The permanent tide and the International Height Reference Frame IHRF

The International Height Reference System (IHRS), adopted by International Association of Geodesy (IAG) in its Resolution No. 1 at the XXVI General Assembly of the International Union of Geodesy and Geophysics (IUGG) in Prague in 2015, contains two novelties. Firstly, the mean-tide concept is adopted for handling the permanent tide. While many national height systems continue to apply the mean-tide concept, this was the first time that the IAG officially introduced it for a potential field quantity. Secondly, the reference level of the height system is defined by the equipotential surface where the geopotential has a conventional value W0 = 62,636,853.4 m2 s–2. This value was first determined empirically to provide a good approximation to the global mean sea level and then adopted as a reference value by convention. I analyse the tidal aspects of the reference level based on W0. By definition, W0 is independent of the tidal concept that was adopted for the equipotential surface, but for different concepts, different functions are involved in the W of the equation W = W0. I find that, in the empirical determination of the adopted estimate W0, the permanent tide is treated inconsistently. However, the consistent estimate from the same data rounds off to the same value. I discuss the tidal conventions and formulas for the International Height Reference Frame (IHRF) and the realisation of the IHRS. I propose a simplified definition of IHRF geopotential numbers that would make it possible to transform between the IHRF and zero-tide geopotential numbers using a simple datum-difference surface. Such a transformation would not be adequate if rigorous mean-tide formulas were imposed. The IHRF should adopt a conventional (best) estimate of the permanent tide-generating potential, such as that which is contained in the International Earth Rotation and Reference Systems Service Conventions, and use it as a basis for other conventional formulas. The tide-free coordinates of the International Terrestrial Reference Frame and tide-free Global Geopotential Models are central in the modelling of geopotential for the purposes of the IHRF. I present a set of correction formulas that can be used to move to the zero-tide model before, during, or after the processing, and finally to the mean-tide IHRF. To reduce the confusion around the multitude of tidal concepts, I propose that modelling should primarily be done using the zero-tide concept, with the mean-tide potential as an add-on. The widespread use of the expression “systems of permanent tide” may also have contributed to the confusion, as such “systems” do not have the properties that are generally associated with other “systems” in geodesy. Hence, this paper mostly uses “concept” instead of “system” when referring to the permanent tide.

Theoretical and Numerical Study on Electrical Resistivity Measurement of Cylindrical Rock Core Samples Using Perimeter Electrodes

The estimation of hydraulic and mechanical properties of bedrock is important for the evaluation of energy-related structures, including high-level nuclear waste repositories, hydraulic fracturing wells, and gas-hydrate production wells. The hydraulic conductivity and stress–strain curves of rocks are conventionally measured through laboratory tests on cylindrical samples. Both ASTM standards for hydraulic conductivity and compressive strength involve the use of the planar bases of a cylindrical sample. Hence, an alternative test method is required for the simultaneous measurement of hydraulic conductivity and stress–strain curves. This study proposes a novel electrical resistivity estimation method using two perimeter electrodes for the estimation of hydraulic properties. The theoretical background for the perimeter electrode setup is derived and the COMSOL MultiPhysics® finite element numerical simulation tool is employed to verify the derived theoretical equation. The accuracy of the numerical simulation tool is first validated by simulating the ASTM standard testing method for electrical resistivity. The electrical resistance values derived from the theoretical equation and numerical simulation are compared for different electrical resistivity and electrode radius. The assumed equidistant, circular equipotential surface results in a theoretical lower bound for the measured electrical resistance in the cylindrical specimen. The introduction of a phenomenological distortion factor to correct for the theoretical equipotential surface results in a good fit with the numerical simulation results. The effects of electrode length and equivalent strap electrodes were investigated to assess the applicability of the suggested method for laboratory testing. Consequently, this study presents an effective alternative theoretical assessment method for the lower bound electrical resistivity of cylindrical rock core samples under confining conditions when the installation of base electrodes is infeasible.

