...Sandra P. Morris; Robert J. Geller abstract Two methods which have been proposed for calculating the normal modes of laterally heterogeneous models of the Earth are first-order degenerate perturbation theory and the Rayleigh-Ritz variational procedure. The purpose of this paper is to compare...

...) where wl and ~ are the lowest and highest eigenvalues of (1), respectively. T o apply the Rayleigh-Ritz procedure to (2), consider {x} to be a vector with elements x~, i -- 1, 2, N and determine the variation in ~x2 as xi is changed infinitesimally about a stationary point of ~x2. This corresponds...

... and eigenfunctions for a string with the average elastic properties; second-order perturbation theory is worse, because the eigenfunction is assumed to be the original eigenfunction plus small correction terms, but actually may be almost completely different. The variational method (Rayleigh-Ritz), using...

... properties of the structure. In SAP2000, mode shapes can be calculated as exact eigenvectors (the default selection) or as load-dependent Ritz vectors. Wilson (2004) recommends that Ritz vectors be selected for NLRHA applications. This leads to a more computationally efficient procedure because mode shapes...

... of the rupture area ( Ritz et al. , 2020 ; Cornou et al. , 2021 ). It corresponds to the most destructive and strongest earthquake in metropolitan France since 1967. This event highlights the issue of the seismic hazard in the region and brings a new dimension to the DARE project (launched and funded before...

...Gabriel J. Bowen; Bruce Wilkinson Abstract Proxy data reflecting the oxygen isotope composition of meteoric precipitation (δ 18 O ppt ) are widely used in reconstructions of continental paleoclimate and paleohydrology. However, actual geographic variation in modern water compositions is difficult...

... Seismic Research Observatory (SRO) or Modified High-Gain Long-Period (ASRO) stations were also searched for 20-sec Rayleigh waves. Seven useful sets of data were obtained by this procedure, but signals were overdriven at three of the digital stations (ANMO, KAAO, and KONO). We also were able to measure...

... information on glacier basal motion, which exerts a primary control on glacier and ice‐sheet dynamics and the associated eustatic sea level rise ( Ritz et al. , 2015 ; Vincent and Moreau, 2016 ). The mechanisms giving rise to stick‐slip sliding as well as its effect on large‐scale ice flow, however, remain...

... combine the finite element in the direction of layering with finite-differences methods for the two integrations over time and wavenumbers, and the direct solution method ( dsm ) of Geller and Ohminato (1984, 1986), which is based on a discrete Rayleigh-Ritz formulation. Boundary-element method, or BEM...

... an eigenvalue/eigenvector solver. Aki and Richards (1980) briefly describe a closely related Rayleigh-Ritz technique for modeling Rayleigh waves. A similar approach has also been described by Karpfinger et al. (2010) for the forward modeling of guided waves in a borehole. The original work by Lysmer...

... variation of permeability. Consequently, the use of a permeability threshold value, or the analysis of a critical Rayleigh number, becomes here useless and our study therefore requires a numerical approach. Heat equation is coupled with Darcy’s law through the use of the Comsol Multiphysics™ software...

... into a series of non-interacting one-electron Kohn–Sham (KS) equations ( Kohn & Sham 1965 ). From these KS equations, the crystal wave-functions were solved, and the electronic structure was obtained using the Rayleigh–Ritz variational principle ( Grosso & Parravicini 2000 ). Three samples...

... for the field of stable isotope hydrology, but is in general agreement with Rayleigh fractionation ( Dansgaard 1964 ; Criss 1999 ). Landais et al. (2012b) investigated seasonal triple oxygen isotope variations in surface snow at the NEEM drilling site, Greenland, where a slight anticorrelation of δ 18 O...

... by color bar at the bottom. The reservoir zone is indicated by the blue brackets. The following subsections describe the procedure for obtaining TL AI variations. The “Background models” subsection describes the derivation of the background velocity model and velocity/density relations required...

..., and that is what we shall do. A problem similar to that treated here has been discussed previously by Geller and Stein (1978). They compared the results obtained from normal mode pertur- bation theory with those obtained from a Rayleigh-Ritz variational method for a particularly simple problem which could...

... approach to propagate focal mechanism uncertainty in stress inversion. Our results significantly improve upon the spatial resolution of available seismological estimates of crustal stress, and resolve spatial variations in stress orientations at an orogenic scale. We show how crustal stresses change from...

... activation for this measurement. An important step toward the use of trace elements was the original work of Boyle (1981) who designed a cleaning procedure to remove the ferro-manganese coating from sedimentary foraminifera and introduced Cd as a tracer for phosphate in the ocean. In a series of papers...

... ( Mats, 2015 ). The fault pattern correlates with crustal stress variations over the rift system. Paleostress reconstructions based on structural measurements and fault slip data indicate that the BRS central segment developed under horizontal extension and transtension ( Sherman and Dneprovsky, 1989...

... being in the conventional shadow zone for infrasound, due to penetrating diffracted/ creeping waves, and the vertical and radial seismic components presents a well-developed crustal Rayleigh wave (Rg). We infer the source characteristics by means of a waveform-based inversion procedure, which rely...

... a steady rise, δ 13 C fluctuates, the most reproducible variations being negative signatures in carbonate caps to glacial units, but more diagenetic work is needed. Four conceptual models for the icehouse conditions are contrasted: Zipper-Rift Earth (diachronous glaciation related to continental rift...