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. 

Establishment of a precise height system using geodetic approaches (such as precise leveling) plays a significant role in precise positioning. However, precise leveling and establishing vertical height reference systems are very important task that requires deep knowledge of land surveying engineering, geometric geodesy, and physical geodesy (potential theory). The beauty of this field lies in its ability to connect geometry, physics, and dynamic properties. Precise leveling in geodesy not only involves using precise instruments to collect data in the field but also emphasizes the importance of data processing, which can sometimes be very time-consumingand needs deep knowledge in geodesy.

The aim of this lecture note is to provide a theoretical background in precise leveling using geodetic methods and technical recommendations, for engineers, students, and researchers. Nowadays, in an era marked by a shortage of experts in geodesy and the crisis of having fewer well-educated professionals in the field, I was motivated to write this compendium. Hopefully, it will help attract more persons to pursue careers in this field. In addition, another motivation behind this effort is the lack of references that adequately present the methods and technical specifications for this purpose.

A special thanks to Mr. Fredrik Dahlström and Docent Jonas Ågren from the National Mapping, Cadastral, and Land Registration Authority of Sweden (Lantmäteriet) for providing and sharingprecise leveling and gravity data.

Studying the Glacial Isostatic Adjustment (GIA) and land uplift modeling can be carried out utilizing geodetic observations (GNSS and precise leveling measurements), and geophysical methods. The Gravity Recovery and Climate Experiment (GRACE) satellite missions’ data has not been formally used in this context in Fennoscandia. If there is insufficient coverage of offshore or onshore data, existing estimates of GIA might be partially biased (by means of spatial pattern and magnitude), particularly over the Gulf of Bothnia where the land uplift rate reaches its maximum. To inspect this issue, we incorporated the GRACE data in estimates of the land uplift rate due to GIA. Despite satellite gravitational information having a low resolution (∼300 km) it can be used for this purpose because the GIA in Fennoscandia has a large-scale regional pattern. Our findings confirmed a bias in existing estimates. According to our results, the maximum land uplift rates reach 9.1 mm/year in the northern part of the Gulf of Bothnia, while previous estimates indicate that the maximum value is shifted westward towards land. Since GRACE data also comprises hydrological signals, we assessed its effect on the satellite gravitational information by applying different hydrological models. Our results ascertained that land uplift estimates in Fennoscandia were not significantly affected by long-term hydrological mass variations. According to our estimates over the period between 2003 and 2017, the hydrological loading effect was approximately 0.1 mm/year or less (in terms of the RMS differences when compared to the reference land uplift model). Hydrological signal variations (over the investigated period of two decades) were, therefore, dominated mainly by seasonal variations without the presence of secular trends. The results show that the land uplift model from GRACE has some discrepancies compared to existing models, so the main idea of this article is to combine land and satellite data. Therefore, we studied a combined land uplift model using GRACE and the latest land uplift model in Fennoscandia.

In this study, we investigate the gravity changes (g˙) associated with the Glacial Isostatic Adjustment (GIA) and their correlation with land uplift rates (h˙) using GRACE and GRACE Follow-on data, spanning 2003 to 2023. We further validate our results using repeated absolute gravity measurements. We analyze the absolute gravity measurements collected in Canada and Fennoscandia since the 1990s and 1960s, respectively. The novelty involves applying the Sjöberg and Bagherbandi (2020) methodology to Canadian data using GRACE-derived upper mantle density and comparing the results with the terrestrial absolute gravimetry results. One of the aims of this study is also to assess the methods effectiveness in Canada, as was previously done in Fennoscandia. In addition, this study explores the similarities and the relationship between the gravity field change with other Earth's interior parameters (e.g. density, viscosity, etc.), that offers new insights into their potential connections and understanding regional geodynamics. For Canada, we process the raw gravity observations, considering instrument precision and weighting, applying systematic corrections (Earth tides, polar motion, ocean loading, and atmospheric pressure effects), standardizing measurements to a uniform height using the vertical gravity gradient correction, implementing a rigorous drop selection and filtering process, and using a filter to minimize short-term environmental variations. For Fennoscandia, we use the processed absolute gravity data published by Olsson et al. (2019). We also estimate upper mantle density, associated with viscous mass flow in the mantle, and temporal variations in surface gravity changes (g˙) using GRACE and GRACE-FO data. Finally, the obtained g˙/h˙ ratio is calculated to be compared with terrestrial absolute gravity measurements and also compared with the previously published results to assess the obtained outcomes. The results derived from GRACE and GRACE Follow-on data show that the ratios between surface gravity and height changes are −0.152 ± 0.010 μGal/mm in Canada and −0.156 ± 0.016 μGal/mm in Fennoscandia aligning closely with findings from terrestrial gravity observations. These values correspond to upper mantle densities of approximately 3736 ± 239 kg/m3 and 3641 ± 382 kg/m3 in Canada and Fennoscandia, respectively.

