An experiment for estimating Moho depth is carried out based on satellite altimetry and topographic information using the Vening Meinesz-Moritz gravimetric isostatic hypothesis. In order to investigate the possibility and quality of satellite altimetry in Moho determination, the DNSC08GRA global marine gravity field model and the DTM2006 global topography model are used to obtain a global Moho depth model over the oceans with a resolution of 1 degrees x 1 degrees. The numerical results show that the estimated Bouguer gravity disturbance varies from 86 to 767 mGal, with a global average of 747 mGal, and the estimated Moho depth varies from 3 to 39 km with a global average of 19 km. Comparing the Bouguer gravity disturbance estimated from satellite altimetry and that derived by the gravimetric satellite-only model GOGRA04S shows that the two models agree to 13 mGal in root mean square (RMS). Similarly, the estimated Moho depths from satellite altimetry and GOGRA04S agree to 0.69 km in RMS. It is also concluded that possible mean dynamic topography in the marine gravity model does not significantly affect the Moho determination.

We present a combined method for estimating a new global Moho model named KTH15C, containing Moho depth and Moho density contrast (or shortly Moho parameters), from a combination of global models of gravity (GOCO05S), topography (DTM2006) and seismic information (CRUST1.0 and MDN07) to a resolution of 1° × 1° based on a solution of Vening Meinesz-Moritz’ inverse problem of isostasy. This paper also aims modelling of the observation standard errors propagated from the Vening Meinesz-Moritz and CRUST1.0 models in estimating the uncertainty of the final Moho model. The numerical results yield Moho depths ranging from 6.5 to 70.3 km, and the estimated Moho density contrasts ranging from 21 to 650 kg/m³, respectively. Moreover, test computations display that in most areas estimated uncertainties in the parameters are less than 3 km and 50 kg/m³, respectively, but they reach to more significant values under Gulf of Mexico, Chile, Eastern Mediterranean, Timor sea and parts of polar regions. Comparing the Moho depths estimated by KTH15C and those derived by KTH11C, GEMMA2012C, CRUST1.0, KTH14C, CRUST14 and GEMMA1.0 models shows that KTH15C agree fairly well with CRUST1.0 but rather poor with other models. The Moho density contrasts estimated by KTH15C and those of the KTH11C, KTH14C and VMM model agree to 112, 31 and 61 kg/m³ in RMS. The regional numerical studies show that the RMS differences between KTH15C and Moho depths from seismic information yields fits of 2 to 4 km in South and North America, Africa, Europe, Asia, Australia and Antarctica, respectively.

According to Vening Meinesz-Moritz (VMM) global inverse isostatic problem, either the Moho density contrast (crust-mantle density contrast) or the Moho geometry can be estimated by solving a non-linear Fredholm integral equation of the first kind. Here solutions to the two Moho parameters are presented by combining the global geopotential model (GOCO-03S), topography (DTM2006) and a seismic crust model, the latter being the recent digital global crustal model (CRUST1.0) with a resolution of 1A(0)x1A(0). The numerical results show that the estimated Moho density contrast varies from 21 to 637 kg/m(3), with a global average of 321 kg/m(3), and the estimated Moho depth varies from 6 to 86 km with a global average of 24 km. Comparing the Moho density contrasts estimated using our leastsquares method and those derived by the CRUST1.0, CRUST2.0, and PREM models shows that our estimate agrees fairly well with CRUST1.0 model and rather poor with other models. The estimated Moho depths by our least-squares method and the CRUST1.0 model agree to 4.8 km in RMS and with the GEMMA1.0 based model to 6.3 km.

In this study, the Moho depth is estimated based on the refined spherical Bouguer gravity disturbance and DTM2006 topographic data using the Vening Meinesz-Moritz gravimetric-isostatic hypothesis. In this context, we compute the refined spherical Bouguer gravity disturbances in a set of 1° × 1° blocks. The spherical terrain correction, a residual correction to each Bouguer shell, is computed using rock heights and ice sheet thicknesses from the DTM2006 and Earth2014 models. The study illustrates that the defined simple Bouguer gravity disturbance corrected for the density variations of the oceans, ice sheets and sediment basins and also the non-isostatic effects needs a significant terrain correction to become the refined Bouguer gravity disturbance, and that the isostatic gravity disturbance is significantly better defined by the latter disturbance plus a compensation attraction. Our study shows that despite the fact that the lateral variation of the crustal depth is rather smooth, the terrain affects the result most significantly in many areas. The global numerical results show that the estimated Moho depths by the simple and refined spherical Bouguer gravity disturbances and the seismic CRUST1.0 model agree to 5.6 and 2.7 km in RMS, respectively. Also, the mean value differences are 1.7 and 0.2 km, respectively. Two regional numerical studies show that the RMS differences between the Moho depths estimated based on the simple and refined spherical Bouguer gravity disturbance and that using CRUST1.0 model yield fits of 4.9 and 3.2 km in South America and yield 3.2 and 3.4 km in Fennoscandia, respectively.

Multidecadal sea level variation in the Baltic Sea is investigated from 1900 to 2020 deploying satellite and in situ datasets. As a part of this investigation, nearly 30 years of satellite altimetry data are used to compare with tide gauge data in terms of linear trend. This, in turn, leads to validation of the regional uplift model developed for the Fennoscandia. The role of North Atlantic Oscillation (NAO) in multidecadal variations of the Baltic Sea is also analyzed. Although NAO impacts the Baltic Sea level on seasonal to decadal time scales according to previous studies, it is not a pronounced factor in the multidecadal variations. The acceleration in the sea level rise of the basin is reported as statistically insignificant in recent studies or even decelerating in an investigation of the early 1990s. It is shown that the reason for these results relates to the global warming hiatus in the 1950s−1970s, which can be seen in all eight tide gauges used for this study. To account for the slowdown period, the acceleration in the basin is investigated by fitting linear trends to time spans of six to seven decades, which include the hiatus. These results imply that the sea level rise is accelerated in the Baltic Sea during the period 1900–2020.

