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1.
The kriging update equations and their application to the selection of neighboring data 总被引：1，自引：0，他引：1
Xavier Emery 《Computational Geosciences》2009,13(3):269-280
A key problem in the application of kriging is the definition of a local neighborhood in which to search for the most relevant
data. A usual practice consists in selecting data close to the location targeted for prediction and, at the same time, distributed
as uniformly as possible around this location, in order to discard data conveying redundant information. This approach may
however not be optimal, insofar as it does not account for the data spatial correlation. To improve the kriging neighborhood
definition, we first examine the effect of including one or more data and present equations in order to quickly update the
kriging weights and kriging variances. These equations are then applied to design a stepwise selection algorithm that progressively
incorporates the most relevant data, i.e., the data that make the kriging variance decrease more. The proposed algorithm is
illustrated on a soil contamination dataset. 相似文献
2.
By definition, kriging with a moving neighborhood consists in kriging each target point from a subset of data that varies with the target. When the target moves, data that were within the neighborhood are suddenly removed from the neighborhood. There is generally no screen effect, and the weight of such data goes suddenly from a non-zero value to a value of zero. This results in a discontinuity of the kriging map. Here a method to avoid such a discontinuity is proposed. It is based on the penalization of the outermost data points of the neighborhood, and amounts to considering that these points values are spoiled with a random error having a variance that increases infinitely when they are about to leave the neighborhood. Additional details are given regarding how the method is to be carried out, and properties are described. The method is illustrated by simple examples. While it appears to be similar to continuous kriging with a smoothing kernel, it is in fact based on a much simpler formalism. 相似文献
3.
Maria Vigsnes Odd Kolbjørnsen Vera Louise Hauge Pål Dahle Petter Abrahamsen 《Mathematical Geosciences》2017,49(5):619-634
Unknown values of a random field can be predicted from observed data using kriging. As data sets grow in size, the computation times become large. To facilitate kriging with large data sets, an approximation where the kriging is performed in sub-segments with common data neighborhoods has been developed. It is shown how the accuracy of the approximation can be controlled by increasing the common data neighborhood. For four different variograms, it is shown how large the data neighborhoods must be to get an accuracy below a chosen threshold, and how much faster these calculations are compared to the kriging where all data are used. Provided that variogram ranges are small compared to the domain of interest, kriging with common data neighborhoods provides excellent speed-ups (2–40) while maintaining high numerical accuracy. Results are presented both for data neighborhoods where the neighborhoods are the same for all sub-segments, and data neighborhoods where the neighborhoods are adapted to fit the data densities around the sub-segments. Kriging in sub-segments with common data neighborhoods is well suited for parallelization and the speed-up is almost linear in the number of threads. A comparison is made to the widely used moving neighborhood approach. It is demonstrated that the accuracy of the moving neighborhood approach can be poor and that computational speed can be slow compared to kriging with common data neighborhoods. 相似文献
4.
Kriging in a global neighborhood 总被引：1，自引：0，他引：1
The kriging estimator is usually computed in a moving neighborhood; only the data near the point to be estimated are used. This moving neighborhood approach creates discontinuities in mapping applications. An alternative approach is presented here, whereby all points are estimated using all the available data. To solve the resulting large linear system the kriging estimator is expressed in terms of the inverse of the covariance matrix. The covariance matrix has the advantage of being positive definite and the size of system which can be solved without encountering numerical instability is substantially increased. Because the kriging matrix does not change, the estimator can be written in terms of scalar products, thus avoiding the more time-consuming matrix multiplications of the standard approach. In the particular case of a covariance which is zero for distances greater than a fixed value (the range), the resulting banded structure of the covariance matrix is shown to lead to substantial computational savings in both run time and storage space. In this case the calculation time for the kriging variance is also substantially reduced. The present method is extended to the nonstationary case. 相似文献
5.
Geostatistics provides a suite of methods, summarized as kriging, to analyze a finite data set to describe a continuous property of the Earth. Kriging methods consist of moving window optimum estimation techniques, which are based on a least-squares principle and use a spatial structure function, usually the variogram. Applications of kriging techniques have become increasingly wide-spread, with ordinary kriging and universal kriging being the most popular ones. The dependence of the final map or model on the input, however, is not generally understood. Herein we demonstrate how changes in the kriging parameters and the neighborhood search affect the cartographic result. Principles are illustrated through a glaciological study. The objective is to map ice thickness and subglacial topography of Storglaciären, Kebnekaise Massif, northern Sweden, from several sets of radio-echo soundings and hot water drillings. New maps are presented. 相似文献
6.
