Pseudo Cross-variograms Research Articles

Covariance models for multivariate random fields resulting from pseudo cross-variograms

So far, the pseudo cross-variogram is primarily used as a tool for the structural analysis of multivariate random fields. Mainly applying recent theoretical results on the pseudo cross-variogram, we use it as a cornerstone in the construction of valid covariance models for multivariate random fields. In particular, we extend known univariate constructions to the multivariate case, and generalize existing multivariate models. Furthermore, we provide a general construction principle for conditionally negative definite matrix-valued kernels, which we use to reinterpret previous modeling proposals.

Characterization theorems for pseudo cross-variograms

AbstractPseudo cross-variograms appear naturally in the context of multivariate Brown–Resnick processes, and are a useful tool for analysis and prediction of multivariate random fields. We give a necessary and sufficient criterion for a matrix-valued function to be a pseudo cross-variogram, and further provide a Schoenberg-type result connecting pseudo cross-variograms and multivariate correlation functions. By means of these characterizations, we provide extensions of the popular univariate space–time covariance model of Gneiting to the multivariate case.

Geostatistics in the presence of geological boundaries: Exploratory tools for contact analysis

Identification and categorization of geological, geotechnical, or geometallurgical domains is a common practice in the modeling of mineral deposits, in order to account for the controls exerted by categorical variables (ore type, rock type or facies) on the quantitative variables of interest. The definition of a suitable geostatistical model for spatial prediction or simulation is commonly based on a contact analysis, aimed at determining whether or not these quantitative variables are spatially continuous across the domain boundaries. In this paper, we present several tools such as cross-to-direct variogram ratios, cross-correlograms, pseudo cross-variograms and lagged scatter plots that are useful for such an analysis and discuss their strengths and weaknesses. The tools are illustrated with applications to synthetic case studies.

Geostatistics in the presence of geological boundaries: Application to mineral resources modeling

This paper addresses the problem of simulating the copper grade and rock type in a Chilean porphyry copper deposit recognized by a set of exploration drill holes. The application of contact analysis tools (cross-to-direct variogram ratios, cross-correlograms, pseudo cross-variograms and lagged scatter plots) reveals the coexistence of hard and soft boundaries between rock type domains. A quadrivariate coregionalization model is developed to fit the spatial correlation structure of the rock type and grade, based on indicator and normal scores direct and cross-variograms. A two-stage approach is first used for simulating the intrusive porphyry and the copper grade in it, while the other two rock type domains and their copper grade are jointly simulated in the space not occupied by porphyry. Such a mixed approach is shown to reproduce the nature of the rock type boundaries and is validated by split-sample validation.

Parametric variogram matrices incorporating both bounded and unbounded functions

We construct a flexible class of parametric models for both traditional and pseudo variogram matrix (valued functions), where the off-diagonal elements are the traditional cross variograms and pseudo cross variograms, respectively, and the diagonal elements are the direct variograms, based on the method of latent dimensions and the linear model of coregionalization. The entries in the parametric variogram matrix allow for a smooth transition between boundedness and unboundedness by changing the values of parameters, and thus between joint second-order and intrinsically stationary vector random fields, or between multivariate geometric Gaussian processes and multivariate Brown–Resnick processes in spatial extreme analysis.

Geostatistical characterization of local accuracies in remotely sensed land cover change categorization with complexly configured reference samples

