Earth’s Rotation: A Challenging Problem in Mathematics and Physics

A suitable knowledge of the orientation and motion of the Earth in space is a common need in various fields. That knowledge has been ever necessary to carry out astronomical observations, but with the advent of the space age, it became essential for making observations of satellites and predicting and determining their orbits, and for observing the Earth from space as well. Given the relevant role it plays in Space Geodesy, Earth rotation is considered as one of the three pillars of Geodesy, the other two being geometry and gravity. Besides, research on Earth rotation has fostered advances in many fields, such as Mathematics, Astronomy and Geophysics, for centuries. One remarkable feature of the problem is in the extreme requirements of accuracy that must be fulfilled in the near future, about a millimetre on the tangent plane to the planet surface, roughly speaking. That challenges all of the theories that have been devised and used to-date; the paper makes a short review of some of the most relevant methods, which can be envisaged as milestones in Earth rotation research, emphasizing the Hamiltonian approach developed by the authors. Some contemporary problems are presented, as well as the main lines of future research prospected by the International Astronomical Union/International Association of Geodesy Joint Working Group on Theory of Earth Rotation, created in 2013.This work has been partially supported by the Spanish government under Grants AYA2010-22039-C02-01 and AYA2010-22039-C02-02 from Ministerio de Economía y Competitividad (MINECO), the University of Alicante under Grant GRE11-08 and the Generalitat Valenciana, Grant GV/2014/072

Partial-Data Superimposed Training with Data Precoding for OFDM Systems

Superimposed training (ST) is a recently addressed technique used for channel estimation where known training sequences are arithmetically added to data symbols, avoiding the use of dedicated pilot subcarriers, and thus, increasing the available bandwidth compared with traditional pilot symbol assisted modulation schemes. However, the system handles data interference over channel estimation as a result of the ST process; also, data detection is degraded by pilot interference. Recent ST methods have analyzed the data interference and presented schemes that deal with it. We propose a novel superimposed model over a precoded data scheme, named partial-data superimposed training (PDST), where an interference control factor assigns the adequate information level to be added to the training sequence in orthogonal frequency division multiplexing systems. Also, a data detection method is introduced to improve the symbol error rate performance. Moreover, a capacity analysis of the system has been derived. Finally, simulation results confirm that performance of PDST is superior to previous proposals

The Rotation of the Nonrigid Earth at the Second Order. II. The Poincaré Model: Nonsingular Complex Canonical Variables and Poisson Terms

[EN] We develop a Hamiltonian analytical theory for the rotation of a Poincaré Earth model (rigid mantle and liquid core) at the second order with respect to the lunisolar potential and moving ecliptic term. Since the Andoyer variables considered in the first-order solution present virtual singularities, i.e., vanishing divisors, we introduce a set of nonsingular complex canonical variables. This choice allows for applying the Hori canonical perturbation method in a standard way. We derive analytical expressions for the first- and second-order solutions of the precession and nutation of the angular momentum axis (Poisson terms). Contrary to first-order theories, there is a part of the Poisson terms that does depend on the Earth's structure. The resulting numerical amplitudes, not incorporated in the International Astronomical Union nutation standard, are not negligible considering current accuracies. They are at the microarcsecond level for a few terms, with a very significant contribution in obliquity of about 40 μas for the nutation argument with period −6798.38 days. The structure-dependent amplitudes present a large amplification with respect to the rigid model due to the fluid core resonance. The features of such resonance, however, are different from those found in first-order solutions. The most prominent is that it does not depend directly on the second-order nutation argument but rather on the combination of first-order arguments generating it. It entails that some first-order approaches, like those based on the transfer function, cannot be applied to obtain the second-order contributions.S

Limitations of the IAU2000 nutation model accuracy due to the lack of Oppolzer terms of planetary origin

[EN] Context. The current IAU2000 nutation model performed different approximations, one of them being that the Oppolzer terms associated to the planetary perturbations of the nutations were assumed to be smaller than 5 μas and thus were neglected. At present, the uncertainties of the amplitudes of individual components of the observed nutations are better, and the conventional nutation model does not fit the accuracy requirements pursued by the International Astronomical Union (IAU) and the International Association of Geodesy (IAG). Aims. The objective of this work is to estimate the magnitude of the lacking Oppolzer terms of the planetary nutations and find out whether they are still negligible or not. Methods. The Oppolzer terms resulting from the direct and indirect planetary perturbations of the Earth’s rotation have been computed for a two-layer Earth model following the Hamiltonian theory of the non-rigid-Earth. Results. The planetary Oppolzer terms for the non-rigid Earth are not really negligible as believed, and some of them have amplitudes larger than 10 μas, therefore significantly above the current level of uncertainty of individual harmonic constituents. Conclusions. In the short term, the IAU2000 nutation model must be supplemented with suitable corrections accounting for those missing components; its planetary component must be thoroughly revised in the medium term.SIThis work was partly supported by grant AYA2016-79775-P (AEI/FEDER, UE)

Power Allocation and Capacity Analysis for FBMC-OQAM With Superimposed Training

Superimposed training (ST) is a semiblind channel estimation technique, proposed for orthogonal frequency division multiplexing (OFDM), where training sequences are added to data symbols, avoiding the use of dedicated pilot-subcarriers, and increasing the available bandwidth compared with pilot symbol assisted modulation (PSAM). Filter bank multicarrier offset quadrature amplitude modulation (FBMC-OQAM) is a promising waveform technique considered to replace the OFDM, which takes advantage of well-designed filters to avoid the use of cyclic prefix and reduce the out-band-emissions. In this paper, we provide the expressions of the average channel capacity of the FBMC-OQAM combined with either PSAM or ST schemes, considering imperfect channel estimation and the presence of the pilot sequences. In order to compute the capacity expression of our proposal, ST-FBMC-OQAM, we analyze the channel estimation error and its variance. The average channel capacity is deduced considering the noise, data interference from ST, and the intrinsic self-interference of the FBMC-OQAM. Additionally, to maximize the average channel capacity, the optimal value of data power allocation is also obtained. The simulation results confirm the validity of the capacity analysis and demonstrate the superiority of the ST-FBMC-OQAM over existing proposals

