Conversion of time-varying Stokes coefficients into mass anomalies at the Earth’s surface considering the Earth’s oblateness

Time-varying Stokes coefficients estimated from GRACE satellite data are routinely converted into mass anomalies at the Earth’s surface with the expression proposed for that purpose by Wahr et al. (J Geophys Res 103(B12):30,205–30,229, 1998). However, the results obtained with it represent mass transport at the spherical surface of 6378 km radius. We show that the accuracy of such conversion may be insufficient, especially if the target area is located in a polar region and the signal-to-noise ratio is high. For instance, the peak values of mean linear trends in 2003–2015 estimated over Greenland and Amundsen Sea embayment of West Antarctica may be underestimated in this way by about 15%. As a solution, we propose an updated expression for the conversion of Stokes coefficients into mass anomalies. This expression is based on the assumptions that: (i) mass transport takes place at the reference ellipsoid and (ii) at each point of interest, the ellipsoidal surface is approximated by the sphere with a radius equal to the current radial distance from the Earth’s center (“locally spherical approximation”). The updated expression is nearly as simple as the traditionally used one but reduces the inaccuracies of the conversion procedure by an order of magnitude. In addition, we remind the reader that the conversion expressions are defined in spherical (geocentric) coordinates. We demonstrate that the difference between mass anomalies computed in spherical and ellipsoidal (geodetic) coordinates may not be negligible, so that a conversion of geodetic colatitudes into geocentric ones should not be omitted.     

Earth’s gravity field modelling based on satellite accelerations derived from onboard GPS phase measurements

GPS data collected by satellite gravity missions can be used for extracting the long-wavelength part of the Earth’s gravity field. We propose a new data processing method which makes use of the ‘average acceleration’ approach to gravity field modelling. In this method, satellite accelerations are directly derived from GPS carrier phase measurements with an epoch-differenced scheme. As a result, no ambiguity solutions are needed and the systematic errors that do not change much from epoch to epoch are largely eliminated. The GPS data collected by the Gravity Field and Steady-State Ocean Circulation Explorer (GOCE) satellite mission are used to demonstrate the added value of the proposed method. An analysis of the residual accelerations shows that accelerations derived in this way are more precise, with noise being reduced by about 20 and 5% at the cross-track component and the other two components, respectively, as compared to those based on kinematic orbits. The accelerations obtained in this way allow the recovery of the gravity field to a slightly higher maximum degree compared to the solution based on kinematic orbits. Furthermore, the gravity field solution has an overall better performance. Errors in spherical harmonic coefficients are smaller, especially at low degrees. The cumulative geoid height error is reduced by about 15 and 5% up to degree 50 and 150, respectively. An analysis in the spatial domain shows that large errors along the geomagnetic equator, which are caused by a high electron density coupled with large short-term variations, are substantially reduced. Finally, the new method allows for a better observation of mass transport signals. In particular, sufficiently realistic signatures of regional mass anomalies in North America and south-west Africa are obtained.     

Correction on Effect of Earth’s Oblateness in Inversion of GPS Occultation Data

IntroductionRadio occultation technique was firstly appliedin the astronomic research in the 18th century .It has developed greatly by the scientists in JetPropulsion Laboratory (JPL) and Stanford Uni-versity since the 1960’s and widely usedin manymissions for planets exploration, which providedus with i mportant information about the at mos-phere of these planets[1]. In 1965 , it wasbrought forward by Fischbach that this tech-nique can be used in the research of the Earth’sat mosphere[2…     

Correction on effect of Earth’s oblateness in inversion of GPS occultation data

By using observed CHAMP orbit ephemeredes and MSISE-90 dry air model and regarding the earth as a sphere and an ellipsoid respectively, phase delays are simulated and the simulated data are retrieved under different schemes. The comparison between the inverted temperature profiles and the model temperature profiles shows that by inverting observed data, we will get temperature results with large errors if the effect of Earth’s oblateness is omitted. The correction method is proved to be effective because the temperature errors decreased obviously with this method.     

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.     

Correction on Effect of Earth＇s Oblateness in Inversion of GPS Occultation Data

By using observed CHAMP orbit ephemeredes and MSISE-90 dry air model and regarding the earth as a sphere and an ellipsoid respectively, phase delays are simulated and the simulated data are retrieved under different schemes. The comparison between the inverted temperature profiles and the model temperature profiles shows that by inverting observed data, we will get temperature results with large errors if the effect of Earth＇s oblateness is omitted. The correction method is proved to be effective because the temperature errors decreased obviously with this method.     

