共查询到20条相似文献，搜索用时 15 毫秒
1.
Normal cross-variograms cannot be estimated from data in the usual way when there are only a few points where both variables have been measured. But the experimental pseudo cross-variogram can be computed even where there are no matching sampling points, and this appears as its principal advantage. The pseudo cross-variogram may be unbounded, though for its existence the intrinsic hypothesis alone is not a sufficient stationarity condition. In addition the differences between the two random processes must be second order stationary. Modeling the function by linear coregionalization reflects the more restrictive stationarity condition: the pseudo cross-variogram can be unbounded only if the unbounded correlation structures are the same in all variograms. As an alternative to using the pseudo cross-variogram a new method is presented that allows estimating the normal cross variogram from data where only one variable has been measured at a point. 相似文献
2.
Normal cross-variograms cannot be estimated from data in the usual way when there are only a few points where both variables have been measured. But the experimental pseudo cross-variogram can be computed even where there are no matching sampling points, and this appears as its principal advantage. The pseudo cross-variogram may be unbounded, though for its existence the intrinsic hypothesis alone is not a sufficient stationarity condition. In addition the differences between the two random processes must be second order stationary. Modeling the function by linear coregionalization reflects the more restrictive stationarity condition: the pseudo cross-variogram can be unbounded only if the unbounded correlation structures are the same in all variograms. As an alternative to using the pseudo cross-variogram a new method is presented that allows estimating the normal cross variogram from data where only one variable has been measured at a point. 相似文献
3.
Donald E. Myers 《Mathematical Geology》1991,23(6):805-816
Cokriging allows the use of data on correlated variables to be used to enhance the estimation of a primary variable or more generally to enhance the estimation of all variables. In the first case, known as the undersampled case, it allows data on an auxiliary variable to be used to make up for an insufficient amount of data. Original formulations required that there be sufficiently many locations where data is available for both variables. The pseudo-cross-variogram, introduced by Clark et al. (1989), allows computing a related empirical spatial function in order to model the function, which can then be used in the cokriging equations in lieu of the cross-variogram. A number of questions left unanswered by Clark et al. are resolved, such as the availability of valid models, an appropriate definition of positive-definiteness, and the relationship of the pseudo-cross-variogram to the usual cross-variogram. The latter is important for modeling this function. 相似文献
4.
A tripartite earthquake response spectrum exhibits peak values of acceleration, velocity, and displacement in concert, each as a function of natural frequency of vibration. Because cokriging is a regionalized variable technique for multiple variable estimation, it was employed to jointly estimate peak values of acceleration, velocity, and displacement through a frequency discretization process. The objective of this application was to develop a technique for estimation of response spectra at uninstrumented locations. Because magnitudes of the three response variables differed, these data were normalized prior to cokriging. This process simply involved dividing each type of variable by its maximum value for a given frequency; hence, values of each type of variable ranged between zero and one. This allowed better accuracy in developing cross-variograms. Cokriging proved to be an efficient and accurate technique to use for the estimation of tripartite response spectra. 相似文献
5.
This paper presents a new application of the cokriging technique for constructing maps of aquifer transmissivity from field measurements of transmissivity and specific capacity. The technique is illustrated using data from Yolo Basin, California. Cokriging is well-suited for estimating undersampled variables. To improve the accuracy of the estimation, cokriging considers the spatial auto-correlation of the variable to be estimated and the spatial cross-correlation between the variable to be estimated and other, better-sampled variables. Consequently, in regions that lack data of the variable to be estimated, accurate estimation can still be made on the basis of auto- and cross-correlation. In addition, estimation variances can be obtained with a little additional computation effort. 相似文献
6.
Spatial and temporal study of nitrate concentration in groundwater by means of coregionalization 总被引：1，自引：0，他引：1
Spatial and temporal behavior of hydrochemical parameters in groundwater can be studied using tools provided by geostatistics.
The cross-variogram can be used to measure the spatial increments between observations at two given times as a function of
distance (spatial structure). Taking into account the existence of such a spatial structure, two different data sets (sampled
at two different times), representing concentrations of the same hydrochemical parameter, can be analyzed by cokriging in
order to reduce the uncertainty of the estimation. In particular, if one of the two data sets is a subset of the other (that
is, an undersampled set), cokriging allows us to study the spatial distribution of the hydrochemical parameter at that time,
while also considering the statistical characteristics of the full data set established at a different time.
This paper presents an application of cokriging by using temporal subsets to study the spatial distribution of nitrate concentration
in the aquifer of the Lucca Plain, central Italy. Three data sets of nitrate concentration in groundwater were collected during
three different periods in 1991. The first set was from 47 wells, but the second and the third are undersampled and represent
28 and 27 wells, respectively. Comparing the result of cokriging with ordinary kriging showed an improvement of the uncertainty
in terms of reducing the estimation variance. The application of cokriging to the undersampled data sets reduced the uncertainty
in estimating nitrate concentration and at the same time decreased the cost of the field sampling and laboratory analysis.