Sea level changes over time because of water mass exchange among the oceans and continents, ice sheets, and atmosphere. It fluctuates also due to variations of seawater salinity and temperature known as the steric contributor. GRACE-based Stokes coefficients provide a valuable source of information, about the water mass exchange as the main contributor to the Earth’s gravity field changes, within decadal scales. Moreover, measuring seawater temperature and salinity at different layers of ocean depth, Argo floats help to model the steric component of Global Mean Sea Level. In this study, we evaluate the Global Mean Sea Level (GMSL) budget closure using satellite altimetry, GRACE, and Argo products. Hereof, considering the most recent released GRACE monthly products (RL06), we examine an iterative remove-restore method to minimize the effect of artifact leaked large signal from ice sheets and land hydrology. In addition, the effect of errors and biases in geophysical model corrections, such as GIA, on the GMSL budget closure is estimated. Moreover, we quantify the influence of spatial and decorrelation filtering of GRACE data on the GMSL budget closure. In terms of the monthly fluctuations of sea level, our results confirm that closing the GMSL budget is highly dependent on the choice of the spatial averaging filter. In addition, comparing the trends and variations for both the global mean sea level time series and those estimated for mass and steric components, we find that spatial averaging functions play a significant role in the sea level budget closure.

Time-varying spherical harmonic coefficients determined from the Gravity Recovery and Climate Experiment (GRACE) data provide a valuable source of information about the water mass exchange that is the main contributor to the Earth’s gravity field changes within a period of less than several hundred years. Moreover, by measuring seawater temperature and salinity at different layers of ocean depth, Argo floats help to measure the steric component of global mean sea level (GMSL). In this study, we quantify the rate of barystatic sea-level change using both GRACE RL05 and RL06 monthly gravity field models and compare the results with estimates achieved from a GMSL budget closure approach. Our satellite altimetry-based results show a trend of 3.90 ± 0.14 mm yr−1 for the GMSL rise. About 35% or 1.29 ± 0.07 mm yr−1 of this rate is caused by the thermosteric contribution, while the remainder is mainly due to the barystatic contribution. Our results confirm that the choice of decorrelation filters does not play a significant role in quantifying the global barystatic sea-level change, and spatial filtering may not be needed. GRACE RL05 and RL06 solutions result in the barystatic sea-level change trends of 2.19 ± 0.13 mm yr−1 and 2.25 ± 0.16 mm yr−1, respectively. Accordingly, the residual trend, defined as the difference between the altimetry-derived GMSL and sum of the steric and barystatic components, amounts to 0.51 ± 0.51 and 0.45 ± 0.44 mm yr−1 for RL05 and RL06-based barystatic sea-level changes, respectively, over January 2005 to December 2016. The exclusion of the halosteric component results in a lower residual trend of about 0.36 ± 0.46 mm yr−1 over the same period, which suggests a sea-level budget closed within the uncertainty. This could be a confirmation on a high level of salinity bias particularly after about 2015. Moreover, considering the assumption that the GRACE-based barystatic component includes all mass change signals, the rather large residual trend could be attributed to an additional contribution from the deep ocean, where salinity and temperature cannot be monitored by the current observing systems. The errors from various sources, including the model-based Glacial Isostatic Adjustment signal, independent estimation of geocenter motion that are not quantified in the GRACE solutions, as well as the uncertainty of the second degree of zonal spherical harmonic coefficients, are other possible contributors to the residual trend.

The geoid, according to the classical Gauss–Listing definition, is, among infinite equipotential surfaces of the Earth’s gravity field, the equipotential surface that in a least squares sense best fits the undisturbed mean sea level. This equipotential surface, except for its zero-degree harmonic, can be characterized using the Earth’s global gravity models (GGM). Although, nowadays, satellite altimetry technique provides the absolute geoid height over oceans that can be used to calibrate the unknown zero-degree harmonic of the gravimetric geoid models, this technique cannot be utilized to estimate the geometric parameters of the mean Earth ellipsoid (MEE). The main objective of this study is to perform a joint estimation of W0, which defines the zero datum of vertical coordinates, and the MEE parameters relying on a new approach and on the newest gravity field, mean sea surface and mean dynamic topography models. As our approach utilizes both satellite altimetry observations and a GGM model, we consider different aspects of the input data to evaluate the sensitivity of our estimations to the input data. Unlike previous studies, our results show that it is not sufficient to use only the satellite-component of a quasi-stationary GGM to estimate W0. In addition, our results confirm a high sensitivity of the applied approach to the altimetry-based geoid heights, i.e., mean sea surface and mean dynamic topography models. Moreover, as W0 should be considered a quasi-stationary parameter, we quantify the effect of time-dependent Earth’s gravity field changes as well as the time-dependent sea level changes on the estimation of W0. Our computations resulted in the geoid potential W0 = 62636848.102 ± 0.004 m2 s−2 and the semi-major and minor axes of the MEE, a = 6378137.678 ± 0.0003 m and b = 6356752.964 ± 0.0005 m, which are 0.678 and 0.650 m larger than those axes of GRS80 reference ellipsoid, respectively. Moreover, a new estimation for the geocentric gravitational constant was obtained as GM = (398600460.55 ± 0.03) × 106 m3 s−2.

According to the classical Gauss–Listing definition, the geoid is the equipotential surface of the Earth’s gravity field that in a least-squares sense best fits the undisturbed mean sea level. This equipotential surface, except for its zero-degree harmonic, can be characterized using the Earth’s Global Gravity Models (GGM). Although nowadays, the satellite altimetry technique provides the absolute geoid height over oceans that can be used to calibrate the unknown zero-degree harmonic of the gravimetric geoid models, this technique cannot be utilized to estimate the geometric parameters of the Mean Earth Ellipsoid (MEE). In this study, we perform joint estimation of W0, which defines the zero datum of vertical coordinates, and the MEE parameters relying on a new approach and on the newest gravity field, mean sea surface, and mean dynamic topography models. As our approach utilizes both satellite altimetry observations and a GGM model, we consider different aspects of the input data to evaluate the sensitivity of our estimations to the input data. Unlike previous studies, our results show that it is not sufficient to use only the satellite componentof a quasi-stationary GGM to estimate W0. In addition, our results confirm a high sensitivity of the applied approach to the altimetry-based geoid heights, i.e. mean sea surface and mean dynamic topography models. Moreover, as W0 should be considered a quasi-stationary parameter, we quantify the effect of time-dependent Earth’s gravity field changes as well as the time-dependent sea-level changes on the estimation of W0. Our computations resulted in the geoid potential W0 = 62636848.102 ± 0.004 m2s-2 and the semi-major and –minor axes of the MEE,a = 6378137.678 ± 0.0003 m and b = 6356752.964 ± 0.0005 m, which are 0.678 and 0.650 m larger than those axes of the GRS80 reference ellipsoid, respectively. Moreover, a new estimation for the geocentric gravitational constant was obtained as GM = (398600460.55 ± 0.03) × 106 m3s-2.

