New Robust Cross-Variogram Estimators and Approximations of Their Distributions Based on Saddlepoint Techniques

Let Z(s)=(Z1(s),…,Zp(s))t be an isotropic second-order stationary multivariate spatial process. We measure the statistical association between the p random components of Z with the correlation coefficients and measure the spatial dependence with variograms. If two of the Z components are correlated, the spatial information provided by one of them can improve the information of the other. To capture this association, both within components of Z(s) and across s, we use a cross-variogram. Only two robust cross-variogram estimators have been proposed in the literature, both by Lark, and their sample distributions were not obtained. In this paper, we propose new robust cross-variogram estimators, following the location estimation method instead of the scale estimation one considered by Lark, thus extending the results obtained by García-Pérez to the multivariate case. We also obtain accurate approximations for their sample distributions using saddlepoint techniques and assuming a multivariate-scale contaminated normal model. The question of the independence of the transformed variables to avoid the usual dependence of spatial observations is also considered in the paper, linking it with the acceptance of linear variograms and cross-variograms.

Co-Kriging Method for Form Error Estimation Incorporating Condition Variable Measurements

The form error estimation under various machining conditions is an essential step in the assessment of product surface quality generated in machining processes. Coordinate measuring machines (CMMs) are widely used to measure complicated surface form error. However, considering measurement cost, only a few measurement points are collected offline by a CMM for a part surface. Therefore, spatial statistics is adopted to interpolate more points for more accurate form error estimation. It is of great significance to decrease the deviation between the interpolated height value and the real one. Compared to univariate spatial statistics, only concerning spatial correlation of height value, this paper presents a method based on multivariate spatial statistics, co-Kriging (CK), to estimate surface form error not only concerning spatial correlation but also concerning the influence of machining conditions. This method can reconstruct a more accurate part surface and make the estimation deviation smaller. It characterizes the spatial correlation of machining errors by variogram and cross-variogram, and it is implemented on one of the common features: flatness error. Simulated datasets as well as actual CMM data are applied to demonstrate the improvement achieved by the proposed multivariate spatial statistics method over the univariate method and other interpolation methods.

Magnetotelluric static shift: Estimation and removal using the cokriging method

We present and test a new method to correct for the static shift affecting magnetotelluric (MT) apparent resistivity sounding curves. We use geostatistical analysis of apparent resistivity and phase data for selected periods. For each period, we first estimate and model the experimental variograms and cross variogram between phase and apparent resistivity. We then use the geostatistical model to estimate, by cokriging, the corrected apparent resistivities using the measured phases and apparent resistivities. The static shift factor is obtained as the difference between the logarithm of the corrected and measured apparent resistivities. We retain as final static shift estimates the ones for the period displaying the best correlation with the estimates at all periods. We present a 3D synthetic case study showing that the static shift is retrieved quite precisely when the static shift factors are uniformly distributed around zero. If the static shift distribution has a nonzero mean, we obtained best results when an apparent resistivity data subset can be identified a priori as unaffected by static shift and cokriging is done using only this subset. The method has been successfully tested on the synthetic COPROD-2S2 2D MT data set and on a 3D-survey data set from Las Cañadas Caldera (Tenerife, Canary Islands) severely affected by static shift.