Mapping the anthropic backfill thickness of the historical center of Rome (Italy) by using kriging with external drift (original) (raw)
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Basic Steps in Geostatistics: The Variogram and Kriging
SpringerBriefs in Agriculture, 2015
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The historical centre of Rome is characterized by the presence of high thickness of anthropic cover with scarce geotechnical characteristics. This anthropic backfill could induce damages in urban areas, i.e. mainly differential settlements and seismic amplifications. About 1400 measurements from boreholes stored in the UrbiSIT database have been used to re-construct the anthropic backfill bottom surface by geostatistical techniques. The Intrinsic Random Functions of order k (IRF-k) was employed and compared with other interpolation methods (i.e. ordinary kriging and kriging with external drift) to determine the best spatial predictor. Furthermore, IRF-k allows to estimate by using an external drift as secondary information. The advantage of this method is that the modeling of the optimal generalized covariance is performed by using an automatic procedure avoiding the time-consuming modeling of the variogram. Furthermore, IRF-k allows the modeling of non stationary variables.
Geostatistical Estimation: Kriging
Geostatistics for Environmental and Geotechnical Applications
The most important step in a feasibility study of a mining project is the determination of reserves in situ. But the question which arises that no method of estimation can give exact value since there is inevitably an associated error when it is important to know the order of magnitude. Decision-makers need to know if the estimated content is correct or not. Geostatistics can help the mining engineer to obtain from the available information, and thus, enable him to decide if the project deserves more investments. Geostatistics provides not only the estimated value but also with the kriging variance, a measure of the accuracy of the estimate. This is one of the superiorities of geostatistics over traditional methods of estimating reserves. In this article, the Kieselguhr deposit reserve of the Sig mine (western Algeria) was evaluated according to the linear geostatistical method with ordinary Kriging (OK), to know the economic value of this mine that allows the exploitation of this deposit, and to carry out study cartography accompanied by 2D modeling of deposit. Finally, we obtained maps in which we could estimate exploitable and geological reserves, and know the characteristics of the deposit.
This paper presents a special scientific analytical methodology to conduct geostatistical spatial analysis, Variogram modeling and interpolation by kriging method using terrain elevation data measured over geographical spatial unit, while accounting for anisotropic behavior of terrain within this unit. The methodology which includes the design of surface interpolation that gives weights to all data points, starts by performing geostatistical analysis and building the Variogram chart. The Variogram models that best representing the data is computed by using standard mathematical regression functions. The modeling process is achieved by using iterative methods and nonlinear least squares optimization process. The coherence between Variogram models constraint and the weights used in the kriging system ensures statistically the best unbiased estimators as well as minimum variances for the interpolated values. Kriging reduces the unrealistic smoothing surfaces inherited in other interpolation methods. It is also robust with respect to very small spatial differences in data points positions, where they are included in the process. There are a large number of semi-Variogram models that could be employed, although different models may lead to different interpolations. The study focuses on the ten most popular models (some of them recently discovered). The mean value of absolute variances provides valuable information help us to select which model is the best from several candidates. If anisotropy exists in variography according to different directions, then several Variogram models needs to be determined. Special Matlab programs were written by the author for implementing all stages of the above methodology. The study has shown that the interpolation process by Kriging fails in some cases and inaccurate in other cases due to several reasons: the size accuracy of data, its representation, misunderstanding the nature of the phenomenon, the goodness of Variogram fitting model and sometimes may be for some unknown reasons. Thus we need easy and fast computational tools performing many experiments at the same time giving clear representation results and final error analysis, so that the best solution is reached at last. This was the main and most important achievement of this study.
Geographical Analysis, 2010
Many branches within geography deal with variables that vary not only in space but also in time. Therefore, conventional geostatistics needs to be extended with methods that estimate and quantify spatiotemporal variation and use it in spatiotemporal interpolation and stochastic simulation. This article briefly summarizes the main concepts of space-time geostatistics. Kriging in space and time can be done in much the same way as it is in a purely spatial setting. The main difficulties are in defining a realistic stochastic model that is assumed to have generated data and in characterizing and estimating the space-time correlation of that model. This article uses a modelbased geostatistical approach to characterize space-time variability. The space-time variable of interest is treated as a sum of independent stationary spatial, temporal, and spatiotemporal components, which leads to a sum-metric space-time variogram model. Methods are illustrated with a case study of space-time interpolation of monthly averages of detected background radiation for a 5-year period in four German states.
