Least Squares collocation

We present the official land uplift model NKG2016LU of the Nordic Commission of Geodesy (NKG) for northern Europe. The model was released in 2016 and covers an area from 49° to 75° latitude and 0° to 50° longitude. It shows a maximum absolute uplift of 10.3 mm/a near the city of Umeå in northern Sweden and a zero-line that follows the shores of Germany and Poland. The model replaces the NKG2005LU model from 2005. Since then, we have collected more data in the core areas of NKG2005LU, specifically in Norway, Sweden, Denmark and Finland, and included observations from the Baltic countries as well. Additionally, we have derived an underlying geophysical glacial isostatic adjustment (GIA) model within NKG as an integrated part of the NKG2016LU project. A major challenge is to estimate a realistic uncertainty grid for the model. We show how the errors in the observations and the underlying GIA model propagate through the calculations to the final uplift model. We find a standard error better than 0.25 mm/a for most of the area covered by precise levelling or uplift rates from Continuously Operating Reference Stations and up to 0.7 mm/a outside this area. As a check, we show that two different methods give approximately the same uncertainty estimates. We also estimate changes in the geoid and derive an alternative uplift model referring to this rising geoid. Using this latter model, the maximum uplift in Umeå reduces from 10.3 to 9.6 mm/a and with a similar reduction ratio elsewhere. When we compare this new NKG2016LU with the former NKG2005LU, we find the largest differences where the GIA model has the strongest influence, i.e. outside the area of geodetic observation. Here, the new model gives from -3 to 4 mm/a larger values. Within the observation area, similar differences reach -1.5 mm/a at the northernmost part of Norway and -1.0 mm/a at the north-western coast of Denmark, but generally within the range of -0.5 to 0.5 mm/a.

The primary objective of the 1-cm geoid experiment in Colorado (USA) is to compare the numerous geoid computation methods used by different groups around the world. This is intended to lay the foundations for tuning computation methods to achieve the sought after 1-cm accuracy, and also evaluate how this accuracy may be robustly assessed. In this experiment, (quasi)geoid models were computed using the same input data provided by the US National Geodetic Survey (NGS), but using different methodologies. The rugged mountainous study area (730 km × 560 km) in Colorado was chosen so as to accentuate any differences between the methodologies, and to take advantage of newly collected GPS/leveling data of the Geoid Slope Validation Survey 2017 (GSVS17) which are now available to be used as an accurate and independent test dataset. Fourteen groups from fourteen countries submitted a gravimetric geoid and a quasigeoid model in a 1 ×1 grid for the study area, as well as geoid heights, height anomalies, and geopotential values at the 223 GSVS17 marks. This paper concentrates on the quasigeoid model comparison and evaluation, while the geopotential value investigations are presented as a separate paper (Sánchez et al. in J Geodesy 95(3):1.

Luiz Paulo Fortes holds an M.Sc. in Computer Science applied to Geomatics and is currently a Ph.D. candidate at the University of Calgary. He has been working the last 19 years for IBGE Department of Geodesy (the "Geodetic Survey" of Brazil) where he was responsible for the Research and Analysis division most of the time and as the head of the department in the last two years before enrolling in the Ph.D. program. Bolsista da CAPES -Brasília/Brasil.

Diplomityö käsittelee epävarmuutta sisältävien korkeusmallien valuma-alueiden laskentamenetelmiä. Työssä perehdyttiin hilamuotoisten korkeusmallien valuma-alueanalyysiin, tilastollisiin korkeusmallin epävarmuuden mallintamistapoihin ja Monte Carlo simulaatioon. Olemassa olevien menetelmien pohjalta kehitettiin hajautettuun laskentaan sopivat laskenta-algoritmit, joiden toimintaa ja skaalautuvuutta testattiin Tieteellisen laskennan CSC:n klusterilaskentaympäristössä. Simulaation tulosten tarkkuuden arvioimiseksi johdettiin korkeaulotteiselle aineistolle ja simulaatiomenetelmässä lasketuille valuma-aluekartoille sopiva konvergenssin arviointimenetelmä. Diplomityö koostuu seitsemästä luvusta. Johdannossa esitellään työn taustaa ja käsitteitä, sekä kuvataan työssä ratkaistut ongelmat. Työn pohjana oleva aiempi tutkimus ja sen tulokset esitellään luvussa kaksi. Työn toteutusosassa johdetaan aluksi menetelmä Monte Carlo simulaation tarkkuuden arvioimiselle simulaatiokierrosten määrän kasv...

Support vector machines (SVM) with wavelet kernel has been applied to the correcting gravimetric geoid using GPS/levelling data. These data were divided into a training and a validation set in order to ensure the extendability of the approximation of the corrector surface. The optimal parameters of the SVM were considered as a trade-off between accuracy and extendability of the solution in order to avoid overlearning. Employing 194 training points and 110 validation points, SVM provided an approximation with less than 3 cm standard deviation of the error and nearly perfect extendability.

The Azores are of volcanic origin and the volcanic activities are still occurs in the area. The main tectonic features in the eastern part of Azores are the Gloria Fault (GF) and São Miguel volcanic Island. The GF is an E-W strike-slip fault and can be traced by bathymetry. In the past decade many geological and geophysical investigations were dedicated to the study of tectonic features in the eastern part of Azores. Two of these cruises were organized by the Institute of Geophysics, University of Hamburg, Germany, in the years 2009 and 2012. In 2009 during the Meteor cruise M79-2 a total of 5500 km new Gravity and 2000 km new magnetic data were collected along some 60 Profiles. During the Poseidon cruise in the year 2012 some 2000 km new gravity and magnetic data were collected along two E-W profiles in the eastern part of Azores. The new gravity data were recorded with the modern Air-Sea-Gravimeter of Bodenseewerk KSS 31M and the new magnetic date with the Gradiometer SeaSpy. All new potential date were combined with the available data of the data base GEODAS and the new gravity anomaly maps (Free-Air and Bouguer) and the new magnetic anomaly map were produced. The maps show clearly the tectonic features in the area. The GF can be traced very well on both gravity and magnetic anomaly maps. Most of the small hills around the São Miguel Island are shown up in the magnetic anomaly map as strong magnetic anomaly. The new gravity and magnetic maps and the interpretation of them will be presented. The results of some 2-D modeling along some interesting profiles will be also presented and discussed.