Accuracy is an integral component in consumption of land cover and change information. Accuracy characterization in change categorization is more complicated than that in single-date classifications, especially when pursued at a local or per-pixel basis. In this paper, reference sample data consisting of reference class labels at sample locations are referred to as (model) training and validation sample data, respectively, when they are used for building and testing predictive models of local accuracy. They are preferably collocated to verify land cover and change types at the same locations across multiple single-date classifications. With collocated training data, methods devised for local accuracy mapping were usually implemented as direct extensions to those designed for accuracy predictions in single-date land cover maps, given that both response and explanatory variables are defined properly based on change maps. However, alternative methods are required when collocated training sample data are unavailable or sparse but non-collocated data collected at sample units not aligned across time exist or can be made available more conveniently and flexibly. This is typically the case for change-classification accuracy analyses, whereby prior sampling design and sample data may not remain informative in time but can be made adaptive by restructuring and augmentation to better represent strata of transitions as well as persistence. Such adaptability leads to great cost-saving in sampling than undertaking collocated sampling anew each time performing or revising accuracy mapping for a specific change analysis over a particular period. With a coherent review of related research, this paper proposes a geostatistical method for predictive mapping of local accuracies in any land-cover change maps created properly through adaptive use of complexly configured training sample data. This method works on a functional relation between accuracies in change-categorization and single-date classifications whereby temporal correlation (cross-correlation) between bi-temporal classification correctness is accommodated. In this cross-correlation-adjusted product method (named PXCOV), quantification of single-date classification accuracies is based on logistic regression. Cross-correlation is estimated via cross-validation cokriging through exploiting a useful relation between traditional and pseudo cross-variograms in the absence of collocated sample data or calculated more easily on the basis of collocated sample data if available. Studies were undertaken to test this method and compare it with direct logistic-regression-kriging (named LogRK direct), its simpler version, direct logistic regression without kriging (named LogR direct), and the baseline product method (named Product, being a simpler option of PXCOV without accounting for cross-correlation), using GlobeLand30 2000 and 2010 land cover at five sites in China. For each site, eleven training samples of equal sizes but differing configurations and one independent validation sample were collected. Logistic regression was performed on map class occurrence pattern indices quantified in size-optimized moving windows for mapping local accuracies in single-date classifications (as necessary intermediate steps in Product and PXCOV) and bi-temporal change-categorization (LogR direct and LogRK direct). In LogRK direct, geostatistical modeling and kriging were based on standardized residuals after logistic regression (based on change maps directly). In PXCOV, cross-correlation between single-date classification correctness was estimated and accounted for properly, with their non-stationarity in change vs. no-change strata of sample pixels accommodated, after estimating aforementioned single-date local accuracies. Statistical testing of relative performances of alternative methods was undertaken based on validation samples for individual study sites. It was confirmed that the proposed method PXCOV is more accurate than LogR direct and LogRK direct when there are relatively few collocated sample pixels (i.e., the typical scenarios justifying use of PXCOV). PXCOV is generally more accurate than Product and should be pursued when enough collocated data are available for estimating cross-correlation, although Product is sometimes adequate for practical purposes. LogR direct is recommended over LogRK direct because collocated sample data are often sparse due to low sampling intensities as in large-area applications. Therefore, as a method complementing LogR direct and similar methods, PXCOV facilitates adaptive use of training sample data of complex configurations for increased accuracy and efficiency in predictive mapping of local change-categorization accuracies.

Spatial interpolation of precipitation from multiple rain gauge networks and weather radar data for operational applications in Alpine catchments

Increasing meteorological data availability and quality implies an adaptation of the interpolation methods for data combination. In this paper, we propose a new method to efficiently combine weather radar data with data from two heated rain gauge networks of different quality. The two networks being non-collocated (no common location between the two networks), pseudo cross-variograms are used to compute the linear model of coregionalization for kriging computation. This allows considering the two networks independently in a co-kriging approach. The methodology is applied to the Upper Rhône River basin, an Alpine catchment in Switzerland with a complex topography and an area of about 5300 km2. The analysis explores the newly proposed Regression co-kriging approach, in which two independent rain gauge networks are considered as primary and secondary kriging variables. Regression co-kriging is compared to four other methods, including the commonly applied Inverse distance weighting method used as baseline scenario. Incorporation of additional networks located within and around the target region in the interpolation computation is also explored. The results firstly demonstrate the added value of the radar information as compared to using only ground stations. As compared to Regression kriging using only the network of highest quality, the Regression co-kriging method using both networks slightly increases the performance. A key outcome of the study is that Regression co-kriging performs better than Inverse distance weighting even for the data availability scenario when the radar network was providing lower quality radar data over the studied basin. The results and discussion underline that combining meteorological information from different rain gauge networks with different equipments remains challenging for operational purposes. Future research in this field should in particular focus on additional pre-processing of the radar data to account for example for areas of low visibility of the weather radars due to the topography.

Early stage variable rate nitrogen fertilization of silage maize driven by multi-temporal clustering of archive satellite data

Nitrogen fertilization of silage maize in Central Italy is typically carried out with two applications at early stages of crop development: 2nd (V2) and 6th (V6) leaf respectively. In such conditions, the crop has not yet fully covered the soil and proximal or remote sensing of the canopy is hindered by the strong soil background signal. There is thus great interest in rapid and inexpensive approaches to N fertilization prescription. Therefore, an indirect method for inferring information on yield potential and soil variability, through a field-based clustering of multi-temporal satellite data, has been developed using archive Landsat images to identify temporally constant patterns. This method is potentially useful for the creation of prescription maps. The usefulness of the method was evaluated during an N fertilisation field trial in Maccarese (Central Italy), in 2016. At the V2 stage, both uniform and variable rate applications were performed and compared. A pseudo-cross variogram and a standardized ordinary co-kriging methodology was used to highlight spatially variable significant differences among the treatments.

