Fitting matrix-valued variogram models by simultaneous diagonalization (Part I: Theory)

1995 ◽  
Vol 27 (7) ◽  
pp. 867-875 ◽  
Author(s):  
Tailiang Xie ◽  
Donald E. Myers
Keyword(s):  
Simultaneous Diagonalization ◽  
Variogram Models

Fitting matrix-valued variogram models by simultaneous diagonalization (Part II: Application)

1995 ◽  
Vol 27 (7) ◽  
pp. 877-888 ◽  
Author(s):  
Tailiang Xie ◽  
Donald E. Myers ◽  
Andrew E. Long
Keyword(s):  
Simultaneous Diagonalization ◽  
Variogram Models

scholarly journals A necessary and sufficient condition for simultaneous diagonalization of two hermitian matrices and its application

1970 ◽  
Vol 11 (1) ◽  
pp. 81-83 ◽  
Author(s):  
Yik-Hoi Au-Yeung
Keyword(s):  
Complex Numbers ◽  
Single Element ◽  
Singular Matrix ◽  
Necessary And Sufficient Condition ◽  
Simultaneous Diagonalization ◽  
Hermitian Matrices ◽  
Skew Field ◽  
Conjugate Transpose ◽  
Necessary And Sufficient ◽  
Real Quaternions

We denote by F the field R of real numbers, the field C of complex numbers, or the skew field H of real quaternions, and by Fn an n dimensional left vector space over F. If A is a matrix with elements in F, we denote by A* its conjugate transpose. In all three cases of F, an n × n matrix A is said to be hermitian if A = A*, and we say that two n × n hermitian matrices A and B with elements in F can be diagonalized simultaneously if there exists a non singular matrix U with elements in F such that UAU* and UBU* are diagonal matrices. We shall regard a vector u ∈ Fn as a l × n matrix and identify a 1 × 1 matrix with its single element, and we shall denote by diag {A1, …, Am} a diagonal block matrix with the square matrices A1, …, Am lying on its diagonal.

scholarly journals Geostatistical interpolation of daily rainfall at catchment scale: the use of several variogram models in the Ourthe and Ambleve catchments, Belgium

2011 ◽  
Vol 15 (7) ◽  
pp. 2259-2274 ◽  
Author(s):  
S. Ly ◽  
C. Charles ◽  
A. Degré
Keyword(s):  
Ordinary Kriging ◽  
Spatial Interpolation ◽  
Gaussian Model ◽  
Daily Rainfall ◽  
Spherical Model ◽  
Daily Basis ◽  
Catchment Scale ◽  
Multivariate Geostatistics ◽  
Variogram Models ◽  
Thiessen Polygon

Abstract. Spatial interpolation of precipitation data is of great importance for hydrological modelling. Geostatistical methods (kriging) are widely applied in spatial interpolation from point measurement to continuous surfaces. The first step in kriging computation is the semi-variogram modelling which usually used only one variogram model for all-moment data. The objective of this paper was to develop different algorithms of spatial interpolation for daily rainfall on 1 km2 regular grids in the catchment area and to compare the results of geostatistical and deterministic approaches. This study leaned on 30-yr daily rainfall data of 70 raingages in the hilly landscape of the Ourthe and Ambleve catchments in Belgium (2908 km2). This area lies between 35 and 693 m in elevation and consists of river networks, which are tributaries of the Meuse River. For geostatistical algorithms, seven semi-variogram models (logarithmic, power, exponential, Gaussian, rational quadratic, spherical and penta-spherical) were fitted to daily sample semi-variogram on a daily basis. These seven variogram models were also adopted to avoid negative interpolated rainfall. The elevation, extracted from a digital elevation model, was incorporated into multivariate geostatistics. Seven validation raingages and cross validation were used to compare the interpolation performance of these algorithms applied to different densities of raingages. We found that between the seven variogram models used, the Gaussian model was the most frequently best fit. Using seven variogram models can avoid negative daily rainfall in ordinary kriging. The negative estimates of kriging were observed for convective more than stratiform rain. The performance of the different methods varied slightly according to the density of raingages, particularly between 8 and 70 raingages but it was much different for interpolation using 4 raingages. Spatial interpolation with the geostatistical and Inverse Distance Weighting (IDW) algorithms outperformed considerably the interpolation with the Thiessen polygon, commonly used in various hydrological models. Integrating elevation into Kriging with an External Drift (KED) and Ordinary Cokriging (OCK) did not improve the interpolation accuracy for daily rainfall. Ordinary Kriging (ORK) and IDW were considered to be the best methods, as they provided smallest RMSE value for nearly all cases. Care should be taken in applying UNK and KED when interpolating daily rainfall with very few neighbourhood sample points. These recommendations complement the results reported in the literature. ORK, UNK and KED using only spherical model offered a slightly better result whereas OCK using seven variogram models achieved better result.

Kriging and Variogram Models

2009 ◽  
pp. 45-51
Author(s):  
Catherine A. Calder ◽  
Noel Cressie
Keyword(s):  
Variogram Models

scholarly journals Description of the Dependence Strength of Two Variogram Models of a Spatial Structure Using Archimedean Copulas

2017 ◽  
Vol 9 (1) ◽  
pp. 117
Author(s):  
Moumouni Diallo ◽  
Diakarya Barro
Keyword(s):  
Spatial Structure ◽  
Dependence Structure ◽  
Archimedean Copulas ◽  
Variogram Models ◽  
Measure Of Similarity

Variogram is a geostatistical tool which describes how the spatialcontinuity changes with a given separating distance between pairs of stations. In this paper, we study the dependence structure within a same class of bivariate spatialized archimedean copulas. Specifically, we point out properties of the gaussian variogram and the exponential one. A new measure of similarity of two copulas is computed particularly between the spatial independent copula and full dependence one.

On the Connection of Nonrenormalizable and Renormalizable Unified Nonlinear Spinor Field IVfodels

1984 ◽  
Vol 39 (5) ◽  
pp. 441-446
Author(s):  
H. Stumpf
Keyword(s):  
Field Equation ◽  
Spinor Field ◽  
Order Derivative ◽  
Simultaneous Diagonalization ◽  
Field Equations ◽  
Nonlinear Spinor ◽  
Original Field ◽  
Higher Order Derivative ◽  
Grand Canonical ◽  
Spinor Field Equation

The nonrenormalizable first order derivative nonlinear spinor field equation with scalar interaction possesses two equivalent Hamiltonians. The first is the conventional one while the second is a two-field Hamiltonian with the original field and its parity transform. By quantization the latter leads to an inequivalent representation compared with the former. This is connected with parity symmetry breaking and the loss of simultaneous diagonalization of energy and subfield particle numbers. The corresponding grand canonical Hamiltonian is shown to result equivalently from a renormalizable second order derivative nonlinear spinor field equation. This is achieved by means of a theorem about the decomposition of higher order derivative nonlinear spinor field equations derived previously

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