Surface gravitational field and topography changes induced by the Earth’s fluid core motions

Using a Love number formalism, the elastic deformations of the mantle and the mass redistribution gravitational potential within the Earth induced by the fluid pressure acting at the core–mantle boundary (CMB) are computed. This pressure field changes at a decadal time scale and may be estimated from observations of the surface magnetic field and its secular variation. First, using a spherical harmonic expansion, the poloidal and toroidal part of the fluid velocity field at the CMB for the last 40 years is computed, under the hypothesis of tangential geostrophy. Then the associated geostrophic pressure, whose order of magnitude is about 1000 Pa, is computed. The surface topography induced by this pressure field is computed using Love numbers, and is a few millimetres. The mass redistribution gravitational potential induced by these deformations and, in particular, the zonal components of the related surface gravitational potential perturbation (J₂, J₃ and J₄ coefficients), are calculated. Overall perturbations for the J₂ coefficient of about 10^–10, for J₃ of about 10^–11 and for J₄ are found of about 0.3×10^–11. Finally, these theoretical results are compared with recent observations of the decadal variation of J₂ from satellite laser ranging. Results concerning J₂ can be described as follows: first, they are one order of magnitude too small to explain the observed decadal variation of J₂ and, second, they show a significant linear trend over the last 40 years, whose rate of decrease amounts to 7% of the observed value.     

Effects of topographic–isostatic masses on gravitational functionals at the Earth’s surface and at airborne and satellite altitudes

Topographic–isostatic masses represent an important source of gravity field information, especially in the high-frequency band, even if the detailed mass-density distribution inside the topographic masses is unknown. If this information is used within a remove-restore procedure, then the instability problems in downward continuation of gravity observations from aircraft or satellite altitudes can be reduced. In this article, integral formulae are derived for determination of gravitational effects of topographic–isostatic masses on the first- and second-order derivatives of the gravitational potential for three topographic–isostatic models. The application of these formulas is useful for airborne gravimetry/gradiometry and satellite gravity gradiometry. The formulas are presented in spherical approximation by separating the 3D integration in an analytical integration in the radial direction and 2D integration over the mean sphere. Therefore, spherical volume elements can be considered as being approximated by mass-lines located at the centre of the discretization compartments (the mass of the tesseroid is condensed mathematically along its vertical axis). The errors of this approximation are investigated for the second-order derivatives of the topographic–isostatic gravitational potential in the vicinity of the Earth’s surface. The formulas are then applied to various scenarios of airborne gravimetry/gradiometry and satellite gradiometry. The components of the gravitational vector at aircraft altitudes of 4 and 10 km have been determined, as well as the gravitational tensor components at a satellite altitude of 250 km envisaged for the forthcoming GOCE (gravity field and steady-state ocean-circulation explorer) mission. The numerical computations are based on digital elevation models with a 5-arc-minute resolution for satellite gravity gradiometry and 1-arc-minute resolution for airborne gravity/gradiometry.     

Length-of-day and space-geodetic determination of the Earth’s variable gravity field

The temporal variations of the Earth’s gravity field, nowadays routinely determined from satellite laser ranging (SLR) and GRACE (Gravity Recovery And Climate Experiment), are related to changes in the Earth’s rotation rate through the Earth’s inertia tensor. We study this connection from actual data by comparing the traditional length-of-day (LOD) measurements provided by the International Earth Rotation and Reference Systems Service (IERS) to the variations of the degree-2 and order-0 Stokes coefficient of the gravity field determined from fitting the orbits of the LAGEOS-1 and −2 satellites since 1985. The two series show a good correlation (0.62) and similar annual and semi-annual signals, indicating that the gravity-field-derived LOD is valuable. Our analysis also provides evidence for additional signals common to both series, especially at a period near 120 days, which could be due to hydrological effects.     

Estimation of mass change trends in the Earth’s system on the basis of GRACE satellite data, with application to Greenland

The Gravity Recovery and Climate Experiment (GRACE) satellite mission measures the Earth’s gravity field since March 2002. We propose a new filtering procedure for post-processing GRACE-based monthly gravity field solutions provided in the form of spherical harmonic coefficients. The procedure is tuned for the optimal estimation of linear trends and other signal components that show a systematic behavior over long time intervals. The key element of the developed methodology is the statistically optimal Wiener-type filter which makes use of the full covariance matrices of noise and signal. The developed methodology is applied to determine the mass balance of the Greenland ice sheet, both per drainage system and integrated, as well as the mass balance of the ice caps on the islands surrounding Greenland. The estimations are performed for three 2-year time intervals (2003–2004, 2005–2006, and 2007–2008), as well as for the 6-year time interval (2003–2008). The study confirms a significant difference in the behavior of the drainage systems over time. The average 6-year rate of mass loss in Greenland is estimated as 165 ± 15 Gt/year. The rate of mass loss of the ice caps on Ellesmere Island (together with Devon Island), Baffin Island, Iceland, and Svalbard is found to be 22 ± 4, 21 ± 6, 17 ± 9, and 6 ± 2 Gt/year, respectively. All these estimates are corrected for the effect of glacial isostatic adjustment.     

