Physical Geodesy

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Physical Geodesy is the branch of geodesy that studies the Earth's shape, gravitational field, and their variations. It employs mathematical and physical principles to understand the Earth's physical properties and how they influence measurements of position and elevation on the Earth's surface.

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Key research themes

1. How can GPS geodesy contribute to understanding crustal deformation and natural hazard mitigation?

This research theme focuses on leveraging the high-precision positioning capabilities of GPS geodesy to investigate crustal deformation processes and improve natural hazard early warning systems. It encompasses methods of integrating GPS-derived displacement data with geophysical and geological observations, as well as analyzing signal disturbances related to atmospheric and environmental processes. Advancements in this area enhance societal resilience to tectonic hazards by refining models of Earth's surface dynamics.

Physical applications of GPS geodesy: a review

by Diego Melgar and

2016

Key finding: GPS geodesy achieves millimeter-level 3D positioning precision globally, enabling detailed monitoring of crustal deformation related to tectonic and magmatic processes. The propagation delays and multipath effects in GPS... Read more

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The European GeoMetre project: developing enhanced large-scale dimensional metrology for geodesy

by Ulla Kallio

2023, Applied Geomatics

Key finding: The project advances SI-traceable geodetic reference frames by innovating instruments and measurement strategies that reduce uncertainties to the millimeter scale over multi-kilometer baselines. Novel environmental sensing... Read more

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Large-Scale Dimensional Metrology for Geodesy—First Results from the European GeoMetre Project

by Ulla Kallio

2023, International Association of Geodesy Symposia

Key finding: The project develops and validates advanced terrestrial instrumentation and methodologies to provide SI-traceable reference distances with uncertainties below 1 mm over several kilometers. Integrated environmental... Read more

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2. What are the advances in geodesic computations and stability analyses in Riemannian and semi-Riemannian geometries?

This theme addresses computational techniques for calculating geodesics on various manifolds—including Riemannian, semi-Riemannian, and Lorentzian spacetimes—and the mathematical frameworks for their stability analysis. It covers numerical methods, variational principles, and geometric approaches such as differential forms and curvature-based stability functions. Understanding geodesic behavior and their perturbations is essential in mathematical physics and astrophysics, underpinning models in general relativity and theoretical cosmology.

Calculating geodesics

by Harold Vacca González

2022, Ingeniería Solidaria

Key finding: Utilizing numerical methods implemented in SageMath, geodesics on Riemannian and semi-Riemannian manifolds can be explicitly computed where analytical approaches fail. The study demonstrates that geodesic equations, derived... Read more

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Geodesics in semi-Riemannian manifolds: geometric properties and variational tools

by Miguel Sanchez

2023, Recent Developments in Pseudo-Riemannian Geometry

Key finding: The paper synthesizes geometric and variational perspectives on geodesics in semi-Riemannian manifolds, highlighting difficulties arising from indefinite metrics, such as issues of geodesic completeness and causal structure.... Read more

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Jacobi and Lyapunov Stability Analysis of Circular Geodesics around a Spherically Symmetric Dilaton Black Hole

by Paul Blaga

2023, Symmetry

Key finding: By reformulating geodesic equations as a dynamical system, the study applies Lyapunov (linear) and Jacobi (geometro-dynamical) stability analyses to orbits in a charged dilaton black hole spacetime. It derives characteristic... Read more

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The Use of Differential Forms to Linearize a Class of Geodesic Equations

by Yusuf Haruna

2024, Journal of the Nigerian Society of Physical Sciences

Key finding: The research demonstrates that employing differential forms enables the linearization of certain geodesic equations on classical surfaces, such as planes, cones, and spheres. This methodological innovation facilitates... Read more

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3. How can global geopotential models and metrology techniques be optimized for precise geoid and gravity field determination?

This theme explores the evaluation and enhancement of global geopotential models (GGMs) integrated with terrestrial gravity data to improve gravimetric geoid computation accuracy. It assesses methodologies for combining satellite-derived and terrestrial datasets, refractivity corrections in optical metrology, and large-scale baseline establishment. Optimization of GGMs and metrological strategies ensures better local gravity field representation, fundamental for high-precision geodesy and related Earth sciences.

Rigorous evaluation of global geopotential models for geoid modelling: A case study in Kenya

by Jonathan Chivatsi and

2022, Journal of African Earth Sciences

Key finding: Through comprehensive geometric and spectral comparisons involving 31 recent high-resolution GGMs against GNSS-levelled points and terrestrial gravity data across Kenya, the study identifies that EGM2008 and EIGEN-6C4 models... Read more

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ASSESSMENT OF RECENT GOCE-BASED GLOBAL GEOPOTENTIAL MODELS AND EGM2008 IN NIGER REPUBLIC

by Salissou IBRAHIM YAHAYA and

2019, Geodesy and Cartography

Key finding: Evaluation of combined (EGM2008, EIGEN-6C4, GECO) and pure satellite GGMs (GOCE-based and others) against terrestrial gravity and GNSS/levelling data in Niger reveals EIGEN-6C4 as the best combined model for geoid height... Read more

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The European GeoMetre project: developing enhanced large-scale dimensional metrology for geodesy

by Ulla Kallio

2023, Applied Geomatics

Key finding: Development and validation of advanced distance meters capable of 1 mm uncertainty across multi-kilometer baselines, alongside novel refractivity compensation strategies, underpin improvements in geodetic reference frames.... Read more

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Large-Scale Dimensional Metrology for Geodesy—First Results from the European GeoMetre Project

by Ulla Kallio

2023, International Association of Geodesy Symposia

Key finding: Innovative instrument developments and environmental sensing methods are demonstrated to enhance SI traceability of terrestrial distances critical for geodetic reference frame realization. The project reports on improved... Read more

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All papers in Physical Geodesy

Frontiers in Relativistic Celestial Mechanics. Volume 1

by Sergei M Kopeikin

Figure 1.1. Sketch of the correspondence between the quantized energy levels of the real and effective conservative dynamics. n denotes the ‘principal quantum number’ (n = n,; + €+ 1, with nr = 0,1, ... denoting the number of nodes in the radial function), while @ denotes the (relative) orbital angular momentum (L? = ¢(¢ + 1) Kh”). Though the EOB method is purely classical, it is conceptually useful to think in terms of the underlying (Bohr-Sommerfeld) quantization conditions of the action variables Ip and J to motivate the identification between n and @ in the two dynamics. Section 1.3 EOB description of the conservative dynamics of two body systems Figure 1.1. Sketch of the correspondence between the quantized energy levels of the real and effective

The ‘correspondence’ between the real and effective energy levels is illustrated in Fig. 1.1. At this stage, one needs to define a ‘dictionary’ between the real (relative) two-body dynamics, sum- marized in Eq. (1.3.10), and the effective one-body one, summarized in Eq. (1.3.14). As, on both sides, quantum mechanics tells us that the action variables are quantized in integers (reat = nh, Netr = nerrh, etc.) it is most natural to identify n = nem and € = Cem. One then still needs a rule for relating the two different energies Ey{""° and E «. Ref. [39] proposed to look for a general map between the real energy levels and the effective ones (which, as seen when comparing (1.3.10) and (1.3.14), cannot be directly identified because they do not include the same rest-mass contribution’), namely

where A’ = dA/dr. As explained above the EOB metric function A(r) is defined by Padé resumming the Taylor-expanded result (1.3.12) obtained from the matching between the real and effective energy levels (as we were mentioning, one uses a similar Padé resumming for D(r) = A(r) B(r)). One similarly needs to resum i y» Le., the y component of the radiation reaction which has been introduced on the r.h.s. of Eq. (1.4.5). Section 1.4 EOB description of radiation reaction and of the emitted waveform during inspiral Several methods have been tried during the development of the EOB formalism to resum the radiatio reaction F, y (starting from the high-order PN-expanded results that have been obtained in the literature Here, we shall briefly explain the new, parameter-free resummation technique for the multipolar wave form (and thus for the energy flux) introduced in Ref. [78, 79] and perfected in [65]. To be precise, th new results discussed in Ref. [65] are twofold: on the one hand, that work generalized the € = m = resummed factorized waveform of [78, 79] to higher multipoles by using the most accurate current] known PN-expanded results [80-83] as well as the higher PN terms which are known in the test-mas limit [84, 85]; on the other hand, it introduced a new resummation procedure which consists in consic ering a new theoretical quantity, denoted as pem(x), which enters the (¢,m) waveform (together wit other building blocks, see below) only through its ¢-th power: hem o (pem(a))*. Here, and below, denotes the invariant PN-ordering parameter given during inspiral by x = (GMQ/ 3)/3,

Figure 1.2. This figure illustrates the comparison (made in Refs. [66, 113]) between the (NR-completed) EOB waveform (Zerilli-normalized quadrupolar (2 = m = 2) metric waveform (1.5.2) with parameter- free radiation reaction (1.4.16) and with a5 = 0,ag = —20) and one of the most accurate numerical relativity waveform (equal-mass case) nowadays available [112]. The phase difference between the two is Ap < +£0.01 radians during the entire inspiral and plunge, which is at the level of the numerical error.

Figure 1.3. Close up around merger of the waveforms of Fig. 1.2. Note the excellent agreement between both modulus and phasing also during the ringdown phase.

Figure 1.4. Comparison (made in [100]) between various analytical estimates of the energy-angular momentum functional relation and its numerical-relativity estimate (equal-mass case). The standard “Taylor-expanded” 3PN E(j) curve shows the largest deviation from NR results, especially at low j’s, while the two (adiabatic and nonadiabatic) 3PN-accurate, non-NR-calibrated EOB E(j) curves agree remarkably well with the NR one.

