ALBERT

Notes: The pseudo cross-variogram can be used for cokriging two or more soil properties when few or none of the sampling locations have values recorded for all of them. The usual estimator of the pseudo cross-variogram is susceptible to the effects of extreme data (outliers). This will lead to overestimation of the error variance of predictions obtained by cokriging. A solution to this problem is to use robust estimators of the pseudo cross-variogram, and three such estimators are proposed in this paper.The robust estimators were demonstrated on simulated data in the presence of different numbers of outlying data drawn from different contaminating distributions. The robust estimators were less sensitive to the outliers than the non-robust one, but they had larger variances. Outliers tend to obscure the spatial structure of the cross-correlation of the simulated variables as described by the non-robust estimator.The several estimators of the pseudo cross-variogram were applied to a multitemporal data set on soil water content. Since these were obtained non-destructively, direct measurements of temporal change can be made. A prediction subset of the data was subsampled as if obtained by destructive analysis and the remainder used for validation. Estimators of the auto-variogram and pseudo cross-variogram were applied to the prediction data, then used to predict the change in water content at the validation sites by cokriging. The estimation variances of these predictions were best calculated with a robustly estimated model of coregionalization, although the validation set was too small to conclude that the non-robust estimators were unsuitable in this instance.

Notes: A wavelet is a compact analysing kernel that can be moved over a sequence of data to measure variation locally. There are several families of wavelet, and within any one family wavelets of different lengths and therefore smoothness and their corresponding scaling functions can be assembled into a collection of orthogonal functions. Such an assemblage can then be applied to filter spatial data into a series of independent components at varying scales in a single coherent analysis. The application requires no assumptions other than that of finite variance. The methods have been developed for processing signals and remote imagery in which data are abundant, and they need modification for data from field sampling. The paper describes the theory of wavelets. It introduces the pyramid algorithm for multiresolution analysis and shows how it can be adapted for fairly small sets of transect data such as one might obtain in soil survey. It then illustrates the application using Daubechies’s wavelets to two soil transects, one of gilgai on plain land in Australia and the other across a sedimentary sequence in England. In both examples the technique revealed strongly contrasting local features of the variation that had been lost by averaging in previous analyses and expressed them quantitatively in combinations of both scale and magnitude. Further, the results could be explained as the spatial effects of change in topography or geology underlying the variation in the soil.〈section xml:id="abs1-2"〉〈title type="main"〉Analyse et éclairissement sur la variation du sol en utilisant les ondelettes RésuméUne ondelette est un noyau compact d’analyse qu’on peut passer sur une séquence de données pour quantifier la variation localement. Plusieurs familles d’ondelettes existent. Chaque famille est caractérisée par des fonctions d’échelle de longueurs d’ondes et de degrés de lissage différents, le tout constituant un ensemble de fonctions orthogonales. L’application de ces fonctions sur des données spatiales est une méthode d’analyse unique et cohérente qui permet de filtrer des données spatiales en identifiant des composantes indépendentes à différentes échelles. L’application n’éxige que l’hypothése d’une variance bornée. Ces méthodes ont été developpées pour le traitement de signaux qui contiennent énormement de données comme des images de télédétection. Elles necessitent des modifications avant de les appliquer sur des données d’échantillonage provenant du terrain. Cet article décrit la théorie d’ondelettes. Il introduit l’algorithme pyramidal pour une analyse à quelques résolutions. Il montre ensuite comment on peut adapter l’algorithme pour un ensemble de données peu nombreuses comme celles qu’on peut obtenir lors d’une prospection du sol sur des transects.Les resultats montrent l’application des ondelettes de Daubechies à deux transects pédologiques, lepremier situé sur une plaine marquée de gilgaï en Australie, et le second à travers une séquence de sediments jurassiques en Angleterre. Dans les deux exemples, la technique a révélé des contrastes bien marqués de certain traits locaux qui ont été totalement occultés dans des analyses antérieures basées sur le calcul des moyennes. La technique des ondelettes permet aussi une quantification de ces traits locaux, en fonction de l’échelle d’observation et de la magnitude. De plus, ces résultats peuvent être interprétés par les effets d’une variation spatiale de la topographie ou de la géologie qui sont à l’origine de la variation du sol.