Superimposed training-based channel estimation for miso optical-OFDM vlc

In this paper, we investigate a novel channel estimation (CE)method for multiple-input and single-output (MISO) systems in visible lightcommunication (VLC). Direct current biased optical orthogonal frequencydivision multiplexing (DCO-OFDM) is commonly used in VLC where halfof the available subcarriers are spent to guarantee a real-valued outputafter the inverse fast Fourier transform operation. Besides, dedicated subcarriers are typically used for CE, thus, many resources are wasted andthe spectral efficiency is degraded. We propose a superimposed trainingapproach for CE in MISO DCO-OFDM VLC scenarios. Analytical expressions of mean squared error (MSE) and spectral efficiency are derived whenthe least squares estimator is considered. This analysis is valid for outdoorand indoor scenarios. For the CE error, simulation results of MSE showa perfect match with analytical expressions. Moreover, results prove thatthis technique guarantees a larger spectral efficiency than previous schemeswhere dedicated pilots were used. Finally, the optimal data power allocationfactor is also analytically derived.This work was supported in part by the National Secretary of Higher Education, Science, Technology, and Innovation (SENESCYT) in Ecuador and in part by the Spanish National Project TERESA-ADA (TEC2017-90093-C3-2-R) (MINECO/AEI/FEDER, UE). The work of B. G. Guzmán was supported by the Spanish MECD FPU Fellowship Program

Superimposed Training for Channel Estimation in FBMC-OQAM

Proceedings of: 2017 IEEE 86th Vehicular Technology Conference (VTC-Fall)Wireless broadband communication systems are always requiring higher data rates. In order to achieve this goal, we should provide advanced schemes which are capable of improving the spectral efficiency and reusing, in a better way, the available radio spectrum resources. Filter Bank Multi-Carrier Offset Quadrature Amplitude Modulation (FBMC-OQAM) combined with Superimposed Training (ST) is a promising technique with a very high spectral efficiency. This improvement is due to the low out-of-band emissions of FBMC-OQAM, because it uses a well-designed prototype filter, and the lack of dedicated pilot tones owing to ST scheme. However, this combination is not straightforward due to the appearance of the self-interference at receiver side in FBMC-OQAM. In this paper, we provide a novel channel estimation which is capable of dealing with this self-interference in the context of combining these two techniques.This work has been partly supported by Spanish National Projects ELISA (TEC2014-59255-C3-3-R) and MACHINE (TSI-100102-2015-17); and also by National Secretary of Science, Technology and Innovation SENESCYT Ecuador

Application of first-order canonical perturbation method with dissipative Hori-like kernel

Lie-Hori canonical perturbation theory provides asymptotic solutions for conservative Hamiltonian systems. This restriction prevents the canonical method from being applied directly to dissipative mechanical systems. There are, however, two main alternatives to overcome this difficulty, enabling the application of canonical perturbation methods. The first one consists in constructing a time-dependent Hamiltonian, through a generating function, related to the energy dissipation pattern of the system. The second embeds the original phase space into a double dimensional one where the dynamics of the system can be formulated in a Hamiltonian way. In this paper, a modified Lie-Hori method that avoid the disadvantages of the former approaches is proposed. Namely, it is not necessary to find out a time-dependent generating function, nor doubling the number of the canonical variables of the original problem. The new algorithm provides first order analytical solutions for a certain set of dissipative non-linear dynamical systems. It is based on a suitable modification of the Hori kernel in the double-dimensional embedding phase space, allowing the inclusion of the dissipative (or generalized) forces. By means of this redefined auxiliary system, the path-integrals of the method can be performed in a domain of the phase space with the same dimensionality as the original problem.This research has been partially supported by the Spanish government MINECO projects AYA2010-22039-C02-02 and AYA2016-79775-P (AEI/FEDER, UE)

Dynamical adjustments in IAU 2000A nutation series arising from IAU 2006 precession

The adoption of International Astronomical Union (IAU) 2006 precession model, IAU 2006 precession, requires IAU 2000A nutation to be adjusted to ensure compatibility between both theories. This consists of adding small terms to some nutation amplitudes relevant at the microarcsecond level. Those contributions were derived in previously published articles and are incorporated into current astronomical standards. They are due to the estimation process of nutation amplitudes by Very Long Baseline Interferometry (VLBI) and to the changes induced by the J2 rate present in the precession theory. We focus on the second kind of those adjustments, and develop a simple model of the Earth nutation capable of determining all the changes arising in the theoretical construction of the nutation series in a dynamical consistent way. This entails the consideration of three main classes of effects: the J2 rate, the orbital coefficients rate, and the variations induced by the update of some IAU 2006 precession quantities. With this aim, we construct a first order model for the nutations of the angular momentum axis of the non-rigid Earth. Our treatment is based on a Hamiltonian formalism and leads to analytical formulae for the nutation amplitudes in the form of in-phase, out-of-phase, and mixed secular terms. They allow numerical evaluation of the contributions of the former effects. We conclude that the accepted corrections associated with the J2 rate must be supplemented with new, hitherto unconsidered terms of the same order of magnitude, and that these should be incorporated into present standards.This work has been partially supported by the Spanish projects I+D+I AYA2010-22039-C02-02 and AYA2016-79775-P (AEI/FEDER, UE)