Optimal interpolation and isarithmic mapping of soil properties. VI. Disjunctive kriging and mapping the conditional porbability

Disjunctive kriging is a technique for estimating values of spatially distributed random variables by combining data non-linearly and with minimum variance. Data are transformed using Hermite polynomials to a normal distribution, and the resultant transformation is assumed to produce a stationary bivariate normal distribution for all pairs of data. Conditional probabilities that the true values exceed or are less than a specified critical threshold can then be calculated. For soil survey both the estimated values of soil properties and their associated probabilities can be mapped isarithmically as aids to decisions on land management.
The technique is described and illustrated from two case studies. The pH, exchangeable potassium and available phosphorus are mapped over 77 ha of the Broom's Barn Farm, and in the eastern Border Region of Scotland the available copper and cobalt in the topsoil are mapped. The estimates are similar to those obtained by simple (linear) kriging. The conditional probabilities were estimated for values falling short of the minimum recommended thresholds, and maps of them showed where farmers would be advised to remedy or forestall deficiencies in lime and plant nutrients and to take prophylactic measures for their livestock.     

An amended functional leaching model applicable to structured soils. II. Model application

The performance of a functional leaching model (Hall, 1993) is compared with leaching data from two lysimeter experiments with soils of contrasting texture using sodium bromide and potassium chloride as the non-reactive solutes. The model parameters are optimized using the solute elution curves as standards and compared with the physical properties of the soil. A good match with the measured discharge of both water and solute was achieved for both soils using the moisture release characteristics to define the pore volume available for mobile and immobile water. The results indicate that preferential flow takes place through even coarse-textured soils but that there is negligible diffusive exchange of solute between water passing through the macropores and the rest of the soil.     

An analysis of crack pattern in clay soil: its density and orientation

The structure of much clay soil is angular blocky, and the pore space consists of a three-dimensional network of narrow, approximately planar cracks. This paper presents a means of describing such a network in terms of its density and orientation. Undisturbed samples are impregnated with resin containing a fluorescent dye. Plane sections of known azimuth and elevation are cut through them and photographed under ultra-violet light. The photographs are projected onto a screen and the spacings between successive cracks along linear probes are measured in several directions. The sets of distances obtained are then analysed. The method was applied to subsoil of the Windsor series. The distribution of inter-crack distances appeared to be exponential but with a dearth of distances less than about 1 mm. The crack pattern was judged to be the outcome of a Poisson process, but one in which some of the very narrow wedges of soil had collapsed. The crack network was perceptibly anisotropic, and was modelled as two superimposed patterns, one isotropic and the other a set of approximately parallel cracks. The orientation of the parallel component was estimated from the sections. The cracks were found to be aligned parallel to the contour of the land and to dip into the hill slope at angles ranging from about 25° to 75° to the horizontal. The intensities in the soil of the two components were estimated as 0.18 mm^?1 for the isotropic component and 0.2 mm^?1 normal to the parallel component, equivalent to 0.1 mm^?1 for a random direction.     

OPTIMAL INTERPOLATION AND ISARITHMIC MAPPING OF SOIL PROPERTIES III CHANGING DRIFT AND UNIVERSAL KRIGING

Universal kriging is a form of interpolation that takes account of local trends in data when minimizing the error associated with estimation. The presence of such trends, or drifts as they are known, is identified qualitatively, and their form found quantitatively by structural analysis, which simultaneously estimates semi-variances of the differences between the drift and actual data. The resulting semi-variograms are then used for the interpolation. The method was applied to measurements of electrical resistivity made in the soil at 1 m intervals at Bekesbourne, Kent. Analysis showed that the data could be adequately represented as a series of linear drifts over distances of 4 m to 8 m and with negligible nugget variance. Semi-variances of residuals from the drift were computed, and used to krige missing values and so complete an isarithmic map of the site. The method is by no means universally applicable in soil survey, mainly because of the large nugget variances usually encountered. These effectively prevent any distinction between constant and changing drift. They arise in part because measurements are made on small widely separated volumes of soil. Universal kriging is likely to be profitable only where measurements are made on contiguous volumes of soil or after substantial bulking.     