Statistical post-processing of forecasts for extremes using bivariate brown-resnick processes with an application to wind gusts

To improve the forecasts of weather extremes, we propose a joint spatial model for the observations and the forecasts, based on a bivariate Brown-Resnick process. As the class of stationary bivariate Brown-Resnick processes is fully characterized by the class of pseudo cross-variograms, we contribute to the theorical understanding of pseudo cross-variograms refining the knowledge of the asymptotic behaviour of all their components and introducing a parsimonious, but flexible parametric model. Both findings are of interest in classical geostatistics on their own. The proposed model is applied to real observation and forecast data for extreme wind gusts at 119 stations in Northern Germany.

Mapping Spatial Distribution of Larch Plantations from Multi-Seasonal Landsat-8 OLI Imagery and Multi-Scale Textures Using Random Forests

The knowledge about spatial distribution of plantation forests is critical for forest management, monitoring programs and functional assessment. This study demonstrates the potential of multi-seasonal (spring, summer, autumn and winter) Landsat-8 Operational Land Imager imageries with random forests (RF) modeling to map larch plantations (LP) in a typical plantation forest landscape in North China. The spectral bands and two types of textures were applied for creating 675 input variables of RF. An accuracy of 92.7% for LP, with a Kappa coefficient of 0.834, was attained using the RF model. A RF-based importance assessment reveals that the spectral bands and bivariate textural features calculated by pseudo-cross variogram (PC) strongly promoted forest class-separability, whereas the univariate textural features influenced weakly. A feature selection strategy eliminated 93% of variables, and then a subset of the 47 most essential variables was generated. In this subset, PC texture derived from summer and winter appeared the most frequently, suggesting that this variability in growing peak season and non-growing season can effectively enhance forest class-separability. A RF classifier applied to the subset led to 91.9% accuracy for LP, with a Kappa coefficient of 0.829. This study provides an insight into approaches for discriminating plantation forests with phenological behaviors.

Random Forest classification of Mediterranean land cover using multi-seasonal imagery and multi-seasonal texture

A Random Forest (RF) classifier was applied to spectral as well as mono- and multi-seasonal textural features extracted from Landsat TM imagery to increase the accuracy of land cover classification over a complex Mediterranean landscape, with a large number of land cover categories and low inter-class separability. Five different types of geostatistical textural measure for three different window sizes and three different lags were applied creating a total of 972 potential input variables. Madograms, rodograms and direct variograms for the univariate case and cross- and pseudo-cross variograms for the multivariate case, together with multi-seasonal spectral information, were used in a RF classifier to map the land cover types. The pseudo-cross and cross variograms were used specifically to incorporate important seasonal/temporal information. Incorporating multi-scale textural features into the RF models led to an increase in the overall index of 10.71% and, for the most accurate classification, the increase was up to 30% in some classes. The differences in the kappa coefficient for the textural classification models were evaluated statistically using a pairwise Z-test, revealing a significant increase in per-class classification accuracy compared to GLCM-based texture measures. The pseudo-cross variogram between the visible and near-infrared bands was the most important textural features for general classification, and the multi-seasonal pseudo-cross variogram had an outstanding importance for agricultural classes. Overall, the RF classifier applied to a reduced subset of input variables composed of the most informative textural features led to the highest accuracy. Highly reliable classification results were obtained when the 16 most important textural features calculated at single scales (window sizes) were selected and used in the classification. The proposed methodology significantly increased the classification accuracy achieved with a spectral maximum likelihood classifier (ML). The kappa values for the textural RF and ML classifications were equal to 0.92 and 0.83, respectively.

Land cover classification using CHRIS/PROBA images and multi-temporal texture

Most existing multi-temporal classification studies use spectral information alone and ignore the temporal correlation between two-date images. This article proposes a new method to characterize the local temporal correlation using multi-temporal texture measured with a geostatistical function called the pseudo cross variogram (PCV). The derived multi-temporal texture, as an additional band, was combined with the spectral information in multi-temporal classification. The performance of the multi-temporal texture was evaluated and compared with the use of multi-temporal spectral data alone and plus the traditional variogram texture in land cover classification using bitemporal hyperspectral Compact High Resolution Imaging Spectrometer/Project for On Board Autonomy (CHRIS/PROBA) images. The results show that although land cover classification using spectral information from bitemporal CHRIS/PROBA data alone had an acceptable overall accuracy of 85.66%, the inclusion of multi-temporal texture in land cover classification led to significant increases (at the 95% confidence level) in both overall accuracy (3.3–4.3% improvement) and the kappa coefficient (4.9–6.6% improvement), particularly for vegetation classes. The incorporation of multi-temporal texture into multi-temporal land cover classification also outperformed the incorporation of the traditional variogram texture. The proposed method provides a new way to exploit the temporal correlation between bitemporal images for improved land cover classification.

