共查询到20条相似文献，搜索用时 46 毫秒
1.
Ordinary Cokriging Revisited 总被引：12，自引：0，他引：12
P. Goovaerts 《Mathematical Geology》1998,30(1):21-42
This paper sets up the relations between simple cokriging and ordinary cokriging with one or several unbiasedness constraints. Differences between cokriging variants are related to differences between models adopted for the means of primary and secondary variables. Because it is not necessary for the secondary data weights to sum to zero, ordinary cokriging with a single unbiasedness constraint gives a larger weight to the secondary information while reducing the occurrence of negative weights. Also the weights provided by such cokriging systems written in terms of covariances or correlograms are not related linearly, hence the estimates are different. The prediction performances of cokriging estimators are assessed using an environmental dataset that includes concentrations of five heavy metals at 359 locations. Analysis of reestimation scores at 100 test locations shows that kriging and cokriging perform equally when the primary and secondary variables are sampled at the same locations. When the secondary information is available at the estimated location, one gains little by retaining other distant secondary data in the estimation. 相似文献
2.
This paper sets up the relations between simple cokriging and ordinary cokriging with one or several unbiasedness constraints. Differences between cokriging variants are related to differences between models adopted for the means of primary and secondary variables. Because it is not necessary for the secondary data weights to sum to zero, ordinary cokriging with a single unbiasedness constraint gives a larger weight to the secondary information while reducing the occurrence of negative weights. Also the weights provided by such cokriging systems written in terms of covariances or correlograms are not related linearly, hence the estimates are different. The prediction performances of cokriging estimators are assessed using an environmental dataset that includes concentrations of five heavy metals at 359 locations. Analysis of reestimation scores at 100 test locations shows that kriging and cokriging perform equally when the primary and secondary variables are sampled at the same locations. When the secondary information is available at the estimated location, one gains little by retaining other distant secondary data in the estimation. 相似文献
3.
R. J. Yao J. S. Yang P. Gao H. B. Shao G. M. Liu S. P. Yu 《Environmental Earth Sciences》2014,71(1):233-243
Soil salinity has been known to be problematic to land productivity and environment in the lower Yellow River Delta due to the presence of a shallow, saline water table and marine sediments. Spatial information on soil salinity has gained increasing importance for the demand of management and sustainable utilization of arable land in this area. Apparent electrical conductivity, as measured by electromagnetic induction instrument in a fairly quick manner, has succeeded in mapping soil salinity and many other soil physical and chemical properties from field to regional scales. This was done based on the correlation that existed between apparent electrical conductivity and many other soil properties. In this paper, four spatial prediction methods, i.e., local polynomial, inverse distance weighed, ordinary kriging and universal kriging, were employed to estimate field-scale apparent electrical conductivity with the aid of an electromagnetic induction instrument (type EM38). The spatial patterns estimated by the four methods using EM38 survey datasets of various sample sizes were compared with those generated by each method using the entire sample size. Spatial similarity was evaluated using difference index (DI) between the maps created using various sample sizes (i.e., target maps) and the maps generated with the entire sample size (i.e., the reference map). The results indicated that universal kriging had the best performance owing to the inclusion of residuals and spatial detrending in the kriging system. DI showed that spatial similarity between the target and reference maps of apparent electrical conductivity decreased with the reduction in sample size for each prediction method. Under the same reduction in sample size, the method retaining the most spatial similarity was universal kriging, followed by ordinary kriging, inverse distance weighed, and local polynomial. Approximately, 70 % of total survey data essentially met the need for retaining 90 % details of the reference map for universal kriging and ordinary kriging methods. This conclusion was that OK and UK were two most appropriate methods for spatial estimation of apparent electrical conductivity as they were robust with the reduction in sample size. 相似文献
4.
Large cokriging systems arise in many situations and are difficult to handle in practice. Simplifications such as simple kriging,
strictly collocated and multicollocated cokriging are often used and models under which such simplifications are, in fact,
equivalent to cokriging have recently received attention. In this paper, a two-dimensional second-order stationary random
process with known mean is considered and the redundancy of certain components of the data at certain locations vis-à-vis
the solution to the simple cokriging system is examined. Conditions for the simple cokriging weights of these components at
these locations are set to zero. The conditions generalise the notion of the autokrigeability coefficient and can, in principle,
be applied to any data configuration. In specific sampling situations such as the isotopic and certain heterotropic configurations,
models under which simple kriging, strictly collocated, multicollocated and dislocated cokriging are equivalent to simple
cokriging are readily identified and results already available in the literature are obtained. These are readily identified
and the results are already available in the literature. The advantage of the approach presented here is that it can be applied
to any data configuration for analysis of permissible simplifications in simple cokriging. 相似文献
5.