Geostatistics provides a suite of methods, summarized as kriging, to analyze a finite data set to describe a continuous property of the Earth. Kriging methods consist of moving window optimum estimation techniques, which are based on a least-squares principle and use a spatial structure function, usually the variogram. Applications of kriging techniques have become increasingly wide-spread, with ordinary kriging and universal kriging being the most popular ones. The dependence of the final map or model on the input, however, is not generally understood. Herein we demonstrate how changes in the kriging parameters and the neighborhood search affect the cartographic result. Principles are illustrated through a glaciological study. The objective is to map ice thickness and subglacial topography of Storglaciären, Kebnekaise Massif, northern Sweden, from several sets of radio-echo soundings and hot water drillings. New maps are presented. 相似文献
7.
Ordinary kriging, in its common formulation, is a discrete estimator in that it requires the solution of a kriging system for each point in space in which an estimate is sought. The dual formulation of ordinary kriging provides a continuous estimator since, for a given set of data, only a kriging system has to be estimated and the resulting estimate is a function continuously defined in space. The main problem with dual kriging up to now has been that its benefits can only be capitalized if a global neighborhood is used. A formulation is proposed to solve the problem of patching together dual kriging estimates obtained with data from different neighborhoods by means of a blending belt around each neighborhood. This formulation ensures continuity of the variable and, if needed, of its first derivative along neighbor borders. The final result is an analytical formulation of the interpolating surface that can be used to compute gradients, cross-sections, or volumes; or for the quick evaluation of the interpolating surface in numerous locations. 相似文献
8.
Geostatistical Mapping with Continuous Moving Neighborhood 总被引：1，自引：0，他引：1
An issue that often arises in such GIS applications as digital elevation modeling (DEM) is how to create a continuous surface using a limited number of point observations. In hydrological applications, such as estimating drainage areas, direction of water flow is easier to detect from a smooth DEM than from a grid created using standard interpolation programs. Another reason for continuous mapping is esthetic; like a picture, a map should be visually appealing, and for some GIS users this is more important than map accuracy. There are many methods for local smoothing. Spline algorithms are usually used to create a continuous map, because they minimize curvature of the surface. Geostatistical models are commonly used approaches to spatial prediction and mapping in many scientific disciplines, but classical kriging models produce noncontinuous surfaces when local neighborhood is used. This motivated us to develop a continuous version of kriging. We propose a modification of kriging that produces continuous prediction and prediction standard error surfaces. The idea is to modify kriging systems so that data outside a specified distance from the prediction location have zero weights. We discuss simple kriging and conditional geostatistical simulation, models that essentially use information about mean value or trend surface. We also discuss how to modify ordinary and universal kriging models to produce continuous predictions, and limitations using the proposed models. 相似文献
9.
A stationary specification of anisotropy does not always capture the complexities of a geologic site. In this situation, the
anisotropy can be varied locally. Directions of continuity and the range of the variogram can change depending on location
within the domain being modeled. Kriging equations have been developed to use a local anisotropy specification within kriging
neighborhoods; however, this approach does not account for variation in anisotropy within the kriging neighborhood. This paper
presents an algorithm to determine the optimum path between points that results in the highest covariance in the presence
of locally varying anisotropy. Using optimum paths increases covariance, results in lower estimation variance and leads to
results that reflect important curvilinear structures. Although CPU intensive, the complex curvilinear structures of the kriged
maps are important for process evaluation. Examples highlight the ability of this methodology to reproduce complex features
that could not be generated with traditional kriging. 相似文献
10.
Cross validation of kriging in a unique neighborhood 总被引：1，自引：0，他引：1
Olivier Dubrule 《Mathematical Geosciences》1983,15(6):687-699
Cross validation is an appropriate tool for testing interpolation methods: it consists of leaving out one data point at a time, and determining how well this point can be estimated from the other data. Cross validation is often used for testing “moving neighborhood” kriging models; in this case, each unknown value is predicted from a small number of surrounding data. In “unique neighborhood” kriging algorithms, each estimation uses all the available data; as a result, cross validation would spend much computer time. For instance, with ndata points it would cost at least the resolution of nsystems of n × nlinear equations (each with a different matrix).Here, we present a much faster method for cross validation in a unique neighborhood. Instead of solving nsystems n × n,it only requires the inversion of one n × nmatrix. We also generalized this method to leaving out several points instead of one. 相似文献
11.
