Nonlinear diffusion filtering of data on the Earth’s surface

The paper deals with data filtering on closed surfaces using linear and nonlinear diffusion equations. We define a surface finite-volume method to approximate numerically parabolic partial differential equations on closed surfaces, namely on a sphere, ellipsoid or the Earth’s surface. The closed surface as a computational domain is approximated by a polyhedral surface created by planar triangles and we construct a dual co-volume grid. On the co-volumes we define a weak formulation of the problem by applying Green’s theorem to the Laplace–Beltrami operator. Then the finite-volume method is applied to discretize the weak formulation. Weak forms of elliptic operators are expressed through surface gradients. In our numerical scheme we use a piece-wise linear approximation of a solution in space and the backward Euler time discretization. Furthermore, we extend a linear diffusion on surface to the regularized surface Perona–Malik model. It represents a nonlinear diffusion equation, which at the same time reduces noise and preserves main edges and other details important for a correct interpretation of the real data. We present four numerical experiments. The first one has an illustrative character showing how an additive noise is filtered out from an artificial function defined on a sphere. Other three examples deal with the real geodetic data on the Earth’s surface, namely (i) we reduce a stripping noise from the GOCE satellite only geopotential model up to degree 240, (ii) we filter noise from the real GOCE measurements (the component $T_{zz})$ , and (iii) we reduce a stripping noise from the satellite only mean dynamic topography at oceans. In all experiments we focus on a comparison of the results obtained by both the linear and nonlinear models presenting advantages of the nonlinear diffusion.     

Constants of the Moon's free libration on the basis of heliometric observations from the years 1841–1945

The results of simultaneous and homogeneous reduction of 8 libration series are presented. On this basis all 6 free libration constants have been estimated using some new formulae.     

Small- and medium-scale effects of high-flying aircraft exhausts on the atmospheric composition

Following numerous model studies of the global impacts of sub- and supersonic aircraft on the atmosphere, this paper assesses the separate aircraft engine exhaust effects of the 45°N cruise flight and at the 10- and 18-km levels of the July atmosphere. A box diffusion photochemical model in the cross-section plane of the flight trajectory is used to compute the effects of gas-phase and heterogeneous reactions on the condensation trail particles in the troposphere, and on the sulphate aerosols in the stratosphere. The enhanced horizontal dispersion of the exhaust plume is considered in the model.A significant but short term depletion of ozone is predicted, which is 99% restored in about 1 h in the wide plume with enhanced horizontal dispersion, but requires more than 24 h in the narrow plume without it. The oxidation rate of NO and NO₂ into the HNO₃ depends on the OH content in the exhausts and varies in all the cases. The heterogeneous photochemistry has only a small influence on the initial evolution of N₂O₅ and HO₂ in the plume.     

Interannual variability of tropospheric NO2 column over central Europe — observations from SCIAMACHY and GEM-AQ model simulations

Spatial and temporal variability of NO₂ tropospheric column over Europe was analyzed for a 3 year period (2008–2010) based on monthly average observations from SCIAMACHY and the GEM-AQ model results. The GEM-AQ model was run in a global variable configuration with a resolution of ～15 km over Central Europe. Spatial averaged time series were calculated for two selected regions in Western and Central Europe in order to assess the seasonal and interannual variability of the tropospheric NO₂. The spatial pattern is similar near large emission sources for consecutive years. However, in remote regions there are differences due to interannual variability of meteorological conditions. Highest tropospheric NO₂ column values (over 150 × 10¹⁵ molecules/cm²) were persistent over the Benelux and over most of the European agglomerations. The general agreement between modelled and observed NO₂ column is good. In the remote areas, the modelled NO₂ column shows weaker gradients than the observed field.     