The next-to-leading order spin-orbit Hamiltonian reads, [29], and the leading order spin(1)-spin(1) dynamics is given by, going beyond linear order in spin, e.g. [47,

Finally, the next-to-leading order spin(1)-spin(1) dynamics reads, [31], Also this Hamiltonian goes beyond linear order in spin. For its derivation the Tulczyjew energy-

Chapter 5 General Relativistic Theory of Light Propagation in Multipolar Gravitational Fields

Section 5.5 Gravitational Perturbations of the Light Ray

Integration of the light-ray propagation equations (5.3.60) — (5.3.63) is fairly straightforward. Perform- ing one integration of these equations with respect to time yields 5.5.2 Relativistic perturbation of the coordinate velocity of light

The leading order spin(1)-spin(2) expression of the spin-orbit center-of-energy vector renders zero. T] contribution to the center-of-energy vector from the spin(a)-spin(a) interaction can be found in [57]. To summarize, the following pN-Hamiltonians with spin are known fully explicitly, As examples, the up to the next-to-leading order spin-orbit center-of-energy vector is given by, [29, 56, 57), momentum tensor was needed one step beyond pole-dipole-particle approximation. All the previous Hamiltonians, spinless and spinning, were valid for both neutron stars and black holes with the excep- tion of the next-to-leading order spin(1)-spin(1) one which is valid for black holes only. Its generalization to neutron stars were achieved in [49, 50]. The next-to-leading order Hamiltonians found confimations through [49, 51-54], see [55]. Fe er i ee ee er ee ee oe

where 7 is the Euler number. The Inz-term has been calculated for the first time in [65, 66] and later verified in [28, 67]. The terms in e4 proportional to v* and v® have been obtained for the first time in [28] and were later confirmed by [68], and the term proportional to v? has been obtained only quite recently, [19]. The same holds with c, which has been derived in [42]. A numerical value of c; was known before, [41], coinciding, however, only in the first five digits, 153.88, with the analytic result. In the test-mass limit v = 0, the exact ISCO occurs for xjsco = 1/6 corresponding to the minimum

The given parametrization of the orbits is based on the 1pN parametrization by [72]. The latter parametriza tion is different from the one in [1] but identical to the one in [73] in the test-body limit. Through IpN order the parametrization given above plays an important role in current observations of binary pulsar systems, e.g. [74].

where the dimensionless spin-parameter a = Sc/GM? has the range a € [—1,1]. The canonica variables are (p,, L, Lz;1r, v, Q), where r and p,. denote the radial coordinate and its canonical conjugat« momentum, respectively, v being the true anomaly, i.e., the angle between the position vector r = rn and the direction of the ascending node N = (ez x L) /|ez x L|: cosv = n-N and sinv = n- W witt W =LxN/L. Qis the angle of the ascending node as measured from the x-axis of the non-rotatins orthonormal basis (ez,€,,e2): cosQ = N- e,; and sinQ = N - e,, measuring the precession of the orbital plane about L. which is tilted from the equatorial plane with the inclination angle 2, the angle between the background z-axis and the orbital angular momentum vector L, so Lz = Losi. The canonical Poisson-bracket relations read Chapter 2 Hamiltonian Dynamics of Spinning Compact Binaries

next step is to calculate the remaining integrals in the action (2.5.33), resulting in

Hereof the two periastron shifts, AU and A®, the intrinsic and the one related with the precession of th orbital plane, respectively, result in the form

This quantity can also be deduced from [78] apart from the higher pN-order term linear in a which has not been treated therein. As @ deviates from L only at the order a?/c*, in the obtained results, C may always be replaced with L. These periastron advances agree with corresponding results in [39, 47] in the test-mass limit, apart from the higher pN-order term linear in a which does not appear therein. For the orbital period we get 2.5.3 Two-body systems with spinning components The motion of spins and precession of orbits of spinning binaries are important indicators for the grav- itational interaction. From an observational point of view, only the leading pN-order interactions are important, so in the following we will restrict ourselves to those interactions, particularly to the leading order spin-orbit and spin(1)-spin(2) interactions. Also, we shall restrict ourselves to the center-of-energy systems.

In the case of equatorial motion the resulting shift, say A®, is given by

where = 7”, 0, is the D’Alembert (wave) operator of the Minkowski spacetime, h = hi; = 65"7hij, and the angular brackets around tensor indices in h<ijs = hij — 6i;h/3, denote the symmetric trace-free (STF) part of the spatial components of the metric tensor [the STF tensors which are explained in Chapter 5 of this book. They are also thoroughly discussed by Thorne [146], Blanchet and Damour [11] and in a review article by Poisson et al. [127]. Equations (3.3.30)-(3.3.33) are valid in the class of coordinates defined by the gauge condition (3.3.22).

where fi) = ee vi = dw /du is a coordinate velocity of the body’s matter element with respect to the origin of the local coordinates w®. It is worth to emphasize that the integrals given in this section, are taken over a hypersurface of the coordinate time wu. It does not coincide with the hypersurface of the coordinate time ¢, which is used in the integrals (3.4.13)—(3.4.20) defining gravitational potentials in the global coordinates x®. This remark is important for making the post-Newtonian transformations of the potentials from one frame to another as it requires to use the Lie transport of the integrand function between the hypersurfaces [93, Section 5.2.3]. Chapter 3. Covariant Theory of the Post-Newtonian Equations of Motion of Extended Bodies

Section 3.4 Parametrized Post-Newtonian Coordinates Vector Ci; in (3.4.43)-(3.4.45) is the angular velocity of the kinematic rotation of the spatial axes of the local coordinates with respect to the global coordinates. We keep its contribution only up to quadratic terms which is sufficient for our goal. We also assume that C” has the post-Newtonian order of magnitude being comparable with the rate of the geodetic precession. The external solution contains monopole terms, Q = Q(u) and A = A(u), entering Ae and hes respectively. Function @ defines the unit of measurement of the coordinate time wu at the origin of the local coordinates, and function A defines the unit of measurement of spatial distances on the hypersuface of constant time wu. Physical meaning of the external multipoles @z can be understood if one writes down the Newtonian equation of motion of freely-falling test particle [93, 94]. It turns out that @; is an acceleration of the local frame with respect to the geodesic world line of the particle, and Qz (J > 2) describes the tidal force with multipolarity / (quadrupole, octupole, etc.).

After fixing the gauge freedom as indicated above, the external metric tensor assumes in the local coordinates the following form

where Vz denotes a volume of the body B. This completes derivation of the metric tensor in the local coordinates. The Mixed Terms The mixed terms entering the metric tensor in the local coordinates are given as a particular solution of the inhomogeneous field equation (3.3.31) with the right side taken as a product of the internal and external solutions found on the previous step of the post-Newtonian iterations. Solving (3.3.31) yields where a double dot above @, denotes a second derivative with respect to time u. We have excluded the monopole, Q, and dipole, Q' terms from the non-linear part of the local metric tensor because they are absorbed to (yet unknown) Q and Q in the linear part of the metric (3.4.46). We could also decompose the product of the two sums in (3.4.49) into irreducible parts and absorb the STF piece of this decomposition to multipoles Q; (J > 2). This procedure is done later in the process of matching the internal and external solutions.

given by two equations [93, 101], is the (global) coordinate distance taken on the hypersurface of constant time ¢ between the point of matching, x’, and the origin of the local coordinates, 73 = x(t). ag ve oe ian ww - a a ae | wee Ps e i -y oe ae py aie ae

that describe the post-Newtonian transformation between the coordinate time wu of the local coordinates and the coordinate time ¢ of the global coordinates. The other functions entering (3.5.16), (3.5.17) are defined by algebraic relations

where the external scalar field @ coincides in this approximation with the external Newtonian potential U and is computed at the origin of the local coordinates, x’;(t), at the instant of time t. We remind that the scalar field perturbation is coupled with the factor 7 — 1, so that all physically-observed scalar- field effects must be proportional to this factor. We also notice that the external monopole (J = 0) and dipole (7 = 1) of the scalar field can not be removed from the observable gravitational effects by making coordinate transformation to a freely-falling frame because if a scalar field presents in one coordinate frame it must be present in any other coordinates. In other words, scalar fields do not obey the principle of equivalence. It was the primary reason why Einstein abandoned a pure scalar theory of gravity. External mass-type multipoles Q” for 1 > 2 are defined by the following equation ? where the dot denotes the time derivative with respect to time t. The external multipole moments @;, and Cy are analogues of Dixon’s multipole moments Aq,...a,;.7 and Bay...c;uv respectively [see (3.1.13) and (3.1.14)]. We shall use the above expressions for the external multipoles in derivation of equations of motion of extended bodies in next section.

where the metric tensor perturbations hoo. loos hoi, hij and h, = hii in the local coordinates have been

Thus, we define the post-Newtonian center of mass of the body by making use of (3.4.54) for the multipolar index 1 = 1. After some simplifications and re-arrangements of terms in the integrand, the conformal dipole moment reads Functional form of equations of motion of extended bodies in N-body system depends crucially on the choice of the reference point inside the body defining its center of mass. Position of the center of mass of the body is determined by the value of the internal dipole moment of the body. In scalar-tensor theory of sravity there are two possible definitions of the dipole moment: active dipole moment J’ and conformal dipole moment J’. It is difficult to foresee which definition is the best. Only after completing direct calculation of equations of motion it becomes clear that it is the conformal dipole moment yields the most optimal choice of the post-Newtonian center of mass of each body [93, 94, 158]. This is because after double differentiation with respect to time, only the conformal dipole moment leads to conservation of the individual body’s linear momentum, p’, while the post-Newtonian scalar or active dipole moments do not bear such a property.

where the dot denotes the time derivative with respect to wu. After taking the derivative and using the local equations of motion of matter (3.6.3), we obtain: Until now the point a(t) has represented a location of the origin of the local coordinate system in the global coordinates taken at the time t. In general, the origin of the local coordinates may not coincide with the center of mass of the body B which can move with respect to the local coordinates with some velocity and acceleration. In order to be able to keep the center of mass of the body at the origin of the local coordinates we have to prove that for any instant of time the dipole moment defined by (3.6.10) and its time derivative that is, the linear momentum of the body) given by (3.6.13, can be kept equal to / The post-Newtonian linear momentum of the body, p’ , is defined as the first derivative of the dipole moment (3.6.10) with respect to the local time u, coordinates coincides with the center OF Mass OF the body, the dipole moment vanishes £ = VU The post-Newtonian definitions of mass and the center of mass of the body depend not only on the distribution of density of matter, its velocity and stresses inside the body but also on other terms describing mutual gravitational coupling of internal and external multipoles. Thorne and Hartle [147] believed that these terms introduce ambiguity to the definitions of mass and the center of mass as they can be either included or excluded to these definitions. We have shown [93, 94, 158] that this ambiguity is fictitious and does not exists. The coupling terms must be included to the definitions of mass, center of mass and other internal multipoles to achieve the most economic form of the equations of motion depending only on the mass-type and spin-type internal multipole moments of the bodies. It is also true that only under such definitions we can get EIH equations of motion in the limiting case of spherically- symmetric and non-rotating bodies. —_ oe 7 z a 7 ioe ow i4@ & «¢ Sf & aaah a =< sk: ee Qe,

Translational equations of motion of the center of mass of body B with respect to the local coordinates w, which cover the body and its neighbourhood, are derived by making use of definition (3.6.13) of its linear momentum, p’. Differentiating (3.6.13) one time with respect to the local coordinate time wu, making use of the microscopic equations of motion (3.6.1)-(3.6.3), and integrating by parts to re-arrange a number of terms, one obtains [93, 158]

where M and M are the conformal and active gravitational masses of the body B and we again omitted terms which are proportional to the dipole moment J‘ and linear momentum p’ of the body like we did it in (3.6.16). The two masses, M and M, are not equal according to (3.6.9). The difference between them plays a role of a “charge” of the scalar field ¢ which couples with the dipole moment P’ = U,; of the external scalar field leading to the Dicke-Nordtvedt effect [52, 54, 153]. In general relativity, 1 = M, It is important to notice that the right side of (3.6.16) is the force exerted on the body B due to the gravitational coupling of its internal active multipole moments, J”, S” with the external multipole moments @r, Pz, Cr. The right side of (3.6.16) also contains the inertial force, J1@;, due to the non- geodesic motion of the origin of the local coordinates. Hence, in the most general case, p’ 4 0 which means that the center of mass of the body B accelerates with respect to the origin of local coordinates and the world line Z of the center of mass does not coincide with the world line W of the origin of the local coordinates. However, the external dipole moment Q; is yet unconstrained function of time and we can always reach consensus with the condition (3.6.15) by choosing @; in such a way that it compensates the gravitational coupling force. To eliminate acceleration of the world line Z with respect to W and to fix the center of mass of the body B at the origin of the local coordinates we, first, impose condition (3.6.15) and solve (3.6.16) with respect to Q;. It constrains the inertial force and yields

Multipolar expansion of the other external potentials is given by External vector potential is expressed in terms of the internal multipole moments as follows external potentials that are shown below. The “bad” multipoles, N% and R4, will not show up in the final form of equations of motion for the same reason — the mutual cancellation with similar terms. External vector potential is expressed in terms of the internal multipole moments as follows

where we have also used index D to numerate the bodies in (3.7.16). One more potential Chapter 3. Covariant Theory of the Post-Newtonian Equations of Motion of Extended Bodies The above expansions contain a number of integrals depending on the internal structure of the bodies explicitly. They will mutually cancel out by similar terms after substitution of these expansions to equations of motion (3.5.26).

where a dot denotes a covariant derivative defined later in (3.8.41) and we have introduced the following where ¢ is the scalar field perturbation caused by external with respect to the body B masses. In the global coordinates ¢ = U defined in (3.5.24a).

