Local North-oriented Reference Frame Research Articles

Cap integration in spectral gravity forward modelling up to the full gravity tensor

Cap-modified spectral gravity forward modelling is extended in this paper to the full gravity vector and tensor expressed in the local north-oriented reference frame. This is achieved by introducing three new groups of altitude-dependent Molodensky’s truncation coefficients. These are given by closed form and infinite spectral relations that are generalized for (i) an arbitrary harmonic degree, (ii) an arbitrary topography power, (iii) an arbitrary radial derivative, (iv) any radius larger than the radius of the reference sphere, and (v) for both near- and far-zone gravity effects. Thanks to the generalization for an arbitrary radial derivative, the cap-modified technique can efficiently be combined with the gradient approach for harmonic synthesis on irregular surfaces. In a numerical study, we exemplarily apply the new technique by forward modelling Earth’s degree-2159 topography up to degree 21,590, employing 30 topography powers. The experiment shows that near- and far-zone gravity effects can be synthesized on the topography with an accuracy (RMS) of 0.005–0.03 m $$^2$$ s $$^{-2}$$ (potential), 0.8–20 $$\upmu \mathrm {Gal}$$ (gravity vector), and 0.1 mE–1 E (gravity tensor). The numerical experiment also shows that the divergence effect of spherical harmonics comes into play around degree 10,795 when evaluating the series on the Earth’s surface. The difficult-to-compute truncation coefficients that are employed in the study are made freely available at http://edisk.cvt.stuba.sk/~xbuchab/ and are accompanied by MATLAB-based routines to evaluate them. Enclosed is also a MATLAB-based package to perform ultra-high-degree surface spherical harmonic analysis, a step of central importance in spectral gravity forward modelling techniques.

Spherical integral transforms of second-order gravitational tensor components onto third-order gravitational tensor components

New spherical integral formulas between components of the second- and third-order gravitational tensors are formulated in this article. First, we review the nomenclature and basic properties of the second- and third-order gravitational tensors. Initial points of mathematical derivations, i.e., the second- and third-order differential operators defined in the spherical local North-oriented reference frame and the analytical solutions of the gradiometric boundary-value problem, are also summarized. Secondly, we apply the third-order differential operators to the analytical solutions of the gradiometric boundary-value problem which gives 30 new integral formulas transforming (1) vertical-vertical, (2) vertical-horizontal and (3) horizontal-horizontal second-order gravitational tensor components onto their third-order counterparts. Using spherical polar coordinates related sub-integral kernels can efficiently be decomposed into azimuthal and isotropic parts. Both spectral and closed forms of the isotropic kernels are provided and their limits are investigated. Thirdly, numerical experiments are performed to test the consistency of the new integral transforms and to investigate properties of the sub-integral kernels. The new mathematical apparatus is valid for any harmonic potential field and may be exploited, e.g., when gravitational/magnetic second- and third-order tensor components become available in the future. The new integral formulas also extend the well-known Meissl diagram and enrich the theoretical apparatus of geodesy.

2D Fourier series representation of gravitational functionals in spherical coordinates

2D Fourier series representation of a scalar field like gravitational potential is conventionally derived by making use of the Fourier series of the Legendre functions in the spherical harmonic representation. This representation has been employed so far only in the case of a scalar field or the functionals that are related to it through a radial derivative. This paper provides a unified scheme to represent any gravitational functional in terms of spherical coordinates using a 2D Fourier series representation. The 2D Fourier series representation for each individual point is derived by transforming the spherical harmonics from the geocentric Earth-fixed frame to a rotated frame so that its equator coincides with the local meridian plane of that point. In the obtained formulation, each functional is linked to the potential in the spectral domain using a spectral transfer. We provide the spectral transfers of the first-, second- and third-order gradients of the gravitational potential in the local north-oriented reference frame and also those of some functionals of frequent use in the physical geodesy. The obtained representation is verified numerically. Moreover, spherical harmonic analysis of anisotropic functionals and contribution analysis of the third-order gradient tensor are provided as two numerical examples to show the power of the formulation. In conclusion, the 2D Fourier series representation on the sphere is generalized to functionals of the potential. In addition, the set of the spectral transfers can be considered as a pocket guide that provides the spectral characteristics of the functionals. Therefore, it extends the so-called Meissl scheme.

Non-singular expressions for the spherical harmonic synthesis of gravitational curvatures in a local north-oriented reference frame

Third-order gradients of the gravitational potential (gravitational curvatures) have already found some applications in geosciences. Observability of these parameters, describing the Earth's gravitational field in a more complex way than any other currently available gravitational parameter, such as gravitational acceleration (first-order gradient) or gravitational (second-order) gradient, is currently discussed by physicists. Moreover, first designs of observational devices (sensors) have already been proposed. The spherical harmonic analysis and synthesis are the common tools used by geoscientists to study spectral properties of various functionals of the Earth's gravitational potential. However, the conventional spherical harmonic expansions of the gravitational curvatures in the local north-oriented reference frame have rather complicated forms that depend on the first-, second- and third-order derivatives of the associated Legendre functions. Moreover, some of these expansions also contain singular terms at the poles. In this paper, the conventional series are transformed to new simpler and non-singular forms based on relations between the associated Legendre functions and their derivatives. Numerical experiments demonstrate the applicability and correctness of the new expressions.

