal of Soil Science 40, 223_234.** (5) J.J. de Gruijter and A.B. M^cBratney (1988). A modified fuzzy k_means method for predictive classification. pp. 97_104. In H.H. Bock (Ed.), Classification and Related Methods of Data Analysis. Elsevier, Amsterdam.** (6) I.O.A. Odeh, D.J. Chittleborough and A.B. M^cBratney (1991). Elucidation of soil_landform interrelationships by canonical ordination analysis. Geoderma 49, 1_32.** (7) A.B. M^cBratney and J.J. de Gruijter (1992). A continuum approach to soil classification by modified fuzzy k_means with extragrades. Journal of Soil Science 43, 159_175.*** (8) I.O.A. Odeh, A.B. M^cBratney and D.J. Chittleborough. (1992). Soil pattern recognition with fuzzy c_means: application to classification and soil_landform interrelationships. Soil ScienceSociety of America Journal 56, 505_516. ** (9) I.O.A. Odeh, A.B. M^cBratney and D.J. Chittleborough (1992). Fuzzy c-means and kriging for mapping soil as a continuous system. Soil Science Society of America Journal 56, 1848_1854.** (10) A.B. M^cBratney, J.J. de Gruijter and D.J. Brus (1992). Spacial prediction and mapping of continuous soil classes. Geoderma 54, 39_64.*** (11) A.W. Ward, W.T. Ward, A.B. M^cBratney and J.J. de Gruijter (1992). MacFUZZY A program for data analysis by generalised fuzzy k_means on the Macintosh. Division of Soils Divisional Report 116, CSIRO Australia, Melbourne. 49 pp. + disk.* (12) A.B. M^cBratney (1993). Some remarks on soil horizon classes. Catena 20, 427_430. (13) A.B. M^cBratney (1994). Allocation of new individuals to continuous soil classes. Australian Journal of Soil Research 32, 623_633.  Numerical soil classification (b) Applied (14) B.Powell, A.B. M^cBratney and D.A. MacLeod (1991). The application of fuzzy classification to soil pH profiles in the Lockyer Valley, Queensland, Australia. Catena 18, 409_420.** (15) B.Powell, A.B. M^cBratney and D.A. MacLeod (1992). Fuzzy classification of soil profiles and horizons from the Lockyer Valley, Queensland, Australia. Geoderma 52, 173_197.** (16) A.B. M^cBratney and J. Triantafilis (1993). Fuzzy soil layer, profile and suitability classification in the lower Namoi valley, New South Wales, Australia. pp. 515_517. In H.J.P. Eijsackers and T. Hamers (Eds), Integrated Soil and Sediment Research: A Basis for Proper Protection, Selected Proceedings of the First European Conference on Integrated Research for Soil and Sediment Protection and Remediation (EUROSOL), 6_12 September 1992, Maastricht, The Netherlands. Kluwer Academic Publishers, Dordrecht.** (17) J. Triantafilis and A.B. M^cBratney (1993). Application of continuous methods of soil classification and land suitability assessment in the lower Namoi valley. Chapter 3: Soil layer classes. pp. 28_64. Division of Soils Divisional Report 121, CSIRO Australia, Melbourne.** (18) J. Triantafilis and A.B. M^cBratney (1993). Application of continuous methods of soil classification and land suitability assessment in the lower Namoi valley. Chapter 5: Land suitability assessment. pp.96_123. Division of Soils Divisional Report 121, CSIRO Australia, Melbourne.** (19) S.A. Mazaheri, A.J. Koppi and A.B. M^cBratney. (1995) A fuzzy allocation scheme for the Australian Great Soil Groups Classification system. European Journal of Soil Science 46, 601_612. **

	(b) Fine-scale variation and soil structure (a) Theoretical and methodological (20) C.J. Moran, A.B. M^cBratney and A.J. Koppi (1988). A micromorphometric method for estimating change in the volume of soil induced by weathering. Journal of Soil Science 39, 357_373.** (21) C.J. Moran, A.B. M^cBratney and A.J. Koppi (1989). A rapid method for analysis of soil macropore structure. I. Specimen preparation and digital binary image production. Soil Science Society of America Journal 53, 921_928.** (22) A.B. M^c Bratney and C.J. Moran (1990). A rapid method for analysis of soil macropore structure. II. Stereological model, statistical analysis and interpretation. Soil Science Society of America Journal 54, 509_515.*** (23)^§ C.J. Moran and A.B. M^cBratney (1991). STRUCTURA: A C program for estimating attributes of two_phase, heterogeneous structures digitized from planar specimens. Computers and Geosciences 17, 335_350.** (24) P.J. Commins, A.B. M^cBratney and A.J. Koppi (1991). Development of a technique for the measurement of root geometry in the soil using resin_impregnated blocks and image analysis. Journal of Soil Science 42, 237_250. ** (25) A.J. Koppi and A.B. M^cBratney (1991). A basis for soil mesomorphological analysis. Journal of Soil Science 42,139_146.