Kriging

Summary

In statistics, originally in geostatistics, kriging or Kriging, (pronounced /ˌˈkɹiːɡɪŋ/) also known as Gaussian process regression, is a method of interpolation based on Gaussian process governed by prior covariances. Under suitable assumptions of the prior, kriging gives the best linear unbiased prediction (BLUP) at unsampled locations. Interpolating methods based on other criteria such as smoothness (e.g., smoothing spline) may not yield the BLUP. The method is widely used in the domain of spatial analysis and computer experiments. The technique is also known as Wiener–Kolmogorov prediction, after Norbert Wiener and Andrey Kolmogorov. The theoretical basis for the method was developed by the French mathematician Georges Matheron in 1960, based on the master's thesis of Danie G. Krige, the pioneering plotter of distance-weighted average gold grades at the Witwatersrand reef complex in South Africa. Krige sought to estimate the most likely distribution of gold based on samples fro

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POD-Kriging reduced method’s application in Tandem Cylinders’ flow

Jan Sickmann Hesthaven, Han Xiao

The configuration of Tandem Cylinders is a typical one in researching noise and interactions between different components, where the Distance-diameter ratio is a very significant parameter with which the flow form varies a lot. To clearly research the ratio’s effect on Tandem Cylinders flow as well as the unsteady flow physics, a reduced-order model is built where Proper orthogonal Decomposition(POD) is used to extract reduced basis modes, and Kriging model is used to train sample data and interpolate projection coefficents for each reduced basis. To ensure the snapshots’ accuracy, the scale-adaptive simulation (SAS) model based on k-SST turbulence model is used to predict time-dependent flow field. By comparing with solutions provided by the numerical solver as well as experimental data, the reduced model presents a good accuracy and efficiency. And then based on the reduced basis, the change regulation of the time-average flow is summarized.

2018

Switzerland has about 6,000 shooting ranges, of which 2,000 are still in use. These activities generate a significant lead pollution. FOEN estimated 200 tons of lead is fired each year. This Master Project has been realized with the consultancy office CSD Ingénieurs Conseils SA in Lausanne. The project is divided into three parts related to investigations and remediation of shooting ranges. Eleven shooting ranges of the army and two communal shooting ranges have formed the field study of this project. On these sites, 20 samples were taken for laboratory analysis. Nearly 350 field tests with a portable X-ray fluorescence (XRF) were also performed. The first part of the study focused on sample preparation methods. Based on several assumptions, they often give conflicting results. It has been shown that not taking into account the size fraction between 1 cm and 2 mm leads to underestimating the lead concentration by 10% on average. Taking into account the bullets, and how, is even more important. The original content of lead in standard military bullets is 74% of the weight. A common use is to assume this content for the bullets found in soils samples. The lead content measured in bullets is from 64% for those in good shape to 17% for fragments. Lead seems to spread more easily than the steel jacket of the bullets. These results are of major importance for the calculation of the lead content of samples. A strong correlation between lead and bullet amount in soils has not been found. The condition of the bullets seemed to be a significant factor, but the representativeness of the samples was not sufficient due too few mass samples. Taking into account these results, a preparation method was proposed. The second part of the study concern leachate analysis. In particular, OSites leachate and OTD1 (acid) leachate was studied. These leachates are used respectively for polluted sites risk assessment and for choosing treatment solution of excavated materials. Correlations were looked for, between concentrations in leachate and soils parameters. CEC, clay content, organic carbon content and aquous pH has been used as soils parameters. Leachate OTD1 (neutral) has been preferred to leachate Osites for economic reasons, these two tests provide similar results for lead. Good correlations were obtained with non-linear models taking into account all the soils parameters. Models based only on the lead content and aqueous pH (better correlation) give interesting results, but have greater error. The third part focused on the study of geostatistics shooting ranges with fixed targets. Variogram were made from field measurements. A metric sampling grid metric was used. The variogram showed a remedy made from a range of 3 meters. Although the site shows a clear anisotropy, an isotropic spherical variogram with nugget effect has been adjusted. The data were insufficient to fit an anisotropic model. Calculations showed that the kriging interpolation provides better results than those obtained with conventional interpolation. The study also showed that the uncertainty of the kriging interpolation decreases when the mesh sampling is lower. An optimum between accuracy and number of tests could not be found, because the variogram used has too much variability. The results are very promising and will continue to be worked out in order to produce the means of zoning and estimate for the remediation of shooting range.

2009

Measurement of Precipitation in the Alps Using Dual-Polarization C-Band Ground-Based Radars, the GPM Spaceborne Ku-Band Radar, and Rain Gauges

Alexis Berne, Marco Gabella, Peter James Speirs

The complex problem of quantitative precipitation estimation in the Alpine region is tackled from four different points of view: (1) the modern MeteoSwiss network of automatic telemetered rain gauges (GAUGE); (2) the recently upgraded MeteoSwiss dual-polarization Doppler, ground-based weather radar network (RADAR); (3) a real-time merging of GAUGE and RADAR, implemented at MeteoSwiss, in which a technique based on co-kriging with external drift (CombiPrecip) is used; (4) spaceborne observations, acquired by the dual-wavelength precipitation radar on board the Global Precipitation Measuring (GPM) core satellite. There are obviously large differences in these sampling modes, which we have tried to minimize by integrating synchronous observations taken during the first 2 years of the GPM mission. The data comprises 327 wet overpasses of Switzerland, taken after the launch of GPM in February 2014. By comparing the GPM radar estimates with the MeteoSwiss products, a similar performance was found in terms of bias. On average (whole country, all days and seasons, both solid and liquid phases), underestimation is as large as -3.0 (-3.4) dB with respect to RADAR (GAUGE). GPM is not suitable for assessing what product is the best in terms of average precipitation over the Alps. GPM can nevertheless be used to evaluate the dispersion of the error around the mean, which is a measure of the geographical distribution of the error inside the country. Using 221 rain-gauge sites, the result is clear both in terms of correlation and in terms of scatter (a robust, weighted measure of the dispersion of the multiplicative error around the mean). The best agreement was observed between GPM and CombiPrecip, and, next, between GPM and RADAR, whereas a larger disagreement was found between GPM and GAUGE. Hence, GPM confirms that, for precipitation mapping in the Alpine region, the best results are obtained by combining ground-based radar with rain-gauge measurements using a geostatistical approach. The GPM mission is adding significant new coverage to mountainous areas, especially in poorly instrumented parts of the world and at latitudes not previously covered by the Tropical Rainfall Measuring Mission (TRMM). According to this study, one could expect an underestimation of the precipitation product by the dual-frequency precipitation radar (DPR) also in other mountainous areas of the world.

2017