On the Equivalence of the Cokriging and Kriging Systems 总被引：2，自引：0，他引：2
Simple cokriging of components of a p-dimensional second-order stationary random process is considered. Necessary and sufficient conditions under which simple cokriging is equivalent to simple kriging are given. Essentially this condition requires that it should be possible to express the cross-covariance at any lag series h using the cross-covariance at |h|=0 and the auto-covariance at lag series h. The mosaic model, multicolocated kriging and the linear model of coregionalization are examined in this context. A data analytic method to examine whether simple kriging of components of a multivariate random process is equivalent to its cokriging is given 相似文献
6.
3D Property Modeling of Void Ratio by Cokriging 总被引：2，自引：0，他引：2
Void ratio measures compactness of ground soil in geotechnical engineering. When samples are collected in certain area for mapping void ratios, other relevant types of properties such as water content may be also analyzed. To map the spatial distribution of void ratio in the area based on these types of point, observation data interpolation is often needed. Owing to the variance of sampling density along the horizontal and vertical directions, special consideration is required to handle anisotropy of estimator. 3D property modeling aims at predicting the overall distribution of property values from limited samples, and geostatistical method can he employed naturally here because they help to minimize the mean square error of estimation. To construct 3D property model of void ratio, cokriging was used considering its mutual correlation with water content, which is another important soil parameter. Moreover, K-D tree was adopted to organize the samples to accelerate neighbor query in 3D space during the above modeling process. At last, spatial configuration of void ratio distribution in an engineering body was modeled through 3D visualization, which provides important information for civil engineering purpose. 相似文献
7.
Masoomeh Delbari 《Arabian Journal of Geosciences》2014,7(10):4221-4233
This paper presents the incorporation of a digital elevation model into the spatial prediction of water table elevation in Mazandaran province (Iran) using a range of interpolation techniques. The multivariate methods used are: linear regression (LR), cokriging (COK), kriging with an external drift (KED) and regression kriging (RK). The analysis is performed on 3 years (1987, 1997 and 2007) of water table elevation data from about 260 monitoring wells. Prediction performances of the different algorithms are compared with two univariate techniques, i.e. inverse distance weighting and ordinary kriging (OK), through cross validation and examination of the consistency of the generated maps with the natural phenomena. Significantly smaller prediction errors are obtained for four multivariate algorithms but, in particular, KED and RK outperform LR and COK for 3 years. The results show the potential for using elevation for a more precise mapping of water table elevation. 相似文献
8.
The problem of estimating a regionalized variable in the presence of other secondary variables is encountered in spatial investigations. Given a context in which the secondary variable is known everywhere (or can be estimated with great precision), different estimation methods are compared: regression, regression with residual simple kriging, kriging, simple kriging with a mean obtained by regression, kriging with an external drift, and cokriging. The study focuses on 19 pairs of regionalized variables from five different datasets representing different domains (geochemical, environmental, geotechnical). The methods are compared by cross-validation using the mean absolute error as criterion. For correlations between the principal and secondary variable under 0.4, similar results are obtained using kriging and cokriging, and these methods are superior slightly to the other approaches in terms of minimizing estimation error. For correlations greater than 0.4, cokriging generally performs better than other methods, with a reduction in mean absolute errors that can reach 46% when there is a high degree of correlation between the variables. Kriging with an external drift or kriging the residuals of a regression (SKR) are almost as precise as cokriging. 相似文献
9.
Compositional data are very common in the earth sciences. Nevertheless, little attention has been paid to the spatial interpolation of these data sets. Most interpolators do not necessarily satisfy the constant sum and nonnegativity constraints of compositional data, nor take spatial structure into account. Therefore, compositional kriging is introduced as a straightforward extension of ordinary kriging that complies with these constraints. In two case studies, the performance of compositional kriging is compared with that of the additive logratio-transform. In the first case study, compositional kriging yielded significantly more accurate predictions than the additive logratio-transform, while in the second case study the performances were comparable. 相似文献
10.