The purpose of this paper is twofold, namely (i) to present the developed dynamic geoid model based on the most recent satellite GRACE data and terrestrial data over Canada and (ii) to provide a methodology for incorporating the dynamic geoid as the new height datum in Canada. The move towards a modernized geoid-based vertical datum to replace the existing (and outdated) official height datum CGVD28 involves incorporating the dynamic nature of the regional geoid model, which includes secular changes of up to +1.5 mm/year as a result of the ongoing mass transport beneath the uplifting/subsiding Earth's crust. Modelling the regional geoid dynamic variations has been facilitated due to the improved accuracy of the regional geoid model, which is computed using the latest high quality and resolution global gravity models.

The area bounded by the Alpine chain to the North and Mediterranean/Ionian sea to the South is characterized by a wide range of tectonic phenomena, such as the indentation of the Adria block into the Eastern Alp, lateral extrusion of the Tauern Window, unbending of the Adriatic lithosphere, opening of the Thyrrenian sea and subduction of the Ionian lithosphere beneath the Calabrian arc. This ongoing tectonics is accompanied by a relatively intense volcanism and seismicity, which justify the expectation of small but measurable horizontal and vertical displacements. We report on surface horizontal displacements determined at discrete locations by analyzing time series of permanent GPS stations with data coverage from one to five years of continuous operation. The horizontal velocities are defined consistently with the ITRF97 velocity datum. We show that the residual velocities relative to rigid rotations about Eulerian poles are never larger than 5 mm yr-1 and in several cases do reflect qualitatively the expected kinematics. Areas characterized by fracturing and faults with orientations changing on a short scale (Tauern window, Southern Alps) exhibit a more irregular distribution of velocities, probably associated with local phenomena. Eigenvalues and eigenvectors of a mean strain rate tensor are computed by optimally-in a least squares collocation sense-interpolating the station velocities to locations baricentric to clusters of stations. The estimated strain is everywhere smaller than 41 10-9 yr-1 with a mean uncertainty of 20 10-9 yr-1 (1σ). The areas with largest strain rate are the Central Apennines and Eastern Alps, while in the Western Alps the estimated strain rate is smaller than 15-20 10-9 yr-1 , hence comparable with its uncertainty. The azimuths of the strain rate ellipse are qualitatively compared with the directions of the stress estimated from fault plane mechanisms and borehole breakouts, and with the strike of major faults. The orientations of the strain rate and stress ellipses show remarkable consistency, within the estimated uncertainties. We conclude in favor of evidence of a yet qualitative but significant correlation between broad scale (~300 km) stress and strain rate patterns, and orientation of large scale active lineaments.

Approximately nine-year data from ERS1, ERS2 and TOPEX/POSEIDON (T/P) satellite altimetry missions have been used for the recovery of gravity anomalies over the Black Sea. The Corrected Sea Surface Height product of Aviso/Altimetry has been proven to be homogeneous after a cross-over adjustment. The Least Squares Collocation method was applied in a so-called "remove-restore" procedure. The residual geoid heights, obtained by subtracting EGM96 geoid heights from cross-over adjusted sea surface heights, were inverted to recover residual gravity anomalies in a grid structure over the Black Sea. Finally, EGM96 free air gravity anomalies were added to the predicted residual gravity anomalies to obtain the free air gravity anomalies. In order to check the consistency with respect to an external source, these computed free air gravity anomalies were compared to ship gravity observations, and to alternative satellite altimetry derived gravity anomalies. Comparisons with the observed gravity data yielded that external consistency of the gravity anomalies computed in this study is about 3 mGal for individual ship tracks. Overall external consistency in the test area is 4.8 mGal. Comparison with other satellite altimetry derived gravity anomalies presented a good agreement.

A high-resolution gravimetric quasigeoid model for Vietnam and its surrounding areas was determined based on new gravity data. A set of 29,121 land gravity measurements was used in combination with fill-in data where no gravity data existed. Global Gravity field Models plus Residual Terrain Model effects and gravity field derived from altimetry satellites were used to provide the fill-in information over land and marine areas. A mixed model up to degree/order 719 was used for the removal of the long and medium wavelengths and the calculation of the quasigeoid restore effects. The residual height anomalies have been determined employing the Stokes integral using the Fast Fourier Transform approach and deterministic kernel modification proposed by Wong-Gore, as well as by means of Least-Squares Collocation. The accuracy of the resulting quasigeoid models was evaluated by comparing with height anomalies derived from 812 co-located GNSS/levelling points. Results are very similar; both local quasigeoid models have a standard deviation of 9.7 cm and 50 cm in mean bias when compared to the GNSS/levelling points. This new local quasigeoid model for Vietnam represents a significant improvement over the global models EIGEN-6C4 and EGM2008, which have standard deviations of 19.2 and 29.1 cm, respectively, for this region.

One approach for unification of height systems relates to the solution of a Geodetic Boundary Value Problem (GBVP). In particular, the fixed GBVP, using gravity disturbances as boundary values, is solved at GNSS/leveling benchmarks, whereupon height datum offsets can be estimated by least squares adjustment. In spherical approximation, the solution of the fixed GBVP is obtained by Hotine’s spherical integral formula. However, this method relies on the global availability of gravity data. In practice, gravity data of the necessary resolution and accuracy is not accessible globally. Thus, the integration is restricted to an area within the vicinity of the computation points. The resulting truncation error can reach several meters in height, making height system unification without further consideration of this effect unfeasible.

Recently, a variety of gravity observations in Antarctica has become available through extensive efforts of airborne surveys. Aircrafts serving as multi-instrumentation platforms provide measurements on gravity, bedrock topography, ice surface topography and ice thickness. Collected datasets are valuable in terms of resolution and homogeneity, which make them suitable for studying regional geoid determination in selected Antarctic regions. Within this context the German joint project VISA provided an excellent database for improving the regional geoid by combining gravity and topographic data from aerogeophysical observations with long-wavelength information from global gravity field models. Using the remove-compute-restore technique in conjunction with least-squares collocation a regional geoid for Dronning Maud Land, East Antarctica, will be presented. A signal threshold of up to 6 m added to the global model that was used as a basis can be expected. The accuracy of the regional geoid will be estimated to be at the level of 15 cm.