Incorporating Spatial Variability Measures in Land-cover Classification using Random Forest

The spatial variability of remotely sensed image values provides important information about the arrangement of objects and their spatial relationships within the image. The characterisation of spatial variability in such images, for example, to measure of texture, is of great utility for the discrimination of land cover classes. To this end, the variogram, a function commonly applied in geostatistics, has been used widely to extract image texture for remotely sensed data classification.The aim of this study was to assess the increase in accuracy that can be achieved by incorporating univariate and multivariate textural measures of Landsat TM imagery in classification models applied to large heterogeneous landscapes. Such landscapes which difficult to classify due to the large number of land cover categories and low inter-class separability. Madogram, rodogram and direct variogram for the univariate case, and cross- and pseudocross variograms for the multivariate one, together with multi-seasonal spectral information were used in a Random Forest classifier to map land cover types.The addition of spatial variability into multi-seasonal Random Forest models leads to an increase in the overall accuracy of 8%, and to an increase in the Kappa index of 9%, respectively. The increase in per categories Kappa for the textural Random Forest model reached 30% for certain categories. This study demonstrates that the use of information on spatial variability produces a fundamental increase in per class classification accuracy of complex land-cover categories.

REMOTE SENSING CHANGE DETECTION BY INCLUSION OF MULTITEMPORAL TEXTURE

Change detection based on remote sensing is one of the important aspects in remote sensing application.Traditional remote sensing change detection methods usually use spectral information alone and ignore the correlation between multitemporal images.This paper proposes to quantitatively express the temporal correlation between multitemporal images by multivariate texture,which is measured by pseudo cross variogram,a geostatistical tool.The obtained multitemporal texture,as an additional band,is incorporated into multitemporal classification for change detection.The results show that,compared with the result of using spectral information alone,the inclusion of multitemporal classification in change detection can significantly improve the overall accuracy.The experimental results also validate the effectiveness of the proposed method.

Enhancing the value of field experimentation through whole-of-block designs

Precision agriculture (PA) offers opportunities for the development of new approaches to on-farm experimentation to assist farmers with site-specific management decisions. Traditional agricultural experiments are usually implemented in fields with the least possible soil heterogeneity under the assumption that responses to inputs and inherent variation of the soil are additive components of yield variation. However, because the soil in typical fields is not homogeneous, PA has much to offer. Farmers faced with variable conditions need to optimize their management to the variation over space and time on their farm, a problem that is not solved by conventional approaches to experimentation. New designs for on-farm experiments were developed in the 1990s for cereal production in which the whole field was used for the experiment rather than small plots. We explore the extension of this type of experiment to a vineyard in the Clare Valley of South Australia aiming to evaluate options to increase grape yield and vine vigour. Manually sampled indices of vine performance measured on georeferenced ‘target’ grapevines were analysed geostatistically. The major advantage of such an approach is that the spatial variation in response to experimental treatments can be examined. Linear models of coregionalization, pseudo cross-variograms and standardized ordinary cokriging are used to map treatment responses over the experimental area and also the differences between them. The results indicate that both treatment responses and the significance of differences between them are spatially variable. Thus, we conclude that whole-of-block on-farm trials are useful in vineyards.

The geostatistical analysis of experiments at the landscape-scale

In conventional field experiments inherent variability is managed by design. In all cases the inherent variation is treated as additive random variables (a residual and any block and/or covariate effects). This assumption is generally reasonable in most field experiments where our basic units are plots within a relatively small and uniform region of a field. It is less plausible when we wish to conduct experiments across heterogeneous landscapes, with inherent variation of the soil over a wide range of many variables. In this paper we present a geostatistical approach to the design and analysis of landscape-scale experiments. We no longer regard the treatment response as a fixed effect, but rather as a random variable. Therefore, at any target site we can estimate the response to different treatments and treatment contrasts. This could be done by ordinary kriging, or by cokriging after we model the treatment responses as coregionalized variables. The advantage of cokriging is that contrasts and their confidence limits may be estimated optimally, and are coherent with estimates of the responses (i.e. the difference between two optimal estimated responses is the optimal estimate of the contrast between them). Since only one treatment response can be observed at a particular location we use the pseudo cross-semivariogram [Papritz, A., Kunsch, H.R., Webster, R., 1993. On the pseudo cross-variogram. Mathematical Geology 25, 1015–1026] to model the cross-covariances of the responses. We compared ordinary kriging and three variants of cokriging; standardized ordinary cokriging, ordinary cokriging and the method of Papritz and Fluhler [Papritz, A., Fluhler, H., 1994. Temporal change of spatially autocorrelated soil properties: optimal estimation by cokriging. Geoderma 62, 29–43]. Standardized ordinary cokriging was found to be the best predictor of the treatment contrast and responses under different levels of spatial auto-correlation and structural correlation, and different experimental designs. The results showed that the plots of different treatments should be placed as close together as possible to give the best predictions of treatment contrasts. A field-scale nitrogen-response experiment was used to illustrate the proposed methods.