Which Models for Collocated Cokriging? 总被引：1，自引：0，他引：1
Jacques Rivoirard 《Mathematical Geology》2001,33(2):117-131
When a target variable is sparsely sampled, compared to a densely sampled auxiliary variable, cokriging requires simplifications. In its strict sense, collocated cokriging makes use of the auxiliary variable only at the current point where the target variable is to be estimated; in the multicollocated form, it also makes use of the auxiliary variable at all points where the target variable is available. This paper looks for the models that support these collocated cokrigings, i.e., the models in which the simplification resulting from the collocated forms does not result in any loss of information. In these models, the cross-structure between the two variables is shown to be proportional to the structure of the auxiliary variable, not to the structure of the target variable as is often assumed (except, of course, when all structures are proportional). The target variable depends on the auxiliary variable and on a spatially uncorrelated residual. Collocated cokriging simplifies to the simple method, which consists in kriging this residual. The strictly collocated cokriging corresponds to the particular case where the residual has a pure nugget structure, but it is then reduced to the single regression at the target point. Except for this trivial case, there are no models in which strictly collocated cokriging is exactly a cokriging. 相似文献
11.
Soil pH plays an important role in biogeochemical processes in soils. The spatial distribution of soil pH provides basic and useful information relevant to soil management and agricultural production. To obtain an accurate distribution map of soil pH on the Loess Plateau of China, 382 sampling sites were investigated throughout the region and four interpolation methods, i.e., inverse distance weighting (IDW), splines, ordinary kriging, and cokriging, were applied to produce a continuous soil pH surface. In the study region, soil pH values ranged from 6.06 to 10.76, with a mean of 8.49 and a median of 8.48. Land use type had a significant effect (p < 0.01) on soil pH; grassland soils had higher pHs than cropland and forestland soils. From a regional perspective, soil pH showed weak variation and strong spatial dependence, indicated by the low values of the coefficient of variation (0.05) and the nugget-to-sill ratios (<0.25). Indices of cross-validation, i.e., average error, mean absolute error, root mean square error, and model efficiency coefficient were used to compare the performance of the four different interpolation methods. Kriging methods interpolated more accurately than IDW and splines. Cokriging performed better than ordinary kriging and the accuracy was improved using soil organic carbon as an auxiliary variable. Regional distribution maps of soil pH were produced. The southeastern part of the region had relatively low soil pH values, probably due to higher precipitation, leaching, and higher soil organic matter contents. Areas of high soil pH were located in the north of the central part of the region, possibly associated with the salinization of sandy soils under inappropriate irrigation practices in an arid climate. Map accuracy could be further improved using new methods and incorporating other auxiliary variables, such as precipitation, elevation, terrain attributes, and vegetation types. 相似文献
12.
Indicator cokriging is an alternative to disjunctive kriging for estimation of spatial distributions. One way to determine which of these techniques is more accurate for estimation of spatial distributions is to apply each to a particular type of data. A procedure is developed for evaluation of disjunctive kriging and indicator cokriging for such an application. Application of this procedure to earthquake ground motion data found disjunctive kriging to be at least as accurate as indicator cokriging for estimation of spatial distributions of peak horizontal acceleration. Indicator cokriging was superior for all other types of earthquake ground motion data. 相似文献
13.
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
14.
Universal cokriging is used to obtain predictions when dealing with multivariate random functions. An important type of nonstationarity is defined in terms of multivariate random functions with increments which are stationary of orderk. The covariance between increments of different variables is modeled by means of the pseudo-cross-covariance function. Criteria are formulated to which the parameters of pseudo-cross-covariance functions must comply so as to ensure positive-definiteness. Cokriging equations and the induced cokriging equations are given. The study is illustrated by an example from soil science. 相似文献
15.
Comparative performance of indicator algorithms for modeling conditional probability distribution functions 总被引：1，自引：0，他引：1
P. Goovaerts 《Mathematical Geology》1994,26(3):389-411
This paper compares the performance of four algorithms (full indicator cokriging. adjacent cutoffs indicator cokriging, multiple indicator kriging, median indicator kriging) for modeling conditional cumulative distribution functions (ccdf).The latter three algorithms are approximations to the theoretically better full indicator cokriging in the sense that they disregard cross-covariances between some indicator variables or they consider that all covariances are proportional to the same function. Comparative performance is assessed using a reference soil data set that includes 2649 locations at which both topsoil copper and cobalt were measured. For all practical purposes, indicator cokriging does not perform better than the other simpler algorithms which involve less variogram modeling effort and smaller computational cost. Furthermore, the number of order relation deviations is found to be higher for cokriging algorithms, especially when constraints on the kriging weights are applied. 相似文献
16.