Notes: Complex spatial variation in soil can be analysed by wavelets into contributions at several scales or resolutions. The first applications were to data recorded at regular intervals in one dimension, i.e. on transects. The theory extends readily to two dimensions, but the application to small sets of gridded data such as one is likely to have from a soil survey requires special adaptation. This paper describes the extension of wavelet theory to two dimensions. The adaptation of the wavelet filters near the limits of a region that was successful in one dimension proved unsuitable in two dimensions. We therefore had to pad the data out symmetrically beyond the limits to minimize edge effects.With the above modifications and Daubechies's wavelet with two vanishing moments the analysis is applied to soil thickness, slope gradient, and direct solar beam radiation at the land surface recorded at 100-m intervals on a 60 × 101 square grid in south-west England. The analysis revealed contributions to the variance at several scales and for different directions and correlations between the variables that were not evident in maps of the original data. In particular, it showed how the thickness of the soil increasingly matches the geological structure with increasing dilation of the wavelet, this relationship being local to the strongly aligned outcrops. The analysis reveals a similar pattern in slope gradient, and a negative correlation with soil thickness, most clearly evident at the coarser scales. The solar beam radiation integrates slope gradient and azimuth, and the analysis emphasizes the relations with topography at the various spatial scales and reveals additional effects of aspect on soil thickness.

Notes: The general linear model encompasses statistical methods such as regression and analysis of variance (anova) which are commonly used by soil scientists. The standard ordinary least squares (OLS) method for estimating the parameters of the general linear model is a design-based method that requires that the data have been collected according to an appropriate randomized sample design. Soil data are often obtained by systematic sampling on transects or grids, so OLS methods are not appropriate.Parameters of the general linear model can be estimated from systematically sampled data by model-based methods. Parameters of a model of the covariance structure of the error are estimated, then used to estimate the remaining parameters of the model with known variance. Residual maximum likelihood (REML) is the best way to estimate the variance parameters since it is unbiased. We present the REML solution to this problem. We then demonstrate how REML can be used to estimate parameters for regression and anova-type models using data from two systematic surveys of soil.We compare an efficient, gradient-based implementation of REML (ASReml) with an implementation that uses simulated annealing. In general the results were very similar; where they differed the error covariance model had a spherical variogram function which can have local optima in its likelihood function. The simulated annealing results were better than the gradient method in this case because simulated annealing is good at escaping local optima.

Notes: This paper shows how the wavelet transform can be used to analyse the complex spatial covariation of the rate of nitrous oxide (N2O) emissions from the soil with soil properties that are expected to control the evolution of N2O. We use data on N2O emission rates from soil cores collected at 4-m intervals on a 1024-m transect across arable land at Silsoe in England. Various soil properties, particularly those expected to influence N2O production in the soil, were also determined on these cores.We used the adapted maximal overlap discrete wavelet transform (AMODWT) coefficients for the N2O emissions and soil variables to compute their wavelet covariances and correlations. These showed that, over the transect as a whole, some soil properties were significantly correlated with N2O emissions at fine spatial scales (soil carbon content), others at intermediate scales (soil water content) and others at coarse spatial scales (soil pH). Ammonium did not appear to be correlated with N2O emissions at any scale, suggesting that nitrification was not a significant source of N2O from these soils in the conditions that pertained at sampling.We used a procedure to detect changes in the wavelet correlations at several spatial scales. This showed that certain soil properties were correlated with N2O emissions only under certain conditions of topography or parent material. This is not unexpected given that N2O is generated by biological processes in the soil, so the rate of emission may be subject to one limiting factor in one environment and a different factor elsewhere. Such changes in the relationship between variables from one part of the landscape to another is not consistent with the geostatistical assumption that our data are realizations of coregionalized random variables.

Notes: Variograms of soil properties are usually obtained by estimating the variogram for distinct lag classes by the method-of-moments and fitting an appropriate model to the estimates. An alternative is to fit a model by maximum likelihood to data on the assumption that they are a realization of a multivariate Gaussian process. This paper compares the two using both simulation and real data.The method-of-moments and maximum likelihood were used to estimate the variograms of data simulated from stationary Gaussian processes. In one example, where the simulated field was sampled at different intensities, maximum likelihood estimation was consistently more efficient than the method-of-moments, but this result was not general and the relative performance of the methods depends on the form of the variogram. Where the nugget variance was relatively small and the correlation range of the data was large the method-of-moments was at an advantage and likewise in the presence of data from a contaminating distribution. When fields were simulated with positive skew this affected the results of both the method-of-moments and maximum likelihood.The two methods were used to estimate variograms from actual metal concentrations in topsoil in the Swiss Jura, and the variograms were used for kriging. Both estimators were susceptible to sampling problems which resulted in over- or underestimation of the variance of three of the metals by kriging. For four other metals the results for kriging using the variogram obtained by maximum likelihood were consistently closer to the theoretical expectation than the results for kriging with the variogram obtained by the method-of-moments, although the differences between the results using the two approaches were not significantly different from each other or from expectation. Soil scientists should use both procedures in their analysis and compare the results.