Simulating Large Gaussian Random Vectors Subject to Inequality Constraints by Gibbs Sampling 总被引：4，自引：2，他引：2
The Gibbs sampler is an iterative algorithm used to simulate Gaussian random vectors subject to inequality constraints. This algorithm relies on the fact that the distribution of a vector component conditioned by the other components is Gaussian, the mean and variance of which are obtained by solving a kriging system. If the number of components is large, kriging is usually applied with a moving search neighborhood, but this practice can make the simulated vector not reproduce the target correlation matrix. To avoid these problems, variations of the Gibbs sampler are presented. The conditioning to inequality constraints on the vector components can be achieved by simulated annealing or by restricting the transition matrix of the iterative algorithm. Numerical experiments indicate that both approaches provide realizations that reproduce the correlation matrix of the Gaussian random vector, but some conditioning constraints may not be satisfied when using simulated annealing. On the contrary, the restriction of the transition matrix manages to satisfy all the constraints, although at the cost of a large number of iterations. 相似文献
12.
Denis Marcotte 《Mathematical Geology》1995,27(6):749-762
Conditional simulation with data subject to measurement error has received little attention in the geostatistical literature. The treatment of measurement error in simulation must be different from its treatment in estimation. Two approaches are examined: pre- and post-simulation filtering of data measurement error. The pre-simulation filtering is shown to be inefficient. The post-simulation filtering performs best. It is done by factorial kriging and a modified version of factorial kriging which ensures predetermined theoretical variance for the filtered data. It also is shown that the theoretical variogram of the filtered data reproduces the underlying variogram (i.e., without noise) almost perfectly. A simulation with a high level of correlated noise is used for validation and comparison. The post-simulation filtered values show an experimental variogram in agreement with the previously identified underlying variogram. Moreover, the filtered image compares well with the true image. The theoretical variogram corresponding to the post-simulation filter can be computed beforehand. Thus, the size of the simulation grid and of the filter neighborhood can be adjusted to ensure good reproduction of the underlying variogram. 相似文献
13.
Compensating for estimation smoothing in kriging 总被引：2，自引：0，他引：2
Smoothing is a characteristic inherent to all minimum mean-square-error spatial estimators such as kriging. Cross-validation can be used to detect and model such smoothing. Inversion of the model produces a new estimator—compensated kriging. A numerical comparison based on an exhaustive permeability sampling of a 4-ft2 slab of Berea Sandstone shows that the estimation surface generated by compensated kriging has properties intermediate between those generated by ordinary kriging and stochastic realizations resulting from simulated annealing and sequential Gaussian simulation. The frequency distribution is well reproduced by the compensated kriging surface, which also approximates the experimental semivariogram well—better than ordinary kriging, but not as well as stochastic realizations. Compensated kriging produces surfaces that are more accurate than stochastic realizations, but not as accurate as ordinary kriging. 相似文献
14.
Clayton V. Deutsch 《Mathematical Geosciences》1993,25(1):41-52
Adopting a random function model {Z(u),u ε study areaA} and using the normal equations (kriging) for estimation amounts to assume that the study areaA is embedded within a infinite domain. At first glance, this assumption has no inherent limitations since all locations outsideA are of no interest and simply not considered. However, there is an interesting and practically important consequence that is reflected in the kriging weights assigned to data contiguously aligned along finite strings; the weights assigned to the end points of a string are large since the end points inform the infinite half-space beyond the string. These large weights are inappropriate when the finite string has been created by either stratigraphic/geological limits or a finite search neighborhood. This problem will be demonstrated with numerical examples and some partial solutions will be proposed. 相似文献
15.
Disjunctive kriging, universal kriging, or no kriging: Small sample results with simulated fields 总被引：3，自引：0，他引：3
This paper provides a comparison between linear (universal) and nonlinear (disjunctive) kriging estimators when they are computed from small samples chosen randomly on simulated stationary and nonstationary fields. Point estimation results are reported. In all cases considered, kriging estimators were found better than a local mean estimator, with universal kriging either better than or as good as disjunctive kriging. The latter, which is suited to handle stationary fields, did not provide more accurate estimates because the use of small samples led to inconsistencies in the assumed bivariate model. Universal kriging was particularly better with nonstationary fields. 相似文献
16.
Marzeihe Shademan Khakestar Hassan Madani Hossein Hassani Parvizz Moarefvand 《Journal of the Geological Society of India》2013,81(4):581-585
Ordinary kriging and non-linear geostatistical estimators are now well accepted methods in mining grade control and mine reserve estimation. In kriging, the search volume or ‘kriging neighbourhood’ is defined by the user. The definition of the search space can have a significant impact on the outcome of the kriging estimate. In particular, too restrictive neighbourhood, can result in serious conditional bias. Kriging is commonly described as a ‘minimum variance estimator’ but this is only true when the neighbourhood is properly selected. Arbitrary decisions about search space are highly risky. The criteria to consider when evaluating a particular kriging neighbourhood are the slope of the regression of the ‘true’ and ‘estimated’ block grades, the number of kriging negative weights and the kriging variance. Search radius is one of the most important parameters of search volume which often is determined on the basis of influence of the variogram. In this paper the above-mentioned parameters are used to determine optimal search radius. 相似文献
17.