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.     

Comparisons of atmospheric mass variations derived from ECMWF reanalysis and operational fields,over 2003–2011

There are two spurious jumps in the atmospheric part of the Gravity Recovery and Climate Experiment-Atmosphere and Ocean De-aliasing level 1B (GRACE-AOD1B) products, which occurred in January-February of the years 2006 and 2010, as a result of the vertical level and horizontal resolution changes in the ECMWFop (European Centre for Medium-Range Weather Forecasts operational analysis). These jumps cause a systematic error in the estimation of mass changes from GRACE time-variable level 2 products, since GRACE-AOD1B mass variations are removed during the computation of GRACE level 2. In this short note, the potential impact of using an improved set of 6-hourly atmospheric de-aliasing products on the computations of linear trends as well as the amplitude of annual and semi-annual mass changes from GRACE is assessed. These improvements result from 1) employing a modified 3D integration approach (ITG3D), and 2) using long-term consistent atmospheric fields from the ECMWF reanalysis (ERA-Interim). The monthly averages of the new ITG3D-ERA-Interim de-aliasing products are then compared to the atmospheric part of GRACE-AOD1B, covering January 2003 to December 2010. These comparisons include the 33 world largest river basins along with Greenland and Antarctica ice sheets. The results indicate a considerable difference in total atmospheric mass derived from the two products over some of the mentioned regions. We suggest that future GRACE studies consider these through updating uncertainty budgets or by applying corrections to estimated trends and amplitudes/phases.     

Theory and Methods of the Earth＇s Gravity Field Model Recovery from GOCE Data

研究GOCE卫星测量恢复地球重力场模型的理论与方法。论文的主要工作和创新点有： （1）建立扰动重力梯度张量各分量没有奇异性的详细计算模型,解决重力梯度张量Txx分量在两极地区计算的奇异性难题。     

The relation between degree-2160 spectral models of Earth’s gravitational and topographic potential: a guide on global correlation measures and their dependency on approximation effects

Comparisons between high-degree models of the Earth’s topographic and gravitational potential may give insight into the quality and resolution of the source data sets, provide feedback on the modelling techniques and help to better understand the gravity field composition. Degree correlations (cross-correlation coefficients) or reduction rates (quantifying the amount of topographic signal contained in the gravitational potential) are indicators used in a number of contemporary studies. However, depending on the modelling techniques and underlying levels of approximation, the correlation at high degrees may vary significantly, as do the conclusions drawn. The present paper addresses this problem by attempting to provide a guide on global correlation measures with particular emphasis on approximation effects and variants of topographic potential modelling. We investigate and discuss the impact of different effects (e.g., truncation of series expansions of the topographic potential, mass compression, ellipsoidal versus spherical approximation, ellipsoidal harmonic coefficient versus spherical harmonic coefficient (SHC) representation) on correlation measures. Our study demonstrates that the correlation coefficients are realistic only when the model’s harmonic coefficients of a given degree are largely independent of the coefficients of other degrees, permitting degree-wise evaluations. This is the case, e.g., when both models are represented in terms of SHCs and spherical approximation (i.e. spherical arrangement of field-generating masses). Alternatively, a representation in ellipsoidal harmonics can be combined with ellipsoidal approximation. The usual ellipsoidal approximation level (i.e. ellipsoidal mass arrangement) is shown to bias correlation coefficients when SHCs are used. Importantly, gravity models from the International Centre for Global Earth Models (ICGEM) are inherently based on this approximation level. A transformation is presented that enables a transformation of ICGEM geopotential models from ellipsoidal to spherical approximation. The transformation is applied to generate a spherical transform of EGM2008 (sphEGM2008) that can meaningfully be correlated degree-wise with the topographic potential. We exploit this new technique and compare a number of models of topographic potential constituents (e.g., potential implied by land topography, ocean water masses) based on the Earth2014 global relief model and a mass-layer forward modelling technique with sphEGM2008. Different to previous findings, our results show very significant short-scale correlation between Earth’s gravitational potential and the potential generated by Earth’s land topography (correlation +0.92, and 60% of EGM2008 signals are delivered through the forward modelling). Our tests reveal that the potential generated by Earth’s oceans water masses is largely unrelated to the geopotential at short scales, suggesting that altimetry-derived gravity and/or bathymetric data sets are significantly underpowered at 5 arc-min scales. We further decompose the topographic potential into the Bouguer shell and terrain correction and show that they are responsible for about 20 and 25% of EGM2008 short-scale signals, respectively. As a general conclusion, the paper shows the importance of using compatible models in topographic/gravitational potential comparisons and recommends the use of SHCs together with spherical approximation or EHCs with ellipsoidal approximation in order to avoid biases in the correlation measures.     

