A Glimpse at Set Theory. The Real Numbers. The Topology of Cartesian Spaces. Convergence. Continuous Functions. Functions of One Variable. Infinite Series. Differentiation in RP Integration in RP.

The elements of real analysis (2nd edition), by Robert G. Bartle. Pp xv, 480. £10. 1976. SBM 0 471 05464 X (Wiley)

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The Antarctica component of postglacial rebound model ICE-6G_C (VM5a) based on GPS positioning, exposure age dating of ice thicknesses, and relative sea level histories

Analysis of Global Positioning System (GPS) data demonstrates that ongoing three-dimensional crustal deformation in Fennoscandia is dominated by glacial isostatic adjustment. Our comparison of these GPS observations with numerical predictions yields an Earth model that satisÞes independent geologic constraints and bounds both the average viscosity in the upper mantle (5 x 10^20 to 1 x 10^21 pascal seconds) and the elastic thickness of the lithosphere (90 to 170 kilometers). We combined GPS-derived radial motions with Fennoscandian tide gauge records to estimate a regional sea surface rise of 2.1 +- 0.3 mm/year. Furthermore, ongoing horizontal tectonic motions greater than ~1 mm/year are ruled out on the basis of the GPS-derived three-dimensional crustal velocity field.

http://publications.lib.chalmers.se/records/fulltext/197900/local_197900.pdf

Space-Geodetic Constraints on Glacial Isostatic Adjustment in Fennoscandia

[1] Data collected under the auspices of the BIFROST GPS project yield a geographically dense suite of estimates of present-day, three-dimensional (3-D) crustal deformation rates in Fennoscandia [Johansson et al., 2002]. A preliminary forward analysis of these estimates [Milne et al., 2001] has indicated that models of ongoing glacial isostatic adjustment (GIA) in response to the final deglaciation event of the current ice age are able to provide an excellent fit to the observed 3-D velocity field. In this study we revisit our previous GIA analysis by considering a more extensive suite of forward calculations and by performing the first formal joint inversion of the BIFROST rate estimates. To establish insight into the physics of the GIA response in the region, we begin by decomposing a forward prediction into the three contributions associated with the ice, ocean, and rotational forcings. From this analysis we demonstrate that recent advances in postglacial sea level theory, in particular the inclusion of rotational effects and improvements in the treatment of the ocean load in the vicinity of an evolving continental margin, involve peak signals that are larger than the observational uncertainties in the BIFROST network. The forward analysis is completed by presenting predictions for a pair of Fennoscandian ice histories and an extensive suite of viscoelastic Earth models. The former indicates that the BIFROST data set provides a powerful discriminant of such histories. The latter yields bounds on the ( assumed constant) upper and lower mantle viscosity (nu(UM), nu(LM)); specifically, we derive a 95% confidence interval of 5 x 10(20) less than or equal to nu(UM) less than or equal to 10(21) Pa s and 5 x 10(21) less than or equal to nu(LM) less than or equal to 5 x 10(22) Pa s, with some preference for (elastic) lithospheric thickness in excess of 100 km. The main goal of the ( Bayesian) inverse analysis is to estimate the radial resolving power of the BIFROST GPS data as a function of depth in the mantle. Assuming a reasonably accurate ice history, we demonstrate that this resolving power varies from similar to 200 km near the base of the upper mantle to similar to 700 km in the top portion of the lower mantle. We conclude that the BIFROST data are able to resolve structure on radial length scale significantly smaller than a single upper mantle layer. However, these data provide little constraint on viscosity in the bottom half of the mantle. Finally, elements of both the forward and inverse analyses indicate that radial and horizontal velocity estimates provide distinct constraints on mantle viscosity.

https://publications.lib.chalmers.se/records/fulltext/197898/local_197898.pdf

Continuous GPS measurements of postglacial adjustment in Fennoscandia 1. Geodetic results

The use of Global Positioning System techniques for the continuous monitoring of landslides: application to the Super-Sauze earthflow (Alpes-de-Haute-Provence, France)

Vening Meinesz' inverse problem in isostasy deals with solving for the Moho depth from known Bouguer gravity anomalies and "normal" Moho depth (T-0, known, e. g. from seismic reflection data) using ...

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Solving Vening Meinesz-Moritz inverse problem in isostasy

Today the combination of Stokes’ formula and an Earth gravity model (EGM) for geoid determination is a standard procedure. However, the method of modifying Stokes’ formula varies from author to author, and numerous methods of modification exist. Most methods modify Stokes’ kernel, but the most widely applied method, the remove compute restore technique, removes the EGM from the gravity anomaly to attain a residual gravity anomaly under Stokes’ integral, and at least one known method modifies both Stokes’ kernel and the gravity anomaly. A general model for modifying Stokes’ formula is presented; it includes most of the well-known techniques of modification as special cases. By assuming that the error spectra of the gravity anomalies and the EGM are known, the optimum model of modification is derived based on the least-squares principle. This solution minimizes the expected mean square error (MSE) of all possible solutions of the general geoid model. A practical formula for estimating the MSE is also presented. The power of the optimum method is demonstrated in two special cases.

A general model for modifying Stokes' formula and its least-squares solution

In a modern application of Stokes’ formula for geoid determination, regional terrestrial gravity is combined with long-wavelength gravity information supplied by an Earth gravity model. Usually, several corrections must be added to gravity to be consistent with Stokes’ formula. In contrast, here all such corrections are applied directly to the approximate geoid height determined from the surface gravity anomalies. In this way, a more efficient workload is obtained. As an example, in applications of the direct and first and second indirect topographic effects significant long-wavelength contributions must be considered, all of which are time consuming to compute. By adding all three effects to produce a combined geoid effect, these long-wavelength features largely cancel. The computational scheme, including two least squares modifications of Stokes’ formula, is outlined, and the specific advantages of this technique, compared to traditional gravity reduction prior to Stokes’ integration, are summarised in the conclusions and final remarks.

A computational scheme to model the geoid by the modified Stokes formula without gravity reductions

This study emphasizes that the harmonic downward continuation of an external representation of the Earth's gravity potential to sea level through the topographic masses implies a topographic bias. ...

The topographic bias by analytical continuation in physical geodesy

An argument is put forward in favor of using a model gravity field of a degree and order higher than 2 as a reference in gravity field studies. Stokes's approach to the evaluation of the geoid from gravity anomalies is then generalized to be applicable to a higher than second-order reference spheroid. The effects of truncating Stokes's integration and of modifying the integration kernels are investigated in the context of the generalized approach. Several different modification schemes, starting with a Molodenskij-like modification and ending with the least squares modification, are studied. Particular attention is devoted to looking at both global and local biases and mean square errors of the individual schemes.

Lars E. Sjöberg

Papers

Solving Vening Meinesz-Moritz inverse problem in isostasy

A general model for modifying Stokes' formula and its least-squares solution

A computational scheme to model the geoid by the modified Stokes formula without gravity reductions

The topographic bias by analytical continuation in physical geodesy

Reformulation of Stokes's theory for higher than second‐degree reference field and modification of integration kernels