Spatio-temporal variability of some metal concentrations in the soil of eastern England, and implications for soil monitoring

Previous workers have proposed the use of multivariate geostatistics for the problem of estimating temporal change in soil properties for soil monitoring, but this has yet to be evaluated. We present a case study of this approach from the Humber–Trent region in North East England. We extracted data from two sources on cobalt, nickel and vanadium concentrations in the topsoil on two dates. Auto-variograms were estimated for each metal on each date, and pseudo cross-variograms for each metal on the two dates. It was shown that robust estimators of the auto and pseudo cross-variograms were needed for the analysis of these data. A linear model of coregionalization was then fitted to describe the spatio-temporal variability of each metal. While the concentration of each metal in the soil showed pronounced spatial dependence that we know is driven by parent material, the change over time was only spatially structured for cobalt and vanadium. This shows that information on spatial variability from a single date may be a poor guide to the design of a monitoring scheme. We showed how the cokriging variance of the change in concentration of cobalt and vanadium depends on sampling effort and strategy. The change in these particular variables between two dates is best estimated by sampling with equal intensity at the same sites on both dates; and when resampling an existing baseline survey it is best to sample them at rather than between the original sites. The best strategy in any case depends on how the variable is coregionalized over time.

Robust estimation of the pseudo cross-variogram for cokriging soil properties

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.

Computing geostatistical image texture for remotely sensed data classification

Most classical mathematical algorithms for image classification do not usually consider the spectral dependence existing between a pixel and its neighbours, i.e., spatial autocorrelation. Thus, it would be advisable for discrimination of landcover classes to add to the radiometric bands of the sensor complementary information related to the textural features of an image, which can be analysed from the autocorrelation spatial structure of the digital numbers. In this way, the results obtained from pixel-by-pixel classifiers simultaneously taking into account both radiometric and texture information could be improved. This improvement would arise from the hypothesis that a pixel is not independent of its neighbours and, furthermore, that its dependence can be quantified and incorporated into the classifier. In this paper we present a methodology based on computing a set of univariate and multivariate textural measures of spatial variability based on several variogram estimators. Madogram and direct variogram for the univariate case, and cross and pseudo-cross variograms for the multivariate one, have been proposed. These measures are calculated for a specific lag of distance in a neighbourhood using a moving window on the two most representative principal components of the radiometric bands, enabling us to quantify the spatial variability of radiometric data at a local level. A computer program has been written to create a multiband image texture as output file that can be used within the classification process as additional information. An application of this methodology to lithological discrimination is presented using a Landsat-5 TM image.

The variance-based cross-variogram : You can add apples and oranges

The variance-based cross-variogram between two spatial processes, Z1 (·) and Z2 (·), is var (Z1 ( u ) − Z2 ( v )), expressed generally as a bivariate function of spatial locations uandv. It characterizes the cross-spatial dependence between Z1 (·) and Z2 (·) and can be used to obtain optimal multivariable predictors (cokriging). It has also been called the pseudo cross-variogram; here we compare its properties to that of the traditional (covariance-based) cross-variogram, cov (Z1 ( u ) − Z1 ( v ), Z2 ( u ) − Z2 ( v )). One concern with the variance-based cross-variogram has been that Z1 (·) and Z2 (·) might be measured in different units (“apples” and “oranges”). In this note, we show that the cokriging predictor based on variance-based cross-variograms can handle any units used for Z1 (·) and Z2 (·); recommendations are given for an appropriate choice of units. We review the differences between the variance-based cross-variogram and the covariance-based cross-variogram and conclude that the former is more appropriate for cokriging. In practice, one often assumes that variograms and cross-variograms are functions of uandv only through the difference u − v. This restricts the types of models that might be fitted to measures of cross-spatial dependence.