Precise gravity measurements have been consistently collected in Fennoscandia and Canada since the 1960s and 1990s, respectively, using relative gravimeters and later employing absolute gravimeters (e.g., FG5 and A10 absolute gravimeters) to establish gravity reference system and study temporal changes in gravity, e.g. associated with ongoing glacial isostatic adjustment (GIA). In this study, we utilized monthly data from GRACE and GRACE Follow-on, spanning 2003 to 2023, to estimate temporal variations in surface gravity changes, their relationship with land uplift rates, and to determine the upper mantle density associated with viscous mass flow in the mantle. The main focus of this paper is Canada; however, the results will be compared with our previous studies in Fennoscandia. We used the ICE-6G_D land uplift model for Canada and the NKG2016LU regional land uplift model for Fennoscandia for this purpose. The satellite gravimetry results were compared with terrestrial absolute gravity observations collected at 43 stations across Canada and Fennoscandia, respectively.

The results derived from GRACE and GRACE Follow-on data show that the ratio between surface gravity and height changes is −0.152 ± 0.010 μGal/mm in Canada and −0.156 ± 0.016 μGal/mm in Fennoscandia aligning closely with findings from terrestrial gravity observations. These values correspond to upper mantle densities of approximately 3736 ± 239 kg/m³ and 3641 ± 382 kg/m³ in Canada and Fennoscandia, respectively. In addition, the results were combined with terrestrial absolute gravimetry results. These findings highlight the importance of satellite gravimetry data and are crucial for GIA modeling and the Earth’s interior parameters.

Preface

This short note is related to the basic concepts of physical geodesy and explains the relationship between the Earth’s geometric and physical parameters. The aim of presenting this short note is to provide students with a better understanding of the spectral domain of the Earth’s gravity field using spherical harmonic coefficients. After teaching this concept for several years, I found that students often lack the necessary background to understand the fundamentals of physical geodesy. I hope this short note and the provided examples help students understand the principles more effectively.

A levelling network was readjusted and a new geoid model compiled within the framework of geodetic vertical datum modernization at the Hong Kong territories. To accomplish all project objectives, the quasigeoid model has to be determinedtoo. A quasigeoid model can be obtained from existing geoid model by applying the geoid-to-quasigeoid separation. Thegeoid-to-quasigeoid separation was traditionally computed as a function of the simple planar Bouguer gravity anomaly, whiledisregarding terrain geometry, topographic density variations, and vertical gravity changes due to mass density heterogeneities below the geoid surface. We applied this approximate method because orthometric heights of levelling benchmarksin Hong Kong were determined only approximately according to Helmert’s theory of orthometric heights. Considering afurther improvement of the accuracy of orthometric heights by applying advanced numerical procedures, we determinedthe geoid-to-quasigeoid separation by applying an accurate method. The comparison of the accurately and approximatelycomputed values of the geoid-to-quasigeoid separation revealed signifcant diferences between them. The approximatevalues are all negative and reach -2.8 cm, whereas values from the accurate method vary between -4.1 and+0.2 cm. In addition, we assessed the efect of anomalous topographic density on the geoid-to-quasigeoid separation by employing a newlydeveloped digital rock density model. According to our estimates the efect of anomalous topographic density reaches amaximum value of 1.6 cm, refecting a predominant presence of light volcanic rocks and sedimentary deposits at the HongKong territories. Our numerical fndings indicate that the conversion between geoid and quasigeoid models should be doneaccurately, even in regions with a moderately elevated topography

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).

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.