Volume 4, Issue 1

Space geodetic techniques provide highly accurate methods for estimating bedrock stability at subcentimetre level. We utilize data derived from Satellite Laser Ranging (SLR), Very Long Baseline Interferometry (VLBI) and Global Positioning Systems (GPS) techniques, collocated at the Hartebeesthoek Radio Astronomy Observatory, to characterise local plate motion and compare the solutions from the three techniques. Data from the GNSS station were processed using the GAMIT/GLOBK (version 10.4) software, data from the SLR station (MOBLAS-6)were processed using the Satellite Laser Ranging Data Analysis Software (SDAS) and the VLBI data sets were processed using the Vienna VLBI Software (VieVS) software. Results show that there is a good agreement between horizontal and vertical velocity components with a maximum deviation of 1.7 mm/yr, 0.7 mm/yr and 1.3 mm/yr between the North, East and Up velocity components respectively for the different techniques. At HartRAO there is no significant trend in the vertical component and all the techniques used are consistent with the a-priori velocities when compared with each other. This information is crucial in monitoring the local motion variations since geodetic instruments require a very stable base to minimise measurement errors. These findings demonstrate that station coordinate time-series derived with different techniques and analysis strategies provide comparable results.

The Gravity field and steady state Ocean Circulation Explorer (GOCE) maps variations in the gravity field by observing second order derivatives (gradients) of the Earth gravitational potential. Flying in the low altitude of 255 km and having a spatially dense data distribution of short wavelengths of the gravity field, GOCE may be used to enhance the time varying gravity signal coming fromthe GRACE satellites. The GOCE gradients may potentially be used for the determination of residual masses in local regions. This can be done using Least-Squares Collocation (LSC) or the Reduced Point Mass (RPM) method. In this study, different gravity field solutions are calculated by the use of RPM, LSC and GOCE gradients, respectively. Gravity field time series are created and presented for the six consecutive months of GOCE gradient observations, data being acquired between November 2009 and June 2010. Corresponding gravity anomaly results are used for the calculation of ice mass changes by the use of theRPMmethod. The results are then compared with the computed topographic effect of the ice by the use of a modified topographic correction and the Gravsoft TC program. The maximal gravity changes at the ground predicted from GOCE gradients are between 2 and 4 mGal for the period considered. The gravity anomaly estimation error arising from the GOCE gradient data using only T zz with an associated error of 20 mE is 11 mGal. This analysis shows the potential of using GOCE data for observations of ice mass changes although the GOCE dataset is limited to only six months. We expect four years of GOCE gradient observations to be available by mid-2014. This will increase the accuracy and spatial resolution of the GOCE measurements, which may lead to an accuracy necessary for observing ice mass changes.

Interannual fluctuations of the global mean sea level are highly correlated with El Niño-Southern Oscillation (ENSO) events, with positive/negative anomalies during El Niño/La Niña. In a previous study we showed that during the 1997 - 1998 El Niño, a positive anomaly observed in the global mean sea level was mostly caused by an increase of the ocean mass component rather than by steric (thermal) effects. This result was related to an increase of precipitation over the tropical ocean and a deficit in land water storage. In the present study, we investigate the effect of the recent 2008 and 2011 La Niña events on the satellite altimetry-based global mean sea level. We find that the large global mean sea level drop associated with the 2011 La Niña results from the combined decrease of the steric and ocean mass components, with a slightly dominant contribution from the latter. We show that the ocean mass contribution to the global mean sea level drop is spatially confined over the north eastern tropical Pacific (just as was found previously for the 1997 - 1998 El Niño, but with opposite sign). Corresponding ocean mass spatial pattern is closely correlated to observed sea level and steric spatial patterns over the duration of the La Niña event. This is also observed for previous El Niño and La Niña events. Such a drop in ocean mass during ENSO in the eastern part of the tropical Pacific has not been reported before. It is possibly related to a temporary decrease in the net precipitation over the north eastern Pacific (opposite situation was found during the 1997 - 1998 El Niño).