The utilization of remote sensing observations to monitor essential climate variables (ECVs) has become increasingly important in studying their regional and global impacts, as defined by the Global Climate Observing System (GCOS). Understanding the Earth’s surface conditions, including soil moisture runoff, snow, temperature, precipitation, water vapour, radiation, groundwater and sea surface height (SSH), can positively impact the environment and ecosystems. Here, the authors present an overview of how global navigation satellite systems (GNSS) can be employed for environmental monitoring, with a particular focus on sea surface height monitoring. This includes examination of the advantages and disadvantages of utilizing a network of permanent GNSS stations for monitoring sea level rise along shorelines.

Human-made infrastructure, such as dams, bridges, tunnels, and high towers, requires highly precise geodetic control networks and continuous monitoring to detect potential failure risks and plan civil engineering maintenance works. In classical 2D geodetic networks, reducing slope distances to horizontal ones is an important task for engineers. The common practice for this reduction involves using vertical angles and applying trigonometric rules. However, using vertical angles introduces systematic errors, primarily due to air refraction, deflections of the vertical (DOV), and the geometric effects of the reference surface, whether it is a sphere or an ellipsoid. Therefore, employing vertical angles in establishing geodetic control networks in 2D is challenging due to these systematic errors. To mitigate the refraction and DOV effects, reciprocal observations of vertical angles can be considered, especially if the elevation differences are small. In this study, we quantify these effects and propose an innovative solution to eliminate these systematic errors in small-scale geodetic networks. Specifically, we propose a new technique that does not rely on vertical angles for the reduction of distances, which is called the network-aided method. Thus, the geometric, physical, and refraction effects cancel out in this method. The results of this study hold significant importance for surveying guidelines. The main advantage of the proposed method is less fieldwork and, hence cost reduction since there is no need for different OFF-construction (reference) and ON-construction (monitoring) networks. Consequently, the number of network points will be less than in traditional networks. There is no need for reciprocal observations since vertical angles are not utilized, while the precision remains equal or even superior (in terms of quality factors i.e., higher redundancy numbers and smaller error ellipses).

This study aims to investigate the ability of different least squares adjustment techniques for detecting deformation. A simulated geodetic netwo rk is used for this purpose. The observations are collected using the Total Station instrument in three epochs and different least squares adjustment methods are used to analyze the simulated network. The applied methods are adjustment-byelement, using variance-covariance components and Tikhonov regularization. For numerical computation, we utilized exist geodetic network around the simulated network and the deformation (changes in the simulated network) imposes to the object using a simulator in each epoch. The obtained results demonstrate that more accurate outcome for detection of small deformation is possible by estimating variance-covariance components. The difference of the estimated and the simulated deformations in the best scenario, i.e., applying variance-covariance components, is 0.2 and 0.1 mm in x and y directions. In comparison with adjustment by element and Tikhonov regularization methods the differences are 1.1 and 0.1 in x direction and 1.4 and 1.1 mm in y direction, respectively. In addition, it is also possible to model the deformation and therefore it can be seen that how the calculated displacement will affect the result of deformation modelling. It has been demonstrated that determining reasonable variance-covariance components is very important to estimate realistic deformation model and monitoring the geodetic networks. 

Most human-made infrastructures require regular deformation monitoring to detect failure risks and plan maintenance works. Continuous health monitoring is crucial for assessing infrastructure stability and plays a key role in mitigating damages and disasters within various environmental and engineering contexts. Structural deformation monitoring methods can be divided into two methods: geodetic and non-geodetic. Geodetic techniques enable the detection of displacements with respect to an external geodetic reference system, while non-geodetic methods can detect relative, internal changes within the monitored object. Both methods will be covered in this lecture note. In addition, after presenting the theoretical background and principle of the least-squares approach in Chapter 1, the necessary recommendations and guidelines for deformation monitoring using geodetic and non-geodetic methods will be provided.

The aim of this lecture note is to provide a theoretical background in the field of deformation monitoring, specifically using geodetic methods, for engineers, students, and researchers. One of the motivations behind this effort is the lack of references that adequately present the methods and recommendations for this purpose.

Airborne mobile mapping is one of the most important data acquisitionmethods for producing topographical maps and extracting terrain featuresfrom aerial images. The interest in 3D geospatial data is expanding, andtechnology is growing at an unprecedented speed with new digitalcamera mapping systems. Different sensors are used for data acquisitionin modern airborne photogrammetry. GNSS/INS (Inertial NavigationSystems) applications are developing, especially for direct georeferencingin airborne photogrammetry. Achieving accurately georeferencedproducts from the integration of GNSS and INS requires removingexisting systematic errors/bias, due to different reference systems, in themobile mapping systems. The collected data should refer to the samereference system; otherwise, it can impose a systematic shift in theresults. In this presentation, we assess the impact of the deflection ofverticals (i.e. the angle between the plumb line and normal to thereference ellipsoid) on the obtained horizontal and vertical coordinates.