EQUIPOTENTIAL SURFACES FOR UNIFORMLY CHARGED SYMMETRIC POLYHEDRAL CONDUCTORS

<p>In this paper, it has been explained how the equipotential surface is influenced by the shape of the symmetric polyhedral conductor and how it finally becomes spherical as it would be if the polyhedral conductor is replaced by a point charge placed at the centre of that conductor. As we move away from the polyhedral conductor the consecutive equipotential surfaces curve more at the sharp bends. A patter n is observed when the diagonal distance from the vertex of the conductor to the first occurrence of its spherical equipotential surface for all symmetric polyhedral conductors is mathematically calculated which is half of the length of its side.</p>

EQUIPOTENTIAL SURFACES FOR UNIFORMLY CHARGED SYMMETRIC POLYHEDRAL CONDUCTORS

<p>In this paper, it has been explained how the equipotential surface is influenced by the shape of the symmetric polyhedral conductor and how it finally becomes spherical as it would be if the polyhedral conductor is replaced by a point charge placed at the centre of that conductor. As we move away from the polyhedral conductor the consecutive equipotential surfaces curve more at the sharp bends. A patter n is observed when the diagonal distance from the vertex of the conductor to the first occurrence of its spherical equipotential surface for all symmetric polyhedral conductors is mathematically calculated which is half of the length of its side.</p>

Восстановление топографии поверхностных дефектов ферромагнетиков при нормальном намагничивающем поле

A technique for reconstructing the topography of defects in ferromagnetic materials in a normal magnetizing field is considered. It is shown that with such a magnetization, the surface of a soft magnetic ferromagnetic material is an equipotential surface. An approximation is proposed that makes it possible to obtain its topography from the results of measuring the three components of the magnetic field at a small distance from the defect. The reconstruction accuracy was estimated from the results of calculating the field from the defect by the finite element method and reconstructing the topography of the defect using the proposed approximation.

Towards a Global Unified Physical Height System

&lt;p&gt;The International Association of Geodesy (IAG), as the organisation responsible for advancing Geodesy, introduced in 2015 the International Height Reference System (IHRS) as the global conventional reference system for the determination of gravity field-related vertical coordinates. The definition of the IHRS is given in terms of potential parameters: the vertical coordinates are geopotential numbers (C&lt;sub&gt;P&lt;/sub&gt; = W&lt;sub&gt;0&lt;/sub&gt; &amp;#8208; W&lt;sub&gt;P&lt;/sub&gt;) referring to an equipotential surface of the Earth's gravity field realised by the conventional value W&lt;sub&gt;0&lt;/sub&gt; = 62 636 853.4 m&lt;sup&gt;2&lt;/sup&gt;s&lt;sup&gt;&amp;#8208;2&lt;/sup&gt;. The spatial reference of the position P for the potential W&lt;sub&gt;P&lt;/sub&gt; = W(&lt;strong&gt;X&lt;/strong&gt;) is given by coordinates &lt;strong&gt;X&lt;/strong&gt; of the International Terrestrial Reference Frame (ITRF). At present, the main challenge is the realisation of the IHRS; i.e., the establishment of the International Height Reference Frame (IHRF): a global network with regional and national densifications, whose geopotential numbers referring to the global IHRS are known. According to the objectives of the IAG Global Geodetic Observing System (GGOS), the target accuracy of these global geopotential numbers is 3 x 10&lt;sup&gt;-2&lt;/sup&gt; m&lt;sup&gt;2&lt;/sup&gt;s&lt;sup&gt;-2&lt;/sup&gt;. In practice, the precise realisation of the IHRS is limited by different aspects; for instance, there are no unified standards for the determination of the potential values W&lt;sub&gt;P&lt;/sub&gt;; the gravity field modelling and the estimation of the position vectors &lt;strong&gt;X&lt;/strong&gt; follow different conventions; the geodetic infrastructure is not homogeneously distributed globally, etc. This may restrict the expected accuracy of 3 x 10&lt;sup&gt;-2&lt;/sup&gt; m&lt;sup&gt;2&lt;/sup&gt;s&lt;sup&gt;-2 &lt;/sup&gt;to some orders lower (from 10 x 10&lt;sup&gt;-2&lt;/sup&gt; m&lt;sup&gt;2&lt;/sup&gt;s&lt;sup&gt;-2&lt;/sup&gt; to 100 x 10&lt;sup&gt;-2&lt;/sup&gt; m&lt;sup&gt;2&lt;/sup&gt;s&lt;sup&gt;-2&lt;/sup&gt;). This contribution summarises advances and present challenges in the establishment of the IHRS/IHRF.&lt;/p&gt;

Strategy for the realisation of the International Height Reference System (IHRS)

