The multivariate total least-squares (MTLS) approach aims at estimating a matrix of parameters, Ξ, from a linear model (Y−EY = (X−EX) · Ξ) that includes an observation matrix, Y, another observation matrix, X, and matrices of randomly distributed errors, EY and EX. Two special cases of the MTLS approach include the standard multivariate least-squares approach where only the observation
matrix, Y, is perturbed by random errors and, on the other hand, the data least-squares approach where only the coefficient matrix
X is affected by random errors. In a previous contribution, the authors derived an iterative algorithm to solve the MTLS problem
by using the nonlinear Euler–Lagrange conditions. In this contribution, new lemmas are developed to analyze the iterative
algorithm, modify it, and compare it with a new ‘closed form’ solution that is based on the singular-value decomposition.
For an application, the total least-squares approach is used to estimate the affine transformation parameters that convert
cadastral data from the old to the new Israeli datum. Technical aspects of this approach, such as scaling the data and fixing
the columns in the coefficient matrix are investigated. This case study illuminates the issue of “symmetry” in the treatment
of two sets of coordinates for identical point fields, a topic that had already been emphasized by Teunissen (1989, Festschrift
to Torben Krarup, Geodetic Institute Bull no. 58, Copenhagen, Denmark, pp 335–342). The differences between the standard least-squares
and the TLS approach are analyzed in terms of the estimated variance component and a first-order approximation of the dispersion
matrix of the estimated parameters. 相似文献
The viscosity of synthetic peridotite liquid has been investigated at high pressures using in-situ falling sphere viscometry by combining a multi-anvil technique with synchrotron radiation. We used a newly designed capsule containing a small recessed reservoir outside of the hot spot of the heater, in which a viscosity marker sphere is embedded in a forsterite + enstatite mixture having a higher solidus temperature than the peridotite. This experimental setup prevents spheres from falling before a stable temperature above the liquidus is established and thus avoids difficulties in evaluating viscosities from velocities of spheres falling through a partially molten sample.
Experiments have been performed between 2.8 and 13 GPa at temperatures ranging from 2043 to 2523 K. Measured viscosities range from 0.019 (± 0.004) to 0.13 (± 0.02) Pa s. At constant temperature, viscosity increases with increasing pressure up to 8.5 GPa but then decreases between 8.5 and 13 GPa. The change in the pressure dependence of viscosity is likely associated with structural changes of the liquid that occur upon compression. By combining our results with recently published 0.1 MPa peridotite liquid viscosities [D.B. Dingwell, C. Courtial, D. Giordano, A. Nichols, Viscosity of peridotite liquid, Earth Planet. Sci. Lett. 226 (2004) 127–138.], the experimental data can be described by a non-Arrhenian, empirical Vogel-Fulcher-Tamman equation, which has been modified by adding a term to account for the observed pressure dependence of viscosity. This equation reproduces measured viscosities to within 0.08 log10-units on average. We use this model to calculate viscosities of a peridotitic magma ocean along a liquid adiabat to a depth of 400 km and discuss possible effects on viscosity at greater pressures and temperatures than experimentally investigated. 相似文献