Long and short-range air navigation on spherical Earth

Journal of geodesy, 2019

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 zerodegree 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 W 0 , 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 0 . 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 0 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 W 0 . Our computations resulted in the geoid potential W 0 62636848.102 ± 0.004 m 2 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) × 10 6 m 3 s -2 . Geodetic reference system • Geoid potential W 0 • Global vertical datum • Mean Earth ellipsoid • Reference ellipsoid B Hadi Amin

Persistent and pervasive scientific theories provide as much predictive power as they do explanatory and this power is captured in some of our oldest and most utilised theories of trigonometry and geometry. The relationships between the angles and sides of a triangle and the radius, circumference and areas of spheres and circles are some of the mathematical basics we learn in primary school and given their broad use and understanding, we can use some readily observed data, such as the distance between the North and South Poles, to demonstrate the mathematical absurdity that a Flat Earth theory represents. In practical terms, if we calculate distance on the globe between the North and the South Poles and apply this as the radius of the flat earth, the circumference and surface area of the flat earth would be too large to reconcile with anything we observe and measure on our spherical planet. By applying this simple maths we will explore this absurdity and provide a desktop deconstruction of the Flat Earth mythology.

Stochastic Environmental Research and Risk Assessment, 2013

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Journal of Geodetic Science, 2013

Earth Gravitational Models (EGMs) describe the Earth's gravity field including the geoid, except for its zero-degree harmonic, which is a scaling parameter that needs a known geometric distance for its calibration. Today this scale can be provided by the absolute geoid height as estimated from satellite altimetry at sea. On the contrary, the above technique cannot be used to determine the geometric parameters of the Mean Earth Ellipsoidal (MEE), as this problem needs global data of both satellite altimetry and gravimetric geoid models, and the standard technique used today leads to a bias for the unknown zero-degree harmonic of the gravimetric geoid height model. Here we present a new method that eliminates this problem and simultaneously determines the potential of the geoid (W 0) and the MEE axes. As the resulting equations are non-linear, the linearized observation equations are also presented.

Journal of Geodesy, 2018

Traditionally a laterally homogeneous and spherical base Earth model (e.g., the PREM model) is considered as input when computing the Earth's equipotential surfaces, which are then resulted to be in symmetric shape. However, the Earth, known with a complex distribution of interior material and density, especially in the upper mantle and the crust, cannot be treated as a symmetric sphere. Recently, a CRUST1.0 model of crust layer is published and well accepted. But the effect caused by the asymmetric crust (and mantle) on equilibrium figures of the Earth cannot be analyzed by the traditional theories. A generalized theory of the figure of the Earth to third-order precision is firstly proposed in this paper, as well as the iterative calculation strategy to solve the complex equation system. In order to validate this generalized theory, the degeneration of this generalized theory with the PREM model is made and is compared with traditional theories, and it is shown that the result of this generalized theory, after degeneration, is consistent very well with traditional theory. Meanwhile, the effect (including both the direct and indirect effects) of the crust layer, from the CRUST1.0 model, on the figures of equipotential surfaces of the Earth's interior, as well as their effects on the global dynamics flattening, will be presented as an application of this theory in accompanying paper.

Contributions to Geophysics and Geodesy, 2012

For decades now the geodetic community has been split down the middle over the question as to whether geoid or quasigeoid should be used as a reference surface for heights. The choice of the geoid implies that orthometric heights must be considered, the choice of the quasigeoid implies the use of the so-called normal heights. The problem with the geoid, a physically meaningful surface, is that it is sensitive to the density variations within the Earth. The problem with the quasigeoid, which is not a physically meaningful surface, is that it requires integration over the Earth's surface.

2018

2018

In 'How to Measure the Earth's circumference' Michael Baizerman mentions the modern equatorial measure in conjunction with the meridian degree length. Confusing. The Greek methodologies invariably were conducted in a north/south direction. They were attempting to measures a distance along a meridian. This distance was then expanded to a count of diverse stade values applied to the meridian circumference. At no time was an equatorial circumference given in the ancient world. What emerges from an assessment of these values when evaluated with knowledge of the measures of the ancient world is that these diverse stadia values all ultimately gave the same dimension for the circumference of Earth in a meridian direction. It was John Michell who extending and adapting the works of previous scholars including Berryman, Stechini and Petrie to name but three, revealed the ancient measuring system. While I disagree with some of his conclusions, his measures were correct. Michell claimed that various differences in values were due to the difference in degree lengths around the meridian quadrant whereas I argue that the same values can be found via counts of time such as days of the month in its various formats. In fact the measures applied greatly predate the interpretation of the Earth's shape from a pure globe to a flattened version. It was not until well into the Christian era that reasonably accurate knowledge of this flattening became available. The unit measures, whether cubits, reeds or stadia, among other denominations are related to the meridian circumference of Earth as accepted before the French determinations for the metric system. Some of these divide accurately into that meridian while others work via specific factors. The Earth's diameter was also derived from this via a pi value of 3.1418181818. This diameter of course was 7920 miles. The circumference measure can be seen as 24883.2 British miles. It was Michell who calculated this value and it and its component parts are verified via ancient cubit rods, measures of buildings etc. Here is an extract from my work Measurements of the Gods showing the values applied by Michell, the French calculation for the metric values and NASA [early 1990s] :

Pure and Applied Geophysics, 2014

Mathematics plays a fundamental role in all scientific fields and, of course, in the Geosciences. This has been especially well known since, for example, the beginning of geodesy in the Greek era (VANÍČ EK and KRAKIWSKY 1992) or in the pioneering studies of planet Earth's interior and the description of the Earth's potential fields (geomagnetic and gravimetric). We can remember that Johann Carl Friedrich Gauss (1777-1855), who is recognized as one of the history's most influential mathematicians, had a remarkable influence in many fields of mathematics and science. He contributed significantly to the development of modern geophysics. In the field of geomagnetism, he developed the first method to obtain absolute geomagnetic measurements, worked out the mathematical theory for separating the inner and outer sources of Earth's magnetic field, and founded the ''Magnetischer Verein'' (Magnetic Union). Another pioneering natural topic was exploring Earth's interior using seismic waves for analyzing the Earth's inner structure and discovering underground resources. For instance, in 1936, the Danish mathematician Inge Lehman was the first to interpret P wave arrivals, which inexplicably appeared in the P wave shadow of the Earth's core as reflections at an inner core, and discovered the Earth's solid inner core by studying these anomalies. Now it is natural to use geophysical methods and remote sensing in exploring our soils to find water, oil, minerals, etc., and to use advanced mathematical tools for interpreting observation results. Therefore, as described by CAMACHO et al. (2008a, b), mathematics is one of the branches of science, together with physics, chemistry, and Information Technology that studies and furthers knowledge of the Earth's structure and dynamics. This research is conducted in multidisciplinary studies that use and incorporate the most advanced methods of those sciences, in the framework of, or in close cooperation with, the different branches of Earth sciences such as geology, geophysics, and geodesy. The international dimension of this collaboration is evident (see, e.g., IUGS 2014; IUGG 2014). This was the environment in which the Complutense International Seminar on ''Earth Sciences and Mathematics'' was organized and held in Madrid in 2006. The presentations at that meeting were published in two Pure and Applied Geophysics topical issues (CAMACHO et al. 2008c, d). Since 2006, Earth sciences and, therefore, their ''mathematical needs'' have evolved, marked to a very significant extent by the consolidated use of artificial satellites for the observation of Earth, which plays a more important role everyday (e.g.