The Earth's mass redistribution due to deglaciation and recent ice sheet melting causes changes in the Earth's gravity field and vertical land motion in Greenland. The changes are because of ongoing mass redistribution and related elastic (on a short time scale) and viscoelastic (on time scales of a few thousands of years) responses. These signatures can be used to determine the mantle viscosity. In this study, we infer the mantle viscosity associated with the glacial isostatic adjustment (GIA) and long-wavelength geoid beneath the Greenland lithosphere. The viscosity is determined based on a spatio-spectral analysis of the Earth's gravity field and the land uplift rate in order to find the GIA-related gravity field. We used different land uplift data, that is, the vertical land motions obtained by the Greenland Global Positioning System (GPS) Network (GNET), gravity recovery and climate experiment (GRACE) and glacial isostatic adjustment (GIA) data, and also combined them using the Kalman filtering technique. Using different land uplift rates, one can obtain different GIA-related gravity fields. As shown in this study, the mantle viscosities of 1.9 × 1022 Pa s and 7.8 × 1021 Pa s for a depth of 200–700 km are obtained using ICE-6G (VM5a) model and the combined land uplift model, respectively, and the GIA-related gravity potential signal

The Earth’s mass redistribution due to deglaciation and recent ice sheet melting causes changes in the Earth’s gravity field and vertical land motion in Greenland. The changes are because of ongoing mass redistribution and related elastic (on a short time scale) and viscoelastic (on time scales of a few thousands of years) responses. These signatures can be used to determine the mantle viscosity. In this study, we infer the mantle viscosity associated with the glacial isostatic adjustment (GIA) and long-wavelength geoid beneath the Greenland lithosphere. The viscosity is determined based on a spatio-spectral analysis of the Earth’s gravity field and the land uplift rate in order to find the GIA-related gravity field. We used and evaluated different land uplift data, i.e. the vertical land motions obtained by the Greenland Global Positioning System (GPS) Network (GNET), GRACE and Glacial Isostatic Adjustment (GIA) data. In addition, a  combined land uplift rate using the Kalman filtering technique is presented in this study. We extract the GIA-related gravity signals by filtering the other effects due to the deeper masses i.e. core-mantle (related to long-wavelengths) and topography (related to short-wavelengths). To do this, we applied correlation analysis to detect the best harmonic window. Finally, the mantle viscosity using the obtained GIA-related gravity field is estimated. Using different land uplift rates, one can obtain different GIA-related gravity fields. For example, different harmonic windows were obtained by employing different land uplift datasets, e.g. the truncated geoid model with a harmonic window between degrees 10 to 39 and 10 to 25 showed a maximum correlation with the GIA model ICE-6G (VM5a) and the combined land uplift rates, respectively. As shown in this study, the mantle viscosities of 1.6×1022 Pa s and 0.9×1022 Pa s for a depth of 200  to 650  km are obtained using ICE-6G (VM5a) model and the combined land uplift model, respectively, and the GIA-related gravity potential signal.

Gravimetric methods applied for Moho recovery in areas with sparse and irregular distribution of seismic data often assume only a constant crustal density. Results of latest studies, however, indicate that corrections for crustal density heterogeneities could improve the gravimetric result, especially in regions with a complex geologic/tectonic structure. Moreover, the isostatic mass balance reflects also the density structure within the lithosphere. The gravimetric methods should therefore incorporate an additional correction for the lithospheric mantle as well as deeper mantle density heterogeneities. Following this principle, we solve the Vening Meinesz-Moritz (VMM) inverse problem of isostasy constrained by seismic data to determine the Moho depth of the South American tectonic plate including surrounding oceans, while taking into consideration the crustal and mantle density heterogeneities. Our numerical result confirms that contribution of sediments significantly modifies the estimation of the Moho geometry especially along the continental margins with large sediment deposits. To account for the mantle density heterogeneities we develop and apply a method in order to correct the Moho geometry for the contribution of the lithospheric thermal state (i.e., the lithospheric thermal-pressure correction). In addition, the misfit between the isostatic and seismic Moho models, attributed mainly to deep mantle density heterogeneities and other geophysical phenomena, is corrected for by applying the non-isostatic correction. The results reveal that the application of the lithospheric thermal-pressure correction improves the RMS fit of the VMM gravimetric Moho solution to the CRUST1.0 (improves ∼ 1.9 km) and GEMMA (∼1.1 km) models and the point-wise seismic data (∼0.7 km) in South America.

In this study, we develop three estimators to optimally combine seismic and gravimetric models of Moho surface. The first estimator combines them by their special harmonic coefficients; the second one uses the spherical harmonic coefficients of the seismic model and use integral formula for the gravimetric one. The kernel of the integral terms of this estimator shows that a cap size of 20◦ is required for the integration, but since this integral is presented to combine the low frequencies of the gravimetric model, a low resolution model is enough for the integration. The third estimator uses the gravity anomaly and converts its low frequencies to those of the gravimetric Moho model, meanwhile combining them with those of seismic one. This integral requires an integration domain of 30◦ for the gravity anomalies but since the maximum degree of this kernel is limited to a specific degree, the use of its spectral form is recommended. The kernel of the integral involving the gravity anomalies, developed for recovering high frequencies of Moho, is written in a closed-from formula and its singularity is investigated. This kernel is well-behaving and decreases fast, meaning that it is suitable for recovering the high frequencies of Moho surface.

One of the GOCE satellite mission goals is to study the Earth's interior structure including its crustal thickness. A gravimetric-isostatic Moho model, based on the Vening Meinesz-Moritz (VMM) theory and GOCE gradiometric data, is determined beneath Iran's continental shelf and surrounding seas. The terrestrial gravimetric data of Iran are also used in a nonlinear inversion for a recovering-Moho model applying the VMM model. The newly-computed Moho models are compared with the Moho data taken from CRUST2.0. The root-mean-square (RMS) of differences between the CRUST2.0 Moho model and the recovered model from GOCE and that from the terrestrial gravimetric data are 3.8 km and 4.6 km, respectively.

Remote sensing observations of Essential Climate Variables (ECVs) provide a means of studying their global and regional impacts. Coastal GNSS stations measure water levels using GNSS Reflectometry (GNSS-R) technique by determining the vertical distance between the antenna and the water surface. In this study, GNSS-R data from four stations over three months were used to estimate sea surface heights (SSH) and assess accuracy using nearest tide gauge observations. Results showed that GNSS signals from GPS, GLONASS, Galileo, and BeiDou were accurate for the SSH estimation. In addition, 145 significant tidal frequencies were extracted using the GNSS-R and tide gauge time series by employing the Least Square Harmonic Estimation (LS-HE) approach. The study demonstrates the usefulness of GNSS-R for tide studies and its potential use alongside tide gauge measurements in coastal locations.

