The anisotropy of magnetic susceptibility in biotite, muscovite and chlorite single crystals

The anisotropy of magnetic susceptibility (AMS) of single crystals of biotite, muscovite and chlorite has been measured in order to provide accurate values of the magnetic anisotropy properties for these common rock-forming minerals. The low-field AMS and the high-field paramagnetic susceptibility are defined. For the high-field values, it is necessary to combine the paramagnetic deviatoric tensor obtained from the high-field torque magnetometer with the paramagnetic bulk susceptibility measured from magnetization curves of the crystals. This leads to the full paramagnetic susceptibility ellipsoid due to the anisotropic distribution of iron cations in the silicate lattice. The ellipsoid of paramagnetic susceptibility, which was obtained for the three phyllosilicates, is highly oblate in shape and the minimum susceptibility direction is subparallel to the crystallographic c-axes. The anisotropy of the susceptibility within the basal plane of the biotite has been evaluated and found to be isotropic within the accuracy of the instrumental measurements. The degree of anisotropy of biotite and chlorite is compatible with previously reported values while for muscovite the smaller than previously published values. The shape of the chlorite AMS ellipsoid for all the samples is near-perfect oblate in contrast with a wide distribution of oblate and prolate values reported in earlier studies. Reliable values are important for deriving models of the magnetic anisotropy where it reflects mineral fabrics and deformation of rocks.     

Finite strain patterns and their significance in Permian rocks of the Alpes Maritimes (France)

More finite strain data has been obtained from autochthonous Permian mudstones of the Alpes Maritimes, S.E. France. These new data were computed from field measurements of green spots on all available sections, deformed mudcracks and from the quantitative correlation between magnetic susceptibility anisotropy and finite strain in these rocks. Previously published finite strain data and the new results are presented on a series of structural maps and cross-sections for the Dôme de Barrot, the Tinée and Vionène region and the Roya region. As in previous studies difficulties arise in explaining the apparently variable extension parallel with the 100°, subhorizontal bedding-cleavage intersection: either this is real or there were large volume changes during the tectonic deformation. Study of quartz fibres, developed in deformed mudcracks in the Tinée valley, suggest that early in the tectonic history incremental stretching directions were parallel with the bedding—cleavage intersection, while later they were down-dip in the 100° trending cleavage. Since these Permian rocks have remained stuck to the Argentera basement they also record displacements and deformations in the basement. The early 100°, subhorizontal stretching is consistent with NW-SE dextral, strike-slip basement faulting, while later, down-dip stretching in the cleavage is consistent with contraction faults in the basement. This information and new palaeomagnetic data on the same samples are combined with recent geophysical evidence and regional tectonic studies, to provide a new precision to the tectonic history of this part of the Western Alpine External Zone.     

Efficient and accurate high-degree spherical harmonic synthesis of gravity field functionals at the Earth’s surface using the gradient approach

Spherical harmonic synthesis (SHS) of gravity field functionals at the Earth’s surface requires the use of heights. The present study investigates the gradient approach as an efficient yet accurate strategy to incorporate height information in SHS at densely spaced multiple points. Taylor series expansions of commonly used functionals quasigeoid heights, gravity disturbances and vertical deflections are formulated, and expressions of their radial derivatives are presented to arbitrary order. Numerical tests show that first-order gradients, as introduced by Rapp (J Geod 71(5):282–289, 1997) for degree 360 models, produce cm- to dm-level RMS approximation errors over rugged terrain when applied with EGM2008 to degree 2190. Instead, higher-order Taylor expansions are recommended that are capable of reducing approximation errors to insignificance for practical applications. Because the height information is separated from the actual synthesis, the gradient approach can be applied along with existing highly efficient SHS routines to compute surface functionals at arbitrarily dense grid points. This confers considerable computational savings (above or well above one order of magnitude) over conventional point-by-point SHS. As an application example, an ultra-high resolution model of surface gravity functionals (EurAlpGM2011) is constructed over the entire European Alps that incorporates height information in the SHS at 12,000,000 surface points. Based on EGM2008 and residual topography data, quasigeoid heights, gravity disturbances and vertical deflections are estimated at ~200m resolution. As a conclusion, the gradient approach is efficient and accurate for high-degree SHS at multiple points at the Earth’s surface.     

Combining EGM2008 and SRTM/DTM2006.0 residual terrain model data to improve quasigeoid computations in mountainous areas devoid of gravity data

A global geopotential model, like EGM2008, is not capable of representing the high-frequency components of Earth’s gravity field. This is known as the omission error. In mountainous terrain, omission errors in EGM2008, even when expanded to degree 2,190, may reach amplitudes of 10 cm and more for height anomalies. The present paper proposes the utilisation of high-resolution residual terrain model (RTM) data for computing estimates of the omission error in rugged terrain. RTM elevations may be constructed as the difference between the SRTM (Shuttle Radar Topography Mission) elevation model and the DTM2006.0 spherical harmonic topographic expansion. Numerical tests, carried out in the German Alps with a precise gravimetric quasigeoid model (GCG05) and GPS/levelling data as references, demonstrate that RTM-based omission error estimates improve EGM2008 height anomaly differences by 10 cm in many cases. The comparisons of EGM2008-only height anomalies and the GCG05 model showed 3.7 cm standard deviation after a bias-fit. Applying RTM omission error estimates to EGM2008 reduces the standard deviation to 1.9 cm which equates to a significant improvement rate of 47%. Using GPS/levelling data strongly corroborates these findings with an improvement rate of 49%. The proposed RTM approach may be of practical value to improve quasigeoid determination in mountainous areas without sufficient regional gravity data coverage, e.g., in parts of Asia, South America or Africa. As a further application, RTM omission error estimates will allow refined validation of global gravity field models like EGM2008 from GPS/levelling data.     

Error sources and data limitations for the prediction of surface gravity: a case study using benchmarks

Gravity-based heights require gravity values at levelled benchmarks (BMs), which sometimes have to be predicted from surrounding observations. We use the Earth Gravitational Model 2008 (EGM2008) and the Australian National Gravity Database (ANGD) as examples of model and terrestrial observed data respectively to predict gravity at Australian National Levelling Network (ANLN) BMs. The aim is to quantify errors that may propagate into the predicted BM gravity values and then into gravimetric height corrections (HCs). Our results indicate that an approximate ±1 arc-min horizontal position error of the BMs causes maximum errors in EGM2008 BM gravity of ~22 mGal (~55 mm in the HC at ~2200 m elevation) and ~18 mGal for ANGD BM gravity because the values are not computed at the true location of the BM. We use RTM (residual terrain modelling) techniques to show that ~50% of EGM2008 BM gravity error in a moderately mountainous region can be accounted for by signal omission. Non-representative sampling of ANGD gravity in this region may cause errors of up to 50 mGals (~120 mm for the Helmert orthometric correction at ~2200 m elevation). For modelled gravity at BMs to be viable, levelling networks need horizontal BM positions accurate to a few metres, while RTM techniques can be used to reduce signal omission error. Unrepresentative gravity sampling in mountains can be remedied by denser and more representative re-surveys, and/or gravity can be forward modelled into regions of sparser gravity.