On Some Simplifications of Cokriging Neighborhood 总被引：2，自引：0，他引：2
Jacques Rivoirard 《Mathematical Geology》2004,36(8):899-915
Choosing the cokriging neighborhood is often difficult. A poor choice, ignoring influent data, can result in a loss of information as well as in artifacts in simulations based on cokriging. Then it is convenient to use if possible, or to refer to models that lead to simplified cokriging neighborhood. We essentially consider the case of two stationary variables, a target variable and an auxiliary one. By examining possible simplifications, we set up a list of models (essentially models with residuals) that, in general or under specific configurations, lead to simplifications of cokriging neighborhood. Collocated, dislocated, and other types of neighborhood are identified, that are optimal in some models and configurations. Possible extensions to cokriging with unknown means, and to more variables, are included. 相似文献
17.
Universal cokriging is used to obtain predictions when dealing with multivariate random functions. An important type of nonstationarity is defined in terms of multivariate random functions with increments which are stationary of orderk. The covariance between increments of different variables is modeled by means of the pseudo-cross-covariance function. Criteria are formulated to which the parameters of pseudo-cross-covariance functions must comply so as to ensure positive-definiteness. Cokriging equations and the induced cokriging equations are given. The study is illustrated by an example from soil science. 相似文献
18.
Goovaerts P 《Mathematical Geosciences》2010,42(5):535-554
A common issue in spatial interpolation is the combination of data measured over different spatial supports. For example,
information available for mapping disease risk typically includes point data (e.g. patients’ and controls’ residence) and
aggregated data (e.g. socio-demographic and economic attributes recorded at the census track level). Similarly, soil measurements
at discrete locations in the field are often supplemented with choropleth maps (e.g. soil or geological maps) that model the
spatial distribution of soil attributes as the juxtaposition of polygons (areas) with constant values. This paper presents
a general formulation of kriging that allows the combination of both point and areal data through the use of area-to-area,
area-to-point, and point-to-point covariances in the kriging system. The procedure is illustrated using two data sets: (1)
geological map and heavy metal concentrations recorded in the topsoil of the Swiss Jura, and (2) incidence rates of late-stage
breast cancer diagnosis per census tract and location of patient residences for three counties in Michigan. In the second
case, the kriging system includes an error variance term derived according to the binomial distribution to account for varying
degree of reliability of incidence rates depending on the total number of cases recorded in those tracts. Except under the
binomial kriging framework, area-and-point (AAP) kriging ensures the coherence of the prediction so that the average of interpolated
values within each mapping unit is equal to the original areal datum. The relationships between binomial kriging, Poisson
kriging, and indicator kriging are discussed under different scenarios for the population size and spatial support. Sensitivity
analysis demonstrates the smaller smoothing and greater prediction accuracy of the new procedure over ordinary and traditional
residual kriging based on the assumption that the local mean is constant within each mapping unit. 相似文献
19.
J. A. Martín-Fernández R. A. Olea-Meneses V. Pawlowsky-Glahn 《Mathematical Geology》2001,33(8):889-909
The additive logratio (alr) transformation has been used in several case studies to predict regionalized compositions using standard geostatistical estimation methods such as ordinary kriging and ordinary cokriging. It is a simple method that allows application to transformed data all the body of knowledge available for geostatistical analysis of coregionalizations without a constant sum constraint. To compare the performance of methods, it is customary to use a univariate crossvalidation approach based on the leaving-one-out technique to evaluate the performance for each attribute separately. For multivariate observations this approach is difficult to interpret in terms of overall performance. Therefore, we propose using appropriate distances in real space and in the simplex, to improve the crossvalidation approach and, going a step forward, to adapt the concept of stress from multidimensional scaling to obtain a global measure of performance for each method. The Lyons West oil field of Kansas is used to illustrate the impactof using different distances in the performance of ordinary kriging versus ordinary cokriging. 相似文献
20.
Multivariable spatial prediction 总被引：1，自引：0，他引：1
For spatial prediction, it has been usual to predict one variable at a time, with the predictor using data from the same type of variable (kriging) or using additional data from auxiliary variables (cokriging). Optimal predictors can be expressed in terms of covariance functions or variograms. In earth science applications, it is often desirable to predict the joint spatial abundance of variables. A review of cokriging shows that a new cross-variogram allows optimal prediction without any symmetry condition on the covariance function. A bivariate model shows that cokriging with previously used cross-variograms can result in inferior prediction. The simultaneous spatial prediction of several variables, based on the new cross-variogram, is then developed. Multivariable spatial prediction yields the mean-squared prediction error matrix, and so allows the construction of multivariate prediction regions. Relationships between cross-variograms, between single-variable and multivariable spatial prediction, and between generalized least squares estimation and spatial prediction are also given. 相似文献