Volume 9, Issue 1

In this work, the geodesic equations and their numerical solution in Cartesian coordinates on an oblate spheroid, presented by Panou and Korakitis (2017), are generalized on a triaxial ellipsoid. A new exact analytical method and a new numerical method of converting Cartesian to ellipsoidal coordinates of a point on a triaxial ellipsoid are presented. An extensive test set for the coordinate conversion is used, in order to evaluate the performance of the two methods. The direct geodesic problem on a triaxial ellipsoid is described as an initial value problem and is solved numerically in Cartesian coordinates. The solution provides the Cartesian coordinates and the angle between the line of constant λ and the geodesic, at any point along the geodesic. Also, the Liouville constant is computed at any point along the geodesic, allowing to check the precision of the method. An extensive data set of geodesics is used, in order to demonstrate the validity of the numerical method for the geodesic problem. We conclude that a complete, stable and precise solution of the problem is accomplished.

The Central Andes experienced major earthquake (Mw =8.2) in April 2014 in a region where the giant 1877 earthquake (Mw=8.8) occurred. The 2014 Iquique earthquake did not break the entire seismic gap zones as previously predicted. Geodetic and seismological observations indicate a highly coupled plate interface. To assess the locking mechanism of plate interfaces beneath Central Andes, a 2.5-D gravity model of the crust and upper mantle structure of the central segment of the subduction zone was developed based on terrestrial and satellite gravity data from the LAGEOS, GRACE and GOCE satellite missions. The densities and major structures of the gravity model are constrained by velocity models from receiver function and seismic tomography. The gravity model defined details of crustal and slab structure necessary to understand the cause of megathrust asperity generation. The densities of the upper and lower crust in the fore-arc (2970 – 3000 kg m −3 ) are much higher than the average density of continental crust. The high density bodies are interpreted as plutonic or ophiolitic structures emplaced onto continental crust. The plutonic or ophiolitic structures may be exerting pressure on the Nazca slab and lock the plate interfaces beneath the Central Andes subduction zone. Thus, normal pressure exerted by high density fore-arc structures and buoyancy force may control plate coupling in the Central Andes. However, this interpretation does not exclude other possible factors controlling plate coupling in the Central Andes. Seafloor roughness and variations in pore-fluid pressure in sediments along subduction channel can affect plate coupling and asperity generation.

The present study computes B-W extension model (extended Bursa-Wolf model) coordinate transformation parameters from World Geodetic System 1984 (WGS-84) to the Everest datum namely Everest (1830) and Everest (1956) using records of coordinate measurements from Global Positioning System (GPS) observable across Nepal region. Synthetic or modeled coordinates were determined by using the Artificial Neural Network (ANN) and Singular Value Decomposition (SVD) methods. We studied 9-transformation parameters with the help of the ANN technique and validated the outcomes with the SVD method. The comparative analysis of the ANN, as well as SVD methods, was done with the observed output following one way ANOVA test. The analysis showed that the null hypothesis for both datums were acceptable and suggesting all models statistically significantly equivalent to each other. The outcomes from this study would complement a relatively better understanding of the techniques for coordinate transformation and precise coordinate assignment while assimilating data sets from different resources.

The AntGrav project, funded by the German Research Foundation (DFG) has the main objective to homogenize and optimize Antarctic gravity field information. Within this project an evaluation procedure is needed to inspect all different kind of gravity field surveys available in Antarctica. In this paper a suitable methodology is proposed. We present an approach for fast 3D gravity point data reduction in different spectral bands. This is achieved through pre-calculating a fine 3D mesh of synthesized gravity functionals over the entirety of the Antarctic continent, for which two different global models are used: the combined satellite model GOCO05s for the long-wavelength part, and the topographic model Earth2014 for the shorter wavelengths. To maximize the applicability separate meshes are calculated for different spectral bands in order to specifically reduce a certain band or a selected combination. All meshes are calculated for gravity anomalies as well as gravity disturbances. Utilizing these meshes, synthesized gravity data at arbitrary positions is computed by conventional 3D interpolation methods (e.g. linear, cubic or spline). It is shown that the applied approach can reach a worst-case interpolation error of less than 1 mGal. Evaluation results are presented for the AntGG grid and exemplary for the in-situ measurements of the AGAP and BAS-LAND campaigns. While general properties, large-scale errors and systematic effects can usually be detected, small-scale errors (e.g. of single points) are mostly untraceable due to the uncertainties within the topographic model.

