TY - JOUR

T1 - Fourth order Taylor–Kármán structured covariance tensor for gravity gradient predictions by means of the Hankel transformation

AU - Grafarend, Erik W.

AU - You, Rey Jer

N1 - Funding Information:
The authors are grateful to the reviewers for the idea to use our isotropic-homogeneous-potential type variance-covariance matrix as prior information for data fit of realistic data. Rey-Jer You wants to thank DAAD (Deutscher Akademischer Austauschdienst) for the support of his stay at Stuttgart University, Department of Geodesy and Geoinformatics, namely to Erik W. Grafarend as his host, and the Ministry of Science and Technology (MOST), Taiwan, for the support of this research under contracts 102-2911-I-006-507 and 104-2911-I-006-518.
Publisher Copyright:
© 2015, Springer-Verlag Berlin Heidelberg.

PY - 2015/11/1

Y1 - 2015/11/1

N2 - Vector-valued stochastic processes have been used for data processing, prediction, filtering, collocation, even network design and analysis in Geodesy, Physical Geodesy, and navigation etc. Recently, LEO satellite missions, such as CHAMP, GRACE and GOCE, provide a number of measurements to study the gravitational field of the Earth. When the gravity gradients, i.e. the second derivatives of the gravitational potential, are used for prediction and filtering by Kolmogorov–Wiener/Gauss–Markov concept, we will need the fourth-order covariance/correlation matrices of the gravity gradient signals. With the assumptions of homogeneous and isotropic field and the random functions of “potential type”, the paper aims at the development of the fourth order tensor-valued Taylor–Kármán structured covariance/correlation matrices. The characteristic functions of these tensor-valued covariance/correlation matrices, namely the lateral and longitudinal components, will be derived for n-dimensional spaces, here specified for n=3 dimensions in the paper. A special part is devoted to the Hankel transformation for gravity gradients and their variance-covariance in order to guarantee consistency well-known from problems in using Fourier transformations. We use the variance-covariance function of type (i) isotropic, (ii) homogeneous and (iii) potential as prior information for fitting the discrete data of variances and covariances estimated from observations.

AB - Vector-valued stochastic processes have been used for data processing, prediction, filtering, collocation, even network design and analysis in Geodesy, Physical Geodesy, and navigation etc. Recently, LEO satellite missions, such as CHAMP, GRACE and GOCE, provide a number of measurements to study the gravitational field of the Earth. When the gravity gradients, i.e. the second derivatives of the gravitational potential, are used for prediction and filtering by Kolmogorov–Wiener/Gauss–Markov concept, we will need the fourth-order covariance/correlation matrices of the gravity gradient signals. With the assumptions of homogeneous and isotropic field and the random functions of “potential type”, the paper aims at the development of the fourth order tensor-valued Taylor–Kármán structured covariance/correlation matrices. The characteristic functions of these tensor-valued covariance/correlation matrices, namely the lateral and longitudinal components, will be derived for n-dimensional spaces, here specified for n=3 dimensions in the paper. A special part is devoted to the Hankel transformation for gravity gradients and their variance-covariance in order to guarantee consistency well-known from problems in using Fourier transformations. We use the variance-covariance function of type (i) isotropic, (ii) homogeneous and (iii) potential as prior information for fitting the discrete data of variances and covariances estimated from observations.

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U2 - 10.1007/s13137-015-0071-y

DO - 10.1007/s13137-015-0071-y

M3 - Article

AN - SCOPUS:84945178605

VL - 6

SP - 319

EP - 342

JO - GEM - International Journal on Geomathematics

JF - GEM - International Journal on Geomathematics

SN - 1869-2672

IS - 2

ER -