The following Theorem is not restricted to the static case. the symmetric Cartesian (every index takes the values x,y,z) tensor yi, ...i, 18 trace-free, i.e., a Cartesian STF-tensor. By induction one can prove that

Note, that the conservation of angular momentum implies that the orbit is confined to a coordinate plane. Tleino nolar coordinates in the arhital nlane with For the PN Schwarzschild field the Lagrangian takes the form the specific energy € and the absolute value of the orbital angular momentum J = |J| can be written as

This implies that the post-Newtonian orbit is that of a precessing ellipse, the secular drift of the argument of periastron per revolution being given by

Theorem 4.5: From the tetrad condition (4.6.9) one infers that 4.6.4 The standard transformation between global and local coordinates The gravitational influence of external bodies is effaced and relation (4.6.10) is sometimes called weak effacement condition.

Figure 4.1. One global and N local coordinate systems are used for the description of the gravitational N-body problem (from [44]).

Figure 4.2. Three events of importance for the inversion of the coordinate transformation, ex, ez: and

where, by convention, we are assuming that any term which contains an undefined tidal moment has to be replaced by zero. E.g., the term containing the factor (71 — 4)/(21 + 1) vanishes for | = 0. where, by convention, we are assuming that any term which contains an undefined tidal moment has Theorem 4.6: The external gravito-electric and gravito-magnetic fields, E and B, are completely deter- mined by the corresponding sets of tidal moments, G, and Hz, to post-Newtonian order. Explicitly they are given by

The tidal moments, GZ and Hf, can be split into two parts: a part resulting from the gravitational action of some external body A¥ E, and some inertial part related with the inhomogeneous part in the transformation of metric potentials: With similar techniques one can show that apart from simple gauge-terms, given by a single function A, also the external potentials, W and Wg, can be expressed in terms of the tidal-moments G'z and Hz, (remember that W(X* = 0) = W.(X*% = 0) = 0).

From this we get the Blanchet-Damour-Schafer (BDS) form [45] of the post-Newtonian Euler-equation: Since VG =1+ 2W/c? + Ou, the right hand side of the last equation reads:

So far the central elements of Damour et al., [2] were exhibited and complemented with some so far unpublished material devoted to certain aspects of the whole issue. The DSX-formalism was further developed by three following papers [3-5]. a, In the second paper [3] the problem of translational Jaws of motion for a system of N arbitrarily composed and shaped, weakly self-gravitating, rotating deformable bodies are obtained at the first post- Newtonian approximation to Einstein’s theory of gravity. The full set of mass- and spin- multipole moments, M7, and S'z, was taken into account for each of the bodies. First complete and explicit results for the laws of motion of each body of the system were presented as an infinite series exhibiting the coupling of all the BD-moments to the post-Newtonian tidal moments, G; and Hz, felt by this body. Finally explicit expressions of these tidal moments in terms of BD-moments of the other bodies are derived. For a derivation of corresponding equations of motion assumptions about the time dependence of BD-moments for each of the bodies have to be made. A rigidly rotating multipole model, that leads to a closed set of dynamical equations of motion, was presented in Klioner et al., [47]. eet, = ueizecced cose wc oS ees coms = ees: lig weds = cds cee ecciiee cee eee ees = oc cece: sds: aise a oe ces Gees = ees

Figure 5.1. The Penrose diagram shows a worldline of the isolated system (binary star) originating at past timelike infinity J~ and ending at the future timelike infinity /*. Both light and gravity propagate along null geodesics going from the past null infinity J~ to the future null infinity J*. A particular light geodesic is emanating from the event (to,ao0) and ending at the event (t,a). Extrapolation of this light geodesic to the past null infinity defines the null wave vector k* = (k°,k) of the electromagnetic wave under consideration. Chapter 5 General Relativistic Theory of Light Propagation in Multipolar Gravitational Fields

re JVU and §S; are the total mass and spin (angular momentum) of the system, and J4, and Sa, > two independent sets of mass-type and spin-type multipole moments, N' = «*/r is a unit vector ected from the origin of the coordinate system to the field point. Because the origin of the coordinate stem has been chosen at the center of mass, the expansions (5.2.5) — (5.2.8) do not depend on the iss-type dipole moment, J;, which is equal to zero by definition. We emphasize that in the linearised proximation the total mass JVl and spin 8; of the astronomical system are constant while all other iltipoles are functions of time which temporal behaviour obeys the equations of motion derived from > law of conservation of the stress-energy tensor of the system [1, 2]. Gravitational waves emitted by > system reduce its energy, linear and angular momenta. This effect does not appear in the linearised neral relativity but in higher order approximations we would obtain where the mass, spin, and linear ymentum of the system must be considered as functions of time like any other multipole. Higher-order avitational perturbations in the metric going beyond (5.2.5)-(5.2.8) are shown in a review paper by anchet [97]. They are not of concern in the present chapter. The canonical metric tensor (5.2.5)-(5.2.8) depends on the multipole moments J 4, (t—r) and 8 4, (t— taken at the retarded instant of time. The retardation is explained by the finite speed of propagation where we are, the so-called, gauge functions describing the freedom in the choice of coordinates cover- can. ing the manifold. The canonical perturbation, h2'3, obeys the homogeneous wave equation in vacuum

These functions, after they are substituted to equation (5.2.3), transform the canonical metric tensor perturbation to a remarkably simple form The most convenient choice simplifying the structure of the metric tensor perturbations, is given by the following gauge functions

where Wa,, X4,, Yia,_,, and 2a, _, are Cartesian tensors depending on the retarded time. Their specific form is a matter of computational convenience (or the boundary conditions) for derivation and interpretation of observable effects but it does not affect the invariant quantities like the phase of elec- tromagnetic wave propagating through the field of the multipoles.

Equation (5.3.57) is linear with respect to the perturbation of the metric tensor, hag. Hence, it can be linearly decomposed in the equation for perturbations of the light-ray trajectory caused separately by mass and spin multipole moments. Substitution of the metric tensor (5.2.5) — (5.2.8) to Eq. (5.3.57) and replacement of spatial derivatives with respect to x’ with those with respect to parameters €' and T by making use of (5.3.30) yield the following linear superposition

Chapter 5 General Relativistic Theory of Light Propagation in Multipolar Gravitational Fields Reference tetrad

which is written down for the case of weak gravitational field, gag = Nag + hag. Here the unit spatial vectors a = (a',a”,a*), b = (b',b*,b°), and k = (k',k?,k®) are orthonormal in the Euclidean sense (6;j;a°b? = isa’ ko = = Oig b’k? = 0 and dija'ad = dij bb? = 047 kik) = 1) with vector k pointing to the direction of propagation of the light ray at infinity as given in (5.3.18). These basis vectors are convenient to track the changes in the parameters of the electromagnetic wave as it travels from the point of emission of light to the point of its observation. Let us define at each point of spacetime a coordinate basis of static observers t is worth noticing that the observer’s tetrad eG has two group of indices. The indices without round rackets run from 0 to 3 and are associated with time and space coordinates. The indices enclosed in the ound brackets numerate vectors of the tetrad and also run from 0 to 3. The coordinate-type indices of the strad have no relation to the tetrad indices. If one changes spacetime coordinates (passive coordinate ‘ansformation) it does not affect the tetrad indices while the coordinate indices of the tetrad change 1 accordance with the transformation law for vectors. On the other hand, one can change the tetrad ectors at each point in spacetime by doing the Lorentz transformation (active coordinate transformation) vithout changing the coordinate chart [106].

where t = t — t* and we used equation (5.3.25) explicitly while expressing the distance r = ry (T) as a function of time 7. The constant distance r; was introduced to make the argument of the logarithmic function dimensionless. This constant is not important for calculations as it always cancel out in final formulas. However, in the case of gravitational lensing it is convenient to identify the scale constant rz with the radius of the Einstein ring [62-65] “ In the case when mass and spin are constant, the integrals that we need to carry out are reduced to (1/r]'-4 and [1/r]!-?! which are formally divergent at the lower limit of integration at the past null infinity when time t + —oo. However, one must bear in mind that these integrals do not enter equa- tions of light-ray geodesics (5.3.57) alone but appear in these equations after taking at least one partial derivative with respect to either €' or 7 parameters. This differentiation effectively eliminates divergent parts of the integrals from the final result. Hence, in what follows, we drop out the formally divergent terms so that the integrals under discussion assume the following form

However, taking one more (a third) derivative eliminates all integrals from equation (5.4.15) completely. More specifically, we have Dealing with the double integrals of type [F'(t—1) / ri ~?] is more Sophisticated. We notice that taking the first and second derivatives from the double integral [F(t — r)/r]'~7! do not eliminate the integration along the light ray trajectory and making use of equations (5.4.12), (5.4.15) for explicit calculation of partial derivatives in the right part of equation (5.4.16) proves that all integrations disappear, that is

where H(p—q) is the Heaviside function defined by the expression (5.1.21), Ci(p,q) are the polynomial coefficient given by equation (5.1.20) and F)(t — r) denotes 0? F(t — r)/Ot” where r is considered as constant. Section 5.5 Gravitational Perturbations of the Light Ray

Figure 5.2. Astronomical coordinates used in calculation of light propagation. The origin of the coor- dinates is at the center-of-mass of the source of gravitational waves. The bundle of light rays is defined by a vector field k*. Vector K' = —k’ + O(c~*) is directed from observer towards the source of light. Vector Kis directed from the observer towards the source of gravitational waves. We define Ki = —N' = —a' /r, where x’ are the coordinates of observer with respect to the source of gravita- tional waves, and r = ||. The picture shows the plane of the sky being orthogonal to vector K’. Section 5.5 Gravitational Perturbations of the Light Ray

There exists a relation between the relativistic perturbations of the eikonal and the light-ray trajectory Equation (5.6.1) defines the light time delay effect in the global time ¢ of the ADM-harmonic reference frame. In order to convert it to observable proper time T’ of the observer, we assume for simplicity that the observer is in a state of free fall and moves with velocity V’ with respect to the reference frame of the isolated system. Transformation from the ADM-harmonic coordinate time ¢ to the proper time T’ is made with the help of the standard formula [1, 2] wave but also along the light rays citepfrolov. Both derivations lead, of course, to the same result

These expressions do not contain any explicit integration along the light ray trajectory because all ex- plicit integrals are either eliminated after taking partial derivatives with respect to the upper limit of the integrals or they are vanish because the numerical coefficient in front of them become nil.