Direct BEM for high-resolution global gravity field modelling

The paper presents a high-resolution global gravity field modelling by the boundary element method (BEM). A direct BEM formulation for the Laplace equation is applied to get a numerical solution of the linearized fixed gravimetric boundary-value problem. The numerical scheme uses the collocation method with linear basis functions. It involves a discretization of the complicated Earth’s surface, which is considered as a fixed boundary. Here 3D positions of collocation points are simulated from the DNSC08 mean sea surface at oceans and from the SRTM30PLUS_V5.0 global topography model added to EGM96 on lands. High-performance computations together with an elimination of the far zones’ interactions allow a very refined integration over the all Earth’s surface with a resolution up to 0.1 deg. Inaccuracy of the approximate coarse solutions used for the elimination of the far zones’ interactions leads to a long-wavelength error surface included in the obtained numerical solution. This paper introduces an iterative procedure how to reduce such long-wavelength error surface. Surface gravity disturbances as oblique derivative boundary conditions are generated from the EGM2008 geopotential model. Numerical experiments demonstrate how the iterative procedure tends to the final numerical solutions that are converging to EGM2008. Finally the input surface gravity disturbances at oceans are replaced by real data obtained from the DNSC08 altimetryderived gravity data. The ITG-GRACE03S satellite geopotential model up to degree 180 is used to eliminate far zones’ interactions. The final high-resolution global gravity field model with the resolution 0.1 deg is compared with EGM2008.     

Differences in GPS coordinate time series resulting from the use of individual instead of type-mean antenna phase center calibration model

It is well-known that the phase center of a Global Navigation Satellite System (GNSS) antenna is not a stable point coinciding with a mechanical reference. The phase center position depends on the direction of the received signal, and is antenna-and signaldependent. Phase center corrections (PCC) models of GNSS antennas have been available for several years. The first method to create antenna PCC models was the relative field calibration procedure. Currently only absolute calibration models are generally recommended for use. In this study we investigate the differences between position estimates obtained using individual and type-mean absolute antenna calibrations in order to better understand how receiver antenna calibration models contribute to the Global Positioning System (GPS) positioning error budget. The station positions were estimated with two absolute calibration models: the igs08.atx model, which contains typemean calibration results, and individual antenna calibration models. Continuous GPS observations from selected Polish European Permanent Network (EPN) stations were used for these studies. The position time series were derived from the precise point positioning (PPP) technique using the NAPEOS scientific GNSS software package. The results show that the differences in the calibrations models propagate directly into the position domain, affecting daily as well sub-daily results. In daily solutions, the position offsets, resulting from the use of individual calibrations instead of type-mean igs08.atx calibrations, can reach up to 5 mm in the Up component, while in the horizontal one they generally stay below 1 mm. It was found that increasing the frequency of sub-daily coordinate solutions amplifies the effects of type-mean vs individual PCC-dependent differences, and also gives visible periodic variations in time series of GPS position differences.     

Statistical analysis and comparison of external factor effects on the total ozone field over the Russian territory in 1973–2007

The analysis of external factors, which are most significant for the formation of the monthly mean total ozone (TO) field and ozone transport over the Russian Federation, based on observation data obtained from about 30 ground-based stations of the ozonometric network averaged over a year, December through March and June through August, over five climatic regions, is considered. Performed spectrum and discriminant analysis allowed obtaining quantitative estimates of the impact of the Arctic Oscillation, deviation of the winter temperature of the lower polar stratosphere, quasi-biennial oscillations (QBO), 11-year solar cycle, El Niño-Southern Oscillation (ENSO) on the TO and to assess the regional differences in the effects of these factors. In December–March, in the years with a negative Arctic Oscillation phase, warm stratosphere, and the easterly QBO phase (QBO-E), the ozone content increases significantly relative to the opposite phases of oscillations on average by 35, 28, and 26 Dobson units (DU), respectively. The spectra, similar to the discriminant function, demonstrate strong influence of the 11-year solar cycle and QBO on the TO even in the summer months, while the QBO is more pronounced in the eastern part of the Russian Federation. The ENSO effect was not singled out against the general “noisy” background of the cold six-month period, when many atmospheric processes become active: however, during the summer months, in warm periods of the ENSO, the TO, at the 97% significance level, increases over most of the Russian area. The rest of the obtained results are significant at the 95–99.9% level.