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

Suppose that ¯(x₁),...,¯Z(x_n). are observations of vector-valued random function ¯(x). In the isotropic situation, the sample variogram γ^*(h) for a given lag h is $$\bar \gamma ^ * (h) = \frac{1}{{2N(h)}}\mathop \sum \limits_{s(h)} (\overline Z (x_1 ) - \overline Z (x_1 )) \overline {(Z} (x_1 ) - \overline Z (x_1 ))^T $$ where s(h) is a set of paired points with distance h and N(h) is the number of pairs in s(h).. For a selection of lags h₁, h₂, .... h_k such that N (h₁) > O. we obtain a ktuple of (semi) positive definite matrices $\bar \gamma ^ * (h_{ 1} ),. . . ., \bar \gamma ^ * (h_{ k} )$ . We want to determine an orthonormal matrix B which simultaneously diagonalizes the $\bar \gamma ^ * (h_{ 1} ),. . . ., \bar \gamma ^ * (h_{ k} )$ or nearly diagonalizes them in the sense that the sum of squares of offdiagonal elements is small compared to the sum of squares of diagonal elements. If such a B exists, we linearly transform $\overline Z (x)$ by $\overline Y (x) = B\overline Z (x)$ . Then, the resulting vector function $\overline Y (x)$ has less spatial correlation among its components than $\overline Z (x)$ does. The components of $\overline Y (x)$ with little contribution to the variogram structure may be dropped, and small crossvariograms fitted by straightlines. Variogram models obtained by this scheme preserve the negative definiteness property of variograms (in the matrix-valued function sense). A simplified analysis and computation in cokriging can be carried out. The principles of this scheme arc presented in this paper.     

Comparison of Seismic Dispersion and Attenuation Models

The frequency-dependent attenuation of seismic waves causes decreased resolution of seismic images with depth, and the difference in transmission losses induces amplitude variations with offset. Transmission losses may occur due to friction or fluid movement, or may result from scattering in thin-layer. Whatever the physical mechanism, they can often be conveniently described using an empirical formulation wherein the elastic moduli and propagation velocity are complex functions of frequency.We have compiled and compared algebraically and numerically eight different models involving complex velocity: the Kolsky-Futterman model, the power-law model, Kjartansson's model, Müller's model, Azimi's second and third model, the Cole-Cole model, and the standard linear-solid model.For two different parameter sets, the attenuation and phase velocity are computed in the seismic frequency band, and the plane-wave propagation of a Ricker wavelet for the other models is compared with that for the Kolsky-Futterman model. The first parameter set consists of parameters for each of the models calculated from expressions given in the appendix. These expressions make the different models behave similarly to the KF model. The second parameter set consists of model parameters that are numerically adapted to the KF model.By selecting proper parameters, all models, except the standard linear-solid model, show behavior similar to that of the Kolsky-Futterman model. The SLS model behaves differently from the other models as the frequency goes to zero or infinity. Broadband measurement data is needed to select a specific model for a given seismic experiment.     

Erratum: Comparison of Seismic Dispersion and Attenuation Models: Stud. Geophys. Geod., 46(2002), 293-320

Studia Geophysica et Geodaetica -     

Constraints of Housing Age and Migration on Residential Mobility

This paper investigates the role of housing age in constraining residential mobility, measured as the percent of households that have moved into their homes in the past 15 months. The leading explanation for why mobility rates differ so much among regions of the United States has been the overall level of growth. The present analysis shows that the growth effect operates through both the newness of population (migration) and the newness of housing available for occupancy by all local residents. The posited explanation for this housing age effect is that progressively older units contain increasingly settled occupants, yielding fewer opportunities for in-movers in areas with older housing. It is empirically demonstrated that households in older housing have lower likelihood of recent mobility even after controlling for age, tenure, migration status, and state location of residence. The analysis reveals the temporal interdependency of mobility, migration, person age, and housing age.     

A new form of the cokriging equations

Myers developed a matrix form of the cokriging equations, but one that entails the solution of a large system of linear equations. Large systems are troublesome because of memory requirements and a general increase in the matrix condition number. We transform Myers’s system into a set of smaller systems, whose solution gives the classical kriging results, and provides simultaneously a nested set of lower dimensional cokriging results. In the course of developing the new formulation we make an interesting link to the Cauchy-Schwarz condition for the invertibility of a system, and another to a simple situation of coregionalization. In addition, we proceed from these new equations to a linear approximation to the cokriging results in the event that the crossvariograms are small, allowing one to take advantage of a recent results of Xie and others which proceeds by diagonalizing the variogram matrix function over the lag classes.