Received: 23 July 1997 · Accepted: 31 March 1998 相似文献
7.
The variance-based cross-variogram between two spatial processes, Z1
(·) and Z2
(·), is var (Z1
(
u
) – Z2
(
v
)), expressed generally as a bivariate function of spatial locations
uandv. It characterizes the cross-spatial dependence between Z1
(·) and Z2
(·) and can be used to obtain optimal multivariable predictors (cokriging). It has also been called the pseudo cross-variogram; here we compare its properties to that of the traditional (covariance-based) cross-variogram, cov (Z1
(
u
) – Z1
(
v
), Z2
(
u
) – Z2
(
v
)). One concern with the variance-based cross-variogram has been that Z1
(·) and Z2
(·) might be measured in different units (apples and oranges). In this note, we show that the cokriging predictor based on variance-based cross-variograms can handle any units used for Z1
(·) and Z2
(·); recommendations are given for an appropriate choice of units. We review the differences between the variance-based cross-variogram and the covariance-based cross-variogram and conclude that the former is more appropriate for cokriging. In practice, one often assumes that variograms and cross-variograms are functions of
uandv
only through the difference
u – v. This restricts the types of models that might be fitted to measures of cross-spatial dependence. 相似文献
8.
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
9.
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
10.
《Mathematical Geology》1997,29(6):779-799
Generalized cross-covariances describe the linear relationships between spatial variables observed at different locations.
They are invariant under translation of the locations for any intrinsic processes, they determine the cokriging predictors
without additional assumptions and they are unique up to linear functions. If the model is stationary, that is if the variograms
are bounded, they correspond to the stationary cross-covariances. Under some symmetry condition they are equal to minus the
usual cross-variogram. We present a method to estimate these generalized cross-covariances from data observed at arbitrary
sampling locations. In particular we do not require that all variables are observed at the same points. For fitting a linear
coregionalization model we combine this new method with a standard algorithm which ensures positive definite coregionalization
matrices. We study the behavior of the method both by computing variances exactly and by simulating from various models. 相似文献
11.
Comparison of kriging techniques in a space-time context 总被引：1，自引：0，他引：1
Patrick Bogaert 《Mathematical Geology》1996,28(1):73-86
Space-time processes constitute a particular class, requiring suitable tools in order to predict values in time and space, such as a space-time variogram or covariance function. The space-time co-variance function is defined and linked to the Linear Model of Coregionalization under second-order space-time stationarity. Simple and ordinary space-time kriging systems are compared to simple and ordinary cokriging and their differences for unbiasedness conditions are underlined. The ordinary space-time kriging estimation then is applied to simulated data. Prediction variances and prediction errors are compared with those for ordinary kriging and cokriging under different unbiasedness conditions using a cross-validation. The results show that space-time kriging tend to produce lower prediction variances and prediction errors that kriging and cokriging. 相似文献
12.
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
13.
Multivariable variogram and its application to the linear model of coregionalization 总被引：1，自引：0，他引：1
In this article, we present the multivariable variogram, which is defined in a way similar to that of the traditional variogram, by the expected value of a distance, squared, in a space withp dimensions. Combined with the linear model of coregionalization, this tool provides a way for finding the elementary variograms that characterize the different spatial scales contained in a set of data withp variables. In the case in which the number of elementary components is less than or equal to the number of variables, it is possible, by means of nonlinear regression of variograms and cross-variograms, to estimate the coregionalization parameters directly in order to obtain the elementary variables themselves, either by cokriging or by direct matrix inversion. This new tool greatly simplifies the procedure proposed by Matheron (1982) and Wackernagel (1985). The search for the elementary variograms is carried out using only one variogram (multivariable), as opposed to thep(p + 1)/2 required by the Matheron approach. Direct estimation of the linear coregionalization model parameters involves the creation of semipositive definite coregionalization matrices of rank 1. 相似文献
14.
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
15.
This study presents a new geostatistical approach to characterization of the geometry and quality of a multilayer coal deposit using the data of seam thickness as a geometric property and the contents of ash, sodium, total sulphur, and the heating value as quality properties. A coal deposit in East Kalimantan (Borneo), Indonesia, which has a synclinal geological structure, was chosen as the study site. Semivariogram analysis clarified the strong dependence of heating value on ash content in the top and bottom parts of each seam and the existence of a strong correlation with sodium content over the sub-seams in the same location. The correlations between the geometry and quality of the seams were generally weak. A linear coregionalization model was used to derive the spatial correlation coefficients of two variables at each scale component from the single- and cross-semivariogram matrices. Because the data were correlated spatially in the same seam or over different seams, multivariate techniques (ordinary cokriging and factorial cokriging) were mainly used and the resultant spatial estimates were compared to those derived using a univariate technique (ordinary kriging). A factorial cokriging was effective to decompose the spatial correlation structures with different scales. Another important characteristic was that the sodium content shows distinct segregation: the low zones are concentrated near the boundary of the sedimentary basin, while the high zones are concentrated in the central part. The main component of sodium originates from the abundance of saline water. Therefore, it can be inferred that seawater had stronger effects on the coal depositional process in the central basin than in the border part. The geostatistical modeling results suggest that the thicknesses of all the major seams were controlled by the syncline structure, while the coal qualities chiefly were originated from the coal depositional and diagenetic processes. 相似文献
16.