When deriving these equations we assumed that the time evolution of both the mass JV and the angular momentum 8° of the isolated system can be neglected. In opposite case, the right side of equations (5.6.50) and (5.6.51) would contain also time derivatives from the mass and the angular momentum. Terms of the order |€|/R in both formulas are produced by the time derivatives of vector k.

use of equations from section 5.3.3. The result is as follows

The Skrotskii effect due to the mass-type multipoles of the isolated system is given by

where Cx is given by equation (5.7.5) and ¢* is the time of the closest approach of the light ray to the barycenter of the isolated system. product of two parameters. Therefore, the convergence requires satisfaction of two conditions where the numerical value of the parameter ¢, must be taken for the smallest wavelength, Amin, in the spectrum of the gravitational radiation emitted by the isolated system. Conditions (5.7.10), (5.7.11) en- sure convergence of the post-Newtonian series in (5.7.8) and (5.7.9) respectively. If the source of light rays and observer are at infinite distances from the isolated system then « ~ 0 and €o ~ O, and the requirements (5.7.10), (5.7.11) are satisfied automatically, irrespective of the structure of the Fourier spectrum (5.4.5) of the gravitational radiation emitted by the isolated system. In a real astronomical practice such an assumption may not be always satisfied. In such cases it is more natural to avoid the

If we assume that the mass and angular momentum of the isolated system are conserved the asymp- totic expansions of integrals (5.4.1), (5.4.2) of the stationary part of the metric tensor have the following form Next several equations yield the estimates for the partial derivatives with respect to the parameters €' and 7 from functions of the retarded time (of gravity). In these estimates the numbers m and n depend on 1; we give the estimates from below for m and n for / > 1. post-Newtonian expansions of the metric tensor and/or observable effects and operate with the func- tions of the retarded time. It is important to notice that the retarded time s = ¢ — r which enters the result of the calculation of the is a characteristic of the Einstein equations of gravitational field, not the Maxwell equations. Therefore, measuring the effect of the gravitational deflection of light caused by time-dependent gravitational field allows us to measure the speed of gravity with respect to the speed of light. This type of experiments have been proposed in our paper [118] and successfully performed in 2003 [119]. There were several publications (see review [150]) arguing that the speed of gravity is irrelevant in the light-ray deflection experiments by moving bodies. Unfortunately, all the authors of those publications operated with the post-Newtonian expansion of the gravitational field which replaces the retarded time s = t — r of the gravitational field with the time ¢* of the closest approach of light to the light-ray deflecting body like in equation (5.7.6). This explains why the effect of the retardation of gravity was confused with the retardation of light in L180). wee i i wee me i

The relativistic time delay is given by The observable unit vector in the direction from the observer to the source of light is given by the following expression This section provides the reader with the asymptotic expressions for potentially observable relativistic effects by taking into account only the leading terms and neglecting all terms which are proportional to the parameter e.

One more asymptotic expression is given for the integral taken from an STF partial derivative Taking into account only the leading order terms in (5.8.15), we obtain the asymptotic expansion Using this expression we can write the Taylor expansion for the functions of the retarded time s = t — 7

In this section we give expressions for the relativistic effects of the time delay, bending of light and the Skrotskii effect in the gravitational plane-wave approximation. In this approximation we neglect all terms of the order of 67, 62 and 52, and higher. For the time delay we have 5.8.2 Asymptotic expressions for observable effects

The observed astrometric direction from observer to the source of light in the plane gravitational-wave annroximation is The observed astrometric direction from observer to the source of light in the plane gravitational-wav

Figure 5.4. Relative configuration of observer (O), a source of light (S), and a localized system emitting of gravitational waves (D). The gravitational waves deflect light rays which are emitted at the moment to at the point S and received at the moment ¢ at the point O. The point E on the line OS corresponds to the moment of the closest approach of light ray to the system D. Notations for distances are OS = R, DO =r, DS =17r0, DE = d- the impact parameter, OF = 7 = rcos6, ES = 7 = 7 — R. The distance R is much smaller than both r and ro. There is no limitation on the impact parameter d which can be or may be not small as compared to all other distances.

where fg = P= lq in accordance with definition (7.3.25). The potential energy of the scalar field, W = W/(®), is kept arbitrary.

where we used the superposition € = €m + €, P = Pm + Pq. Second step is to transform the linear differential operator F’i,, in (7.5.37) to a more convenient form that will allow us to single out the gauge- dependent vector denoted by Linearized field equations for gravitational field variables, J,,,, are obtained from (7.5.28) after neglect- ing in its right side the non-linear source J,,,, and making a series of transformations to sort out similar terms. First, let us make use of equations (7.5.40) and (7.5.41) to find out

Here, H is a conformal Hubble parameter (7.4.4), a prime means a derivative with respect to 7, and we denoted big — FP | eB. q= (loo + lik) /2, len = liitleetlss, lays. = bag (1/3) dijlek, and the same index notations are applied to the tensor of energy-momentum %;; of the localized astronomical system, Deis = Liz — (1/3)bij Tex. These equations are clearly decoupled from one another, thus, demonstrating the advantage of the gauge sendition Be =0.

The Laplace-Beltrami operator for function q has been derived in (7.6.25). Now, we make use of equa- tions (7.6.25), (7.6.28a), (7.6.33) in calculating the right side of (7.6.32). After a significant amount of algebra, we find out that all terms explicitly depending on q and the gauge functions B® cancel out, so that equation for Vm becomes The Laplace-Beltrami operator for function ym is given in equation (7.6.28a) which is not gauge- invariant. Taking the covariant derivative from this equation and contracting it with &@* brings about the first term in the right side of equation (7.6.32),

This equation can be also derived by the procedure being similar to that used in the previous subsection in deriving equation for Vm. We followed this procedure and confirm that it leads to (7.6.37) as expected. Equation (7.6.37) is clearly gauge-invariant. Besides Vq it depends on variable x and should be solved along with equation (7.6.29). which is obtained after subtraction of equation (7.6.13a) from (7.6.13b). Equation for Vq is derived directly from (7.6.36) and equations (7.6.34) and (7.6.29) for Vm and y respectively. We obtain, This equation can be also derived by the procedure being similar to that used in the previous subsection

of the wave operators for the scalar, vector and tensor fields in the conformal spacetime and in arbi- trary coordinates. Notice that although the conformal spacetime coincides, in case of k = 0, with the Minkowski spacetime, the metric fg is not the diagonal Minkowski metric 7. unless the coordinates are Cartesian. Of course, the covariant derivative from a scalar must be understood as a partial derivative, thatis Pg = Fg.

Similar procedure is applied to equation (7.6.37) and leads to a wave equation for function V, de-

This is a wave equation with the speed of sound v, which determines the speed of propagation of the scalar perturbations in the dark matter considered as an ideal fluid. These perturbations can be interpreted as acoustic or sound waves of different wavelengths propagating in spacetime. Solution of homogeneous equation (7.7.37) describes the propagation of primordial scalar perturbations of dark matter. A particu- lar solution of the inhomogeneous equation (7.7.37) tells us how the perturbation of dark matter caused by the isolated astronomical system propagate. Dark matter and dark energy are described by scalar fields ® and V. The fields themselves are not gauge- invariant. Therefore, physical meaning have only the equations for the gauge-invariant perturbations of these fields which are Vn, Vy, and y. We consider, first, equation (7.6.34) for the gauge-invariant scalar Vn. We convert the covariant derivatives taken with respect to the background metric, gag, to the partial derivatives of the conformally-flat metric, fg and use equation (7.7.31a) for the Laplace- Beltrami operator along with the expressions for various cosmological parameters given in section 7.7.2. After arranging terms with respect to the powers of the Hubble parameter J, we obtain the scalar equation for function Vm describing the perturbations of dark matter,

The reader can observe that equations (7.7.42a)-(7.7.42d) for linearized metric perturbations are decou- pled from each other. Moreover, equations (7.7.42b-(7.7.42d) are decoupled from the matter perturba- tions Vn, Xm, etc. Only equation (7.7.42a) for q is coupled with the matter perturbations governed by Post-Newtonian equations for gravitational perturbations in physical spacetime are (7.6.25), (7.6.27), (7.6.38) and (7.6.40). These equations are gauge-dependent. In order to fix the gauge we imposed the gauge conditions (7.5.64), (7.6.43). In this gauge, equations for the conformal metric tensor perturba- tions become

This equation describes propagation of the density perturbation of dark matter, dm, in the form of sound waves with velocity v;. Equation (7.8.6) is decoupled from any other perturbation and can be solved separately after the boundary conditions are specified. For this reason we call (7.8.6) master equation.

where the scale factor a in front of the integrals depends on the coordinates of the field point a = a [n(t,a)], and the functions in the integrand depend on the retarded time A particular solution of the wave equations satisfying condition (7.8.28), is the retarded integral [75. where 27? = (2°,a'), 7 = X° is the conformal time in isotropic coordinates connected to the coordinates x by a Lorentz boost 7 = n(27) = L°gx%, the null vector n® = {1,2'/r}, and r = 6;j;2'«? is the radial distance. This condition ensures that there is no infalling gravitational radiation arriving to the localized astronomical system from the future null infinity J‘. Effectively, it singles out the retarded solution of the wave equation. Whether the boundary condition (7.8.28) is valid or not, we do not know for sure because our knowledge of the universe is limited by the existence of the cosmological (also known as light or particle) horizon [81] that represents the boundary between the observable and the unobservable regions of the universe. Nonetheless, in case of spatially flat (k = 0) universe, the condition (7.8.28) seems to be highly plausible. ey EY CS Ae hs a ae Sea Cae et ee Sa dy MS ig tt eS A Se A Si a ey Me Bae a , because gravity propagates with finite speed. Equation (7.8.30) describes characteristic of the null cone in the conformal Minkowski spacetime that determines the causal nature of the gravitational field in the expanding universe with k = 0 []. Solutions (7.8.29a)—(7.8.29d) are Lorentz-invariant as shown by calculations in section 7.8.4.

and y = 1/,/1— £? is the Lorentz factor. Equation (7.8.90) has two roots corresponding to the ad- vanced and retarded times. The root corresponding to the retarded-time solution of (7.8.90) is

where the coefficients A,B,C of the quadratic form are,

Chapter 2 Hamiltonian Dynamics of Spinning Compact Binaries The 6 constants C411, C412, C413, C211, C212, Co1 are still unknown. Their calculations will need an in- tricate treatment of spacelike infinity in a post-Newtonian setting. The both articles [41, 42] clearly indicate this intricate problem.