With the official launch of Romanian Position Determination System ROMPOS, whose basic infrastructure consists of a network of 74 active geodetic reference station, the main problem of surveyors in Romania is to calculate a precise quasigeoid model. The topic was addressed in several large studies, particularly doctoral thesis, noting that the absence of gravity measurements, gravimetric method was treated for only a limited area, namely the Bucharest zone. Therefore, this article presents a comparative study between the geometric and gravimetric determination methods for a quasigeoid model, test area being limited to Bucharest.

the geoid and the heights also need gravity values to be known and used in their determination. Normal height system consists of quasigeoid as a datum and "Normal heights", H N , measured along normal gravity plumb line. Gravity values are needed to convert observed height differences into normal height differences. More about these issues later. How is a (practical) height system realized in practice? There are two issues here: realization of the datum, i.e., the "practical datum" and transformation of observed (levelled) heights to "proper heights" (in a transformation that makes use of real gravity values).

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A new gravimetric geoid model, the KW-FLGM2021, is developed for Kuwait in this study. This new geoid model is driven by a combination of the XGM2019e-combined global geopotential model (GGM), terrestrial gravity, and the SRTM 3 global digital elevation model with a spatial resolution of three arc seconds. The KW-FLGM2021 has been computed by using the technique of Least Squares Collocation (LSC) with Remove-Compute-Restore (RCR) procedure. To evaluate the external accuracy of the KW-FLGM2021 gravimetric geoid model, GPS/leveling data were used. As a result of this evaluation, the residual of geoid heights obtained from the KW-FLGM2021 geoid model is 2.2 cm. The KW-FLGM2021 is possible to be recommended as the first accurate geoid model for Kuwait.

Satellite gradiometry is studied as a means to improve the geoid in local areas from a limited data coverage. Least-squares collocation is used for this purpose because it allows to combine heterogeneous data in a consistent way and to estimate the integrated effect of the attenuated spectrun~ In this way accuracy studies can be performed in a general and reliable manner. It is shown that only three second-order gradients contribute significantly to the estimation of the geoidal undulations and that it is suffia'ent to have gradiometer data in a 5 ~ x 5 ~ area around the estimation point. The accuracy of the geoid determination is strongly dependent on the degree and order of the reference field used. An accuracy of about + I m can be achieved with a reference field of (12, 12). There is an optimal satellite altitude for each reference field and this altitude may be higher than 300 km for a field of low degree and order. The influence of measuring errors is discussed and it is shown that only gradiometer data with accuracies better than + 0.05 E will give a sign~Fu~nt improvement of the geoM. Finally, some results on the combination of satellite gradiometry and terrestrial gravity measurements are giverL The proposed method seems to be well suited for local geoid determinations down to the meter range. It is especially interesting for unsurveyed and difficult areas because no terrestrial measurements are necesm,y. Furthermore, it has the practical advantage that only a local data coverage is needed.

The U.S. Navy satellite, GEOSAT (Figure 1) is third in the series of microwave altimeter satellites which make, or made, precise, direct observations of the marine geoid. GEOSAT, launched March 1985, is active today. Its two predecessors are NASA's Geos‐3 mission which operated from 1975 through 1978 and NASA's SEASAT mission which functioned only during the summer of 1978. Each of the three has provided highly significant, abundant, new information about the gravity field of the oceans. As such, these missions have begun a revolution in marine geodesy and geophysics. One exemplary result of this ongoing revolution is the global map of marine gravity constructed from SEASAT altimeter data by Haxby [1987]. In fact, members of the marine geophysical community were, and, in some cases, still are, largely unaware of the spate of gravity field information coming from these satellites. Our purpose here is to help assure that this same community is aware of the very major increase ...

Three different approaches for determining global gravity field solutions from GRACE satellites are presented and compared. Gravity field solutions - the so-called GRACE level-2 data - are mainly spherical harmonic expansions of the Earth's gravitational potential and are widely used by the geosciences community. Level-2 data are obtained via the functional modeling of GRACE level-1 data which are in principle the GRACE orbit, observed by GPS high-low and K-band low-low satellite-to-satellite tracking as well as on-board accelerometry. There are several approaches to connect the Earth's gravitational potential to the level-1 observations. In this research study we compare three different approaches using simulated GRACE level-1 data. The methods being considered here are the acceleration approach, the energy balance approach and the integral equation method. This work is part of the cooperation between Institut für Erdmessung (IfE) and Albert Einstein Institut (AEI) at Leibn...

Spherical harmonic expansion is a commonly applied mathematical representation of the earth's gravity eld. This representation is implied by the potential coe cients determined by using elements/parameters of the eld observed on the surface of the earth and/or in space outside the earth in the spherical harmonic expansion of the eld. International Centre for Gravity Earth Models (ICGEM) publishes, from time to time, Global Gravity Field Models (GGMs) that have been developed. These GGMs need evaluation with terrestrial data of di erent locations to ascertain their accuracy for application in those locations. In this study, Bouguer gravity anomalies derived from a total of eleven (11) recent GGMs, using sixty sample points, were evaluated by means of Root-Mean-Square difference and correlation coe cient. The Root-Mean-Square di erences of the computed Bouguer anomalies from ICGEM website compared to their positionally corresponding terrestrial Bouguer anomalies range from 9.530 mgal to 37.113 mgal. Additionally, the correlation coe cients of the structure of the signal of the terrestrial and GGM-derived Bouguer anomalies range from 0.480 to 0.879. It was observed that GECO derived Bouguer gravity anomalies have the best signal structure relationship with the terrestrial data than the other ten GGMs. We also discovered that EIGEN-6C4 and GECO derived Bouguer anomalies have enormous potential to be used as supplements to the terrestrial Bouguer anomalies for Enugu State, Nigeria.