The principle of isostasy plays an important role to understand the relation between different geodynamic processes. Although, it is difficult to find an exact method that delivers a complete image of the Earth structure. However, gravimetric methods are alternative to provide images of the interior of the Earth. The Earth’s crust parameters, i.e. crustal depth and crust-mantle density contrast, can reveal adequate information about the solid Earth system such as volcanic activity, earthquake and continental rifting. Hence, in this study, a combine Moho model using seismic and gravity data is determined to investigate the relationship between the isostatic state of the lithosphere and seismic activities in East Africa. Our results show that isostatic equilibrium and compensation states are closely correlated to the seismicity patterns in the study area. For example, several studies suggest that African superplume causes the rift valley, and consequently differences in crustal and mantle densities occur. This paper presents a method to determine the crustal thickness and crust-mantle density contrast and consequently one can observe low-density contrast (about 200 kg/m3 ) and thin crust (about 30 km) near the triple junction plate tectonics in East Africa (Afar Triangle), which confirms the state of overcompensation in the rift valley areas. Furthermore, the density structure of the lithosphere shows a large correlation with the earthquake activity, sub-crustal stress and volcanic distribution across East Africa.

There is an exceptional opportunity of achieving simultaneous and complementary data from a multitude of geoscience and environmental near-earth orbiting artificial satellites to study phenomena related to the climate change e.g. sea level change, ice melting, soil moisture variation, temperature changes, and earth surface deformations. In this study, we focus on permafrost thawing and its associated gravity change, and organic material changes using GRACE data and other satellite- and ground-based observations. The estimation of permafrost changes requires combining information from various sources, particularly using the gravity field change, surface temperature change, and GIA. The most significant factor for careful monitoring of the permafrost thawing is the fact that this process could be responsible for releasing an additional enormous amount of greenhouse gases emitted to the atmosphere, most importantly to mention Carbone dioxide and Methane that are currently stored in the frozen ground. The results of a preliminary numerical analysis reveal a possible existence of a high correlation between the secular trends of greenhouse gases, temperature and equivalent water thickness in the selected regions. Furthermore, according to our estimates based on processing the GRACE data, the groundwater storage attributed to the due to permafrost thawing increased at the annual rates of 3.4, 3.8, 4.4 and 4.0 cm, in Siberia, northern Alaska, and Canada. Despite a rather preliminary character of our results, these findings indicate that the methodology developed and applied in this study should be improved by incorporating the in situ permafrost measurements.

GNSS/INS applications are being developed, especially for direct georeferencing in airborne photogrammetry. Achieving accurately georeferenced products from the integration of GNSS and INS requires removing systematic errors in the mobile mapping systems. The INS sensor's uncertainty is decreasing; therefore, the influence of the deflection of verticals (DOV, the angle between the plumb line and normal to the ellipsoid) should be considered in the direct georeferencing. Otherwise, an error is imposed for calculating the exterior orientation parameters of the aerial images and aerial laser scanning. This study determines the DOV using the EGM2008 model and gravity data in Sweden. The impact of the DOVs on horizontal and vertical coordinates, considering different flight altitudes and camera field of view, is assessed. The results confirm that the calculated DOV components using the EGM2008 model are sufficiently accurate for aerial mapping system purposes except for mountainous areas because the topographic signal is not modelled correctly.

Direct georeferencing of airborne mobile mapping systems is developing with unprecedented speed using GNSS/INSintegration. Removal of systematic errors is required for achieving a high accurate georeferenced product in mobile mappingplatforms with integrated GNSS/INS sensors. It is crucial to consider the deflection of verticals (DOV) in direct georeferencing dueto the recently improved INS sensor accuracy. This study determines the DOV using Sweden’s EGM2008 model and gravity data.The influence of the DOVs on horizontal and vertical coordinates and considering different flight heights is assessed. The resultsconfirm that the calculated DOV components using the EGM2008 model are sufficiently accurate for aerial photogrammetrypurposes except for the mountainous areas because the topographic signal is not modeled correctly.

What are the key challenges in establishing precise geodetic control networks? This is one of the most important tasks of geodesists and land surveyors, since geodetic control networks are essential for the deformation and environmental monitoring of dams, tunnels, high towers, landslides and bridges, among others. This article discusses the main challenges relating to vertical angles and provides some recommendations for how they can be overcome.

Determination of the Earth’s gravity field and geopotential value is one of the fundamental topics in physical geodesy. Traditional terrestrial gravity and precise leveling measurements can be used to determine the geopotential values at a local or regional scale. However, recent developments in optical atomic clocks have not only rapidly improved fundamental science but also contributed to applied research. The latest generation of optical clocks is approaching the accuracy level of 10−18 when facilitating atomic clock networks. These systems allow examining fundamental theories and many research applications, such as atomic clocks applications in relativistic geodesy, to precisely determine the Earth’s gravity field parameters (e.g., geopotential values). According to the theory of relativistic geodesy, the frequency difference measured by an optical clock network is related to the gravity potential anomaly, provided that the effects of disturbing signals (i.e., tidal and non-tidal contributions) are filtered out. The relativistic geodesy principle could be used for a practical realization of global geodetic infrastructure, most importantly, a vertical datum unification or realization of height systems. This paper aims to review the background of relativistic (clock-based) geodesy and study the variations of optical atomic clock measurements (e.g., due to hydrology loading and land motion).