Chapter 2 Hamiltonian Dynamics of Spinning Compact Binaries

Up to 2pN order, these expressions read, [39, 71],

Post-Newtonian expansion of the metric tensor and the scalar field introduces a corresponding expan- sion of the stress-energy-momentum tensor of matter In order to simplify notations, we shall use the following abbreviations:

where y+: = Ox/Ot?, the post-Newtonian potential

Chapter 3 Covariant Theory of the Post-Newtonian Equations of Motion of Extended Bodies

Section 3.7 Post-Newtonian Equations of Motion of Extended Bodies in Global Coordinates

The multipolar tensor coefficients are expressed in terms of the internal (active) multipole moments and their derivatives. They also depend on velocities uv’, viz, Uc of the bodies and on their relative velocities,

Four-dimensional scalar field external multipoles are defined by (3.5.27) and (3.8.31) and reads

which provides the post-Newtonian momentum-velocity relation that has not been established in Dixon’s paper [62] to full extent. Partial progress in finding this relation has been made by Ehlers and Rudolph [67] but their momentum-velocity relation is incomplete and, furthermore, goes beyond the post-Newtonian approximation without justification. It is straightforward to check that the dynamical velocity (3.8.44) is orthogonal to the kinematic ve-

Chapter 3. Covariant Theory of the Post-Newtonian Equations of Motion of Extended Bodies Herein, functions ® and © without indices correspond to definitions (3.8.37) and (3.8.38) with indices

In the following we will frequently encounter such order symbols that we will often abbreviate by For the orbital velocity of the Earth about the Sun we have

For the components of the Riemann curvature tensor one obtains We will now compute the differential geometrical quantities to post-Newtonian accuracy for celestial mechanical problems:

The global equations of motion in the gravitational N-body problem result from the local evolution

Cartesian tensors of the mass-type (mass) multipoles J< 4,s and spin-type (spin) multipoles $< 4,+ entirely describing gravitational field outside of an isolated astronomical system are always STF objects that can be checked by inspection of the definition following from the multipolar decomposition of the metric tensor perturbation hag [5, 56]. For this reason, to avoid the appearance of overcomplicated index notations we shall omit the angular brackets around the spatial indices of these (and only these) Cartesian tensors, that is we adopt: J4, = J<a,> and84, = $<a,>. We shall also use transverse (T) and transverse-traceless (TT) Cartesian tensors in our calculations

Without going into further details of the observational procedure we give an explicit expression fot the total angle of the light deflection a’. We have where the dot between the two vectors denotes the usual Euclidean scalar product. It is worth empha- sizing that the observed direction s’ to each source of light includes relativistic deflection of the light ray in the form of three perturbations. Two of them, a’ and 7’, depend only on the quantities taken at the point of observation, but 6", according to equation (5.5.35), is also sensitive to the strength of the gravitational field taken at the point of emission of light. This remark reveals that according to relation (5.6.24) a single gravitational wave signal may cause different angular displacements and/or time delays for different sources of light located at different distances from the source of gravitational waves even if the directions to the light sources are the same.

Covariant generalization of (7.2.12)—(7.2.14) is not quite straightforward because the covariant derivative of the metric tensor gv; = 0, and we cannot use it as a dynamic variable. In this case, we consider the set of the metric tensor, g,., the Christoffel symbols [,,,, and the Riemann tensor Rg), as a set of independent dynamic variables. The action is given by (7.2.1) where F = /—@f (guv TP uv,R” by) is a scalar density of weight +1. Variation of F is The order, in which the covariant derivatives are taken, is imposed by the procedure of the extracting the total divergence from the variation of the action in (7.2.2). The order of the derivatives is important because the covariant derivatives do not commute.

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An overview manual for the GRAVSOFT Geodetic Gravity Field Modelling Programs

by Tanggo Rastawira

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Toward a Centimeter-Geoid Model for Engineering Surveying in Egypt: Status and Projected Activities

by Gomaa Dawod

With the broad spread of Global Navigation Satellite Systems (GNSS) utilization in geomatics and engineering surveying in Egypt in the last two decades, a precise geoid model becomes an essential demand. This paper investigates the... more

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Frontiers in Relativistic Celestial Mechanics. Volume 2.

by Sergei M Kopeikin

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ASSESSMENT OF GOCE MODELS IN EGYPT

by Ahmed Zaki

The European Space Agency’s dedicated satellite gravity field mission the Gravity Field and Steady-state Ocean Circulation Explorer (GOCE) is planned to achieve 1-2 cm accuracy level for geoid undulation and at 1-2 mGal level for gravity anomalies down to a spatial resolution of 100 km. This thesis attempts to answer the question: What is the accuracy of GOCE satellite-only global geopotential models in Egypt? The main research objectives that were investigated in order to answer this question are included in two steps: First, the four generations of GOCE models are assessed by using GPS/levelling stations, ground gravity anomalies data and the geoid model for Egypt (EGGM08). Secondly, the best GGM model in the previous assessment is complemented with local terrestrial gravity data to create a local gravimetric geoid, then it is tested against the GPS/leveling and the geoid model for Egypt EGGM08.
The main problem with such assessments is the different spectral content of the GGMs and terrestrial observations. Because the spatial resolution of GGMs is limited by their maximum degree, there is always an omission error in the GGM-derived functional. In contrast, terrestrial observations contain the full spectral signal. To overcome the problem of different spectral content, the Spectral Enhancement Method SEM was applied. In the SEM, the spectral gap between the GOCE-GGMs and the terrestrial observations is bridged partially by a combination of the high-degree spectral bands of EGM2008, and omission error estimates sourced from RTM (Residual Terrain Model) data, providing information on the very-short wave length gravity field constituents. To compute RTM omission error estimates, the RTM was constructed as the difference between the 30 arc second Shuttle Radar Topography Mission (SRTM) elevation model and the spherical harmonic expansion of the DTM2006.0 elevation data base, expanded to degree 2,160. Subtraction of DTM2006.0 spherical harmonic elevations from the SRTM elevations removes a large part of the ‘spectral information’ already implied by EGM2008 (expanded to degree 2,190).
The available point gravity anomalies data set consists of 991 stations. The date of these observations and their accuracy vary greatly between the most recent Egyptian National Gravity Standardization Network of 1997 (ENGSN97) that was established by the Survey Research Institute (136 stations) and older gravity surveys (855 stations) carried out by many private organizations several decades ago. The accuracy of ENGSN97 gravity values are 0.022 mGal, while the accuracy estimate for older gravity data are 0.5 mGal on average. The number and distribution of the free-air gravity anomaly stations are very poor, concentrated mainly along the Nile valley and a lot of areas are empty. The statistics of gravity anomalies difference between the GGMs and ground gravity stations after applying the Spectral
xviii
Enhancement Method (SEM) give good agreement in Egypt. The best model shows differences range between -41.61 mGal and 41.65 mGal with an average of 0.63 mGal and a standard deviation of about 12.59 mGal. After performing topographic – isostatic reduction, worse results were obtained. All models are also giving non – centered anomalies, due to the lack data in most of Egypt and the best model shows differences range between -76.18 mGal and 67.45 mGal with an average of -23.01 mGal and a standard deviation of about 21.45 mGal.
The GPS/ leveling data set belongs to the High Accuracy Reference Network (HARN), which furnishes the national Egyptian GPS datum established by the Egyptian Survey Authority (ESA) in 1995. The precision of the HARN network is 0.1 part per million (ppm), which can be written in another form as 1: 10,000,000. Although the HARN network consists of 30 stations, only those 17 points have first order levels. The precision of geoid undulations at these stations may be estimated as 1 cm or better. The distribution of the GPS/ leveling data used in the assessment (17 stations) is not good, a lot of areas are empty and the total is too small in relation to Egypt’s surface area. The differences between the Global Geopotential Models (GGMs) and GPS/levelling stations show incompatible geoid undulation in Egypt. Large difference were obtained, in the Western desert. The best model shows differences range between -1.65 m and 0.06 m with an average of -0.66 m and a standard deviation of about 0.39 m.
The Gravimetric geoid for Egypt have been computed in this thesis using DIR 1 geopotential model in the frame work of remove – restore technique by using Meissl's modified stocke's kernel. The computation were based on 991 land free –air gravity anomaly stations. Airy Heiskanen isostatic model was used to reduce the gravity anomaly. The computed geoid have been fitted to GPS – derived geoid by removing a trend surfaces. A kriging trend function has been computed using only 16 GPS/ leveling stations .The internal precision of the fitted geoid is very good (about 1 cm). The external geoid accuracy at the 3 GPS / leveling stations which were not used for the geoid fitting is 0.58 m (RMS).

Figure 2.3: Picture of the Super STAR accelerometer.

Figure 2.10: Magnetotorque (Rummel, 2005).

Civil Engineering- Public Work Department

Table 2.1: Specifications of the CHAMP mission (Rummel etal., 2002). Figure 2.1 (a) SST-hl concept (Rummel et al., 2002). (b) CHAMP satellite system.

Figure 2.6: Complete gravity gradiometer (Rummel, 2005). Figure 2.5: The star camera sensor to provide the attitude.

Figure 2.7: LAGRANGE GPS Receiver (Rummel, 2005).

Figure 2.2: (a) SST-LL concept (Rummel et al., 2002). (b) GRACE satellite system.

Figure 2.11: Top-down view on the GOCE satellite showing the accommodation of instruments. The acronyms in the figure stand for: star tracker (STR), coarse Earth-Sun sensor (CESS), magneto-torquers (MTR), magnetometers (MGM), The LAGRANGE GPS receiver (GPS), command and data management unit (CDMU), power conditioning and distribution unit (PCDU), and laser retn- reflector (LRR). The Xenon tank contains the fuel for the ion propulsion module while the Nitrogen tank contains the fuel for the cold-gas thrusters.

Figure 2.12: Scientific applications of the global gravity field. (Rummel, 2005)

An improvement of the geoid accuracy will lead to a better understanding of the physics behind orbit perturbations. An improved separation of perturbations from gravity field on the one hand, and on the other hand from other perturbing forces, such as atmospheric drag, solar radiation pressure, etc., will be possible. Thus, a reduction of orbital errors is feasible, leading, for example, to reduce radial orbit errors for altimetric satellites.

Figure 2.18: Overview of GOCE applications (ESA, 2010).

Figure 3.1: Illustration of the geoid, reference ellipsoid, and the vectors of the gravity and normal gravity. If Wp = Wo = Ug , the equations (3.9) can be rewritten as (Bruns’s formula):

Figure 3.2: The geometry of the RTM reduction.

Figure 3.4: The Pratt-Hayford model of isostatic compensation.