The geodetic and geophysical applications of Earth Gravity Field parameters computed from Global Geopotential Models (GGMs) are quite on the increase despite the inherent commission and omission errors of these models. In view of this, this study focuses on refining and quantifying terrain-induced effects on Bouguer gravity anomalies computed directly from a total of seven recent GGMs. In the study, the Residual Terrain Model (RTM) technique was used to estimate the residual terrain effects that were added to the GGM-computed Bouguer gravity anomalies at the sixty test points in Enugu State, Nigeria. The computed residual terrain effects range from −24.6 to 37.5 mgal while the percentage of the omission errors of the GGMs based on their Root-Mean-Square (RMS) differences ranges from 7.8% to 44.7%. It can be concluded that GGM-refined Bouguer gravity anomalies are better in accuracy than the unrefined GGM-computed Bouguer gravity anomalies and hence there is need for accurate height information in the development of GGMs. We, therefore, recommend that refined Bouguer gravity anomalies obtained from HUST-Grace2016s, EIGEN-6C4 and GECO that gave best improvement amongst the seven GGMs under consideration should be used to supplement the available terrestrial Bouguer anomalies for geodetic and geophysical applications within the study area.

Spherical harmonic expansion is a commonly applied mathematical representation of the earth’s gravity field. This representation is implied by the potential coeffcients determined by using elements/parameters of the field observed on the surface of the earth and/or in space outside the earth in the spherical harmonic expansion of the field. International Centre for Gravity Earth Models (ICGEM) publishes, from time to time, Global Gravity Field Models (GGMs) that have been developed. These GGMs need evaluation with terrestrial data of different locations to ascertain their accuracy for application in those locations. In this study, Bouguer gravity anomalies derived from a total of eleven (11) recent GGMs, using sixty sample points, were evaluated by means of Root-Mean-Square difference and correlation coeficient. The Root-Mean-Square differences of the computed Bouguer anomalies from ICGEMwebsite compared to their positionally corresponding terrestrial Bouguer anomalies range from 9....

Here we perform the integration of all available data from the BNDG (Banco Nacional de Dados Gravimétricos) and the BDEP (Banco de Dados de Exploração e Produção da Agência Brasileira de Petróleo) to provide new free-air and Bouguer gravity anomaly maps for the Brazilian territory with newly acquired data over the years, mainly in regions with no data coverage in the past. Quality controls and subsequent gridding processes, in the same system of the whole dataset, are developed on the Oasis Montaj software (OM). Subsets of data from various gravity surveys are gridded and upward continued up to 3000 m to avoid high-frequency noise, allowing them to be gathered. We fill areas with no data coverage in the North Region of Brazil with gravity values from the XGM2019e geopotential model. To join the subset grids of ground and airborne surveys and the geopotential model, we use a collection of grids knitting methods from the OM. To verify the consistency of our grids, we compare them with previously derived gravity anomalies maps and geopotential models. Our new free-air and Bouguer gravity anomalies maps show more detailed short-wavelength geological structures than their predecessors. Therefore, these new gravity anomalies maps may be helpful for the development of recent tectonic, oil, and mining studies.

An important problem in the practical use of optimal spectral methods in gravity field modelling is the stationarity assumption for the data noise and the underlying unknown signals. Such a restriction is required, according to the standard Wiener-Kolmogorov estimation theory, in order to obtain signal approximation algorithms of simple convolution structure that can be evaluated very efficiently through fast Fourier transform (FFT) techniques. Often, the observation errors in the input data have significant spatial variations in their statistical behaviour, thus making the noise stationarity assumption unrealistic for many practical situations. Also, a stochastic interpretation of the true values of the various gravity field signals as random variables with similar statistical parameters is rather questionable, since they describe physical phenomena that are not random (probabilistic) and certainly do not have a uniform (stationary) behaviour over their domain. The aim of this pape...

Marine geoid modeling in the Atlantic coastal region of Argentina is problematic. Firstly, because of the insufficient amount of available shipborne gravity data, which renders a purely gravimetric solution not feasible. Secondly, because of the very strong ocean currents, which affect the quality of satellite altimetry data, so that a purely altimetric model is too noisy even after low-pass filtering the Sea Surface Heights (SSHs) to remove (part of) the influence of the oceanographic signals. Thus, the recommended solution is to employ a combination method and use all the available gravity and altimetry data together. This is a suitable solution since (i) combination methods such as least-squares collocation and Input Output System Theory (IOST) inherently low-pass filter and weigh the data, and (ii) will make use of the altimetric heights to fill the gaps of the shipborne gravity data. Following this idea, purely altimetric, gravimetric and combined (using the IOST method) marine geoid models have been estimated for Argentina employing all available shipborne gravity data, satellite altimetry SSHs and the latest Earth Gravity Models (EGMs) developed from the missions of CHAMP and GRACE. The new EGMs are especially useful to assess the quality of the new geoids, especially against EGM96, which was used in an older ERS1-only solution for the same area. From the comparison of the estimated geoid models with respect to stacked TOPEX/Poseidon SSHs, we found that the altimetric model provides the best agreement while the combined one improves the accuracy (1σ) of the gravimetric solution.

This study describes a methodology of recovery of the Earth's gravity field from CHAMP and GRACE satellites data in Pakistan using least squares collocation (LSC) based downward continuation technique. The CHAMP height anomalies and GRACE gravity disturbances derived from the observed satellite data have been used in combination solution using LSC with observed gravity values at the Earth surface. The combined covariance functions of height anomalies and/or gravity disturbances at satellite altitudes and observed gravity anomalies at Earth surface have been used as the basis for combination and downward continuation solution. The variance of predicted gravity anomalies from GRACE gravity disturbances is relatively lower than the corresponding results of gravity anomalies from CHAMP height anomalies. This fact may be attributed partly to the amplification of noise and partly to the unstable inverse transformation process of height anomalies to gravity anomalies. The impact of data error variance has been studied in the context of smoothing and noise reduction in the final solution of downward continuation using least squares collocation. The raising of data error suppresses the noise and as a result a smooth final solution is obtained. The prediction results appear to be dependent on the quality of data and goodness of combined covariance function, which are fairly comparable for the CHAMP and GRACE data. The recovered gravity field from satellite data appears to contribute mainly to medium and long wavelength parts of total gravity field spectrum. Due to flexibility of data handling in least squares collocation, this procedure is applicable to any observable of gravity field being at different altitudes and with different data spacing.

