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Based on the image method and addition theorem of spherical vector wave functions, an analytical solution of the multiple scattering by a buried medium-coated conducting sphere is proposed in this paper. An iterative process to obtain the scattered electromagnetic field is discussed on the basis of the continuous boundary condition in the plane boundary, the medium inner and the outer surface of a coated conducting sphere, respectively. Applying an image method and the addition theorem of spherical vector wave functions, the scattering electromagnetic fields by the plane in a local coordinate system can be transformed into the globe coordinate, and they can be regarded as the next incident electromagnetic fields to the buried medium-coated conducting sphere. This process does not end until the scattering electrical field on the plane boundary is accurate enough. Numerical results are given and compared with commercial software FEKO, they coincide enough; the calculation times of the present method are very short compared to those of the software FEKO, and some discussion is given at the end of this paper.

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Water use intensity (WUI) reveals water withdrawals with respect to economic output. Decomposing WUI into factors provides inner-system information affecting the indicator. The present study investigates variability in WUI among provinces in China by clustering the principal components of the decomposed factors. Motivated by the index decomposition method, the authors decomposed WUI into seven factors: water use in agricultural, industrial, household and ecological sectors, exploitation rate of water resources, per capita water resources and population intensity. Those seven factors condense into four principal components under application of principal component analysis. Comprehensive WUI is calculated by these four components. Then the cluster analysis is applied to get different patterns in WUI. The principal components and the comprehensive intensity are taken as cluster variables. The number of clusters is determined to be three by applying the k-means clustering method and the F-statistic value. Variability in WUI is detected by implementing three clustering algorithms, namely k-means, fuzzy c-means and the Gaussian mixture model. WUI in China is clustered into three clusters by the k-means clustering method. Characteristics of each cluster are analyzed.