While the Errors-In-Variables (EIV) Model has been treated as a special case of the nonlinear Gauss- Helmert Model (GHM) for more than a century, it was only in 1980 that Golub and Van Loan showed how the Total Least-Squares (TLS) solution can be obtained from a certain minimum eigenvalue problem, assuming a particular relationship between the diagonal dispersion matrices for the observations involved in both the data vector and the data matrix. More general, but always nonsingular, dispersion matrices to generate the “properly weighted” TLS solution were only recently introduced by Schaffrin and Wieser, Fang, and Mahboub, among others. Here, the case of singular dispersion matrices is investigated, and algorithms are presented under a rank condition that indicates the existence of a unique TLS solution, thereby adding a new method to the existing literature on TLS adjustment. In contrast to more general “measurement error models,” the restriction to the EIV-Model still allows the derivation of (nonlinear) closed formulas for the weighted TLS solution. The practicality will be evidenced by an example from geodetic science, namely the over-determined similarity transformation between different coordinate estimates for a set of identical points.

Global degree-1 coefficients are derived by means of the method by Swenson et al. (2008) from a model of ocean mass variability and RL05 GRACE monthly mean gravity fields. Since an ocean model consistent with the GRACE GSM fields is required to solely include eustatic sea-level variability which can be safely assumed to be globally homogeneous, it can be empirically derived from GRACE aswell, thereby allowing to approximate geocenter motion entirely out of the GRACE monthly mean gravity fields. Numerical experiments with a decade-long model time-series reveal that the methodology is generally robust both with respect to potential errors in the atmospheric part of AOD1B and assumptions on global degree-1 coefficients for the eustatic sea-level model. Good correspondence of the GRACE RL05-based geocenter estimates with independent results let us conclude that this approximate method for the geocenter motion is well suited to be used for oceanographic and hydrological applications of regional mass variability from GRACE,where otherwise an important part of the signal would be omitted.

In the classical geodetic data processing, a non- linear problem always can be converted to a linear least squares adjustment. However, the errors in Jacob matrix are often not being considered when using the least square method to estimate the optimal parameters from a system of equations. Furthermore, the identity weight matrix may not suitable for each element in Jacob matrix. The weighted total least squares method has been frequently applied in geodetic data processing for the case that the observation vector and the coefficient matrix are perturbed by random errors, which are zero mean and statistically in- dependent with inequality variance. In this contribution, we suggested an approach that employ the weighted total least squares to solve the nonlinear problems and to mitigate the affection of noise in Jacob matrix. The weight matrix of the vector from Jacob matrix is derived by the law of nonlinear error propagation. Two numerical examples, one is the triangulation adjustment and another is a simulation experiment, are given at last to validate the feasibility of the developed method.

This study formulates regression of vector data that will enable statistical analysis of various geodetic phenomena such as, polar motion, ocean currents, typhoon/hurricane tracking, crustal deformations, and precursory earthquake signals. The observed vector variable of an event (dependent vector variable) is expressed as a function of a number of hypothesized phenomena realized also as vector variables (independent vector variables) and/or scalar variables that are likely to impact the dependent vector variable. The proposed representation has the unique property of solving the coefficients of independent vector variables (explanatory variables) also as vectors, hence it supersedes multivariate multiple regression models, in which the unknown coefficients are scalar quantities. For the solution, complex numbers are used to rep- resent vector information, and the method of least squares is deployed to estimate the vector model parameters after transforming the complex vector regression model into a real vector regression model through isomorphism. Various operational statistics for testing the predictive significance of the estimated vector parameter coefficients are also derived. A simple numerical example demonstrates the use of the proposed vector regression analysis in modeling typhoon paths.

State of the art Precise Point Positioning (PPP) is currently based on dual-frequency processing of GPS and Glonass navigation systems. The International GNSS Service (IGS) is routinely providing the most accurate orbit and clock products for these constellations, allowing point positioning at centimeter-level accuracy. At the same time, the GNSS landscape is evolving rapidly, with the deployment of new constellations, such as Galileo and BeiDou. The BeiDou constellation currently consists of 14 operational satellites, and the 4 Galileo In-Orbit Validation (IOV) satellites are transmitting initial Galileo signals. This paper focuses on the integration of Galileo and BeiDou in PPP, together with GPS and Glonass. Satellite orbits and clocks for all constellations are generated using a network adjustment with observation data collected by the IGS Multi-GNSS Experiment (MGEX), as well as from Fugro proprietary reference station network. The orbit processing strategy is described, and orbit accuracy for Galileo and BeiDou is assessed via orbit overlaps, for different arc lengths. Kinematic post-processed multi-GNSS positioning results are presented. The benefits of multiconstellation PPP are discussed in terms of enhanced availability and positioning accuracy.