Atmospheric water vapour, a major component in weather systems serves as the main source for precipitation, provides latent heat which helps maintain the earth’s energy balance and a major parameter in Numerical Weather Prediction (NWP) models. An observational technique based on the Global Navigation Satellite System (GNSS) has made it possible to easily retrieve Precipitable Water (PW) at station’s antenna position with very high spatial and temporal variabilities. GNSS techniques are superior to ground-based and balloons sensors in terms of accuracy, ease of use, wider coverage and easier assimilation into NWP models. This study sought to use prediction models using daily observational data from Four (4) International GNSS Service stations in West Africa. The best prediction model can be used in cases of station outages and to predict PW over data poor regions using computed Zenith Tropospheric Delays (ZTD). gLAB software was used to process the stations’ data in Precise Point Positioning mode and PW were retrieved using station’s temperature and pressure values. Computed PW were compared against Total Column Water Vapour from ERA-Interim Reanalysis data in 2016. Correlation coefficient (R 2 ) values ranging from 0.947 — 0.995 were obtained for the four stations. With computed PW’s, three regression models were tested to find the best-fit with PW as the dependent variable and ZTD being the independent variable. The quadratic model gave the highest R 2 and lowest RMSE values as against the linear and exponential models. Time series forecasts models such as moving average, autoregressive, exponential smoothing and autoregressive integrated moving average were also employed. The forecasts results were compared against ZTD with autoregressive model reporting the highest R 2 and lowest RMSE amongst the forecast models developed.

As changes in gravity are directly related to mass variability, satellite missions observing the Earth’s time varying gravity field are a unique tool for observing mass transport processes in the Earth system, such as the water cycle, rapid changes in the cryosphere, oceans, and solid Earth processes, on a global scale. The observation of Earth’s gravity field was successfully performed by the GRACE and GOCE satellite missions, and will be continued by the GRACE Follow-On mission. A comprehensive team of European scientists proposed the next-generation gravity field mission MOBILE in response to the European Space Agency (ESA) call for a Core Mission in the frame of Earth Explorer 10 (EE10). MOBILE is based on the innovative observational concept of a high-low tracking formation with micrometer ranging accuracy, complemented by new instrument concepts. Since a high-low tracking mission primarily observes the radial component of gravity-induced orbit perturbations, the error structure is close to isotropic. This geometry significantly reduces artefacts of previous along-track ranging low-low formations (GRACE, GRACE-Follow-On) such as the typical striping patterns. The minimum configuration consists of at least two medium-Earth orbiters (MEOs) at 10000 km altitude or higher, and one low-Earth orbiter (LEO) at 350-400 km. The main instrument is a laser-based distance or distance change measurement system, which is placed at the LEO. The MEOs are equipped either with passive reflectors or transponders. In a numerical closed-loop simulation, it was demonstrated that this minimum configuration is in agreement with the threshold science requirements of 5 mm equivalent water height (EWH) accuracy at 400 km wavelength, and 10 cm EWH at 200 km. MOBILE provides promising potential future perspectives by linking the concept to existing space infrastructure such as Galileo next-generation, as future element of the Copernicus/Sentinel programme, and holds the potential of miniaturization even up to swarm configurations. As such MOBILE can be considered as a precursor and role model for a sustained mass transport observing system from space.

The topographic potential bias at geoid level is the error of the analytically continued geopotential from or above the Earth’s surface to the geoid. We show that the topographic potential can be expressed as the sum of two Bouguer shell components, where the density distribution of one is spherical symmetric and the other is harmonic at any point along the normal to a sphere through the computation point. As a harmonic potential does not affect the bias, the resulting topographic bias is that of the first component, i.e. the spherical symmetric Bouguer shell. This implies that the so-called terrain potential is not likely to contribute significantly to the bias. We present three examples of the geoid bias for different topographic density distributions.

For the calculation of gravity disturbance in the Earth’s external gravity field, the Stokes-Pizzetti integral is a commonly used method. However, when the target point approaches the Earth’s surface, such problems as singularity and discontinuity arise due to the Stokes kernel structure itself. To settle the problems, firstly the reason for singularity and discontinuity was discussed, and then modification was made to the integral formula, by which the singularity at the surface point is eliminated. Finally the non-singular integral formulas for the calculation of disturbing gravity were derived. In numerical experiments, an area in China was selected to test the modified formula. Numerical results show that the modified formula performs much better than classical Stokes-Pizzetti integral formula when dealing with the calculation of the radial component of gravity disturbance near the Earth’s surface.