In classical two-dimensional (2D) geodetic networks, reducing slope distances to horizontal ones is an important task for engineers. These horizontal distances along with horizontal directions are used in 2D geodetic adjustment. The common practice for this reduction is the use of vertical angles to reduce distances using trigonometric rules. However, one faces systematic effects when using vertical angles. These effects are mainly due to refraction, deflection of the vertical (DOV), and the geometric effect of the reference surface (sphere or ellipsoid). To mitigate refraction and DOV effects, one can choose to observe the vertical angles reciprocally if the baseline points’ elevation difference is small. This paper quantifies these effects and proposes a proper solution to eliminate the effects in small-scale geodetic networks (where the longest distances are less than 5 km). The goal is to calculate slope distances into horizontal ones appropriately. For this purpose, we used the SWEN17_RH2000 quasigeoid model (in Sweden) to study the impact of the DOV applying different baseline lengths, azimuths, and vertical angles. Finally, we propose an approach to study the impact of the geometric effect on vertical angles. We illustrate that the DOV and the geometric effects on vertical angles measured reciprocally are significant if the height difference of the start point and endpoint in the baseline is large. Geometric correction should be considered for the measured vertical angles to calculate horizontal distances correctly if the network points are not on the same elevation, even if the vertical angles are measured reciprocally. 

Vertical land motion and the redistribution of masses within and on the surface of the Earth affect the Earth’s gravity field. Hence, studying the ratio between temporal changes of the surface gravity g˙ and height (h˙) is important in geoscience, e.g., for reduction of gravity observations, assessing satellite gravimetry missions, and tuning vertical land motion models. Sjöberg and Bagherbandi (2020) estimated a combined ratio of g˙/h˙ in Fennoscandia based on relative gravity observations along the 63 degree gravity line running from Vågstranda in Norway to Joensuu in Finland, 688 absolute gravity observations observed at 59 stations over Fennoscandia, monthly gravity data derived from the GRACE satellite mission between January 2003 and August 2016, as well as a land uplift model. The weighted least-squares solution of all these data was g˙/h˙ = − 0.166 ± 0.011 μGal/mm, which corresponds to an upper mantle density of about 3402 ± 95 kg/m3. The present note includes additional GRACE data to June 2017 and GRACE Follow-on data from June 2018 to November 2023. The resulting weighted least-squares solution for all data is g˙/h˙ = − 0.160 ± 0.011 μGal/mm, yielding an upper mantle density of about 3546 ± 71 kg/m3. The outcomes show the importance of satellite gravimetry data in Glacial Isostatic Adjustment (GIA) modeling and other parameters such as land uplift rate. Utilizing a longer time span of GRACE and GRACE Follow-on data allows us to capture fine variations and trends in the gravity-to-height ratio with better precision. This will be useful for constraining and adjusting GIA models and refining gravity observations. 

The Earth's gravity field is related to the topographic potential in medium and higher degrees, which is isostatically compensated. Hence, the topographic-isostatic (TI) data are indispensable for extending an available Earth Gravitational Model (EGM) to higher degrees. Here we use TI harmonic coefficients to construct a Synthetic Earth Gravitational Model (SEGM) to extend the EGMs to higher degrees. To achieve a high-quality SEGM, a global geopotential model (EGM96) is used to describe the low degrees, whereas the medium and high degrees are obtained from the TI or topographic potential. This study differes from others in that it uses a new gravimetric-isostatic model for determining the TI potential. We test different alternatives based on TI or only topographic data to determine the SEGM. Although the topography is isostatically compensated only to about degree 40-60, our study shows that using a compensation model improves the SEGM in comparison with using only topographic data for higher degree harmonics. This is because the TI data better adjust the applied Butterworth filter, which bridges the known EGM and the new high-degree potential field than the topographic data alone.

The principle of isostasy is important in different fields of geosciences. Using an isostatic hypothesis for estimating the crustal thickness suffers from the more or less incomplete isostatic model and that the observed gravity anomaly is not only generated by the topographic/isostatic signal but also by non-isostatic effects (NIEs). In most applications of isostatic models the NIEs are disregarded. In this paper, we study how some isostatic models related with Vening Meinez's isostatic hypothesis can be improved by considering the NIE. The isostatic gravity anomaly needs a correction for the NIEs, which varies from as much as 494 mGal to − 308 mGal. The result shows that by adding this correction the global crustal thickness estimate improves about 50% with respect to the global model CRUST2.0, i.e. the root mean square differences of the crustal thickness of the best Vening Meinesz type and CRUST2.0 models are 6.9 and 3.2 km before and after improvement, respectively. As a result, a new global model of crustal thickness using Vening Meinesz and CRUST2.0 models is generated. A comparison with an independent African crustal depth model shows an improvement of the new model by 6.8 km vs. CRUST2.0 (i.e. rms differences of 3.0 and 9.8 km, respectively). A comparison between oceanic lithosphere age and the NIEs is discussed in this study, too. One application of this study can be to improve crustal depth in areas where CRUST2.0 data are sparse and bad and to densify the resolution vs. the CRUST2.0 model. Other applications can be used to infer the viscosity of the mantle from the NIEs signal to study various locations around the Earth for understanding complete, over- and under-compensations of the topography.

The crustal thickness (Moho depth) is of interest in several geosciences applications, such as geography, geophysics and geodesy. Usually the crustal depth and density variations are estimated by seismic survey data. As such data collection is very time-consuming and expensive an attractive option could be to use a gravimetric/isostatic model. In this case, realistic estimates for the crustal density and Moho density contrast (MDC) are important. In this study, we first use the seismic crustal thickness of CRUST2.0 model as a known parameter in combination with gravimetric data in estimating the crust–mantle density contrast by the isostatic model of Vening Meinesz–Moritz. We present different models to estimate the MDC and its impact on the modelling of the gravimetric–isostatic Moho depth. The theory is applied to estimate the Moho depth of the African continental crust by using different models for the MDC: (a) constant value (0.6 g/cm3), (b) Pratt–Hayford’s model, (c) CRUST2.0 as input to three gravimetric/isostatic models based on Vening Meinesz–Moritz theory. The isostatic models agree by 5.8–7.1 km in the rms with the regional seismic model at a resolution of 2 x2, and the smallest rms difference at a resolution of 1x1is of

7.2 km. For comparison, the rms differences of CRUST2.0 and the regional seismic model are 8.8 and 9.1 km at the resolutions of 2 deg (interpolated) and 1 deg respectively. The result suggests that the gravimetric/isostatic Moho model can be used in densification of the CRUST2.0 Moho geometry, and to improve it in areas with poor data.