Figure 3.5: Local and regional compensation. 3.2.5 Rudzki’s Inversion

Figure 3.6: Geometry of Rudzki reduction in spherical approximation.

Figure 3.9: Representation of spherical coordinates.

Figure 3.11: Practical computation of the geoid. The remove-compute-restore technique depends on the combination of the different gravity field wavelength to determine the geoid. The window technique (A bd-Elmotaal and K ithtreiber, 2003) has been suggested to get rid of the double consideration of the topographic-isostatic masses within the data window in the framework of the remove-restore technique. The modified Strokes’ integral kernel has been suggested to possibly combine the local data signals with the global geopotential models. 3.5.1 The Window Technique

The window technique can be overcome this problem by adapt the used global geopotential model. The adapted model can be obtained by subtracting the effect of the topographic-isostatic masses of the data window, in terms of potential coefficients, from the reference field coefficients. Thus, removing the long wavelength part by using this adapted reference field does not lead to a double consideration of a part of the topographic- isostatic masses (no double hatched area in Fig. 3.13) (Abd-Elmotaal and Kiihtreiber, 2003).

Figure 3.13: The window remove-restore technique.

To compute RTM omission error estimates, we construct the RTM as difference between the 30 arc second Shuttle Radar Topography Mission (SRTM) elevation model and the spherical harmonic expansion of the DTM2006.0 elevation data base (Pavlis et al. 2007), expanded to degree 2,160. Subtraction of DTM2006.0 spherical harmonic elevations from the SRTM elevations removes a large part of the “spectral information’ already implied by EGM2008 (expanded to degree 2,190). The SRTM—DTM2006.0 residual topography possesses spectral energy beyond the resolution of EGM2008 (in spectral band nz +1=2,191 to~21,600, which is equivalent to the SRTM resolution of 30 arc second).The computation scheme for applied SEM in evaluation of GGM with GPS/levelling stations, the geoid model for Egypt EGGMO08 and ground gravity anomalies points will discuss in the next sections.

Figure 4.2: Distribution of the free-air gravity anomalies stations in Egypt.

Figure 4.5: The gravity anomalies difference between the DIR1 and ground gravity stations before applying the SEM.

Figure 4.7. Distribution of the 17 GPS/leveling benchmarks in Egypt. The statistics of the comparisons of the GGMs after applying SEM are given in Tables 4,7, 4.8 and 4.9. According to Tables 4.7, 4.8 and 4.9, GPS/leveling comparisons suggest ¢ geoid agreement of 0.48 m (DIR 1) to 0.39 m (ITG-GO) for the highest expansions. Figures 4.8, 4.9, 4.10, 4.11 and 4.12 shows the GPS/leveling differences of the highest expansions of the GGMs for Egypt.

(b) Geoid undulation differences GPS/leveling stations and the DIR 2.

(b) Geoid undulation differences GPS/leveling stations and the GOCO 2.

(c) Geoid undulation differences GPS/leveling stations and the ITG-GO.

Figure 4.13: EGGM08 geoid model for Egypt (A bd-Elmotaal, 2008).

Figure 4.14; Geoid undulations difference between EGGMO08 and DIR 1.

Figure 5.1: DIR 1 gravity anomalies in Egypt. Contour interval: 20 mgal. USA government, and the German and Italian Space Agencies. On February 11, 2000, the Shuttle

Figure 5.3: Distribution of the free-air gravity anomalies stations used in geoid determination.

Figure 5.4; 30 arc second SRTM + for Egypt.

Figure 5.6: The topographic- isostatic corrections for Egypt. Contour interval: 10 mgal. The effect of topographic —isostatic masses according to Airy-Heiskanen model has been computed by tcfour- program (Forsberg et al. 2008) for all masses within a radius of 167 km around the computational point. A constant value for the density of the topography of Po =2.67gcm~3. A value of .4 gcm~? used for the density contrast between the crust and mantle and a normal crustal thickness Ty of 30 km has been used. Figure 5.6 shows the topographic- isostatic corrections for window ( 20° N to 34° N, 24° E to 38 °E).

Figure 5.8: The indirect effect of the topographic isostatic reduction. isostatic reduction. The contribution of the reference geopotential model (DIR 1) Ney was computed according to E.q 3.66. (See. Figure 5.2)

Table5.3: Statistics of the remaining differences at the 16 GPS stations. Figure 5.10: The geoid difference (Ngps — Ngeoia ) before fit. Contour interval: 0.1 m.

Figure 5.12: Geoid difference for Egypt between EGGM08 model and computed geoid undulations.

Figure 5.14: Geoid difference for Egypt between EGM96 model and DIR1.

Table 2.2: Specifications of the GRACE mission (Rummel etal., 2002). GOCE (Gravity-field and steady-state Ocean Circulation Experiment) has been launched on March 17, 2009 by the ESA (European Space A gency), for the purpose of developing the

Table 4.1. The statistics of the gravity anomalies at ground gravity stations. Table 4.2.shows the statistics of gravity anomalies difference between the GGMs and ground gravity stations before applying the SEM. The result shows that the first generation of GOCE time-wise approach is better than the other time wise generations with standard deviation 16.98 mgal. Similarly, the first generation of GOCE combined model (GOCO) is better than the second and third generations. The same previous results the first generation of GOCE direct approach models DIR 1 is better than other models with standard deviation 16.15 mgal. For GOCE space-wise models, the second generation is better than the first generation, the standard deviation of SP2 is 16.62 mgal compared to 18.43 mgal for SP1. Figure 4.5 shows the gravity anomalies difference between the DIR1 and ground gravity stations before applying the SEM.

Table 4.3. The statistics of gravity anomalies difference between the GG Ms and ground gravity stations after applying SEM. Table 4.4 shows the statistics of gravity anomalies difference between the GGMs and ground gravity stations after remove the topographic —isostatic masses. The result shows that the first generation of GOCE time-wise approach is better than the other time wise generations with standard deviation 22.48 mgal. The first generation of GOCE combined model (GOCO) is better than the other generations. The same of previous results the first generation of GOCE direct approach models DIR 1 is better than the other models with standard deviation 21.58 mgal. For GOCE space- wise models, the second generation is better than the other generations.

Table4.5: The evaluation of GOCE models with GPS/leveling over the world (RMS).

Table 4.7; GPS/leveling differences, in m, of the highest expansions of GOCE direct approach and GOCE combined models (GOCO).

The GOCE short arc approach (ITG-GO) have standard deviation 0.39 m, and GOCE combined models (DGM and EIGEN-6S) standard deviation are 0.40 m and 0.42 m respectively. From the previous results the best model in this assessment is ITG-GO model with standard deviation 0.39m. The main differences occurs in the north of the western desert.

Table5.1: Statistics of the gravity anomalies in Egypt (991 gravity stations)

Table5.2: Statistics of the empirical tests for the cap size yy, at the GPS stations.

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Enhancement of Global Geopotential Harmonic Models for Egypt

by Abdelrahim Ruby and

The aim of this research is to develop a new precise and high-resolution geoid model for Egypt by refining the Global Geopotential Models (GGMs) through a process named tailoring, where the existing spherical harmonic coefficients of... more

shows the statistics of this heights without marine areas. In this study, the DTM2006.0 model, complete to degree and order 190, has been used to create a 5’x5’ mean heights for Egypt Fig. (4.6) by sing harmonic synth program (Holmes & Pavlis, 2006), in the area ounded by latitudes 212 to 332 and longitudes 24° and 382, and Table (4.6) haAtare tha ectatchre nf thic haimhte waithait marina araac

Figure (4.8): Distribution of NRIAG Gravity Data.

Table (4.3): Statistics of the 5 Arc-minute Anomaly Data Selected by the Merging Procedure Used to Develop the EGM2008 Model, Unit is mGal [source: Pavlis et al., 2012, Table (2)]. Over areas where only lower resolution gravity data were available

Figure (5.28): Difference Gravimetric minus Combined Geoid Solution Computed by Using Tailored Geopotential Model EG1GOC5s in Remove-Restore Technique. Contour interval 5 cm.

Table (4.2): Parameters of Earth Gravity Field Model EGM2008. The available gravity anomaly data that were necessary for the computation of the 5 arc-minute area mean values to develop the EGM2008 model, divide into three sub-divisions (see Pavlis et al., [2012, sections 3.5]), as shown in Fig. (4.1a):- a) Areas without any restrictions [are colored green in Fig. (4.1a)], most of this area is ocean areas, where the altimetry-derived gravity anomalies data.

Figure (2.1): Attraction and Centrifugal Force.

Figure (2.5): Terrain Correction (Hofmann-Wellenhof & Moritz 2005, p. 136)

Figure (2.7): Pratt-Hayford Isostatic Compensation Model (Sanso &Sideris, 2013, p.349) A anrnminn tn BUnfmann_W allanhnaf and Manty (INN n1A1) tha

Figure (2.8): The geometry of the RTM reduction

Figure (2.9): The Reference Surfaces and Height Systems.

Figure (2.10): Signal (thick lines) and error (thin lines) amplitudes per degree in terms of gravity anomaly for the geopotential harmonic model EGM 2008 (Dotted), EGM96 (Dashed) and GOCO05s (Dash-dot).

Figure (2.11) Cumulative RMS of Gravity Anomaly and Geoid Error.

Figure (2.12): Essential parameters of covariance function

Figure (2.14): The First-order Levelling Network in Egypt (Saad, A., 1993).

the n = m location. We have: Then, the diagonal elements corresponding to the diagonal passing through

Figure (4.1): Geographic Display of the 5 Arc-minute Anomalies Used to Develop the EGM2008 model: (a) Data Availability. (b) Data Source Identification [source: Pavlis et al., 2012, Fig.(3)].

Figure (4.2): Data Availability for Egypt Used to Develop The EGM2008 Model [source: subset from Fig. (4.1)].

Figure (5.29): Difference Gravimetric minus Combined Geoid Solution Computed by Using Tailored Geopotential Model EGTM0818 i Remove-Restore Technique. Contour interval 5 cm.

Figure (4.3): Geographic Locations and Identification of Merged 30 Arc- minute Gravity Anomalies Used to Develop The EGM96. (Source: Lemoine et al., 1998, p.8-21).

Figure (5.24): Gravimetric Geoid Norav-a for Egypt. Contour interval 1.0 m.

Figure (5.25): Gravimetric Geoid Norav-8 for Egypt. Contour interval 1.0 m.

Figure (4.5): Distribution of Egyptian Gravity Data in A frican Gravity Project. [Source: Fairhead at el., 1997, subset from Fig. (1)].

Table (2.2): Numerical Values of Some Parameters of WGS 84 Ellipsoid.

Table (4.1): Parameters of Earth Gravity Field Model GOCO05s.

Table (4.4): Parameters of Earth Gravity Field Model EGM96.

