Wavelets and the generalization of the variogram

The experimental variogram computed in the usual way by the method of moments and the Haar wavelet transform are similar in that they filter data and yield informative summaries that may be interpreted. The variogram filters out constant values; wavelets can filter variation at several spatial scales and thereby provide a richer repertoire for analysis and demand no assumptions other than that of finite variance. This paper compares the two functions, identifying that part of the Haar wavelet transform that gives it its advantages. It goes on to show that the generalized variogram of order k=1, 2, and 3 filters linear, quadratic, and cubic polynomials from the data, respectively, which correspond with more complex wavelets in Daubechies's family. The additional filter coefficients of the latter can reveal features of the data that are not evident in its usual form. Three examples in which data recorded at regular intervals on transects are analyzed illustrate the extended form of the variogram. The apparent periodicity of gilgais in Australia seems to be accentuated as filter coefficients are added, but otherwise the analysis provides no new insight. Analysis of hyerpsectral data with a strong linear trend showed that the wavelet-based variograms filtered it out. Adding filter coefficients in the analysis of the topsoil across the Jurassic scarplands of England changed the upper bound of the variogram; it then resembled the within-class variogram computed by the method of moments. To elucidate these results, we simulated several series of data to represent a random process with values fluctuating about a mean, data with long-range linear trend, data with local trend, and data with stepped transitions. The results suggest that the wavelet variogram can filter out the effects of long-range trend, but not local trend, and of transitions from one class to another, as across boundaries.

Determining the effect of asymmetric data on the variogram. I. Underlying asymmetry

Matheron's usual variogram estimator can result in unreliable variograms when data are strongly asymmetric or skewed. Asymmetry in a distribution can arise from a long tail of values in the underlying process or from outliers that belong to another population that contaminate the primary process. This paper examines the effects of underlying asymmetry on the variogram and on the accuracy of prediction, and the second one examines the effects arising from outliers. Standard geostatistical texts suggest ways of dealing with underlying asymmetry; however, this is based on informed intuition rather than detailed investigation. To determine whether the methods generally used to deal with underlying asymmetry are appropriate, the effects of different coefficients of skewness on the shape of the experimental variogram and on the model parameters were investigated. Simulated annealing was used to create normally distributed random fields of different size from variograms with different nugget:sill ratios. These data were then modified to give different degrees of asymmetry and the experimental variogram was computed in each case. The effects of standard data transformations on the form of the variogram were also investigated. Cross-validation was used to assess quantitatively the performance of the different variogram models for kriging. The results showed that the shape of the variogram was affected by the degree of asymmetry, and that the effect increased as the size of data set decreased. Transformations of the data were more effective in reducing the skewness coefficient in the larger sets of data. Cross-validation confirmed that variogram models from transformed data were more suitable for kriging than were those from the raw asymmetric data. The results of this study have implications for the 'standard best practice' in dealing with asymmetry in data for geostatistical analyses. (C) 2007 Elsevier Ltd. All rights reserved.

Determining the effect of asymmetric data on the variogram. II. Outliers

Asymmetry in a distribution can arise from a long tail of values in the underlying process or from outliers that belong to another population that contaminate the primary process. The first paper of this series examined the effects of the former on the variogram and this paper examines the effects of asymmetry arising from outliers. Simulated annealing was used to create normally distributed random fields of different size that are realizations of known processes described by variograms with different nugget:sill ratios. These primary data sets were then contaminated with randomly located and spatially aggregated outliers from a secondary process to produce different degrees of asymmetry. Experimental variograms were computed from these data by Matheron's estimator and by three robust estimators. The effects of standard data transformations on the coefficient of skewness and on the variogram were also investigated. Cross-validation was used to assess the performance of models fitted to experimental variograms computed from a range of data contaminated by outliers for kriging. The results showed that where skewness was caused by outliers the variograms retained their general shape, but showed an increase in the nugget and sill variances and nugget:sill ratios. This effect was only slightly more for the smallest data set than for the two larger data sets and there was little difference between the results for the latter. Overall, the effect of size of data set was small for all analyses. The nugget:sill ratio showed a consistent decrease after transformation to both square roots and logarithms; the decrease was generally larger for the latter, however. Aggregated outliers had different effects on the variogram shape from those that were randomly located, and this also depended on whether they were aggregated near to the edge or the centre of the field. The results of cross-validation showed that the robust estimators and the removal of outliers were the most effective ways of dealing with outliers for variogram estimation and kriging. (C) 2007 Elsevier Ltd. All rights reserved.