In this paper, we introduce the Micro-g LaCoste Turnkey Airborne Gravity System (TAGS) with Fugro’s improved gravity processing and geoid modeling software package for regional gravity field mapping and geoid determination. Three test areas with different topographic characteristics under the Gravity for the Redefinition of the AmericanVertical Datum (GRAV-D) project of theUSNOAA National Geodetic Surveys (NGS) were used for case studies and determine the available accuracy of the system. The preliminary results of all these test cases show that the system with Fugro’s improved gravity and geoid processing software package is able to achieve a comparable geoid mapping result to traditional terrestrial methods.

This study describes a methodology of recovery of the Earth's gravity field from CHAMP and GRACE satellites data in Pakistan using least squares collocation (LSC) based downward continuation technique. The CHAMP height anomalies and GRACE gravity disturbances derived from the observed satellite data have been used in combination solution using LSC with observed gravity values at the Earth surface. The combined covariance functions of height anomalies and/or gravity disturbances at satellite altitudes and observed gravity anomalies at Earth surface have been used as the basis for combination and downward continuation solution. The variance of predicted gravity anomalies from GRACE gravity disturbances is relatively lower than the corresponding results of gravity anomalies from CHAMP height anomalies. This fact may be attributed partly to the amplification of noise and partly to the unstable inverse transformation process of height anomalies to gravity anomalies. The impact of data error variance has been studied in the context of smoothing and noise reduction in the final solution of downward continuation using least squares collocation. The raising of data error suppresses the noise and as a result a smooth final solution is obtained. The prediction results appear to be dependent on the quality of data and goodness of combined covariance function, which are fairly comparable for the CHAMP and GRACE data. The recovered gravity field from satellite data appears to contribute mainly to medium and long wavelength parts of total gravity field spectrum. Due to flexibility of data handling in least squares collocation, this procedure is applicable to any observable of gravity field being at different altitudes and with different data spacing.

Santorini is located in the central part of the Hellenic Volcanic Arc (South Aegean Sea) and is well known for the Late-Bronze-Age "Minoan" eruption that might had been responsible for the decline of the great Minoan civilization on Crete island. Dionysos Satellite Observatory (DSO) of the National Technical University of Athens (NTUA) carried out terrestrial gravity measurements in December 2012 and in September 2014 at selected locations on Thira, Nea Kameni, Palea Kameni, Thirasia, Aspronisi and Christiana islands. Absolute gravity values were calculated using raw gravity data at every station to a combined dataset (2012-2014) and, consequently, complete Bouguer gravity anomaly maps were produced following the appropriate data corrections and reductions. The results were compared with gravity measurements that took place in July 1976 by DSO/NTUA and gravity variations at selected common locations were revealed. Marine gravity data that were collected during the PROTEUS project in November and December 2015 have been graded from University of Oregon and they fill the gravity dataset. An appropriate Digital Elevation Model (DEM) with topographic and bathymetric data was produced for the calculation of new, local scale, quasigeoid and geoid model in Santorini Volcanic Group using the residual terrain approach and the GRAVSOFT package (Forsberg, 1994; Tscherning et al., 1992). Reliability of the model was assessed with geoid heights from independent measurements such as GPS-leveling in known triangulation points.

This thesis covers the process taken to complete the Oman National Geoid Model (ONGM) project for the Sultanate of Oman. The steps taken to repair poor quality and badly referenced gravity data are explained. Each observation point was assigned a new orthometric height, its position was updated to geodetic coordinates, and its "observed" gravity was inversely calculated. The major biases that existed in the ground dataset were fixed using airborne free air anomalies at altitude. The ground data was merged with downward continued airborne gravity and the merged dataset was used to calculate the gravimetric geoid using the remove-computerestore (RCR) method. The remove step was completed using the residual terrain model modelling technique (RTM), a General Bathymetric Chart of the Oceans (GEBCO) model mixed with a 30" NASA Shuttle Radar Topography Mission (SRTM) digital terrain model (DTM), and an expansion of the EGM08 Global Geopotential Model (GGM) to degree and order 360. The "residual anomalies" were run through the Stokes integral with Wong Gore Tapering using the GRAVSOFT software package, the effects were restored to calculate the quasigeoid. The gravimetric geoid was computed by adding the − separation term to the quasigeoid and was fitted to the GPS-on-benchmarks provided by Oman. The external accuracy of the computed gravimetric geoid is 14 cm below mean sea level (MSL) with a standard deviation of ±30 cm.

The primary objective of the 1-cm geoid experiment in Colorado (USA) is to compare the numerous geoid computation methods used by different groups around the world. This is intended to lay the foundations for tuning computation methods to achieve the sought after 1-cm accuracy, and also evaluate how this accuracy may be robustly assessed. In this experiment, (quasi)geoid models were computed using the same input data provided by the US National Geodetic Survey (NGS), but using different methodologies. The rugged mountainous study area (730 km × 560 km) in Colorado was chosen so as to accentuate any differences between the methodologies, and to take advantage of newly collected GPS/leveling data of the Geoid Slope Validation Survey 2017 (GSVS17) which are now available to be used as an accurate and independent test dataset. Fourteen groups from fourteen countries submitted a gravimetric geoid and a quasigeoid model in a 1 ×1 grid for the study area, as well as geoid heights, height anomalies, and geopotential values at the 223 GSVS17 marks. This paper concentrates on the quasigeoid model comparison and evaluation, while the geopotential value investigations are presented as a separate paper (Sánchez et al. in J Geodesy 95(3):1.

This paper is written as a progression of the ongoing discussion in geodesy about the merits of the Molodensky height system versus the classical height system. It is a rebuttal of a publication in the Proceedings of the IX Hotine-Marussi Symposium on Mathematical Geodesy by Victor Popadyev titled “On the Advantage of Normal Heights: Once More on the Shape of Quasigeoid.” Even though Popadyev’s paper was not presented at the symposium it was published in the proceedings regardless. It purports to address a presentation from the symposium titled “The shape of the quasigeoid”, that applied a set of criteria to judge the suitability of the quasigeoid as a vertical reference surface, ultimately finding it inferior due to its edges and folds. The proceedings paper acknowledges these irregularities in the quasigeoid, but instead argues that the Molodensky system, apart from any vertical reference surface, should be evaluated on two different and more favorable criteria, and finds it super...