** (26) A.B. M^cBratney, C.J. Moran, J.B. Stewart, S.R. Cattle and A.J. Koppi (1992). Modifications to a method of rapid assessment of soil macropore structure by image analysis. Geoderma 53, 255_274. ** (27) C.J. Moran and A.B. M^cBratney (1992. Acquisition and analysis of three_component digital images of soil pore structure. I. Method. Journal of Soil Science 43, 541_549.** (28) C.J. Moran and A.B. M^cBratney (1992). Acquisition and analysis of three_component digital images of soil pore structure. II. Application to seed beds in a fallow management trial. Journal of Soil Science 43, 551_566.** (29) A.B. M^cBratney (1993). Comments on "Fractal dimensions of soil aggregate_size distributions calculated by number and mass". Soil Science Society of America Journal 57, 1393. (30) A.B. M^cBratney and C.J. Moran (1994). Soil pore structure modelling using fuzzy random pseudofractal sets. pp. 495_506. In A.J. Ringrose-Voase and G.S. Humphreys (Eds), Soil Micromorphology: Studies in Management and Genesis. Proceedings of the 9th International Working Meeting on Soil Micromorphology. Elsevier, Amsterdam.** (31) A.N. Anderson and A.B. M^cBratney (1995). Soil aggregates as mass fractals. Australian Journal of Soil Research33, 757_772. ** (32) A.N. Anderson, A.B. M^cBratney and E.A. FitzPatrick. (1996). Soil mass, surface and spectral fractal dimensions estimated from thin section photographs. Soil Science Society of America Journal 60, 962_969. **

	Fine-scale variation and soil structure (b) Applied (33) C.J. Moran, A.J. Koppi, B.W. Murphy and A.B. M^cBratney (1988). Comparison of the macropore structure of a sandy loam surface soil horizon subjected to two tillage treatments. Soil Use and Management 4, 96_102.* (34) C.J. Moran, A.B. M^cBratney, A.J. Ringrose_Voase and C.J. Chartres (1989). A method for the dehydration and impregnation of clay soil. Journal of Soil Science 40, 569_575. ** (35) P.P. Cavanagh, A.J. Koppi and A.B. M^cBratney (1991). The effects of minimum cultivation after three years on some physical and chemical properties of a red_brown earth at Forbes, NSW. Australian Journal of Soil Research 29, 263_270.* (36) M.R. Wild, A.J. Koppi, D.J. M^cKenzie and A.B. M^cBratney (1992). The effect of tillage and gypsum application on the macropore structure of an Australian Vertisol used for irrigated cotton. Soil and Tillage Research 22, 55_71.* (37) C.J. Moran and A.B. M^cBratney (1992). Image measurement and modelling of the two_dimensional spatial distribution of wheat straw. Geoderma53, 201_216.** (38) D.C. McKenzie, A.J. Koppi, C.J. Moran and A.B. M^cBratney (1994). A pragmatic role for image analysis when assessing compaction in vertisols. pp. 669_675. In A.J. Ringrose-Voase and G.S. Humphreys (Eds), Soil Micromorphology: Studies in Management and Genesis. Proceedings of the 9th International Working Meeting on Soil Micromorphology. Elsevier, Amsterdam.* (39) J. B. Stewart, C.J. Moran and A.B. M^cBratney (1994). Measurement of root distribution from sections through undisturbed soil specimens. pp. 507_514. In A.J. Ringrose-Voase and G.S. Humphreys (Eds), Soil Micromorphology: Studies in Management and Genesis. Proceedings of the 9th International Working Meeting on Soil Micromorphology. Elsevier, Amsterdam.** (40) S.R. Cattle, A.J. Koppi and A.B. M^cBratney (1994). The effect of cultivation on the properties of a Rhodoxeralf from the wheat/sheep belt of New South Wales. Geoderma 63, 215_225. * (41) J.A. Lytton-Hitchins, A.J. Koppi and A.B. M^cBratney(1994). The soil condition of adjacent bio-dynamic and conventionally managed dairy pastures in Victoria, Australia. Soil Use and Management 10, 79_87. * (42) B.M. Whelan, A.J. Koppi, A.B. M^cBratney and W.J. Dougherty (1995). An instrument for the in-situ characterisation of soil structural stability based on the relative intrinsic permeabilities to air and water. Geoderma 65, 209_222.*

