Construction of spherical harmonic series for the potential derivatives of arbitrary orders in the geocentric Earth-fixed reference frame

Petrovskaya, M. S.; Vershkov, A. N.

doi:10.1007/s00190-009-0353-y

Cited by 17 publications

(9 citation statements)

References 8 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…x i x i n,m can be obtained from [28]. The specific calculation formula of ∂ 2 1 r /∂x i x j is as follows:…”

Section: Conflicts Of Interestmentioning

confidence: 99%

“…

and

are the harmonic geopotential coefficients with degree

and order

. According to [ 27 ], the calculation formula of gravity gradients reads as follows:

where

and

can be obtained from [ 28 ]. The specific calculation formula of

is as follows:

The transformation of the gravity gradient tensor matrix between the body coordinate system and the navigation coordinate system is as follows:

…”

mentioning

confidence: 99%

“…According to [ 27 ], the calculation formula of gravity gradients reads as follows:

where

and

can be obtained from [ 28 ]. The specific calculation formula of

is as follows:

…”

mentioning

confidence: 99%

See 2 more Smart Citations

An Aided Navigation Method Based on Strapdown Gravity Gradiometer

Gao

Chang

et al. 2021

Sensors

View full text Add to dashboard Cite

The gravity gradient is the second derivative of gravity potential. A gravity gradiometer can measure the small change of gravity at two points, which contains more abundant navigation and positioning information than gravity. In order to solve the problem of passive autonomous, long-voyage, and high-precision navigation and positioning of submarines, an aided navigation method based on strapdown gravity gradiometer is proposed. The unscented Kalman filter framework is used to realize the fusion of inertial navigation and gravity gradient information. The performance of aided navigation is analyzed and evaluated from six aspects: long voyage, measurement update period, measurement noise, database noise, initial error, and inertial navigation system device level. When the parameters are set according to the benchmark parameters and after about 10 h of simulation, the results show that the attitude error, velocity error, and position error of the gravity gradiometer aided navigation system are less than 1 arcmin, 0.1 m/s, and 33 m, respectively.

show abstract

“…x i x i n,m can be obtained from [28]. The specific calculation formula of ∂ 2 1 r /∂x i x j is as follows:…”

Section: Conflicts Of Interestmentioning

confidence: 99%

“…

and

are the harmonic geopotential coefficients with degree

and order

. According to [ 27 ], the calculation formula of gravity gradients reads as follows:

where

and

can be obtained from [ 28 ]. The specific calculation formula of

is as follows:

The transformation of the gravity gradient tensor matrix between the body coordinate system and the navigation coordinate system is as follows:

…”

mentioning

confidence: 99%

See 1 more Smart Citation

An Aided Navigation Method Based on Strapdown Gravity Gradiometer

Gao

Chang

et al. 2021

Sensors

View full text Add to dashboard Cite

show abstract

“…It is based on the elegant Hotine's formulation of the gravitational potential that permits easy rotation of observation equations. Detailed derivation of Hotine's equations can be found in Hotine (1969) or in the papers of Bettadpur (1995); Petrovskaya and Vershkov (2009). Recently, an introduction to the harmonic analysis with Hotine's equations is provided by Petrovskaya and Vershkov (2011), where the recurrences for computing the geopotential coefficients are derived.…”