Notes: We used the wavelet transform to quantify the performance of models that predict the rate of emission of nitrous oxide (N2O) from soil. Emissions of N2O and other soil variables that influence emissions were measured on soil cores collected at 256 locations across arable land in Bedfordshire, England. Rate-limiting models of N2O emissions were constructed and fitted to the data by functional analysis. These models were then evaluated by wavelet variance and wavelet correlations, estimated from coefficients of the adapted maximal overlap discrete wavelet transform (AMODWT), of the fitted and measured emission rates.We estimated wavelet variances to assess whether the partition of the variance of modelled rates of N2O emission between scales reflected that of the data. Where the relative distribution of variance in the model is more skewed to coarser scales than is the case for the observation, for example, this indicates that the model predictions are too smooth spatially, and fail adequately to represent some of the variation at finer scales. Scale-dependent wavelet correlations between model and data were used to quantify the model performance at each scale, and in several cases to determine the scale at which the model description of the data broke down. We detected significant changes in correlation between modelled and predicted emissions at each spatial scale, showing that, at some scales, model performance was not uniform in space. This suggested that the influence of a soil variable on N2O emissions, important in one region but not in another, had been omitted from the model or modelled poorly. Change points usually occurred at field boundaries or where soil textural class changed.We show that wavelet analysis can be used to quantify aspects of model performance that other methods cannot. By evaluating model behaviour at several scales and positions wavelet analysis helps us to determine whether a model is suitable for a particular purpose.

Notes: Emissions of gases from the soil are known to vary spatially in a complex way. In this paper we show how such data can be analysed with the wavelet transform. We analysed data on rates of N2O emission from soil cores collected at 4-m intervals on a 1024-m transect across arable land at Silsoe in England. We used a thresholding procedure to represent intermittent variation in N2O emission from the soil as a sparse wavelet process, i.e. one in which most of the wavelet coefficients are not significantly different from zero. This analysis made clear that the rate of N2O emission varied more intermittently on this transect than did soil pH, for which many more of the wavelet coefficients had to be retained. This account of intermittent variation motivated us to consider a class of random functions, which we call wavelet random functions, for the simulation of spatially intermittent variation. A wavelet random function (WRF) is an inverse wavelet transform of a set of random wavelet coefficients with specified variance at each scale. We generated intermittent variation at a particular scale in the WRF by specifying a binormal process for the wavelet coefficients at this scale. We showed by simulation that adaptive sampling schemes are more efficient than ordinary stratified random sampling to estimate the mean of a spatial variable that is intermittent at a particular scale. This is because the sampling can be concentrated in the more variable regions. When we simulated values that emulate the intermittency of our data on N2O we found that the gains in efficiency from simple adaptive sampling schemes were small. This was because the emission of N2O is intermittent over several disparate scales. More sophisticated adaptive sampling is needed for these conditions, and it should embody knowledge of the relevant soil processes.

Notes: The magnitude of variation in soil properties can change from place to place, and this lack of stationarity can preclude conventional geostatistical and spectral analysis. In contrast, wavelets and their scaling functions, which take non-zero values only over short intervals and are therefore local, enable us to handle such variation. Wavelets can be used to analyse scale-dependence and spatial changes in the correlation of two variables where the linear model of coregionalization is inadmissible.We have adapted wavelet methods to analyse soil properties with non-stationary variation and covariation in fairly small sets of data, such as we can expect in soil survey, and we have applied them to measurements of pH and the contents of clay and calcium carbonate on a 3-km transect in Central England. Places on the transect where significant changes in the variance of the soil properties occur were identified. The scale-dependence of the correlations of soil properties was investigated by calculating wavelet correlations for each spatial scale. We identified where the covariance of the properties appeared to change and then computed the wavelet correlations on each side of the change point and compared them. The correlation of topsoil and subsoil clay content was found to be uniform along the transect at one important scale, although there were significant changes in the variance. In contrast, carbonate content and pH of the topsoil were correlated only in parts of the transect.