Assessment of kriging accuracy in the GIS environment
The 21st Annual ESRI International User …, 2001
The demand for spatial data is on the rise. However, even the latest technology cannot guarantee an error free database in Geographic Information System (GIS). In natural resources the point field sampling is often used for spatially oriented projects and interpolation methods are implemented to predict the values in an unsampled location and to generate maps. In order to evaluate the performance of Kriging interpolation in GIS the Kriging errors were analyzed and compared to the four other interpolation methods using fundanmental statistical parameters. The sensitivity of ordinary Kriging interpolation in the GIS environment was evaluated with respect to the resolution of the predicted grid and conclusions were drawn for applications in spatial analysis. METHODOLOGY AND DATA Interpolation methods estimate the values in unsampled locations. The mapping and spatial analysis often requires converting the field measurements into continuous space. Therefore the point data sets must be converted to a continuous form using an interpolation method. The errors, however, enter the spatial database long before any interpolation method is applied to the data set. The first type of error is associated with sampling design. The magnitude of a sample, as well as the procedure of obtaining it, depend on the objectives of the sampling process, and consequently vary with these objectives (9). Increasing the sample size also improves the accuracy of measurements up to a certain point. In spatial analysis the sampling is often performed on a regular grid or on an irregular set of points however, this might not depict the true variation of studied phenomena in the space.
Interpolation of spatial data: Some theory for Kriging
2006
In the preface to his book, Michael Stein recalls a visit to the University of Chicago library in the autumn of 1985. Browsing among the library books, he comes across Gaussian Random Processes by Ibragimov and Rozanov. His first reaction is to dismiss it as being too difficult for him to read and in any case irrelevant to his research interests. It seems to me that most soil geostatisticians will have similar thoughts in a first encounter with this book by Stein. Stein's book is of an advanced mathematical level. It presumes that readers are familiar with mathematical concepts such as Lebesgue measure, Borel set, characteristic function, Hilbert space, and so on. The average soil geostatistician will have a hard job working his or her way through the many theorems, lemmas and corollaries, let alone to understand their proofs and try the exercises. However, just as Stein experienced with the book by Ibragimov and Rozanov, perseverance turns out to be rewarding. This is because the mathematical results presented in Stein's book have many repercussions on the practice of kriging. Before addressing these repercussions, let me briefly outline the contents of the book. Clearly, as the title indicates, this is a book on kriging. To be more precise: it is a book on some specific aspects of kriging. Stein does not provide a broad overview of kriging. His book is a research monograph, not a textbook. Chapters 1 and 2 are introductory chapters that give a condensed review of the mathematical theory on linear prediction and random fields. The dependence of kriging on the behaviour of the semivariogram at the origin, the contrast between interpolation and extrapolation, and the effect of taking more and more Ž . samples in a fixed domain fixed domain asymptotics are treated in Chapter 3. In the remaining three chapters, the equivalence of Gaussian measures and prediction, integration of random fields and kriging under uncertain semivariograms are discussed. What are the practical implications that make this book a worthwhile read for soil geostatisticians? To start with, there are quite a few on the estimation and modelling of semivariograms. For example, Stein makes an underpinned claim Ž that the current set of most frequently used semivariogram models linear, . spherical, exponential, Gaussian needs revision. The use of the Matern model iś advocated, which allows for great flexibility in the smoothness of the random 0016-7061r00r$ -see front matter q
Kriging approach for local height transformations
Geodesy and Cartography, 2014
In the paper a transformation between two height datums (Kronstadt’60 and Kronstadt’86, the latter being a part of the present National Spatial Reference System in Poland) with the use of geostatistical method - kriging is presented. As the height differences between the two datums reveal visible trend a natural decision is to use the kind of kriging method that takes into account nonstationarity in the average behavior of the spatial process (height differences between the two datums). Hence, two methods were applied: hybrid technique (a method combining Trend Surface Analysis with ordinary kriging on least squares residuals) and universal kriging. The background of the two methods has been presented. The two methods were compared with respect to the prediction capabilities in a process of crossvalidation and additionally they were compared to the results obtained by applying a polynomial regression transformation model. The results obtained within this study prove that the structu...