Non-singular spherical harmonic expressions of geomagnetic vector and gradient tensor fields in the local north-oriented reference frame

Abstract. General expressions of magnetic vector (MV) and magnetic gradient tensor (MGT) in terms of the first- and second-order derivatives of spherical harmonics at different degrees/orders are relatively complicated and singular at the poles. In this paper, we derived alternative non-singular expressions for the MV, the MGT and also the third-order partial derivatives of the magnetic potential field in the local north-oriented reference frame. Using our newly derived formulae, the magnetic potential, vector and gradient tensor fields and also the third-order partial derivatives of the magnetic potential field at an altitude of 300 km are calculated based on a global lithospheric magnetic field model GRIMM_L120 (GFZ Reference Internal Magnetic Model, version 0.0) with spherical harmonic degrees 16–90. The corresponding results at the poles are discussed and the validity of the derived formulas is verified using the Laplace equation of the magnetic potential field.

Rotation of GOCE gravity gradients to local frames

ESA′s GOCE mission aims at improved global and regional gravity field information with high spatial resolution by measuring gravity gradients. A local analysis of the GOCE gravity gradient tensor may benefit from a rotation from the gradiometer reference frame to local reference frames such as the local north oriented reference frame. As the GOCE gravity gradients include accurate and less accurate measured gradients, the point-wise tensor rotation of the GOCE-only measurements may suffer from the projection of the errors of the less accurate gravity gradients onto the accurate gravity gradients. In addition, the GOCE gravity gradients have high accuracy in the measurement bandwidth but low accuracy below, and tensor rotation may cause leakage of the large error below the measurement bandwidth to the measurement bandwidth. Degradation of the rotated gravity gradients is circumvented by replacing the less accurate tensor components, as well as the signal below the measurement bandwidth of the accurate gravity gradients, with model signal. The combination of GOCE and model gravity gradients is performed by determining the effective measurement bandwidth (EMB), that is, the bandwidth in which the integrated signal-to-noise ratio of the GOCE gravity gradients is maximized. We find that the determination of the EMB is relatively independent of the reference gravity field model that is used. The lower bound of the EMB is well below the pre-mission specifications for the four accurate gravity gradients. In addition, we assess how much GOCE contributes to the gravity gradient signal in local frames and how much the model. For the radial gravity gradient the relative GOCE contribution is 98 per cent on average, whereas this is 65-97 per cent for the other gravity gradients. These numbers strongly depend on the local frame under consideration and on the geographical position.

Development of the second-order derivatives of the Earth’s potential in the local north-oriented reference frame in orthogonal series of modified spherical harmonics

The second-order derivatives of the Earth’s potential in the local north-oriented reference frame are expanded in series of modified spherical harmonics. Linear relations are derived between the spectral coefficients of these series and the spectrum of the geopotential. On the basis of these relations, recurrence procedures are developed for evaluating the geopotential coefficients from the spectrum of each derivative and, inversely, for simulating the latter from a known geopotential model. Very simple structure of the derived expressions for the derivatives is convenient for estimating the geopotential coefficients by the least-squares procedure, at a certain step of processing satellite gradiometry data. Due to the orthogonality of the new series, the quadrature formula approach can be also applied, which allows avoidance of aliasing errors caused by the series truncation. The spectral coefficients of the derivatives are evaluated on the basis of the derived relations from the geopotential models EGM96 and EIGEN-CG01C at a mean orbital sphere of the GOCE satellite. Various characteristics of the spectra are studied corresponding to the EGM96 model.

Explicit Construction of the Modified Spherical Harmonic Series for the Gravity Gradients and Analysis of their Characteristics

Explicit Construction of the Modified Spherical Harmonic Series for the Gravity Gradients and Analysis of their CharacteristicsThe present paper complements the research carried out in PV2008 (Petrovskaya and Vershkov, 2008), concerning the expansion of the gravity gradients in the local north-oriented reference frame in orthogonal series of modified spherical harmonics. In PV2008 procedures are developed for recovering the orthogonal bases of these series. Then an idea is briefly described how the spectral relations can be obtained between the gravity gradients and the geopotential. However no explicit procedures are demonstrated for their derivation. In the present paper successive transformations are described for each derivative which convert the initial non-orthogonal expansion into the orthogonal series. The resulting spectral relations are applied for evaluating the harmonic coefficients of these series at different altitudes, on the basis of the geopotential model EGM2008. The corresponding degree variances are estimated. The new simple expressions for the gravity gradients are convenient for various applications. In the present paper they are implemented for constructing digital colored maps for Fennoscandia region which attracts much attention of geophysicists. These maps visually demonstrate an anomalous behavior of the gravity gradients in this area.

Non-singular expressions for the gravity gradients in the local north-oriented and orbital reference frames

The conventional expansions of the gravity gradients in the local north-oriented reference frame have a complicated form, depending on the first- and second-order derivatives of the associated Legendre functions of the colatitude and containing factors which tend to infinity when approaching the poles. In the present paper, the general term of each of these series is transformed to a product of a geopotential coefficient $$\overline{C}_{n,m} $$ and a sum of several adjacent Legendre functions of the colatitude multiplied by a function of the longitude. These transformations are performed on the basis of relations between the Legendre functions and their derivatives published by Ilk (1983). The second-order geopotential derivatives corresponding to the local orbital reference frame are presented as linear functions of the north-oriented gravity gradients. The new expansions for the latter are substituted into these functions. As a result, the orbital derivatives are also presented as series depending on the geopotential coefficients $$\overline{C}_{n,m} $$ multiplied by sums of the Legendre functions whose coefficients depend on the longitude and the satellite track azimuth at an observation point. The derived expansions of the observables can be applied for constructing a geopotential model from the GOCE mission data by the time-wise and space-wise approaches. The numerical experiments demonstrate the correctness of the analytical formulas.