Using the variogram to explore imagery of two different spatial resolutions

The resolution of remotely sensed data is becoming increasingly fine, and there are now many sources of data with a pixel size of 1 m x 1 m. This produces huge amounts of data that have to be stored, processed and transmitted. For environmental applications this resolution possibly provides far more data than are needed: data overload. This poses the question: how much is too much? We have explored two resolutions of data-20 in pixel SPOT data and I in pixel Computerized Airborne Multispectral Imaging System (CAMIS) data from Fort A. P. Hill (Virginia, USA), using the variogram of geostatistics. For both we used the normalized difference vegetation index (NDVI). Three scales of spatial variation were identified in both the SPOT and 1 in data: there was some overlap at the intermediate spatial scales of about 150 in and of 500 m-600 in. We subsampled the I in data and scales of variation of about 30 in and of 300 in were identified consistently until the separation between pixel centroids was 15 in (or 1 in 225pixels). At this stage, spatial scales of about 100m and 600m were described, which suggested that only now was there a real difference in the amount of spatial information available from an environmental perspective. These latter were similar spatial scales to those identified from the SPOT image. We have also analysed I in CAMIS data from Fort Story (Virginia, USA) for comparison and the outcome is similar.:From these analyses it seems that a pixel size of 20m is adequate for many environmental applications, and that if more detail is required the higher resolution data could be sub-sampled to a 10m separation between pixel centroids without any serious loss of information. This reduces significantly the amount of data that needs to be stored, transmitted and analysed and has important implications for data compression.

The Initial Slope of the Variogram, Foundation of the Trabecular Bone Score, Is Not or Poorly Associated With Vertebral Strength

Trabecular bone score (TBS) rests on the textural analysis of DXA to reflect the decay in trabecular structure characterising osteoporosis. Yet, its discriminative power in fracture studies remains incomprehensible as prior biomechanical tests found no correlation with vertebral strength. To verify this result possibly due to an unrealistic set-up and to cover a wide range of loading scenarios, the data from three previous biomechanical studies using different experimental settings was used. They involved the compressive failure of 62 human lumbar vertebrae loaded 1) via intervertebral discs to mimic the in vivo situation (“full vertebra”), 2) via the classical endplate embedding (“vertebral body”) or 3) via a ball joint to induce anterior wedge failure (“vertebral section”). HR-pQCT scans acquired prior testing were used to simulate anterior-posterior DXA from which areal bone mineral density (aBMD) and the initial slope of the variogram (ISV), the early definition of TBS, were evaluated. Finally, the relation of aBMD and ISV with failure load (Fexp) and apparent failure stress (σexp) was assessed and their relative contribution to a multi-linear model was quantified via ANOVA. We found that, unlike aBMD, ISV did not significantly correlate with Fexp and σexp, except for the “vertebral body” case (r2 = 0.396, p = 0.028). Aside from the “vertebra section” set-up where it explained only 6.4% of σexp (p = 0.037), it brought no significant improvement to aBMD. These results indicate that ISV, a replica of TBS, is a poor surrogate for vertebral strength no matter the testing set-up, which supports the prior observations and raises a fortiori the question of the deterministic factors underlying the statistical relationship between TBS and vertebral fracture risk.

Linking riparian vegetation spatial structure in Australian tropical savannas to ecosystem health indicators: semi-variogram analysis of high spatial resolution satellite imagery

Government agencies responsible for riparian environments are assessing the combined utility of field survey and remote sensing for mapping and monitoring indicators of riparian zone health. The objective of this work was to determine if the structural attributes of savanna riparian zones in northern Australia can be detected from commercially available remotely sensed image data. Two QuickBird images and coincident field data covering sections of the Daly River and the South Alligator River - Barramundie Creek in the Northern Territory were used. Semi-variograms were calculated to determine the characteristic spatial scales of riparian zone features, both vegetative and landform. Interpretation of semi-variograms showed that structural dimensions of riparian environments could be detected and estimated from the QuickBird image data. The results also show that selecting the correct spatial resolution and spectral bands is essential to maximize the accuracy of mapping spatial characteristics of savanna riparian features. The distribution of foliage projective cover of riparian vegetation affected spectral reflectance variations in individual spectral bands differently. Pan-sharpened image data enabled small-scale information extraction (< 6 m) on riparian zone structural parameters. The semi-variogram analysis results provide the basis for an inversion approach using high spatial resolution satellite image data to map indicators of savanna riparian zone health.