	(c) Soil variability and geostatistics (a) Theoretical and methodological (43) A.B. M^cBratney and R. Webster (1981). Spatial dependence and classification of the soil along a transect in northeast Scotland. Geoderma 26, 63_82.** (44) A.B. M^cBratney, R. Webster and T.M. Burgess (1981). The design of optimal sampling schemes for local estimation and mapping of regionalised variables. I. Theory and method. Computers and Geosciences 7, 331_334.*** (45)^¶ A.B. M^cBratney and R. Webster (1981). The design of optimal sampling schemes for local estimation and mapping of regionalised variables. II. Program and examples. Computers and Geosciences 7, 335-365. *** (46) T.M. Burgess, R. Webster and A.B. M^cBratney (1981). Optimal interpolation and isarithmic mapping of soil properties. IV. Sampling strategy. Journal of Soil Science 32, 643_659.** (47) A.B. M^cBratney and R. Webster (1983). How many observations are needed for regional estimation of soil properties? Soil Science 135, 177_183.*** (48) A.B. M^cBratney and R. Webster (1983). Optimal interpolation and isarithmic mapping of soil properties. V. Co_regionalization and multiple sampling strategy. Journal of Soil Science 34, 137_162.*** (49) A.B. M^cBratney and R. Webster (1986). Choosing functions for semi_variograms of soil properties and fitting them to sampling estimates. Journal of Soil Science 37, 617_639.*** (50) G.M. Laslett, A.B. M^cBratney, P.J. Pahl and M.F. Hutchinson (1987). Comparison of several spatial prediction methods for soil pH. Journal of Soil Science 38, 325_341.** (51) R. Webster and A.B. M^cBratney (1989). A note on the use of the Akaike Information Criterion for choosing models for variograms of soil properties. Journal of Soil Science 40, 493_496.** (52) I.O.A. Odeh, A.B. M^cBratney and D.J. Chittleborough (1990). Design of optimal sample spacings for mapping soil using fuzzy k_means and regionalized variable theory. Geoderma 47,93_122.** (53) G.M. Laslett and A.B. M^cBratney (1990). A further comparison of spatial methods for predicting soil pH. For Soil Science Society of America Journal 54, 1553_1558.** (54) A.K. Bregt, A.B. M^cBratney and M.C.S. Wopereis (1991). Construction of isolinear maps of soil attributes with empirical confidence limits. Soil Science Society of America Journal 55, 14_19.** (55) A.B. M^cBratney, G. A. Hart and D. M^cGarry (1991). The use of region partitioning to improve the representation of geostatistically mapped soil attributes. Journal of Soil Science 42, 513_531.*** (56) A.N. Pettitt and A.B. M^cBratney (1993). Sampling designs for estimating spatial variance components. Applied Statistics 42, 185_209.** (57) I.O.A. Odeh, A.B. M^cBratney and D.J. Chittleborough (1994). Spatial prediction of soil properties from landform attributes derived from a digital elevation model Geoderma 63, 197_214.** (58) I.O.A. Odeh, A.B. M^cBratney and D.J. Chittleborough (1995). Further results on spatial prediction of soil properties from landform attributes derived from digital elevation models: heterotopic cokriging and regression-kriging models. Geoderma 67, 215_226. ** (59) I.O.A. Odeh, A.B. M^cBratney and B.K. Slater (1997). Predicting soil properties from ancillary information: non-spatial models compared with geostatistical and combined methods. pp. 1008_1019. In E.Y. Baafi and N.A. Schofield (Eds), Geostatistics Wollongong '96. Vol. 2. Kluwer Academic Publishers, Dordrecht, The Netherlands.**

	Soil variability and geostatistics (b) Applied (60) B.K. Slater, K. M^cSweeney, A.B. M^cBratney, S.J. Ventura and B. Irvin (1994). A spatial framework for integrating soil-landscape and pedogenic models. pp. 169-185. In R.B. Bryant and R.W. Arnold (Eds), Pedogenic Modeling. Special Publication, Soil Science Society of America, Madison, Wisconsin.* (61) G.M. Laslett and A.B. M^cBratney (1990). Estimation and implications of instrumental drift, measurement error and nugget variance of spatial soil attributes _ a case study for soil pH. Journal of Soil Science 41, 451_471.** (62) J.A. Markus and A.B. M^cBratney (1996). An urban soil study: heavy metals in Glebe, Australia. Australian Journal of Soil Research 34, 453_465. ** (63) A.B. M^cBratney and G.M. Laslett (1993). Sampling schemes for contaminated soil. pp. 435_439. In H.J.P. Eijsackers and T. Hamers (Eds), Integrated Soil and Sediment Research: A Basis for Proper Protection, Selected Proceedings of the First European Conference on Integrated Research for Soil and Sediment Protection and Remediation (EUROSOL), 6_12 September 1992, Maastricht, The Netherlands. Kluwer Academic Publishers, Dordrecht.*** (64) A.B. M^cBratney, R. Webster, R.G. McLaren and R.B. Spiers (1982). Regional variation of extractable copper and cobalt in the topsoil of south_east Scotland. Agronomie 2, 969_982.*** (65) I.O.A. Odeh, P.E. Gessler, N.J. McKenzie and A.B. M^cBratney (1994). Using attributes derived from digital elevation models for spatial prediction of soil properties. pp. 451_463. In I. Bishop (Ed), Resource Technology '94: New Opportunities _ Best Practice. Proceedings of Resource Technology '94 Conference, 26_30 September 1994, Melbourne, Australia. Centre for Geographic Information Systems and Modelling, University of Melbourne, Australia.*