Detection of temporal changes in the spatial distribution of cancer rates using local Moran’s I and geostatistically simulated spatial neutral models

This paper presents the first application of spatially correlated neutral models to the detection of changes in mortality rates across space and time using the local Morans I statistic. Sequential Gaussian simulation is used to generate realizations of the spatial distribution of mortality rates under increasingly stringent conditions: 1) reproduction of the sample histogram, 2) reproduction of the pattern of spatial autocorrelation modeled from the data, 3) incorporation of regional background obtained by geostatistical smoothing of observed mortality rates, and 4) incorporation of smooth regional background observed at a prior time interval. The simulated neutral models are then processed using two new spatio-temporal variants of the Morans I statistic, which allow one to identify significant changes in mortality rates above and beyond past spatial patterns. Last, the results are displayed using an original classification of clusters/outliers tailored to the space-time nature of the data. Using this new methodology the space-time distribution of cervix cancer mortality rates recorded over all US State Economic Areas (SEA) is explored for 9 time periods of 5 years each. Incorporation of spatial autocorrelation leads to fewer significant SEA units than obtained under the traditional assumption of spatial independence, confirming earlier claims that Type I errors may increase when tests using the assumption of independence are applied to spatially correlated data. Integration of regional background into the neutral models yields substantially different spatial clusters and outliers, highlighting local patterns which were blurred when local Morans I was applied under the null hypothesis of constant risk.This research was funded by grants R01 CA92669 and 1R43CA105819-01 from the National Cancer Institute and R43CA92807 under the Innovation in Biomedical Information Science and Technology Initiative at the National Institute of Health. The views stated in this publication are those of the authors and do not necessarily represent the official views of the NCI. The authors also thank three anonymous reviewers for their comments that helped improve the presentation of the methodology.     

Development of the second-order derivatives of the Earth’s potential in the local north-oriented reference frame in orthogonal series of modified spherical harmonics

The second-order derivatives of the Earth’s potential in the local north-oriented reference frame are expanded in series of modified spherical harmonics. Linear relations are derived between the spectral coefficients of these series and the spectrum of the geopotential. On the basis of these relations, recurrence procedures are developed for evaluating the geopotential coefficients from the spectrum of each derivative and, inversely, for simulating the latter from a known geopotential model. Very simple structure of the derived expressions for the derivatives is convenient for estimating the geopotential coefficients by the least-squares procedure, at a certain step of processing satellite gradiometry data. Due to the orthogonality of the new series, the quadrature formula approach can be also applied, which allows avoidance of aliasing errors caused by the series truncation. The spectral coefficients of the derivatives are evaluated on the basis of the derived relations from the geopotential models EGM96 and EIGEN-CG01C at a mean orbital sphere of the GOCE satellite. Various characteristics of the spectra are studied corresponding to the EGM96 model. Electronic supplementary material  The online version of this article (doi:) contains supplementary material, which is available to authorized users.     

Evaluation of morphometric parameters derived from ASTER and SRTM DEM — A study on Bandihole sub-watershed basin in Karnataka

Morphometric parameters derived from three different sources viz., Survey of India topographic map (1:50,000), SRTM (Shuttle Radar Topographic Mission 90 m) and DEM derived from ASTER (Advanced Spaceborne Thermal Emission and Reflection Radiometer — 30 m) are evaluated to examine any difference within the results for the proper planning and management of the watersheds. Extracting drainage network from DEMs is mainly based on the flow of water from higher to lower elevation and steepest descent in a pixel. Common morphometric parameters are considered for analysis. The results show that the morphometric parameters derived from the SRTM and ASTER data provide good and satisfying results. The results will be more efficient when the DEM cell size is smaller or the resolution of the image is higher.     

Horizontal and vertical accuracy of Google Earth®: comment on ‘Positional accuracy of the Google Earth terrain model derived from stratigraphic unconformities in the Big Bend region,Texas, USA’ by S.C. Benker,R.P. Langford and T.L. Pavlis

Comment on ‘Positional accuracy of the Google Earth terrain model derived from stratigraphic unconformities in the Big Bend region, Texas, USA’ by S.C. Benker, R.P. Langford and T.L. Pavlis (Geocarto Int. 26:291–303, doi: 10.1080/10106049.2011.568125).