The aim of this paper is to show a present state-of-the-art precise gravimetric geoid determination using the UNB Stokes-Helmert’s technique in a simple schematic way. A detailed description of a practical application of this technique in the Auvergne test area is also provided. In this paper, we discuss the most problematic parts of the solution: correct application of topographic and atmospheric effects including the lateral topographical density variations, downward continuation of gravity anomalies from the Earth surface to the geoid, and the optimal incorporation of the global gravity field into the final geoid model. The final model is tested on 75 GNSS/levelling points supplied with normal Molodenskij heights, which for this investigation are transformed to rigorous orthometric heights. The standard deviation of the computed geoid model is 3.3 cm without applying any artificial improvement which is the same as that of the most accurate quasigeoid.

Santorini is located in the central part of the Hellenic Volcanic Arc (South Aegean Sea) and is well known for the Late-Bronze-Age "Minoan" eruption that might had been responsible for the decline of the great Minoan civilization on Crete island. Dionysos Satellite Observatory (DSO) of the National Technical University of Athens (NTUA) carried out terrestrial gravity measurements in December 2012 and in September 2014 at selected locations on Thira, Nea Kameni, Palea Kameni, Thirasia, Aspronisi and Christiana islands. Absolute gravity values were calculated using raw gravity data at every station to a combined dataset (2012-2014) and, consequently, complete Bouguer gravity anomaly maps were produced following the appropriate data corrections and reductions. The results were compared with gravity measurements that took place in July 1976 by DSO/NTUA and gravity variations at selected common locations were revealed. Marine gravity data that were collected during the PROTEUS project in November and December 2015 have been graded from University of Oregon and they fill the gravity dataset. An appropriate Digital Elevation Model (DEM) with topographic and bathymetric data was produced for the calculation of new, local scale, quasigeoid and geoid model in Santorini Volcanic Group using the residual terrain approach and the GRAVSOFT package (Forsberg, 1994; Tscherning et al., 1992). Reliability of the model was assessed with geoid heights from independent measurements such as GPS-leveling in known triangulation points.

Determination of heights with help of GPS in local geodetic networks is still more actual respecting the fact that the GPS technology becames more and more effective with hardware progress, with improvements in measuring and evaluating procedures, and with better modelling of the disturbing influences. In comparison with GPS the employment of classical terrestrial measuring technologies is often more difficult namely in broken mountain environment. In period 1998-2005 authors carried out repeated measurements of GPS baselines of various length and various height differences in local geodynamical network Sněžník and in other experimental areas. On ground of analyses of large GPS data sets the modified procedure for GPS observation was designed. The procedure is based on repetition of shorter static sessions separated by time intervals of optimal length. This technology represents an alternative to the ususal long static sessions, and is offering better effectivity of vertical GPS mea...

Marine gravity anomalies are essential data for determining coastal geoid, investigating tectonics and crustal structures, and offshore explorations. The objective of this study is to present a methodology for estimating marine gravity anomalies from CryoSat-2 and Saral/ AltiKa satellite altimeter data for the Gulf of Tonkin of Vietnam with a high-resolution 2 0 Â 2 0 grid. A total of 15,665 sea surface height (SSH) grid points, including derived from the Cryosat-2 (6842 grid points) and Saral/AltiKa (8823 grid points) satellite altimeter data were used. Then, the remove-restore technique and the crossover adjustment algorithm were used to remove the long-wavelength geoid height, the mean dynamic topography h MDT ð Þ, and time-varying sea-surface topography h t ð Þ. The residual geoid heights DN ð Þ were used to determine the residual gravity anomalies dg ð Þ using the Least-Squares Collocation method, whereas the Earth Geopotential Model was employed to restore the long-wavelength gravity anomalies Dg EIGEN ð Þ. GPS/leveling and tidal gauge of 31 tidal stations were used for assessing and choosing the best Earth Geopotential Model and Mean Dynamic Topography models for the study area (EIGEN6C4 and DTU15MDT models). The accuracy of the final marine gravity anomaly result was assessed using 56,978 marine gravity points, which were distributed in the study area. The result showed that the standard deviation between the satellite-derived gravity anomalies and checked points is ± 3.36 mGal, indicating good accuracy. After improving with ship-measured gravity anomalies, the accuracy of satellite-derived marine gravity anomaly improves to ± 2.63 mGal. The results of this research are useful for geodetic and geophysical applications in the region.

Requirement for a high resolution geoid model has increased substantially during the last few decades especially with the advent of Global Positioning System (GPS). Many countries across the world have already developed their own geoidal model which serve as the means of deriving orthometric heights from GNSS observations. On the other hand, the need for transformation of the ellipsoidal heights derived from GNSS observations to the physical orthometric heights has forced geodesists to determine the high precision local geoid heights. Besides, because orthometric heights are used in engineering and GIS applications, local quasi-geoid determination studies have become especially important. As it is well known, Turkey is situated on a tectonically very active region and earthquakes occurring in different times cause deformations in heights of levelling points of Turkish National Vertical Network. On the other hand, National Mapping Agency of Turkey (GDM: General Directorate of Mapping) which is responsible for the establishment and maintenance of national geodetic networks compute geoid models for whole Turkey and release them for public use in different periods. These GDM-computed geoid models are called "hybrid geoid" models. Hybrid geoid models are computed with the combination of gravimetric geoid models and geoid heights on the GPS_levelling points. And it is also known that the absolute accuracies of these geoid models are nearly at the 10 cm level or even worse. But, the requested accuracy is about better than 5 cm. Hence, in order to prove that the required accuracy levels can be achieved, a local gravimetric and quasi-geoid determination project using the resources of Selçuk University was initiated. Project area has been planned to cover a limited part of Central Anatolia including Konya. Necessary basic data (gravity, levelling etc.) for this project have been obtained in the field by performing observations/measurements. Some other necessary data such as absolute gravity values have been obtained from GDM. And as the next phase, high accuracy (better than 5 cm) local gravimetric and quasi-geoid models will be computed for the limited part of Central Anatolia using the above mentioned data. In order to be able to reach this goal suitable geoid computations softwares must be used. In this project, we will use DFHRS developed by the Karlsruhe Applied Sciences and GRAVSOFT softwares. According to the results to be obtained from this project, a more comprehensive project will be launched and the project area will be expanded. Thus, additional gravity, levelling and GNSS measurements will be carried out in the new project area for the determination of a high accuracy regional gravimetric geoid.