In 2015, the International Association of Geodesy defined the International Height Reference System (IHRS) as the conventional gravity field-related global height system. The IHRS is a geopotential reference system co-rotating with the Earth. Coordinates of points or objects close to or on the Earth’s surface are given by geopotential numbersC(P) referring to an equipotential surface defined by the conventional valueW0 = 62,636,853.4 m2 s−2, and geocentric Cartesian coordinatesXreferring to the International Terrestrial Reference System (ITRS). Current efforts concentrate on an accurate, consistent, and well-defined realisation of the IHRS to provide an international standard for the precise determination of physical coordinates worldwide. Accordingly, this study focuses on the strategy for the realisation of the IHRS; i.e. the establishment of the International Height Reference Frame (IHRF). Four main aspects are considered: (1) methods for the determination of IHRF physical coordinates; (2) standards and conventions needed to ensure consistency between the definition and the realisation of the reference system; (3) criteria for the IHRF reference network design and station selection; and (4) operational infrastructure to guarantee a reliable and long-term sustainability of the IHRF. A highlight of this work is the evaluation of different approaches for the determination and accuracy assessment of IHRF coordinates based on the existing resources, namely (1) global gravity models of high resolution, (2) precise regional gravity field modelling, and (3) vertical datum unification of the local height systems into the IHRF. After a detailed discussion of the advantages, current limitations, and possibilities of improvement in the coordinate determination using these options, we define a strategy for the establishment of the IHRF including data requirements, a set of minimum standards/conventions for the determination of potential coordinates, a first IHRF reference network configuration, and a proposal to create a component of the International Gravity Field Service (IGFS) dedicated to the maintenance and servicing of the IHRS/IHRF.

Indoor height determination of the new absolute gravimetric station of L'Aquila

In this paper we describe all the field operations and the robust post-processing proceduresto determine the height of the new absolute gravimetric station purposely selected to belong to a new absolute gravimetric network and located in the Science Faculty of the L’Aquila University. This site has been realized indoor in the Geomagnetism laboratory, so that the height cannot be measured directly, but linking it to the GNSS antenna of AQUI benchmark located on the roof of the same building, by a classical topographic survey. After the topographic survey, the estimated height difference between AQUI and the absolute gravimetric site AQUIgis 14.9700.003 m. At the epoch of the 2018 gravimetric measures, the height of AQUI GNSS station was 712.9740.003 m, therefore the estimated ellipsoidalheight of the gravimetric site at the epoch of gravity measurements is 698.0040.005 m. Absolute gravity measurements are referred to the equipotential surface of gravity field, so that the knowledge of the geoidal undulation at AQUIg allows us to infer the orthometric height as 649.32 m.

Electrical Resistivity Measurement with Spherical-Tipped Cylindrical Electrode Embedded on Two Layers

Complex geological processes form multiple layers and change pore water chemistry, saturation level, and temperature. Eventually, the strata hinder interpreting electrical resistivity data. There are no studies that theoretically explore the effects of electrode geometries and multiple layered systems on laboratory electrical resistivity measurements. This study formulates a theoretical electrical resistance between half spherical-tipped cylindrical electrodes embedded on two horizontal layers. The electrical resistivity of each layer is considered separately in the general electrical potential equation with different equipotential surface areas. The finite element analysis is conducted to validate the theoretical equation. Further interpretation provides insights into the distribution of electrical current flow under electrical resistivity mismatch for discussion.

A global vertical datum defined by the conventional geoid potential and the Earth ellipsoid parameters

&lt;p&gt;According to the classical Gauss&amp;#8211;Listing definition, the geoid is the equipotential surface of the Earth&amp;#8217;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&amp;#8217;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 W&lt;sub&gt;0&lt;/sub&gt;, 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 W&lt;sub&gt;0&lt;/sub&gt;. 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 W&lt;sub&gt;0&lt;/sub&gt; should be considered a quasi-stationary parameter, we quantify the effect of time-dependent Earth&amp;#8217;s gravity field changes as well as the time-dependent sea-level changes on the estimation of W&lt;sub&gt;0&lt;/sub&gt;. Our computations resulted in the geoid potential W&lt;sub&gt;0 &lt;/sub&gt;= 62636848.102 &amp;#177; 0.004 m&lt;sup&gt;2&lt;/sup&gt;s&lt;sup&gt;-2&lt;/sup&gt; and the semi-major and &amp;#8211;minor axes of the MEE, a = 6378137.678 &amp;#177; 0.0003 m and b = 6356752.964 &amp;#177; 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 &amp;#177; 0.03) &amp;#215; 10&lt;sup&gt;6&lt;/sup&gt; m&lt;sup&gt;3&lt;/sup&gt;s&lt;sup&gt;-2&lt;/sup&gt;.&lt;/p&gt;