Kriging in a finite domain 总被引：2，自引：0，他引：2
Clayton V. Deutsch 《Mathematical Geology》1993,25(1):41-52
Adopting a random function model {Z(u),u study areaA} and using the normal equations (kriging) for estimation amounts to assume that the study areaA is embedded within a infinite domain. At first glance, this assumption has no inherent limitations since all locations outsideA are of no interest and simply not considered. However, there is an interesting and practically important consequence that is reflected in the kriging weights assigned to data contiguously aligned along finite strings; the weights assigned to the end points of a string are large since the end points inform the infinite half-space beyond the string. These large weights are inappropriate when the finite string has been created by either stratigraphic/geological limits or a finite search neighborhood. This problem will be demonstrated with numerical examples and some partial solutions will be proposed. 相似文献
18.
Interpolating geo-data with curvilinear structures using geostatistics is often disappointing. Channels, for example, become
disconnected sets of lakes when interpolated from point data. In order to improve the interpolation of geological structures
(e.g., curvilinear structures), we present a new form of kriging, local anisotropy kriging (LAK). Local anisotropy kriging
combines a gradient algorithm from image analysis with kriging in an iterative way. After an initial standard kriging interpolation,
the gradient algorithm determines the local anisotropy for each cell in the grid using a search area around the cell. Subsequently,
kriging is carried out with the spatially varying anisotropy. The anisotropy calculation and subsequent kriging steps will
then succeed until the result is satisfactory in the way of reproducing the curvilinear structures. Depending on the size
of the search area more or less detail in the geological structures can be reproduced with LAK. Using test examples we show
that LAK interpolates data with curvilinear structures more realistically than standard kriging. In a real world case, using
bathymetric data of the Oosterschelde estuary, LAK also proves to be quantitatively superior to standard kriging. Absolute
interpolation errors are decreased by 23%. Local anisotropy kriging only uses information from point data, which makes the
method very objective, it only presents “what the data can tell.” 相似文献
19.
《Applied Geochemistry》1999,14(1):133-145
Three univariate geostatistical methods of estimation are applied to a geochemical data set. The studied methods are: ordinary kriging (cross-validation), factorial kriging, and indicator kriging. These techniques use the probabilistic and spatial behaviour of geochemical variables, giving a tool for identifying potential anomalous areas to locate mineralization. Ordinary kriging is easy to apply and to interpret the results. It has the advantage of using the same experimental grid points for its estimates, and no additional grid points are needed. Factorial kriging decomposes the raw variable into as many components as there are identified structures in the variogram. This, however, is a complex method and its application is more difficult than that of ordinary or indicator kriging. The main advantages of indicator kriging are that data are used by their rank order, being more robust about outlier values, and that the presentation of results is simple. Nevertheless, indicator kriging is incapable of separating anomalous values and the high values from the background, which have a behaviour different to the anomaly. In this work, the results of the application of these 3 kriging methods to a set of mineral exploration data obtained from a geochemical survey carried out in NW Spain are presented. This area is characterised by the presence of Au mineral occurrences. The kriging methods were applied to As, considered as a pathfinder of Au in this area. Numerical treatment of Au is not applicable, because it presents most values equal to the detection limit, and a series of extreme values. The results of the application of ordinary kriging, factorial kriging and indicator kriging to As make possible the location of a series of rich values, sited along a N–S shear zone, considered a structure related to the presence of Au. 相似文献
20.
Geological data frequently have a heavy-tailed normal-in-the-middle distribution, which gives rise to grade distributions that appear to be normal except for the occurrence of a few outliers. This same situation also applies to log-transformed data to which lognormal kriging is to be applied. For such data, linear kriging is nonrobust in that (1)kriged estimates tend to infinity as the outliers do, and (2)it is also not minimum mean squared error. The more general nonlinear method of disjunctive kriging is even more nonrobust, computationally more laborious, and in the end need not produce better practical answers. We propose a robust kriging method for such nearly normal data based on linear kriging of an editing of the data. It is little more laborious than conventional linear kriging and, used in conjunction with a robust estimator of the variogram, provides good protection against the effects of data outliers. The method is also applicable to time series analysis. 相似文献