We improve the accuracy of global geoid models (GGMs) by utilizing deep convolutional neural networks (CNNs) and empirical rules. The geoid heights obtained from the global geopotential models (GGM), EGM2008, and XG2019, are validated using geometrical geoid heights obtained from GNSS/leveling data. We used AlexNet, LeNet, and VGG16 CNN models to investigate the improvement of geoid accuracy and found that VGG16 model shows the most signiĄcant improvement among all the models. The VGG16 model improves the geoid accuracy of EGM2008 and XGM2019e by about 39 cm. Additionally, we apply empirical rules for outliers detection to eliminate unreliable GNSS/leveling data and found that this further improves the geoid accuracy by 13 cm for EGM2008 and 12 cm for XGM2019e. We also used a 7-parameter stochastic model to remove existing systematic errors, impacting 9 mm on AlexNet and VGG16. Keywords geoid • CNNs • EGM2008 • XGM2019e • AlexNet • LeNet • VGG16 • outliers removal. 1 Introduction The geoid is considered a crucial geodetic reference surface since it represents a connection between different surfaces and height systems. The geoid connects the orthometric and ellipsoidal heights geometrically through a simple mathematical relationship (Heiskanen and Moritz, 1967). However, the computation of each of these heights has its complexity and challenges. For instance, the geoid calculation has different considerations regarding the M. Nasser * Corresponding author

One of the main challenges in physical geodesy today is achieving the 1-centimetre gravimetric quasi-geoid model, since a model of such accuracy could be used in the definition and realisation of the height reference frame. One of the main obstacles in this challenge is the lack of terrestrial gravity data or its uneven distribution. Therefore, the main question arises: what density of gravity points is necessary in order to obtain a gravimetric quasi-geoid model with an accuracy of 1 centimetre or even better? In this simulated study, the results show a trend of decreasing RMS related to the sparser dataset used in computation, leading to the conclusion that the determination of the sub-centimetre quasi-geoid model is no longer a theory, but can be achieved with the exact density of terrestrial gravity data: gravity observation stations have to be distributed no farther than 500 m from one another. When dealing with a very rough topography, the distance should be even shorter.

The geoid heights are needed for determining the orthometric heights from the Global Positioning System (GPS) ellipsoidal heights. There are several methods for geoid height determination. The paper presents a method employing the Artificial Neural Network (ANN) approximation together with the Least Squares Collocation (LSC). The surface obtained by the ANN approximation is used as a trend surface in the least squares collocation. In numerical examples four surfaces were compared: the global geopotential model (EGM96), the European gravimetric quasigeoid 1997 (EGG97), the surface approximated with minimum curvature splines in tension algorithm and the ANN surface approximation. The effectiveness of the ANN surface approximation depends on the number of control points. If the number of well distributed control points is sufficiently large, the results are better than those obtained by the minimum curvature algorithm and comparable to those obtained by the EGG97 model.

We analyzed the seismicity of central and western France, using historical data, a compilation of all recorded earthquakes from 1962 to 2002 (4574 events, relocated), and all published focal mechanisms (119 focal solutions). The aim is to understand what are the causes of earthquakes and stress accumulation in a slowly deforming intraplate region. The distribution of earthquakes and focal mechanisms is first correlated with recognized faults, geological structures and tomographic images. Then, in order to better understand the distribution of hypocenters and seek deeper crustal sources for stress accumulation, Euler solutions are computed from the available Bouguer anomaly data. The analysis of the obtained pattern for heat flow values, provides a better understanding of the concentration of seismicity in some particular zones. Two different behaviors of this slowly deforming intraplate region are evidenced. One is linked to the presence of a hot spot under the Massif Central, the other to reactivation of the Hercynian structural heritage. Our results highlight that several possible sources for earthquake clustering can be invoked in central-western France.

The use of GPS for the estimation of orthometric heights in a given region, with the help of existing levelling data requires the determination of a local geoid and the bias between the local levelling and the one implicitly defined when the geoid is calculated which is generally based on the gravity anomalies data. The heights of new data can be collected swiftly without using the orthometric heights from levelling; it is what one calls commonly levelling by GPS. In this framework, the Least Squares Collocation method (LSC) has been used to evaluate the quality of the available GPS-Levelling data, to determine a gravimetric geoid in the North region of Algeria and to estimate the constant datum bias. The data used in the setting of this study are: The geopotential model EGM96, a total number of 2534 gravity anomalies, as well as 43 GPS points connected to the geodetic network levelling present on the whole North part of Algerian.

Two techniques are commonly used to predict values of a random field u(t) from a vector of observations Y; one, mainly used in geodesy, is collocation, the other, mainly used in hydrology and geology, is kriging. Both techniques are based on the same optimization principle, that is minimizing the mean square prediction error, and use the same class of predictors, namely linear in Y. However, in collocation theory u(t) is assumed to have zero average and the main statistic to be used is covariance; once covariance is estimated, it is easy to switch from one functional to another, through covariance propagation. In kriging theory, on the contrary, the average of the field can be an arbitrary constant (or even a function in a suitable finite space) and the main statistic to be estimated is the variogram. Due to its shape, there is no simple rule for variogram propagation from one functional to another; for this reason kriging has mainly dealt with simple evaluations of u(t), or possibly with block averages. This paper treats the unification of the two techniques under a method called by the authors general kriging. After an introduction discussing some typical problems where such theories can be usefully applied, in Section 2 we formalize our proposal in general mathematical terms. Basically one could say that a proper functional propagation based on variograms is established. A first 1-D simulated example follows. In Section 3, our proposal is contrasted on a conceptual basis with cokriging, showing that this method solves a different problem, and step-wise collocation with parameters, recalling some conceptual drawbacks of this approach. In Section 4, numerical examples taken from the problem of estimating a local gravimetric geoid are presented. In such a problem, in fact, both hypotheses, non-zero average and change of functionals, occur and general kriging has proved to be able to deal with them. Finally a short discussion of the results and ongoing problems closes the paper.