Sustainable development and digitalization need reliable data. Geospatial data becomes a more and more important tool in society for many kinds of research of immediate use, but also for future planning and enterprise. Harmonization of geodata is very important for data producers and organizations, e.g. for mapping agencies. Establishing a uniform horizontal/vertical reference system is a basic prerequisite for combining data from different sources, and for allowing cross-border presentations and analyzes. If we do not use the same reference for positioning, it is not certain that one can compose reliable geodata from different organizations.

The overall aim of this study is to provide a theoretical and practical solution to unifying height systems in order to overcome systematic datum inconsistencies in height data and digital terrain models. The study deals with a variety of issues in physical geodesy such as Earth’s gravity field, sea level rise, sea surface topography and GNSS data. The advent of satellite altimetry in the 1970s provided a tool for the realization of a global vertical datum as being the equipotential surface of the Earth’s gravity field that minimizes the sea-surface topography (SST) all over the oceans in a least-squares sense. This leads to a direct determination of the geoid potential (W₀) from satellite altimetry and an Earth Gravitational Model (EGM).  In contrast, here we will first determine the Mean Earth Ellipsoid parameters and from these follows W₀. This means that once the size of the axes of the globally best-fitting ellipsoid is determined, W₀ follows. A major problem with this method is that satellite altimetry is only successful over the oceans, but the method requires global data. This problem is solved by employing satellite altimetry and the EGM in a practical combination.  

In this paper a new gravimetric-isostatic crustal thickness model (VMM14_FEN) is estimated for Fennoscandia. The main motivation is to investigate the relations between geological and geophysical properties, the Moho depth and crust-mantle density contrast at the crust-mantle discontinuity. For this purpose the Bouguer gravity disturbance data is corrected in two main ways namely for the gravitational contributions of mass density variation due to the different layers of the Earth's crust such as ice and sediments, as well as for the gravitational contribution from deeper masses below the crust. This second correction (for non-isostatic effects) is necessary because in general the crust is not in complete isostatic equilibrium and the observed gravity data are not only generated by the topographic/isostatic masses but also from those in the deep Earth interior. The correction for non-isostatic effects is mainly attributed to unmodeled mantle and core boundary density heterogeneities. These corrections are determined using the recent seismic crustal thickness model CRUST1.0. We compare our modeling results with previous studies in the area and test the fitness. The comparison with the external Moho model EuCRUST-07 shows a 3.3. km RMS agreement for the Moho depth in Fennoscandia. We also illustrate how the above corrections improve the Moho depth estimation. Finally, the signatures of geological structures and isostatic equilibrium are studied using VMM14_FEN, showing how main geological unit structures attribute in isostatic balance by affecting the Moho geometry. The main geological features are also discussed in the context of the complete and incomplete isostatic equilibrium. 

We compare three different numerical schemes of treating the Moho density contrast in gravimetric inverse problems for finding the Moho depths. The results are validated using the global crustal model CRUST2.0, which is determined based purely on seismic data. Firstly, the gravimetric recovery of the Moho depths is realized by solving Moritz’s generalization of the Vening-Meinesz inverse problem of isostasy while the constant Moho density contrast is adopted. The Pratt-Hayford isostatic model is then facilitated to estimate the variable Moho density contrast. This variable Moho density contrast is subsequently used to determine the Moho depths. Finally, the combined least-squares approach is applied to estimate jointly the Moho depths and density contract based on a priori error model. The EGM2008 global gravity model and the DTM2006.0 global topographic/bathymetric model are used to generate the isostatic gravity anomalies. The comparison of numerical results reveals that the optimal isostatic inverse scheme should take into consideration both the variable depth and density of compensation. This is achieved by applying the combined least-squares approach for a simultaneous estimation of both Moho parameters. We demonstrate that the result obtained using this method has the best agreement with the CRUST2.0 Moho depths. The numerical experiments are conducted at the regional study area of New Zealand’s continental shelf.

In gravimetric methods for a determination of the Moho geometry, the constant value of the Moho density contract is often adopted. Results of gravimetric and seismic studies, however, showed that the Moho density contrast varies significantly. The assumption of a uniform density contrast thus might yield large errors in the estimated Moho depths. In this study we investigate these errors by comparing the Moho depths determined globally for the uniform and variable models of the Moho density contrast. These two gravimetric results are obtained based on solving the Vening Meinesz-Moritz’s inverse problem of isostasy. The uniform model of the Moho density contrast is defined individually for the continental and oceanic lithosphere to better reproduce the reality. The global data of the lower crust and upper mantle retrieved from the CRUST1.0 seismic crustal model are used to define the variable Moho density contrast. This seismic model is also used to validate both gravimetric solutions. Results of our numerical experiment reveals that the consideration of the variable Moho density contrast improves the agreement between the gravimetric and seismic Moho models; the RMS of differences is 5.4 km (for the uniform density contrast) and 4.7 km (for the variable density contrast).

The geoid-to-quasigeoid correction has been traditionally computed approximately as a function of the planar Bouguer gravity anomaly and the topographic height. Recent numerical studies based on newly developed theoretical models, however, indicate that the computation of this correction using the approximate formula yields large errors especially in mountainous regions with computation points at high elevations. In this study we investigate these approximation errors at the study area which comprises Himalayas and Tibet where this correction reaches global maxima. Since the GPS-leveling and terrestrial gravity datasets in this part of the world are not (freely) available, global gravitational models (GGMs) are used to compute this correction utilizing the expressions for a spherical harmonic analysis of the gravity field. The computation of this correction can be done using the GGM coefficients taken from the Earth Gravitational Model 2008 (EGM08) complete to degree 2160 of spherical harmonics. The recent studies based on a regional accuracy assessment of GGMs have shown that the combined GRACE/GOCE solutions provide a substantial improvement of the Earth’s gravity field at medium wavelengths of spherical harmonics compared to EGM08. We address this aspect in numerical analysis by comparing the gravity field quantities computed using the satellite-only combined GRACE/GOCE model GOCO02S against the EGM08 results. The numerical results reveal that errors in the geoid-to-quasigeoid correction computed using the approximate formula can reach as much as ~1.5 m. We also demonstrate that the expected improvement of the GOCO02S gravity field quantities at medium wavelengths (within the frequency band approximately between 100 and 250) compared to EGM08 is as much as ±60 mGal and ±0.2 m in terms of gravity anomalies and geoid/quasigeoid heights respectively.