The first triangulation activity on Danish ground was carried out by the astronomer Tycho Brahe who resided on the island Hven. He wanted to determine the longitude difference of his observatory Uraniborg to Copenhagen. A by-product was a map of his island made in 1579. In 1761 the Royal Danish Academy of Sciences and Letters initiated a mapping project which should be based on the principle of triangulation. Eventually 24 maps were printed in varying scales, predominantly in 1:120 000. The last map was engraved in 1842. The Danish GradeMeasurement initiated remeasurements and redesign of the fundamental triangulation network. This network served scientific as well as cartographic purposes in more than a century. Only in the 1960s all triangulation sides were measured electronically. A combined least-squares adjustment followed in the 1970s

Precision, reliability and cost are the major criteriaapplied in optimization and design of geodetic networks.The terrestrial networks are being replaced quicklyby permanent and campaign Global Positioning System(GPS) networks. These networks must be optimized usingthe same three criteria. In this article the optimization ofthe observational plan of local GPS networks (Second OrderDesign (SOD)) is considered using the precision criterion.This study is limited to the selection of optimal numbersand the best distribution of the non-trivial baselinesthroughout the network. This objective is accomplishedbased on the SOD solution through the analytical methodin operational research by the means of quadratic programmingalgorithm. This presented method is tested ona real GPS network and appears to be a useful techniquein terms of cost reduction in the field work by the providedobservational plan and optimal distribution of thebaselines throughout the network. Results indicate thatweights of almost 36% of the baselines are negligiblewhencompared to the weights of the rest of the baselines; therefore,they could be eliminated fromthe observational plan,resulting in a 36% saving in the fieldwork cost.

The processing of measurement data from satelliteconstellations such as Global Navigation Satellite Systems(GNSS), including the well-known Global PositioningSystem (GPS), have been successfully applied to virtuallyall areas of geophysical sciences. In this work, amethod is described where Geographical Information Systems(GIS) are employed to build hourly ionospheric TotalElectron Content (TEC) maps for 2011 over the southernIberian Peninsula. The maps used GPS-derived geometryfreelinear combinations attained from station data fromthe Algarve, Alentejo (Portugal), Andalusia, Murcia andValencia (Spain) regions. Following the construction ofthe ionospheric maps, it was possible to relate these resultsto natural phenomena. The observed phenomena includeddiurnal and seasonal variations: daytime TEC maxima,nighttime TEC peaks, summer TEC value decreases,and spring and fall TEC maxima. After validation of theseperiodic phenomena, detection of non-periodic changes,such as solar flares and tectonic interactions with the ionospherewere attempted. The results showed a TEC increasefollowing a selected solar flare event and a potential TECbuild-up prior to the 2011 Lorca earthquake. Further studiescould open up the possibility of building early warningsystems. The presented methods, based on available softwarepackages, are also of value in monitoring the effect ofthe ionosphere on radio signals, satellite and mobile communication,power grids, and for accurate GNSS navigation.

When processing satellite altimetry data for Mean Sea Surface (MSS) modelling in coastal environments many problems arise. The degradation of the accuracy of the Sea Surface Height (SSH) observations close to the coastline and the usually irregular pattern and variability of the sea surface topography are the two dominant factors which have to be addressed. In the present paper, we study the statistical behavior of the SSH observations in relation to the range from the coastline for many satellite altimetry missions and we make an effort to minimize the effects of the ocean variability. Based on the above concepts we present a process strategy for the homogenization of multi satellite altimetry data that takes advantage ofweighted SSH observations and applies high degree polynomials for the adjustment and their uniffcation at a common epoch. At each step we present the contribution of each concept to MSS modelling and then we develop a MSS, a marine geoid model and a grid of gravity Free Air Anomalies (FAA) for the area under study. Finally, we evaluate the accuracy of the resulting models by comparisons to state of the art global models and other available data such as GPS/leveling points, marine GPS SSH’s and marine gravity FAA’s, in order to investigate any progress achieved by the presented strategy