A note on extension variances in IR2

Matheron (1971) proposed an approximation of the extension variance in IR. We propose in this note an extension of this formula in IR2, based on a MacLaurin formula. Its application is shown in an example, the estimation of the maximum depressional storage of a soil surface.

Application of spatial analysis techniques to adjust for fertility trends and identify interplot competition in early stage sugarcane selection trials

Most sugarcane breeding programs in Australia use large unreplicated trials to evaluate clones in the early stages of selection. Commercial varieties that are replicated provide a method of local control of soil fertility. Although such methods may be useful in detecting broad trends in the field, variation often occurs on a much smaller scale. Methods such as spatial analysis adjust a plot for variability by using information from immediate neighbours. These techniques are routinely used to analyse cereal data in Australia and have resulted in increased accuracy and precision in the estimates of variety effects. In this paper, spatial analyses in which the variability is decomposed into local, natural, and extraneous components are applied to early selection trials in sugarcane. Interplot competition in cane yield and trend in sugar content were substantial in many of the trials and there were often large differences in the selections between the spatial and current method used by the Bureau of Sugar Experiment Stations. A joint modelling approach for tonnes sugar per hectare in response to fertility trends and interplot competition is recommended.

SAKWeb – Spatial autocorrelation and kriging web, a W3 geocomputation perspective

A thesis submitted in partial satisfaction of the requirements for the degree of Doctor of Philosophy in Information Systems

Comparing the performance of geostatistical models with additional information from covariates for sewage plume characterization

In this work, kriging with covariates is used to model and map the spatial distribution of salinity measurements gathered by an autonomous underwater vehicle in a sea outfall monitoring campaign aiming to distinguish the effluent plume from the receiving waters and characterize its spatial variability in the vicinity of the discharge. Four different geostatistical linear models for salinity were assumed, where the distance to diffuser, the west-east positioning, and the south-north positioning were used as covariates. Sample variograms were fitted by the Mat`ern models using weighted least squares and maximum likelihood estimation methods as a way to detect eventual discrepancies. Typically, the maximum likelihood method estimated very low ranges which have limited the kriging process. So, at least for these data sets, weighted least squares showed to be the most appropriate estimation method for variogram fitting. The kriged maps show clearly the spatial variation of salinity, and it is possible to identify the effluent plume in the area studied. The results obtained show some guidelines for sewage monitoring if a geostatistical analysis of the data is in mind. It is important to treat properly the existence of anomalous values and to adopt a sampling strategy that includes transects parallel and perpendicular to the effluent dispersion.

Stochastic simulation of the morphology of fluvial sand channels reservoirs

Dissertação para obtenção do Grau de Mestre em Engenharia Geológica (Georrecursos)

Solució paral·lelitzada d'interpolació kriging amb ajust automatitzat del variograma

El principal objectiu d'aquest treball és proporcionar una metodologia per a reduir el temps de càlcul del mètode d'interpolació kriging sense pèrdua de la qualitat del model resultat. La solució adoptada ha estat la paral·lelització de l'algorisme mitjançant MPI sobre llenguatge C. Prèviament ha estat necessari automatitzar l'ajust del variograma que millor s'adapta a la distribució espacial de la variable d'estudi. Els resultats experimentals demostren la validesa de la solució implementada, en reduir de forma significativa els temps d'execució final de tot el procés.

Trabecular bone score (TBS) the new parameter of 2D texture analysis for the evaluation of 3D bone micro architecture status