In order to model the earth gravity field and its temporal variations, different gravity data with terrestrial, airborne and satellite gathered kinds are necessary. It is possible to recover by them the short, medium and long wavelengths of the gravity field respectively. Terrestrial gravity data, especially for the regions with highly variations, are useful for different purposes, i.e. to estimate

NOAA's National Geophysical Data Center (NGDC) has extensive holdings of airborne and marine magnetic data. Such data have been collected for more than half a century, providing global coverage of the Earth. Due to the changing main field from the Earth's core, and due to differences in quality and coverage, combining these data to a consistent global magnetic anomaly grid is challenging. A key ingredient is the long wavelength magnetic field observed by the low-orbiting CHAMP satellite. To produce a homogeneous grid, the marine and aeromagnetic trackline data are first line-leveled and then merged with the existing grids of continental-scale compilations by Least Squares Collocation. In the final processing step the short-to-intermediate wavelengths of the near-surface grid are merged with a CHAMP satellite magnetic anomaly model. In analogy to NGDC's 2-arcminute resolution ETOPO2 grid, we call our initial 3-arcminute magnetic anomaly grid EMAG3. The grid will be updated regularly. It is available in digital form and as plug-ins for NASA World Wind and Google Earth.

The 2011 Tohoku-Oki earthquake (Mw9.0) provides us the first opportunity to examine the responses of strain accumulation zones and active faults to megathrust earthquakes with dense permanent GPS network. In this presentation, we report the differences and/or similarities between pre, co, and post seismic crustal deformation of the Tohoku-Oki earthquake using GPS data in and around the Atotsugawa fault, located at the central part of Niigata-Kobe Tectonic Zone (NKTZ, Sagiya et al., 2000). We used daily coordinates obtained from the GPS stations operated by university group in addition to GEONET by Geospatial Information Authority (GSI). For the pre-seismic period, we estimated the velocity field by removing annual and semi-annual components from the daily coordinates. For the post-seismic deformation, we extracted the period from 25 November 2014 to 2 July 2016, and estimated the velocity field in the same manner. For the co-seismic displacement, we calculated the average coordinate...

Requirement for a high resolution geoid model has increased substantially during the last few decades especially with the advent of Global Positioning System (GPS). Many countries across the world have already developed their own geoidal model which serve as the means of deriving orthometric heights from GNSS observations. On the other hand, the need for transformation of the ellipsoidal heights derived from GNSS observations to the physical orthometric heights has forced geodesists to determine the high precision local geoid heights. Besides, because orthometric heights are used in engineering and GIS applications, local quasi-geoid determination studies have become especially important. As it is well known, Turkey is situated on a tectonically very active region and earthquakes occurring in different times cause deformations in heights of levelling points of Turkish National Vertical Network. On the other hand, National Mapping Agency of Turkey (GDM: General Directorate of Mapping) which is responsible for the establishment and maintenance of national geodetic networks compute geoid models for whole Turkey and release them for public use in different periods. These GDM-computed geoid models are called "hybrid geoid" models. Hybrid geoid models are computed with the combination of gravimetric geoid models and geoid heights on the GPS_levelling points. And it is also known that the absolute accuracies of these geoid models are nearly at the 10 cm level or even worse. But, the requested accuracy is about better than 5 cm. Hence, in order to prove that the required accuracy levels can be achieved, a local gravimetric and quasi-geoid determination project using the resources of Selçuk University was initiated. Project area has been planned to cover a limited part of Central Anatolia including Konya. Necessary basic data (gravity, levelling etc.) for this project have been obtained in the field by performing observations/measurements. Some other necessary data such as absolute gravity values have been obtained from GDM. And as the next phase, high accuracy (better than 5 cm) local gravimetric and quasi-geoid models will be computed for the limited part of Central Anatolia using the above mentioned data. In order to be able to reach this goal suitable geoid computations softwares must be used. In this project, we will use DFHRS developed by the Karlsruhe Applied Sciences and GRAVSOFT softwares. According to the results to be obtained from this project, a more comprehensive project will be launched and the project area will be expanded. Thus, additional gravity, levelling and GNSS measurements will be carried out in the new project area for the determination of a high accuracy regional gravimetric geoid.

We have evaluated temporal variation in strain rates in the Chubu and Kinki districts, central Japan, between 1997 and 2004 using continuous GPS data of GEONET. We found that strain rates throughout the southern part of Chubu district, including the Yoro fault and Nobi fault zone, have significantly deviated from the secular crustal deformation pattern since at least 2001. The results of an inversion procedure show that the strain rate change can be well explained by a slow slip beneath the Tokai area, suggesting that the Tokai slow event has significantly affected the crustal deformation not only in the Tokai area but also in an area located more than 100 km from the main slip area of the event. We also found that the seismicity rate has clearly decreased by approximately 50% around the Yoro fault since the latter half of 2000. The δCFF inferred from our Tokai slow slip model agrees with the observed reduction in seismicity. These results were interpreted as indicating that the sei...

The primary objective of the 1-cm geoid experiment in Colorado (USA) is to compare the numerous geoid computation methods used by different groups around the world. This is intended to lay the foundations for tuning computation methods to achieve the sought after 1-cm accuracy, and also evaluate how this accuracy may be robustly assessed. In this experiment, (quasi)geoid models were computed using the same input data provided by the US National Geodetic Survey (NGS), but using different methodologies. The rugged mountainous study area (730 km × 560 km) in Colorado was chosen so as to accentuate any differences between the methodologies, and to take advantage of newly collected GPS/leveling data of the Geoid Slope Validation Survey 2017 (GSVS17) which are now available to be used as an accurate and independent test dataset. Fourteen groups from fourteen countries submitted a gravimetric geoid and a quasigeoid model in a 1 ×1 grid for the study area, as well as geoid heights, height anomalies, and geopotential values at the 223 GSVS17 marks. This paper concentrates on the quasigeoid model comparison and evaluation, while the geopotential value investigations are presented as a separate paper (Sánchez et al. in J Geodesy 95(3):1.