In classical isostatic models, a uniform crustal density is typically assumed, while disregarding the crustal density heterogeneities. This assumption, however, yields large errors in the Moho geometry determined from gravity data, because the actual topography is not fully isostatically compensated. Moreover, the sub-crustal density structures and additional geodynamic processes contribute to the overall isostatic balance. In this study we investigate the effects of unmodelled density structures and geodynamic processes on the gravity anomaly and the Moho geometry. For this purpose, we define the residual isostatic topography as the difference between actual topography and isostatic topography, which is computed based on utilizing the Vening Meinesz-Moritz isostatic theory. We show that the isostatic gravity bias due to disagreement between the actual and isostatically compensated topography varies between -382 and 596 mGal. This gravity bias corresponds to the Moho correction term of -16 to 25 km. Numerical results reveal that the application of this Moho correction to the gravimetrically determined Moho depths significantly improves the RMS fit of our result with some published global seismic and gravimetric Moho models. We also demonstrate that the isostatic equilibrium at long-to-medium wavelengths (up to degree of about 40) is mainly controlled by a variable Moho depth, while the topographic mass balance at a higher-frequency spectrum is mainly attained by a variable crustal density.

In classical isostatic models for a gravimetric recovery of the Moho parameters (i.e., Moho depths and density contrast) the isostatic gravity anomalies are usually defined based on the assumption that the topographic mass surplus and the ocean mass deficiency are compensated within the Earth’s crust. As acquired in this study, this assumption yields large disagreements between isostatic and seismic Moho models. To assess the effects not accounted for in classical isostatic models, we conduct a number of numerical experiments using available global gravity and crustal structure models. First, we compute the gravitational contributions of mass density contrasts due to ice and sediments, and subsequently evaluate respective changes in the Moho geometry. Residual differences between the gravimetric and seismic Moho models are then used to predict a remaining non-isostatic gravity signal, which is mainly attributed to unmodeled density structures and other geophysical phenomena. We utilize three recently developed computational schemes in our numerical studies. The apparatus of spherical harmonic analysis and synthesis is applied in forward modeling of the isostatic gravity disturbances. The Moho depths are estimated globally on a 1 arc-deg equiangular grid by solving the Vening-Meinesz Moritz inverse problem of isostasy. The same estimation model is applied to evaluate the differences between the isostatic and seismic models. We demonstrate that the application of the ice and sediment density contrasts stripping gravity corrections is essential for a more accurate determination of the Moho geometry. We also show that the application of the additional non-isostatic correction further improves the agreement between the Moho models derived based on gravity and seismic data. Our conclusions are based on comparing the gravimetric results with the CRUST2.0 global crustal model compiled using results of seismic surveys.

We formulate an error propagation model based on solving the Vening Meinesz-Moritz (VMM) inverse problem of isostasy. The system ofobservation equations in the VMM model defines the relation between theisostatic gravity data and the Moho depth  by means of a second-order Fredholm integralequation of the first kind. The corresponding error model (derived in aspectral domain) functionally relates the Moho depth errors with the commissionerrors of used gravity and topographic/bathymetric models. The error model alsoincorporates the non-isostatic bias which describesthe disagreement, mainly of systematic nature, between the isostatic andseismic models. The error analysis is conducted at the study area of theTibetan Plateau and Himalayas with the world largest crustal thickness. TheMoho depth uncertainties due to errors of the currently available globalgravity and topographic models are estimated to be typically up to 1-2 km,provided that the GOCE gravity gradient observables improved themedium-wavelength gravity spectra. The errors due to disregarding sedimentarybasins can locally exceed ~2 km. The largest errors (which cause a systematic bias betweenisostatic and seismic models) are attributed to unmodeled mantleheterogeneities (including thecore-mantle boundary) and other geophysical processes. These errors aremostly less than 2 km under significant orogens (Himalayas, Ural), but canreach up to ~10 km under the oceanic crust.

Substantial progress has been achieved over the last four decades to better understand a deep structure in the Himalayas and Tibet. Nevertheless, the remoteness of this part of the world still considerably limits the use of seismic data. A possible way to overcome this practical restriction partially is to use products from the Earth’s satellite observation systems. Global topographic data are provided by the Shuttle Radar Topography Mission (SRTM). Global gravitational models have been derived from observables delivered by the gravity-dedicated satellite missions, such as the Gravity Recovery and Climate Experiment (GRACE) and the Gravity field and steady-state Ocean Circulation Explorer (GOCE). Optimally, the topographic and gravity data should be combined with available results from tomographic surveys to interpret the lithospheric structure, including also a Moho relief. In this study, we use seismic, gravity, and topographic data to estimate the Moho depth under orogenic structures of the Himalayas and Tibet. The combined Moho model is computed based on solving the Vening Meinesz-Moritz (VMM) inverse problem of isostasy, while incorporating seismic data to constrain the gravimetric solution. The result of the combined gravimetric-seismic data analysis exhibits an anticipated more detailed structure of the Moho geometry when compared to the solution obtained merely from seismic data. This is especially evident over regions with sparse seismic data coverage. The newly-determined combined Moho model of Tibet shows a typical contrast between a thick crustal structure of orogenic formations compared to a thinner crust of continental basins. The Moho depth under most of the Himalayas and the Tibetan Plateau is typically within 60-70 km. The maximum Moho deepening of similar to 76 km occurs to the south of the Bangong-Nujiang suture under the Lhasa terrane. Local maxima of the Moho depth to similar to 74 km are also found beneath Taksha at the Karakoram fault. This Moho pattern generally agrees with the findings from existing gravimetric and seismic studies, but some inconsistencies are also identified and discussed in this study.