MULTIPLE DISCRIMINANT ANALYSIS IN SOIL SURVEY

Multiple discriminant analysis can be used to distinguish to best advantage several multivariate, more or less polythetic, classes of soil. The method allows profiles to be allocated consistently and most appropriately to existing groups for which representatives have already been chosen. The method is described in geometric terms and is applied to the allocation of profiles in both well known and little known areas, South Central England and Sabah. For the areas in England good separation between soil classes was demonstrated; in Sabah the classes of the reconnaissance survey were less distinct. Representative profiles were chosen for all existing classes and the remainder of the profiles allocated using discriminant functions. In all cases a better classification resulted in that there was less dispersion within classes and hence better separation between them. Provided conformant data can be collected in routine survey the method promises to allow a good working combination of man and machine: the soil surveyor to decide the classes desired, and the computer to handle large quantities of data and make routine allocation.     

Hypokalaemic episodic polymyopathy in cats fed a vegetarian diet

A previously undocumented hypokalaemic condition with a cyclical nature, comprising acute bouts of polymyopathy followed by spontaneous recoveries, is described in the cat. Cats being fed a high protein vegetarian diet developed recurrent episodes of polymyopathy, characterised by ventroflexion of the head and neck, stiff forelimb gait, lateral head-resting and generalised muscle weakness. Plasma potassium concentrations (mean +/- standard deviation) were reduced from 3.28 +/- 0.33 mmol/l at the beginning of the experiment to 2.45 +/- 0.24 mmol/l during bouts of myopathy. This hypokalaemia was associated with increased creatine kinase activities indicative of muscle damage, and decreased urinary potassium concentrations, and was caused by insufficient dietary potassium. Cats that received the same diet supplemented with potassium did not develop hypokalaemic polymyopathy. Spontaneous recoveries of affected cats were not associated consistently with increases in plasma potassium concentrations. Plasma taurine concentrations decreased and glutamic acid increased markedly in all cats fed the experimental diet. There was no evidence of thiamin deficiency associated with the high glutamic acid intake. Veterinarians should be aware that hypokalaemic cats, and in particular those on potassium-deficient diets, may show cyclical disease with episodes of polymyopathy recurring after periods of spontaneous clinical recovery. This condition in cats may be a useful animal model for familial hypokalaemic periodic paralysis in humans.     

Scale-dependent correlation between topsoil copper and cobalt concentrations in Scotland

The soil of south-east Scotland is locally deficient in copper and cobalt. Measurements from nearly 3000 fields for which the soil association is known were analysed to study the coregionalization of the two elements and to assess the influence of parent material on the metals' concentrations. The experimental auto- and cross-variograms revealed distinct local (1.5 km) and regional (20 km) scales of spatial variation. A combination of indicator variograms of the soil associations had the same spatial structures, suggesting that parent material influences the concentrations of the metals. The coregionalization between copper and cobalt was modelled as a linear combination of three spatial structures. The resulting structural correlation coefficients showed the two elements to be fairly strongly positively correlated at the regional scale. Kriging allowed determination and mapping of each spatial component; these maps were then compared with the spatial distribution of soil associations in the region. An analysis of variance was performed before and after filtering out the nugget and short-range spatial components. Classification by soil association (parent material) accounted for a large proportion of the variance at the regional scale, suggesting that the parent material contributes substantially more to the trace element content of the soil than had been thought earlier.     

COMPUTER-BASED SOIL MAPPING OF SMALL AREAS FROM SAMPLE DATA

A 1400 m × 600 m rectangular area of north Berkshire had been sampled at the intersections of a 100 m square grid, and seventeen properties of the soil profile measured. The eighty-four sampling sites were classified numerically to produce a hierarchy, and the classes of the upper part of the hierarchy mapped. The mapped classes became increasingly fragmented as the number of classes increased. At the 3-class level, the classes corresponded to character space clusters and class fragmentation was not serious. Principal-component analysis of the sample data yielded a first component that accounted for 40 per cent of the total variance and well represented the field characters used for soil classification. An isarithm map of the first component shows how the soil changes gradually over the landscape in good agreement with a soil-series map made by free survey.