The signal content and error level of recent GOCE-based high resolution gravity field models is assessed by means of signal degree variances and comparisons to independent GNSS-levelling geoid heights. The signal of the spherical harmonic series of these models is compared to the pre-GOCE EGM2008 model in order to identify the impact of GOCE data, of improved surface and altimetric gravity data and of modelling approaches. Results of the signal analysis show that in a global average roughly 80% of the differences are due to the inclusion of GOCE satellite information, while the remaining 20% are contributed by improved surface data. Comparisons of the global models to GNSS-levelling derived geoid heights demonstrate that a 1 cm geoid from the global model is feasible, if there is a high quality terrestrial gravity data set available. For areas with less good coverage an accuracy of several centimetres to a decimetre is feasible taking into account that GOCE provides now the geoid with a centimetre accuracy at spatial scales of 80 to 100 km. Comparisons with GNSS-levelling geoid heights also are a good tool to investigate possible systematic errors in the global models, in the spirit levelling and in the GNSS height observations. By means of geoid height differences and geoid slope differences one can draw conclusions for each regional data set separately. These conclusions need to be considered for a refined analysis e.g. to eliminate suspicious GNSS-levelling data, to improve the global modelling by using full variance-covariance matrices and by consistently weighting the various data sources used for high resolution gravity field models. The paper describes the applied procedures, shows results for these geoid height and geoid slope differences for some regional data sets and draws conclusions about possible error sources and future work to be done in this context.

The paper presents a new approach to the airborne vector gravimetry problem. The idea of the approach is to take into account spatial correlation of the gravity field to improve observability of horizontal components of the gravity disturbance vector (GDV). We consider the GDV determination problem given airborne data at a set of parallel survey lines assuming that lines are flown in the same direction at a constant height above the reference ellipsoid. We use a 2-D random field model for the gravity field at the flight height. The random field is governed by two autoregressive equations (one in the direction along the lines, the other across the lines). Then we pose the estimation problem simultaneously for the GDV horizontal components and systematic errors of an inertial navigation system at all the lines simultaneously. The developed estimation algorithm is based on 2D Kalman filtering and smoothing techniques. Numerical results obtained from simulated data processing showed improved accuracy of the gravity horizontal component determination.

The NKG2015 geoid model covers the Nordic and Baltic countries and has been computed based on the least-squares modification of Stokes’ formula with additive corrections method. New and precise terrestrial, airborne and shipborne gravimetric measurements, the recent global gravity model of the gravity field and steady-state ocean circulation explorer (GOCE) and detailed digital terrain models over each territory have been used for computing this new geoid model. Some estimates for the error of this model have been roughly presented by comparing it with the global navigation satellite system (GNSS) data over each country. In this paper, our goal is to have a closer look at the relative error of this model by performing some statistical tests and finding the proper corrective surface for absorbing the systematic errors over each country. Our main assumption is realisticity of the errors of GNSS/levelling data and we will investigate its consequences in estimating the error of the geoid model. Our results show that the 4-parameter corrective surface is suitable for modelling the systematic trends of the differences between the gravimetric and GNSS geoid heights in Sweden, Denmark and Finland, but a filtered discrepancies by a confidence interval of 95% should be used for Sweden. A 7-aparameter model is suitable for the filtered discrepancies with the confidence interval of 95% in Norway. Based on the selected corrective surface and our newly developed regional iterative variance estimator, the confidence interval for the error of NKG2015 geoid model in Sweden, Denmark and Norway yielded 0-6.5 mm, 1.8-5.2 mm, 14.8-17.7 mm, respectively with a confidence level of 95%. We could not estimate the geoid error in Finland because the given error of the GNSS/levelling heights is significantly larger than the size of residuals. Based on the selected corrective surfaces and our presented local variance estimator, the average error of geoid becomes 3.6, 2.4, 8.8 and 5.8 mm with a confidence interval of 68%, respectively, over Sweden, Denmark, Norway and Finland.

The GRACE mission has demonstrated a tremendous potential for observing mass changes in the Earth system from space for climate research and the observation of climate change. Future mission should on the one hand extend the already existing time series and also provide higher spatial and temporal resolution that is required to fulfil all needs placed on a future mission. To analyse the applicability of such a Next Generation Gravity Mission (NGGM) concept regarding hydrological applications, two GRACE-FO-type pairs in Bender formation are analysed. The numerical closed loop simulations with a realistic noise assumption are based on the short arc approach and make use of the Wiese approach, enabling a self-de-aliasing of high-frequency atmospheric and oceanic signals, and a NRT approach for a short latency. Numerical simulations for future gravity mission concepts are based on geophysical models, representing the time-variable gravity field. First tests regarding the usability of the hydrology component contained in the Earth System Model (ESM) by the European Space Agency (ESA) for the analysis regarding a possible flood monitoring and detection showed a clear signal in a third of the analysed flood cases. Our analysis of selected cases found that detection of floods was clearly possible with the reconstructed AOHIS/HIS signal in 20% of the tested examples, while in 40% of the cases a peak was visible but not clearly recognisable.