Hence, each gravity stations of the ENGSN97 precisely have a three-dimensional geodetic position (latitude, longitude, and geodetic height) from GPS data, as well as vertical position made by the orthometric height from precise levelling. The standard deviation of the final solution of the ENGSN97 network is range from 0.002 to 0.048 mGal (Dawod, 1998). Figure (4.9) Egyptian National Gravity Standardization Network 1997. (Dawod, 1998)

Figure (4.12): Distribution of Available Measurements of Gravity in the Database of BGI. (Balmino G.at el., 1999).

Figure (4.13): Distribution of BGI Gravity Data.

Figure (4.14): Geographic Locations of Available Gravity Data for Egypt.

Figure (4.16): Components of Deflections of The Vertical Data in Egypt.

In each of the above three GPS project, the WGS84 reference frame was utilized to be the basic datum of the produced coordinates which were gained directly from GPS.

Table (4.11): Statistics of The Egyptian 5'x5' Mean Free-air Gravity Anomalies. Figure (4.21): The Egyptian 5'x5' Mean Free-air Gravity Anomalies Interpolated by Least Squares Collocation.

Figure (4.23): Components of Deflections of The Vertical Used for The Computation.

Differences between tide-free and mean tide system for the ellipsoidal heights er a Figure (4.24): Differences between tide- Free and mean tide system for the ellipsoidal heights of HARN stations

Figure (5.1): Iteration Scheme for Tailored geopotential Model.

Figure (5.3): (a) Difference between The Egyptian 5’ x 5’ Mean Gravity Anomalies and The Computed Gravity Anomalies Using The Oniginal Model GOCO05s (b) Histogram of These Differences in 10 mGal bins. (Iteration Number =0)

Figure (5.4): (a) Difference between The Egyptian 5’ x 5’ Mean Gravity Anomalies and The Computed Gravity Anomalies Using The EG1GOC5s Tailored Model (b) Histogram of These Differences in 10 mGal bins. (Iteration Number =5)

Figure (5.5): The Egyptian 5'x5' Mean Gravity Anomalies Derived by Using EG1GOC5s Model Completed to Degree and Order 280.

Figure (5.7): (a) Difference between The Egyptian 5’ x 5’ Mean Gravity Anomalies and The Computed Gravity Anomalies Using The Original Model EGM2008 (till degree 360) (b) Histogram of These Differences in 10 mGal bins.(Iteration Number =0)

Figure (5.8): (a) Difference between The Egyptian 5’ x 5’ Mean Gravity Anomalies and The Computed Gravity Anomalies Using The EGM08F Tailored Model (b) Histogram of These Differences in 10 mGal bins.(Iteration Number =5)

Figure (5.9): (a) Difference between The Egyptian 5’ x 5’ Mean Gravity Anomalies and The Computed Gravity Anomalies Using The Original Model EGM2008 (Max. degree. 2190) (b) Histogram of These Differences in 15 mGal bins.

Figure (5.10): (a) Difference between The Egyptian 5’ x 5’ Mean Gravity Anomalies and The Computed Gravity Anomalies Using The EGTM0818 Tailored Model (Max. degree. 2190) (b) Histogram of These Differences ir 15 mGal bins.

Figure (5.11): The Egyptian 5’x5’ Mean Gravity Anomalies Derived By Using EGTM0818 Model Complete to Degree and Order 2190.

Table (5.10) shows the differences between the geoid heights derived from 17 GPS/levelling stations of HARN project and those derived from the fitted geoid models after removing trend surface.

Figure (5.16): Geoid Model for Egypt Derived from Tailored Geopotential Model EG1GOC5s (till degree and order 280) after Removing Trend Surface Using The 4-parameter Model and The Weighted-Mean. Contour interval 1.0 m

jure (5.17): 3D - Fitted Geoid Model for Egypt Derived from Tailored Geopotential Model EG1GOC5s.

Figure (5.18): Geoid Model for Egypt Derived from Tailored Geopotential Model EGTM0818 (till degree and order 2190) after Removing Trend Surface Using The 4-parameter Model and The Weighted-Mean. Contour interval 1.0 m

Figure (5.19): 3D - Fitted Geoid Model for Egypt Derived from Tailored Geopotential Model EGTM0818.

Now all tools available for using 3D LSC for geoid solutions; the residual quantities, the covariance function, and the predictions surface area. The rest is to establish the normal equations Eq. (3.52) and compute predictions and error estimates, Eq. (3.53) and (3.56) without parameters (X) i.e. the output of 3D LSC called absolute solution. This may be done using GRAVSOFT program GEOCOL.

Figure (5.22): Difference in Geoid Heights between The Gravimetric Geoid Norav-a and The GPS/Levelling Geoid before Removing Trend Surface. Contour interval = 30 cm.

Figure (5.23): Difference in Geoid Heights between The Gravimetric Geoid Nerav-s and The GPS/Levelling Geoid before Removing Trend Surface. Contour interval =30 cm.

Accordingly, from Fig. (4.3), and referring to Table (4.5), most of Egypt is located in green area file (A), blue area file (B) and a small part in gray area file (D) as shown in Fig. (4.4). This data obtained from African Gravity Project (AGP) (Fairhead & Watts, 1989; Merry, 2003; Merry et al., 2005).

‘7&é PyGravsoft Launcher - Gravity field programs Figure (4.19) Main Programs Package of GRAVSOFT Software.

Table (4.10): Statistics of the selection Egyptian Point Gravity Anomalies.

Figure (4.22): Histogram of Residual between The Gravity Anomalies Derived from Original Points and 5 Arc-minute Gravity Anomalies 4.8.4 Preparation of Deflections of the vertical Data

Figure(5.2): RMS of The Differences of The Egyptian 5 Arc-minute Mean Gravity Anomalies Using EG1GOC5s Tailored Model with respect to The Iteration Number.

Figure(5.6): RMS of The Differences of The Egyptian 5 Arc-minute Mean Gravity Anomalies Using EGMO8F Tailored Model with respect to The Iteration Number.

Table (5.10): Differences between The Geoid Models and The 17 GPS/Levelling of HARN Stations after Removing Trend Surface. In order to validate an external check of the quality of the fitted geoid models, the 9 GPS/levelling stations were used Table (5.11), which was not used for the geoid fitting in the previous section. In addition, the EGM96 geopotential model (cf. Fig. 4.5), complete to degree and order 360, has also been used to compute geoid model for Egypt for comparison

The result of check points shows that the geoid models derived from

Finally, Fig. (5.16) and Fig. (5.18) show the geoid model for Egypt derived from harmonic coefficients of both EG1GOC5s and EGTM0818 tailored geopotential models, respectively, while Fig. (5.17) and Fig. (5.19) show their 3D fitted geoid model for both geoid models Negcicocss and Necrmosis- Table (5.12): Statistics of The 5'x5' Fitted Geoid Models for Egypt.

After the remove-step, the residual height anomalies, as well as residual gravity anomalies, are estimated at the surface of the Earth from the residual gravity Table (5.13) or a combining between a residual gravity

Table (5.20): Statistics of Gravimetric Geoid Ncray-g for Egypt. Table (5.19): Statistics of Gravimetric Geoid Ncrav-a for Egypt.

From Table (5.21), the accuracy of both combined geoids is about

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K07251021114 paper

by Essam Al_krargy and

Fig. 1: The Relation between Ellipsoid Height and Geoid Height

The Earth Geopotential Model 1996 (EGM 96) is a GGM produced through a collaborative effort of the NASA Goddard Space Flight Centre (GSFC), the National Imagery and Mapping A gency (NIMA), and the Ohio State University (OSU). EGM 96 was improved the data holding over many of the world's land area. Furthermore, it has been calculated using a new set of 30’ x 30’ mean free-air gravity anomalies obtained from several ocean gravimetric missions which took place from 1975 to 1990 (Catlao and Sevilla, 1999). In addition, EGM96 incorporated improved surface gravity data, altimeter-derived anomalies from ERS-1 and from the GEOSAT Geodetic Mission, extensive satellite tracking data, the GPS NASA’s Tracking and Data Relay Satellite System (TDRSS), the French DORIS system, the US Navy TRANSET Doppler tracking system, as well as direct altimeter ranges from TOPEX/POSEIDON (T/P), ERS-1, and GEOSAT. The distribution and extent of the surface gravity data in EGM 96 is a major improvement on the data available for the OSU-91A. The geoid undulations of EGM 96 GGM over Egypt are represented as a contour map in Fig. (2b) (Saad and Dawod, 2002).

Fig. 2: (A) A Contour Map of Geoid Undulations of OSU9-1A (Saad and Dawod, 2002), (B) A Contour Map of Geoid Undulations of EGM 96 (Saad and Dawod, 2002), and (C) A Contour Map of Geoid Undulations of EGM 2008 in Egypt (Rabah and Kaloop, 2013)

The study area is located in the northern Egypt from Rosetta to Brullous Fishing port along the Mediterranean coast (Fig. 3).It extends from longitude 30222.40' E to 30955.15' E, and from latitude 312 27.8' N to 31° 33.9' N. The current area of the study area covers 198 square kilometers approximately. The study have been carried out using 31 GPS/leveling data points as shown in Fig. (4). 24 of these points were used as control points in the development of a geoid model for the Rosetta area, while 7 points were used as check points. The selection of control points set was based upon maintaining homogeneous distribution taking into consideration the topography of the region. The majority of this data set comes from research project carried out by the Survey Research Institute (SRI). The spirit levelling observations were performed as closed loops that run between known high-precision benchmarks established by Egyptian survey Authority (ESA) based on the national vertical datum of Egypt, whose origin is based on Mean Sea Level (MSL) at Alexandria tide gauge of 1906. In addition, GPS measurements were carried out relative to the ESA national geodetic reference framework of Egypt Fig. 4: Distribution of C ontrol and C heck Points in the Study Area

Fig. 6: (A) A Contour Map of dN between (NOSU-91A, NObs.), (B) A Contour Map of dN between (NEGM96, NObs.), and(C) (Contour Map of dN between (NEGM 2008, NObs.)

Fig. (8), and (9) along with Table (III) depict the accomplished results. -Finally, a comparison has been carried out between the values of geoid undulations of each local model (NLocal_ Model), and geoid undulations of the GPS/Levelling (NObs) at the check points. -Thus, 10 local geoid models have been obtained as a combination of all possible cases.