Section: Introductionmentioning

confidence: 99%

Short guide to direct gravitational field modelling with Hotine’s equations

Sebera

Wagner

Bezděk

et al. 2012

J Geod

View full text Add to dashboard Cite

This paper presents a unified approach to the least squares spherical harmonic analysis of the acceleration vector and Eötvös tensor (gravitational gradients) in an arbitrary orientation. The Jacobian matrices are based on Hotine's equations that hold in the Earth-fixed Cartesian frame and do not need any derivatives of the associated Legendre functions. The implementation was confirmed through closedloop tests in which the simulated input is inverted in the least square sense using the rotated Hotine's equations. The precision achieved is at the level of rounding error with RMS about 10 −12 −10 −14 m in terms of the height anomaly. The second validation of the linear model is done with help from the standard ellipsoidal correction for the gravity disturbance that can be computed with an analytic expression as well as with the rotated equations. Although the analytic expression for this correction is only of a limited accuracy at the submillimeter level, it was used for an independent validation. Finally, the equivalent of the ellipsoidal correction, called the effect of the normal, has been numerically obtained also for other gravitational functionals and some of their combinations. Most of the numerical investigations are provided up to spherical harmonic degree 70, with degree 80 for the computation time comparison using real GRACE data. The relevant Matlab source codes for the design matrices are provided.

show abstract

“…In the paper PV2010 (Petrovskaya and Vershkov 2010), conventional spherical harmonic series are constructed for a derivative of arbitrary order of the geopotential in the terrestrial reference frame (TRF). In that system the axis z is the Earth mean axis of rotation, x is directed along the line of the intersection of the Greenwich meridian with the equatorial plane and y is directed to the east.…”

Section: Introductionmentioning

confidence: 99%

Basic equations for constructing geopotential models from the gravitational potential derivatives of the first and second orders in the terrestrial reference frame

Petrovskaya

Vershkov

2011

J Geod

Self Cite

View full text Add to dashboard Cite

This research represents a continuation of the investigation carried out in the paper of Petrovskaya and Vershkov (J Geod 84 (3): [165][166][167][168][169][170][171][172][173][174][175][176][177][178] 2010) where conventional spherical harmonic series are constructed for arbitrary order derivatives of the Earth gravitational potential in the terrestrial reference frame. The problem of converting the potential derivatives of the first and second orders into geopotential models is studied. Two kinds of basic equations for solving this problem are derived. The equations of the first kind represent new non-singular non-orthogonal series for the geopotential derivatives, which are constructed by means of transforming the intermediate expressions for these derivatives from the above-mentioned paper. In contrast to the spherical harmonic expansions, these alternative series directly depend on the geopotential coefficients C n,m and S n,m . Each term of the series for the first-order derivatives is represented by a sum of these coefficients, which are multiplied by linear combinations of at most two spherical harmonics. For the second-order derivatives, the geopotential coefficients are multiplied by linear combinations of at most three spherical harmonics. As compared to existing non-singular expressions for the geopotential derivatives, the new expressions have a more simple structure. They depend only on the conventional spherical harmonics and do not depend on the first-and second-order derivatives of the associated Legendre functions. The basic equations of the second kind are inferred from the linear equations, constructed in the cited paper, which express the coefficients of the spherical harmonic series for the first-and second-order derivatives in terms of the geopotential coefficients. These equations are M. S. Petrovskaya (B) · A. N. Vershkov Central (Pulkovo) Astronomical Observatory of the Russian Academy of Sciences, converted into recurrent relations from which the coefficients C n,m and S n,m are determined on the basis of the spherical harmonic coefficients of each derivative. The latter coefficients can be estimated from the values of the geopotential derivatives by the quadrature formulas or the least-squares approach. The new expressions of two kinds can be applied for spherical harmonic synthesis and analysis. In particular, they might be incorporated in geopotential modeling on the basis of the orbit data from the CHAMP, GRACE and GOCE missions, and the gradiometry data from the GOCE mission.Keywords Terrestrial reference frame · Gravitational gradients of the first and second orders · Transformation of the gradients into geopotential models

show abstract

Construction of spherical harmonic series for the potential derivatives of arbitrary orders in the geocentric Earth-fixed reference frame

Cited by 17 publications

References 8 publications

An Aided Navigation Method Based on Strapdown Gravity Gradiometer

An Aided Navigation Method Based on Strapdown Gravity Gradiometer

Short guide to direct gravitational field modelling with Hotine’s equations

Basic equations for constructing geopotential models from the gravitational potential derivatives of the first and second orders in the terrestrial reference frame

Contact Info

Product

Resources

About