Spatial prediction of soil water content in karst area using prime terrain variables as auxiliary cokriging variable 总被引：1，自引：0，他引：1
In karst areas, accurately measuring and managing the spatial variability of soil water content (SWC) is very critical in settling numerous issues such as karst rocky desertification, ecosystem reconstruction, etc. In these areas, SWC exhibits strong spatial dependence, and it is a time and labor consuming procedure to measure its spatial variability. Therefore, estimation of this kind of soil property at an acceptable level of accuracy is of great significance. This study was conducted to evaluate and compare the spatial estimation of SWC by using ordinary kriging (OK) and cokriging (COK) methods with prime terrain variables, tending to predict SWC using limited available sample data for a 2,363.7 km2 study area in Mashan County, Guangxi Zhuang Autonomous Region, Southwest China. The measured SWC ranged from 3.36 to 26.69 %, with a mean of 17.34 %. The correlation analysis between SWC and prime terrain variables indicated that SWC showed significantly positive correlation with elevation (r is 0.46, P < 0.01), and significantly negative correlation with slope (r is ?0.30, P < 0.01); however, SWC was not significantly correlated with aspect in the study area. Therefore, elevation and slope were used as auxiliary data together for SWC prediction using COK method, and mean error (ME) and root mean square error were adopted to validate the prediction of SWC by these methods. Results indicated that COK with prime terrain variables data was superior to OK with relative improvement of 28.52 % in the case of limited available data, and also revealed that such elevation and slope data have the potential to improve the precision and reliability of SWC prediction as useful auxiliary variables. 相似文献
17.
There exist many secondary data that must be considered in in reservoir characterization for resource assessment and performance
forecasting. These include multiple seismic attributes, geological trends and structural controls. It is essential that all
secondary data be accounted for with the precision warranted by that data type. Cokriging is the standard technique in geostatistics
to account for multiple data types. The most common variant of cokriging in petroleum geostatistics is collocated cokriging.
Implementations of collocated cokriging are often limited to a single secondary variable. Practitioners often choose the most
correlated or most relevant secondary variable. Improved models would be constructed if multiple variables were accounted
for simultaneously. This paper presents a novel approach to (1) merge all secondary data into a single super secondary variable,
then (2) implement collocated cokriging with the single variable. The preprocessing step is straightforward and no major changes
are required in the standard implementation of collocated cokriging. The theoretical validity of this approach is proven,
that is, the results are proven to be identical to a “full” approach using all multiple secondary variables simultaneously. 相似文献
18.
协同克立格法,同时兼具化探数据多元性及克立格法表征空间属性的特点。考虑到地质变量两个以上的空间属性,在化探数据处理中运用协同克立格法,可以进一步提高估计精度。运用协同克立格法对广西林旺矿区中金元素成矿进行预测,经相关性分析显示,在矿区中元素的基本组合是Au、Ag、As、Hg,协同克立格插值以Au为主要变量,伴生元素Ag、As、Hg为次要变量。把插值计算后的结果与传统多元统计方法和普通克立格法的计算结果进行比较,结果协同克立格法得到的估值误差较小,预测精度较高,在成矿元素的预测中具有一定程度的优越性。 相似文献
19.
Matrix formulation of co-kriging 总被引：11，自引：0，他引：11
Donald E. Myers 《Mathematical Geology》1982,14(3):249-257
The matrix form of the general co-kriging problem is presented. Matrix solutions are given for SRFs with covariance functions, for IRFs of order zero using variograms and for universal co-kriging. General methods for obtaining cross-covariance or cross-variogram models are given. The relationship of the general co-kriging problem to the problem of one under sampled variable is presented. 相似文献
20.
A regionalized composition is a random vector function whose components are positive and sum to a constant at every point of the sampling region. Consequently, the components of a regionalized composition are necessarily spatially correlated. This spatial dependence—induced by the constant sum constraint—is a spurious spatial correlation and may lead to misinterpretations of statistical analyses. Furthermore, the cross-covariance matrices of the regionalized composition are singular, as is the coefficient matrix of the cokriging system of equations. Three methods of performing estimation or prediction of a regionalized composition at unsampled points are discussed: (1) the direct approach of estimating each variable separately; (2) the basis method, which is applicable only when a random function is available that can he regarded as the size of the regionalized composition under study; (3) the logratio approach, using the additive-log-ratio transformation proposed by J. Aitchison, which allows statistical analysis of compositional data. We present a brief theoretical review of these three methods and compare them using compositional data from the Lyons West Oil Field in Kansas (USA). It is shown that, although there are no important numerical differences, the direct approach leads to invalid results, whereas the basis method and the additive-log-ratio approach are comparable. 相似文献