Recently, many organizations have begun providingonline GNSS (Global Navigation Satellite System)data processing services. Currently, only one of these organizationsi.e., OPUS (On-line Positioning Users Service)provides a rapid static data processing option. In case ofstatic online data processing, the users are required to submitat least two hours of data to get reasonably preciseresults. To provide processing option with less than twohours of data, NGS (National Geodetic Survey) developedOPUS–RS for rapid static data processing so that usersmaysubmit as little as 15 minutes of dual frequency GNSS data.In this study, multiple observation sessions are conductedat the same locations to compare OPUS-RS generated coordinatesamong the different sessions to see whether separatevalues agree with each other. The results indicate thatwith OPUS-RS results the differences in horizontal coordinatesagree with each other within 3.5 cm and vertical coordinatesagree within 7.2 cm. For an independent check,OPUS-RS results are also compared against LGO(Leica GeoOffice) produced Static results; this comparison yieldedup to 4.5 cm variations among horizontal coordinate differencesand variations among vertical coordinate differencesare up to 11.4 cm.

Traditional gravimetric geoid determination relieson Stokes’ formula with removal and restoration of thetopographic effects. It is shown that this solution is in errorof the order of the quasigeoid-to-geoid difference, which ismainly due to incomplete downward continuation (dwc)of gravity from the Earth’s surface to the geoid. A slightlyimproved estimator, based on the surface Bouguer gravityanomaly, is also biased due to the imperfect harmonic dwcthe Bouguer anomaly. Only the third estimator,which usesthe (harmonic) surface no-topography gravity anomaly, isconsistent with the boundary condition and Stokes’ formula,providing a theoretically correct geoid height.The difference between the Bouguer and no-topographygravity anomalies (on the geoid or in space) is the “secondaryindirect topographic effect”, which is a necessarycorrection in removing all topographic signals.

Geodetic and geophysical time series may containsinusoidal oscillations of unknown angular frequency.Often it is required to decide if such sinusoidaloscillations are truly present in a given time series. Herewe pose the decision problem as a statistical hypothesistest, an approach very popular in geodesy and other scientificdisciplines. In the case of unknown angular frequenciessuch a test has not yet been proposed.We restrict ourselvesto the detection of a single sinusoidal oscillationin a one-dimensional time series, sampled at non-uniformtime intervals.We compare two solution methods: the likelihoodratio test for parameters in a Gauss-Markov modeland the analysis of the Lomb-Scargle periodogram. Wheneverneeded, critical values of these tests are computedusing the Monte Carlo method. We analyze an exemplarytime series from an absolute gravimetric observation byvarious tests. Finally, we compare their statistical power.It is found that the results for the exemplary time seriesare comparable. The LR test is more flexible, but alwaysrequires the Monte Carlo method for the computation ofcritical values. The periodogram analysis is computationallyfaster, because critical values can be approximatelydeduced from the exponential distribution, at least if thesampling is nearly uniform.

The working hypothesis of this study is that periodiclunar nodal tides and almost periodic solar radiationvariations influence sea level changes through theirharmonic beating of nearby natural and/or forced broadbandoscillations of the sea level at multi-decadal frequencies.The presence of the harmonics of the lunar nodaltides and the solar radiation variations, including the poletides, is investigated by modeling and estimating the amplitudesof the corresponding periodicities in 27 globallydistributed long tide gauge records. Statistically significantsignatures of sub and superharmonics of lunar nodaltides and forced sea level variations due to solar radiationare detected in all station records.Meta-analysis of the harmonicamplitudes from all stations reveals that the effectsizes are statistically significant and provide evidence forthe harmonic beating of sea level changes as a global phenomenon.Consequently, the compounding of the lunarnodal tides and forced sea level changes due to solar radiationwith other broadband natural and forced sea leveloscillations is a plausible explanation for the recent sealevel accelerations and decelerations detected by satellitealtimetry measurements and long tidal records.

Recently, it has been claimed that the HomogeneousErrors-In-Variables (HEIV) Model, where the lefthandside (LHS) vector is allowed to be multiplied withan unknown scale factor, would represent a generalizationof the regular EIV-Model for which a number of efficientalgorithms already exist. Unfortunately, due to theforced rank deficiency in the case of the HEIV-Model, anadditional constraint needs to be introduced to guaranteeuniqueness of the TLS solution (“datum constraint”).If this constraint is linear, a simple manipulation will reducethe HEIV-Model with one constraint to the regularEIV-Model. But also in the case of a non-linear datum constraint,by introducing parameter ratios as unknowns, anEIV-Model may result that can be treated by standard TLSadjustment, followed by a solution of the datum constraintfor the additional LHS scale parameter. This approach willbe applied to an example where the datum constraint ischosen to be quadratic.