Search Results (11)

This study examines the influence of gravity on different height systems by integrating Global Navigation Satellite Systems (GNSS), leveling, and gravimetric measurements. Although the theoretical influence of gravity on height systems is well known, empirical studies that quantify these effects along steep terrain are rare—particularly within the Croatian reference systems. Geometric leveling, recognized for its precision in geodesy, was employed alongside gravimetric data to analyze the relationship between gravity variations and height differences. The research was conducted along Sljeme Road on Mt. Medvednica, Croatia, where altitude-dependent gravity effects were systematically investigated along an elevation profile with a height difference of about 650 m. GNSS measurements provided positional coordinates referenced to the Croatian Terrestrial Reference System 1996 (HTRS96) (EPSG:4888), while leveling and gravimetric data were analyzed within the Croatian Height Reference System 1971 (HVRS71) (EPSG:5610) and Croatian Gravimetric Reference System 2003 (HGRS03), respectively. The results demonstrate that differences between geometric and normal–orthometric heights become more pronounced at higher elevations but remain at the millimeter level. Notably, the impact of gravity is evident in normal and orthometric heights, with differences from geometric heights reaching up to 3.7 cm at the highest points. Additionally, a comparison between normal and orthometric heights reveals that at the beginning of the leveling line, the difference is around 4 mm. However, as the elevation increases, this difference grows, reaching over 1 cm at the end of the leveling line. The study also confirms the theoretical correlation between the geoid–quasigeoid height difference and terrain elevation, with increasing differences observed at higher altitudes. To examine the consistency of different height determination methods, two approaches were applied: one based on adjustment within the geopotential system, and the other involving direct adjustment in the desired height system, with specific height corrections applied. The results confirmed that the height differences between the two methods were 0, to the tenth of a millimeter, indicating that both methods provided identical results. These findings contribute to a deeper understanding of geodetic height systems and the role of gravity in height determination. Full article

High-accuracy optical clocks have garnered increasing attention for their potential application in various fields, including geodesy. According to the gravitational red-shift effect, clocks at lower altitudes on the Earth’s surface run slower than those at higher altitudes due to the differential gravitational field. Consequently, the geopotential difference can be determined by simultaneously comparing the frequency of two optical clocks at disparate locations. Here, we report geopotential difference measurements conducted using a pair of transportable ⁴⁰Ca⁺ optical clocks with uncertainties at the ${10}^{-17}$ level. After calibrating the output frequencies of two optical clocks in the horizontal position, frequency comparison is realized by moving Clock 2 to two different positions using a high-precision optical fiber time–frequency transmission link with Clock 1. The elevation difference of the two different positions, as processed by ensemble empirical mode decomposition (EEMD), is measured as −88.4 cm ± 16.7 cm and 104.5 cm ± 20.1 cm, respectively, which is consistent with the geometric measurement results within the error range. This experimental result validates the credibility of the optical clock time–frequency comparison used in determining geopotential differences, thereby providing a novel measurement model for the establishment of a global unified elevation datum. Full article

The Hansen problem is a classic and well-known geometric challenge in geodesy and surveying involving the determination of two unknown points relative to two known reference locations using angular measurements. Traditional analytical solutions rely on cumbersome trigonometric calculations and are prone to propagation errors. This paper presents a novel framework leveraging geometric algebra (GA) to formulate and solve the Hansen problem. Our approach utilizes the representational capabilities of Vector Geometric Algebra (VGA) and Conformal Geometric Algebra (CGA) to avoid the need for tedious analytical manipulations and provide an efficient, unified solution. We develop concise geometric formulas tailored for computational implementation. The rigorous analyses and simulations that were completed as part of this work demonstrate that the precision and robustness of this new technique are equal or superior to those of conventional resection methods. The integration of classical concepts like the Hansen problem with modern GA-based spatial computing delivers more intuitive solutions while advancing the mathematical discourse. This work transforms conventional perspectives through methodological innovation, avoiding the limitations of prevailing paradigms. Full article