When compared against observed undulations, this model produces differences range from 0.059 meter to -0.083 meter, with average of -0.010 meter, and a standard deviation equals + 0.050 meter. This level of precision exceeds that of all tested GGMs models over the study area. Fig. 9: (A) A Contour Map of Geoid Undulation (NObs), (B) A Contour Map of Geoid Undulation (NLocal_Model), and (C) Contour Map of Difference Geoid Undulations (dNLocal-M odel)

Table I: Difference of Geoid Undulation (dN) between GPS/Leveling and GGMs

Table III: Statistical Parameters of Geoid Undulations Difference (ANiocal-Model) between (Niocal-Model) and (Nobs) for Four Cases in Meters

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LOCAL GEOID DETERMINATION AROUND LAKE NASSER

by Mostafa Rabah

The strategic importance of the area of Lake Nasser, where the High Dam exists, utilizes an accurate heights control for many aspects such as the determination of the lake level and the sedimentation rate at the bottom of the lake. GPS... more

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On the Accuracy Assessment of the Latest Releases of GOCE Satellite-Based Geopotential Models with EGM2008 and Terrestrial GPS/Levelling and Gravity Data over Egypt

by Gamal El-Fiky and

The Global Geopotential Models (GGMs) of GOCE (Gravity Recovery and steady-state Ocean Circulation Explorer) differ globally as well as regionally in their accuracy and resolution based on the maximum degree and order (d/o) of the fully... more

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AN INVESTIGATION INTO THE GEOMETRIC ACCURACY OF A DRONE ACQUIRED BLOCK OF IMAGES USING ANALYTICAL TWO PLATFORM ORIENTATION AND SPATIAL INTERSECTION

by ABIODUN O ALABI

Advancement in aviation technology has greatly improved the processes of acquiring images for map production. Over the years images acquired by aircraft and satellite technologies have been the only sources of images for map production. This technically made map production by photogrammetric method an esoteric endeavour available to advanced technologies. The huddles in the way of its use in developing countries included cost, time, expertise and others. In recent times, the combination of advancements in computer and aviation technologies has produced robotic aircrafts called drones which though unmanned, can be used to acquire images from the air and almost in any location even impossible for regular aircraft imaging system. This innovation is also cheap to acquire and cheap to deploy and to maintain. So, research focus has turned to investigating the use of drone captured images in mapping large geographic areas. Therefore, this project deals with investigation into the geometric accuracy of a drone acquired block of images using analytical two platform orientation and spatial intersection. The aim of this project is to investigate the geometric accuracy of a drone acquired block of images. The drone technology used is an Unmanned Fixed-Wing Aircraft System. To fulfil the aim of this project, block of images of Faculty of Education, University of Lagos acquired from an Unmanned Fixed-Wing Aircraft System was obtained. A stereo pair was extracted from the block of images with 60% overlap. Measurement of image coordinates of control, check and detail points was achieved using AutoCAD software as a mono-comparator simulator. The ground survey for control and check points establishment was achieved using Promark 3 Differential Global Positioning System (DGPS). MATLAB program was developed to compute the exterior orientation parameters of the stereo image and subsequently the 3D object space coordinates of control, check and detail points were computed. Statistical analysis was done to evaluate the geometric accuracy of the stereo image using coordinates obtained from ground and photogrammetric survey methods. The root mean square error (RMSE) in plan was 20-50cm for positional accuracy while that for height was 40cm. The relative error obtained for distance measurement is approximately 1:100 while that for area measurements are approximately 0.3:100 and 7:100. Based on the findings of this study, it is concluded that 1 drone (UAV) techniques can replace other methods of surveying as the main method of data capture for engineering projects, boundary mapping and topographical surveying.

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Definition of local geoidal potential and datum height difference of Turkish National Vertical Control Network (TUDKA99

by ali kilicoglu

The determination and use of heights require the definition of a reference surface, which is called the vertical datum, preferably an equipotential surface. The definition of the vertical datum is usually made by assigning values to a... more

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A gravimetric quasi-geoid evaluation in the Northern region of Algeria using EGM96 and GPS/Levelling

by Mohamed Aissa Meslem

The use of GPS for the estimation of orthometric heights in a given region, with the help of existing levelling data requires the determination of a local geoid and the bias between the local levelling and the one implicitly defined when... more

Fig.1 Free air anomalies after subtraction of the EGM96 contribution. Units mGal.

The Figure2 shows the empirical and analytical covariance functions.

Table 1. Statistics of the EGM96 contribution on the 2534 free air anomalies.

Table 3. Statistics of the differences between predicted and observed values. Units m.
Table 4. Statistics of the differences between predicted and observed values. Units m.

Table 4. Statistics of the differences between predicted and observed values. Units m.

Table 6. Statistics of the differences between predicted and observed values. Units m.

Table 5. Statistics of the differences between predicted and observed values. Units m.
Table 6. Statistics of the differences between predicted and observed values. Units m.

Table 9. statistics of differences between the residual gravimetric quasi-geoid undulations and GPS-levelling at 37 control points (ir
meters) after fitting.

Table 7. Statistics of differences between the residual gravimetric quasi-geoid undulations and GPS-levelling at 37 control points (i
meters) before fitting.
Table 8. The parameter transformation model.
Table 9. statistics of differences between the residual gravimetric quasi-geoid undulations and GPS-levelling at 37 control points (ir
meters) after fitting.

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ASSESSMENT OF RECENT GOCE-BASED GLOBAL GEOPOTENTIAL MODELS AND EGM2008 IN NIGER REPUBLIC

by Salissou IBRAHIM YAHAYA and

2019, Geodesy and Cartography

In this study, we assessed recent GOCE-based Global Geopotential Models (GGMs) and EGM2008 in Niger. The combined GGMs EIGEN_6C4, GECO and EGM2008 were evaluated up to their maximum degree and order (d/o) 2,190 to select the one for... more

Figure 1. Simplified principle of Spectral Enhancement Method (SEM) (Hirt et al., 2011)

Figure 2. Spatial distribution of terrestrial gravity data The available 39 GNSS/Levelling points are located in the southwest part of Niger (I El Azzab, 2017a). They be evelling distance in km. TI 1 points of Nigerian geod in our study area from (O 2014). The GNSS/Levelling The geoid undulation Neps the difference between the thometric height H: brahim Yahaya, El Brirchi, & ong to the second order and the complementary levelling networks, with the accura- cies of 5VS mm and 5VS cm, respectively, where S is the hey are also measured by dif- ferential GNSS using the static mode. We also retrieved etic network that are located rupabo, Opuaji, & Adekunle, points are plotted on Figure 3. from GNSS/Levelling point is ellipsoid height h and the or-

Figure 3. Spatial distribution of GNSS/Levelling points

‘igure 4. Histograms of gravity anomaly differences for combined GGMs up to d/o 2,190 and normal distributions from the same and o (Red). Before SEM: a - EGM2008; b - EIGEN_6C4; c - GECO. After SEM: d - EGM2008; e - EIGEN_6C4; f - GECO The statistics of the difference between the geoid heights from GNSS/Levelling points and those modelled from the GGMs in Niger and northern Nigeria are presented in Table 3. According to the results obtained with the gravity anomalies, EGM2008 is the candidate GGM for spectral enhancement and densification of the gravity database but it does not contain GOCE data. However, better precision is expected from the other two models in long and me- dium wavelengths because they contain GOCE data and EIGEN-6C4 showed the greatest probability according to

Figure 5. RMS of gravity anomaly differences as function of d/o of spherical harmonics: a - before SEM; b - after SE] The RMS values for gravity anomalies differences as func- tion of d/o after applying SEM are shown in Figure 5b. Up to d/o 180, all models have RMS curves coinciding with that of EIGEN-6C4 up to d/o 2,190 and enhanced by RTM, except GGM05G and EGM2008, which show poor performance compared to other models. The result of EGM2008 is justified by the fact that it does not con- tain GOCE data, on the other hand GGM05G combines The RMS values of differences for gravity anomalies as a function of truncation d/o prior to SEM are shown in Figure 5a. All models showed similar performance from d/o 100 to 215 compared to terrestrial data, RMS val- ues decreased gradually with high d/o. From d/o 215, GGM05G stands out from other GGMs, with the largest gap up to its maximum degree, 240. Similarities between models continue for other GGMs up to d/o 230 where ITU_GGC16 and TIM5 stand out from other models.

Figure 6. RMS of geoid height as function of d/o of spherical harmonics before SEM: a — in Niger; b - in Northern Nigeria; and after SEM: c - Niger; d - in Northern Nigeria RMS values of geoid height differences in Niger as func- tion of d/o prior to SEM are shown in Figure 6a. All models exhibit similar behaviour, except EGM2008 which All pure satellite and combined models containing GOCE data showed a better performance compared to EGM2008 up to d/o ranging from 100 to 200. This con- firms the improvement in the gravity field recovery in these spectral bands by GOCE gravity mission.

Table 1. Models under evaluation S = Satellite, G = Gravity, A = Altimetry 1.6. Assessment of combined models

Table 2. Statistics of differences between terrestrial gravity data and combined GGMs at d/o 2190 (in mGal)

Table 3. Statistics of differences between GNSS/Levelling data and combined models at d/o 2190 (in metres) 3-sigma rule. Also, for global assessments EIGEN-6C4 is presently used as a reference model (Barthelmes & Kohler, 2016). We therefore retained EIGEN-6C4. In order to have an overview of the performance of these combined models in long and medium wavelengths, we included them in the comparison of “pure satellite’ GGMs in the following section. From d/o 240, all pure satellite models become stable. ITU_GGC16 and TIM5 perform similarly up to d/o 280, also for EIGEN_6S4v2 and DIRS up to d/o 300, and SPW5 is more accurate up to this d/o. Beyond d/o 200, the RMS values of differences for the combined models decrease progressively and similarly until d/o 300, GECO is less ac- curate than EGM2008 and EIGEN-6c4. EGM2008 appears similar to EIGEIN-6C4 and is better than other models. In these spectral bands, the combined models are generally better than the pure satellite GGMs due to the integration of terrestrial data that contain the full gravity spectrum.

Table 4. RMS of gravity anomaly differences in mGal for pure satellite GGMs up to d/o 180, 200 and 215 after SEM

Table 5. RMS of geoid height differences for pure satellite GGMs in metres at d/o 200 after SEM

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Airborne Laser Scanning for calibration and validation of inshore satellite altimetry: A proof of concept

by András Zlinszky

A B S T R A C T Recent developments in satellite altimetry are leading to improved spatial resolution, allowing applications in the coastal zone and over inland waters. Validation of these sensors near the shore remains a challenge, since... more

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Evaluation des Surfaces d’Anomalies Gravimétriques à l’Air Libre en Méditerranée Occidentale

by Mohamed Habib

L'objectif de l'article est de vérifier la qualité des surfaces d'anomalies gravimétriques globales en Méditerranée Occidentale. Dans ce cadre, l'ensemble des données marines d'anomalies à l'air libre, acquises auprès du Bureau... more

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Results on Earth Gravity field Models comparison over Algeria

by Mohamed Aissa Meslem

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Earth Tide NRIAG2004

by Mostafa Rabah

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An Integrated Geomatics Approach for Projecting Sea Level Variations and Risks A Case Study in the Nile Delta, Egypt 1

by Essam Al_krargy and

2021, Quest Journals

Sustainable development of coastal areas depends fundamentally on the availability and comprehensive analysis of precise and up-to-date geospatial datasets. This study aims to develop an integrated geomatics approach for collecting and... more

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Physical Geodesy

Key research themes

1. How can GPS geodesy contribute to understanding crustal deformation and natural hazard mitigation?

2. What are the advances in geodesic computations and stability analyses in Riemannian and semi-Riemannian geometries?

3. How can global geopotential models and metrology techniques be optimized for precise geoid and gravity field determination?

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