For the modelling and determination of the Earth’s external gravity potential as well as its second-order radial derivatives in the space near sea surface, the surface layer integral method was discussed in the paper. The reasons for the applicability of the method over sea surface were discussed. From the original integral formula of disturbing potential based on the surface layer method, the expression of the radial component of the gravity gradient tensor was derived. Furthermore, an identity relation was introduced to modify the formula in order to reduce the singularity problem. Numerical experiments carried out over the marine area of China show that, the modi-fied surface layer integral method effectively improves the accuracy and reliability of the calculation of the second-order radial gradient component of the disturbing potential near sea surface.

The continental water storage is significantly in-fluenced by wetlands, which are highly affected by climate change and anthropogenic influences. The Pantanal, located in the Paraguay river basin, is one of the world’s largest and most important wetlands because of the environmental biodiversity that represents. The satellite gravity mission GRACE (Gravity Recovery And Climate Experiment) provided until 2017 time-variable Earth’s gravity field models that reflected the variations due to mass transport processes-like continental water storage changes-which allowed to study environments such as wetlands, at large spatial scales. The water storage variations for the period 2002-2016, by using monthly land water mass grids of Total Water Storage (TWS) derived from GRACE solutions, were evaluated in the Pantanal area. The capability of the GRACE mission for monitoring this particular environment is analyzed, and the comparison of the water mass changes with rainfall and hydrometric heights data at different stations distributed over the Pantanal region was carried out. Additionally, the correlation between the TWS and river gauge measurements, and the phase differences for these variables, were also evaluated. Results show two distinct zones: high correlations and low phase shifts at the north, and smaller correlation values and consequently significant phase differences towards the south. This situation is mainly related to the hydrogeological domains of the area.

Knowledge of vertical crustal movement is fundamental to quantify absolute sea level changes at tide gauge locations as well as for satellite altimetry calibration validations. While GPS measurements at collocated tide gauge stations fulfill this need, currently only few hundred tide gauge stations are equipped with GPS, and their measurements do not span a long period of time. In the past, several studies addressed this problem by calculating relative and geocentric trends from the tide gauge and satellite altimetry measurements respectively, and then difference the two trends to calculate the rate of changes at the tide gauge stations. However, this approach is suboptimal. This study offers an optimal statistical protocol based on the method of condition equations with unknown parameters . An example solution demonstrates the proposed mathematical and statistical models’ optimality in estimating vertical crustal movement and its standard error by comparing them with the results of current methods. The proposed model accounts for the effect of autocorrelations in observed tide gauge and satellite altimetry sea level time series, adjusts observed corrections such as inverted barometer effects, and constraints tide gauge and satellite altimeter measurement to close. The new model can accommodate estimating other systematic effects such as pole tides that are not eliminated by differencing.

Precise knowledge of the oceanic Mean Dynamic Topography (MDT) is crucial for a number of geodetic applications, such as vertical datum unification and marine geoid modelling. The lack of gravity surveys over many regions of the Greek seas and the incapacity of the space borne gradiometry/gravity missions to resolve the small and medium wavelengths of the geoid led to the investigation of the oceanographic approach for computing the MDT. We compute two new regional MDT surfaces after averaging, for given epochs, the periodic gridded solutions of the Dynamic Ocean Topography (DOT) provided by two ocean circulation models. These newly developed regional MDT surfaces are compared to three state-of-theart models, which represent the oceanographic, the geodetic and the mixed oceanographic/geodetic approaches in the implementation of the MDT, respectively. Based on these comparisons, we discuss the differences between the three approaches for the case study area and we present some valuable findings regarding the computation of the regional MDT. Furthermore, in order to have an estimate of the precision of the oceanographic approach, we apply extensive evaluation tests on the ability of the two regional ocean circulation models to track the sea level variations by comparing their solutions to tide gauge records and satellite altimetry Sea Level Anomalies (SLA) data. The overall findings support the claim that, for the computation of the MDT surface due to the lack of geodetic data and to limitations of the Global Geopotential Models (GGMs) in the case study area, the oceanographic approach is preferable over the geodetic or the mixed oceano-graphic/geodetic approaches.