This paper recasts four geodetic curves—the great ellipse, the normal section, the loxodrome, and the curve of alignment—into a parametric form of vector-algebra formula. These formulas allow these curves to be drawn using simple, efficient, and robust algorithms. The curve of alignment, which seems to be quite obscure, ought not to be. Like the great ellipse and the loxodrome, and unlike the normal section, the curve of alignment from point A to point B (both on the same ellipsoid) is the same as the curve of alignment from point B to point A. The algorithm used to draw the curve of alignment is much simpler than any of the others and its shape is quite similar to that of the geodesic, which suggests it would be a practical surrogate when drawing these curves. Full article

In many applications, such as space navigation, metrology, testing, and geodesy, it is necessary to measure accelerations with frequencies ranging from fractions of a hertz to several kilohertz. For this purpose, optomechanical sensors are used. The natural frequency of such sensors should be approximately ten times greater than the frequency of the measured acceleration. In the case of triaxial acceleration measurements, a planar design with two sensors that measure accelerations in two perpendicular in-plane directions and a third sensor that measures out-of-plane acceleration is effective. The mechanical characteristics of the existing designs of both in-plane and out-of-plane types of sensors were analyzed, and the improved designs were elaborated. Using numerical simulation, the dependencies of the natural frequency level in the range from several hertz to tens of kilohertz on the designs and geometric parameters of opto-mechanical accelerometers were modeled. This allows one to select the accelerometer design and its parameters to measure the acceleration at the assigned frequency. It is shown that the opto-mechanical accelerometers of the proposed designs have reduced dissipation losses and crosstalk. Full article

Besides precise levels, precise leveling staffs are a crucial part of the measuring equipment when carrying out geodetic (geometric) leveling measurements. The leveling staffs define the scale of the height reference system, so it is important to calibrate them periodically and when necessary. This paper shortly describes the development of the new method of calibrating leveling staffs in the Laboratory for Measurements and Measuring Technique of the Faculty of Geodesy, University of Zagreb. The existing horizontal comparator was upgraded by installing a servo-motorized positioning drive with a mounted CCD camera and telecentric lens that is used to record graduations of the leveling staffs. The software was developed to support the management of the comparator system, as well as for the analysis and processing of images and measurement data and, most importantly, giving the result in the form of a calibration report. The main subject of this paper is a detailed assessment of the measurement uncertainty of determining the position of the edges of the graduation lines and determining the scale of precise centimeter and coded leveling staffs. The estimates were confirmed by experimental measurements. Full article

Geodesy is the science of accurately measuring the topography of the earth (geometric shape and size), its orientation in space, and its gravity field. With the advances in our knowledge and technology, this scientific field has extended to the understanding of geodynamical phenomena such as crustal motion, tides, and polar motion. This Special Issue is dedicated to the recent advances in modelling geodetic time series recorded using various instruments. Due to the stochastic noise properties inherent in each of the time series, careful modelling is necessary in order to extract accurate geophysical information with realistic associated uncertainties (statistically sufficient). The analyzed data have been recorded with various space missions or ground-based instruments. It is impossible to be comprehensive in the vast and dynamic field that is Geodesy, particularly so-called “Environmental Geodesy”, which intends to understand the Earth’s geodynamics by monitoring any changes in our environment. This field has gained much attention in the past two decades due to the need by the international community to understand how climate change modifies our environment. Therefore, this Special Issue collects some articles which emphasize the recent development of specific algorithms or methodologies to study particular natural phenomena related to the geodynamics of the earth’s crust and climate change. Full article

Satellite altimetry technology has unparalleled advantages in the monitoring of hydrological resources. After decades of development, satellite altimetry technology has achieved a perfect integration from the geometric research of geodesy to the natural resource monitoring research. Satellite altimetry technology has shown great potential, whether solid or liquid. In general, this paper systematically reviews the development of satellite altimetry technology, especially in terms of data availability and program practicability, and proposes a multi-source altimetry data fusion method based on deep learning. Secondly, in view of the development prospects of satellite altimetry technology, the challenges and opportunities in the monitoring application and expansion of surface water changes are sorted out. Among them, the limitations of the data and the redundancy of the program are emphasized. Finally, the fusion scheme of altimetry technology and deep learning proposed in this paper is presented. It is hoped that it can provide effective technical support for the monitoring and application research of hydrological resources. Full article

