Weak-contrast approximation of the elastic scattering matrix in anisotropic media

The plane-wave reflection and transmission coefficients at a plane interface between two anisotropic media constitute the elements of the elastic scattering matrix. For a 1-D anisotropic medium the eigenvector decomposition of the system matrix of the transformed elasto-dynamic equations is used to derive a general expression for the scattering matrix. Depending on the normalization of the eigenvectors, the expressions give scattering coefficients for amplitudes or for vertical energy flux.Computing the vertical slownesses and the corresponding polarizations, the eigenvector matrix and its inverse can be found. We give a simple formula for the inverse, regardless of the normalization of the eigenvectors. When the eigenvectors are normalized with respect to amplitudes of displacement (or velocity), the calculation of the scattering matrix for amplitudes is simplified.When the relative changes in all parameters are small, a weak-contrast approximation of the scattering matrix, based on the exactly determined polarization vectors in an average medium, is obtained. The same approximation is also derived directly from the transformed elasto-dynamic equations for a smooth vertically inhomogeneous medium, proving the consistency of the approximation.For monoclinic media, with the mirror symmetry plane parallel to the interface, the approximative scattering matrix is given in terms of analytic expressions for the non-normalized eigenvectors and vertical slownesses. For transversely isotropic media with a vertical axis of symmetry (VTI) and isotropic media, explicit solutions for the weak-contrast approximations of the scattering matrices have been obtained. The scattering matrix for amplitudes for isotropic media is well known. The scattering matrix for vertical energy flux may have applications in AVO analysis and inversion due to the reciprocity of the reflection coefficients for converted waves.Numerical examples for monoclinic and VTI media provide good agreement between the approximative and the exact reflection matrices. It is, however, expected that the approximations cannot be used when the symmetry properties of the two media are very different. This is because the approximation relies on a small relative contrast between the eigenvectors in the two media.Presented at the Workshop Meeting on Seismic Waves in Laterally Inhomogeneous Media, Castle of Trest, Czech Republic, May 22–27, 1995.     

Adaptive Multiscale Permeability Estimation

With multiscale permeability estimation one does not select parameterization prior to the estimation. Instead, one performs a hierarchical search for the right parameterization while solving a sequence of estimation problems with an increasing parameterization dimension. In some previous works on the subject, the same refinement is applied all over the porous medium. This may lead to over-parameterization, and subsequently, to unrealistic permeability estimates and excessive computational work. With adaptive multiscale permeability estimation, the new parameterization at an arbitrary stage in the estimation sequence is such that new degrees of freedom are not necessarily introduced all over the porous medium. The aim is to introduce new degrees of freedom only where it is warranted by the data. In this paper, we introduce a novel adaptive multiscale estimation. The approach is used to estimate absolute permeability from two-phase pressure data in several numerical examples.     

Late glacial fans in the eastern Skagerrak; depositional environment interpreted from swath bathymetry and seismostratigraphy

The origin of acoustically transparent fan deposits overlying glacial till and ice-proximal sediments on the southern margin of the Norwegian Channel has been studied using high-resolution seismic-reflection profiles and multibeam bathymetry. The first deposits overlying glacigenic sediments are a series of stacked, acoustically transparent submarine fans. The lack of glaciomarine sediments below and between individual fans indicates that deposition was rapid and immediately followed the break up of the Late Weichselian ice cover. The fans are overlain by stratified glaciomarine sediments and Holocene mud. Because of the uniformity of this drape, the upper surface of the fan deposits is mimicked at the present seafloor, and the bathymetric images clearly show the spatial relationship of the fans to bedrock ridges and the presence of braided channel-levee systems on the surface of the youngest fans. The acoustically transparent character of the fan deposits indicates that they comprise silt and clay, and their lobate form and lack of internal stratification indicates that they were deposited by debris flows. The channel-levee morphology indicates deposition from more watery hyperconcentrated fluid flows. The fan sediments were either derived from 1) erosion of Mid Weichselian lake deposits in southern Skagerrak or 2) from Late glacial ice-margin lake deposits, ponded against the Norwegian Channel ice stream, which collapsed catastrophically when the lateral support was removed as the ice disintegrated. Fans composed almost exclusively of fine-grained sediment need not, therefore, rule out an origin in a deglacial setting relatively close to the former margins of glaciers and ice sheets.     

The Ensemble Kalman Filter: theoretical formulation and practical implementation

The purpose of this paper is to provide a comprehensive presentation and interpretation of the Ensemble Kalman Filter (EnKF) and its numerical implementation. The EnKF has a large user group, and numerous publications have discussed applications and theoretical aspects of it. This paper reviews the important results from these studies and also presents new ideas and alternative interpretations which further explain the success of the EnKF. In addition to providing the theoretical framework needed for using the EnKF, there is also a focus on the algorithmic formulation and optimal numerical implementation. A program listing is given for some of the key subroutines. The paper also touches upon specific issues such as the use of nonlinear measurements, in situ profiles of temperature and salinity, and data which are available with high frequency in time. An ensemble based optimal interpolation (EnOI) scheme is presented as a cost-effective approach which may serve as an alternative to the EnKF in some applications. A fairly extensive discussion is devoted to the use of time correlated model errors and the estimation of model bias.Responsible Editor: Jörg-Olaf Wolff     

Is pure groundwater safe to drink?: natural "contamination' of groundwater in Norway

The title of this article is designed to provoke. Naturally occurring parameters are, by definition, not contamination. Nevertheless, nature is not necessarily nice, and naturally occurring trace toxins can be every bit as undesirable as their counterparts derived from human pollution.     

Towards a 4D topographic view of the Norwegian sea margin

The present-day topography/bathymetry of the Norwegian mainland and passive margin is a product of complex interactions between large-scale tectonomagmatic and climatic processes that can be traced back in time to the Late Silurian Caledonian Orogeny. The isostatic balance of the crust and lithosphere was clearly influenced by orogenic thickening during the Caledonian Orogeny, but was soon affected by post-orogenic collapse including overprinting of the mountain root, and was subsequently affected by a number of discrete extensional events eventually leading to continental break-up in Early Eocene time. In the mid-Jurassic the land areas experienced deep erosion in the warm and humid climate, forming a regional paleic surface. Rift episodes in the Late Jurassic and Early Cretaceous, with differential uplift along major fault zones, led to more pronounced topographic contrasts during the Cretaceous, and thick sequences of clastic sediments accumulated in the subsiding basins on the shelf. Following renewed extension in the Late Cretaceous, a new paleic surface developed in the Paleocene. Following break-up the margin has largely subsided thermally, but several Cenozoic shortening events have generated positive contraction structures. On the western side of the on-shore drainage divide, deeper erosion took place along pre-existing weakness zones, creating the template of the present day valleys and fjords. In the Neogene the mainland and large portions of the Barents Sea were uplifted. It appears that this uplift permitted ice caps to nucleate and accumulate during the Late Pliocene northern hemisphere climatic deterioration. The Late Pliocene to Pleistocene glacial erosion caused huge sediment aprons to be shed on to the Norwegian Sea and Barents Sea margins. Upon removal of the ice load the landmass adjusted isostatically, and this still continues today.