X-ray is a technology that is used for numerous applications in the medical field. The process of X-ray projection gives a 2-dimension (2D) grey-level texture from a 3- dimension (3D) object. Until now no clear demonstration or correlation has positioned the 2D texture analysis as a valid indirect evaluation of the 3D microarchitecture. TBS is a new texture parameter based on the measure of the experimental variogram. TBS evaluates the variation between 2D image grey-levels. The aim of this study was to evaluate existing correlations between 3D bone microarchitecture parameters - evaluated from μCT reconstructions - and the TBS value, calculated on 2D projected images. 30 dried human cadaveric vertebrae were acquired on a micro-scanner (eXplorer Locus, GE) at isotropic resolution of 93 μm. 3D vertebral body models were used. The following 3D microarchitecture parameters were used: Bone volume fraction (BV/TV), Trabecular thickness (TbTh), trabecular space (TbSp), trabecular number (TbN) and connectivity density (ConnD). 3D/2D projections has been done by taking into account the Beer-Lambert Law at X-ray energy of 50, 100, 150 KeV. TBS was assessed on 2D projected images. Correlations between TBS and the 3D microarchitecture parameters were evaluated using a linear regression analysis. Paired T-test is used to assess the X-ray energy effects on TBS. Multiple linear regressions (backward) were used to evaluate relationships between TBS and 3D microarchitecture parameters using a bootstrap process. BV/TV of the sample ranged from 18.5 to 37.6% with an average value at 28.8%. Correlations' analysis showedthat TBSwere strongly correlatedwith ConnD(0.856≤r≤0.862; p<0.001),with TbN (0.805≤r≤0.810; p<0.001) and negatively with TbSp (−0.714≤r≤−0.726; p<0.001), regardless X-ray energy. Results show that lower TBS values are related to "degraded" microarchitecture, with low ConnD, low TbN and a high TbSp. The opposite is also true. X-ray energy has no effect onTBS neither on the correlations betweenTBS and the 3Dmicroarchitecture parameters. In this study, we demonstrated that TBS was significantly correlated with 3D microarchitecture parameters ConnD and TbN, and negatively with TbSp, no matter what X-ray energy has been used. This article is part of a Special Issue entitled ECTS 2011. Disclosure of interest: None declared.

Heterogeneous 40Ar* distributions in naturally deformed muscovite: in situ UV-laser ablation evidence for micro structurally controlled intragrain diffusion

Micas are commonly used in Ar-40/Ar-39 thermochronological studies of variably deformed rocks yet the physical basis by which deformation may affect radiogenic argon retention in mica is poorly constrained. This study examines the relationship between deformation and deformation-induced microstructures on radiogenic argon retention in muscovite, A combination of furnace step-heating and high-spatial resolution in situ UV-laser ablation Ar-40/Ar-39 analyses are reported for deformed muscovites sampled from a granitic pegmatite vein within the Siviez-Mischabel Nappe, western Swiss Alps (Penninic domain, Brianconnais unit). The pegmatite forms part of the Variscan (similar to 350 Ma) Alpine basement and exhibits a prominent Alpine S-C fabric including numerous mica `fish' that developed under greenschist facies metamorphic conditions, during the dominant Tertiary Alpine tectonic phase of nappe emplacement. Furnace step-heating of milligram quantities of separated muscovite grains yields an Ar-40/Ar-39 age spectrum with two distinct staircase segments but without any statistical plateau, consistent with a previous study from the same area. A single (3 X 5 mm) muscovite porphyroclast (fish) was investigated by in situ UV-laser ablation. A histogram plot of 170 individual Ar-40/Ar-39 UV-laser ablation ages exhibit a range from 115 to 387 Ma with modes at approximately 340 and 260 Ma. A variogram statistical treatment of the (40)Ad/Ar-39 results reveals ages correlated with two directions; a highly correlated direction at 310 degrees and a lesser correlation at 0 degrees relative to the sense of shearing. Using the highly correlated direction a statistically generated (Kriging method) age contour map of the Ar-40/Ar-39 data reveals a series of elongated contours subparallel to the C-surfaces which where formed during Tertiary nappe emplacement. Similar data distributions and slightly younger apparent ages are recognized in a smaller mica fish. The observed intragrain age variations are interpreted to reflect the partial loss of radiogenic argon during Alpine (similar to 35 Ma) greenschist facies metamorphism. One-dirnensional diffusion modelling results are consistent with the idea that the zones of youngest apparent age represent incipient shear band development within the mica porphyroclasts, thus providing a network of fast diffusion pathways. During Alpine greenschist facies metamorphism the incipient shear bands enhanced the intragrain loss of radiogenic argon. The structurally controlled intragrain age variations observed in this investigation imply that deformation has a direct control on the effective length scale for argon diffusion, which is consistent with the heterogeneous nature of deformation. (C) 2001 Elsevier Science B.V. All rights reserved.