A flexible approach for geometric modelling of point clouds obtained from Terrestrial Laser Scanning (TLS) is by means of B-splines. These functions have gained some popularity in the engineering geodesy as they provide a suitable basis for a spatially continuous and parametric deformation analysis. In the predominant studies on geometric modelling of point clouds by B-splines, uncorrelated and equally weighted measurements are assumed. Trying to overcome this, the elementary errors theory is applied for establishing fully populated covariance matrices of TLS observations that consider correlations in the observed point clouds. In this article, a systematic approach for establishing realistic synthetic variance–covariance matrices (SVCMs) is presented and afterward used to model TLS point clouds by B-splines. Additionally, three criteria are selected to analyze the impact of different SVCMs on the functional and stochastic components of the estimation results. Plausible levels for variances and covariances are obtained using a test specimen of several dm—dimension. It is used to identify the most dominant elementary errors under laboratory conditions. Starting values for the variance level are obtained from a TLS calibration. The impact of SVCMs with different structures and different numeric values are comparatively investigated. Main findings of the paper are that for the analyzed object size and distances, the structure of the covariance matrix does not significantly affect the location of the estimated surface control points, but their precision in terms of the corresponding standard deviations. Regarding the latter, properly setting the main diagonal terms of the SVCM is of superordinate importance compared to setting the off-diagonal ones. The investigation of some individual errors revealed that the influence of their standard deviation on the precision of the estimated parameters is primarily dependent on the scanning distance. When the distance stays the same, one-sided influences on the precision of the estimated control points can be observed with an increase in the standard deviations. Full article

With knowledge of geometry and density-distribution of topography, the residual terrain modelling (RTM) technique has been broadly applied in geodesy and geophysics for the determination of the high-frequency gravity field signals. Depending on the size of investigation areas, challenges in computational efficiency are encountered when using an ultra-high-resolution digital elevation model (DEM) in the Newtonian integration. For efficient and accurate gravity forward modelling in the spatial domain, we developed a new MATLAB-based program called, terrain gravity field (TGF). Our new software is capable of calculating the gravity field generated by an arbitrary topographic mass-density distribution. Depending on the attenuation character of gravity field with distance, the adaptive algorithm divides the integration masses into four zones, and adaptively combines four types of geometries (i.e., polyhedron, prism, tesseroid and point-mass) and DEMs with different spatial resolutions. Compared to some publicly available algorithms depending on one type of geometric approximation, this enables accurate modelling of gravity field and greatly reduces the computation time. Besides, the TGF software allows to calculate ten independent gravity field functionals, supports two types of density inputs (constant density value and digital density map), and considers the curvature of the Earth by involving spherical approximation and ellipsoidal approximation. Further to this, the TGF software is also capable of delivering the gravity field of full-scale topographic gravity field implied by masses between the Earth’s surface and mean sea level. In this contribution, the TGF software is introduced to the geoscience community and its capabilities are explained. Results from internal and external numerical validation experiments of TGF confirmed its accuracy at the sub-mGal level. Based on TGF, the trade-off between accuracy and efficiency, values for the spatial resolution and extension of topography models are recommended. The TGF software has been extensively tested and recently been applied in the SRTM2gravity project to convert the global 3” SRTM topography to implied gravity effects at 28 billion computation points. This confirms the capability of TGF for dealing with large datasets. Together with this paper, the TGF software will be released in the public domain for free use in geodetic and geophysical forward modelling computations. Full article

Coordinate transformation plays an indispensable role in industrial measurements, including photogrammetry, geodesy, laser 3-D measurement and robotics. The widely applied methods of coordinate transformation are generally based on solving the equations of point clouds. Despite the high accuracy, this might result in no solution due to the use of ill conditioned matrices. In this paper, a novel coordinate transformation method is proposed, not based on the equation solution but based on the geometric transformation. We construct characteristic lines to represent the coordinate systems. According to the space geometry relation, the characteristic line scan is made to coincide by a series of rotations and translations. The transformation matrix can be obtained using matrix transformation theory. Experiments are designed to compare the proposed method with other methods. The results show that the proposed method has the same high accuracy, but the operation is more convenient and flexible. A multi-sensor combined measurement system is also presented to improve the position accuracy of a robot with the calibration of the robot kinematic parameters. Experimental verification shows that the position accuracy of robot manipulator is improved by 45.8% with the proposed method and robot calibration. Full article