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

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Abstract

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 closed-loop 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.

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Notes

  1. Although common digits should be in principle integers, we use decimal points to distinguish slight differences in the results.

    Fig. 2
    figure 2

    Test of the rotated observation equations through closed-loop tests for \(\ell =T_i^e\) (left top/bottom panel) and \(\ell =T_{ij}^e\) (right top/bottom panel) and \(N=70\). The top panels depict the common digits of the input geopotential set and the recovered coefficients from the LS SHA. Common digits of the two sets \(P_1, P_2\) are determined by \(\log _{10}\left(\frac{|P_1|}{|P_1-P_2|}\right)\). The bottom panels display coefficient residuals in terms of height anomaly with \(\text{ RMS}=6.2\times 10^{-14}\) m (left) and \(\text{ RMS}=2.1\times 10^{-12}\) m (right)

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Acknowledgments

This work was co-sponsored by ESA/PECS grant n. C98056 and by project CEDR LC506 of Ministry of Education of the Czech Republic. The authors thank to all reviewers and the editor, special thanks go to Dr. Pavel Ditmar for his detailed comments.

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Authors and Affiliations

  1. Faculty of Civil Engineering, CTU in Prague, Thákurova 7, 166-29 , Praha 6, Czech Republic

    Josef Sebera

  2. Astronomical Institute, Academy of Sciences of the Czech Republic, Fričova 298, 25165, Ondřejov, Czech Republic

    Josef Sebera, Aleš Bezděk & Jaroslav Klokočník

  3. 2801 Roesh Way, Vienna, VA, USA

    Carl A. Wagner

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  1. Josef Sebera
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  2. Carl A. Wagner
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Correspondence to Josef Sebera.

Appendices

Appendix A: Hotine’s harmonics and normalization factors

For the reader’s convenience, the relations between Hotine’s harmonics and the ordinary geopotential coefficients are listed. The original Hotine’s equations in Hotine (1969) are non-normalized and the similar functions in Bettadpur (1995); Petrovskaya and Vershkov (2009), (2011) use a different convention. Although in Hotine’s coefficients, the maximum order can formally exceed the degree of the input gravitational model, the true limit in order is given by the maximum degree/order of the gravitational model.

$$\begin{aligned}&\quad V_x: \\&{\left\{ \begin{array}{ll} \bar{C}_{n+1,0}^{x}&=\!\! \frac{1}{2} v_{+1}\bar{C}_{n,1}\\ \bar{S}_{n+1,0}^{x}&\!\! = 0 \\ \end{array}\right.} \\&{\left\{ \begin{array}{ll} \bar{C}_{n+1,1}^{x}&\!\!= -v_{-1}\bar{C}_{n,0} + \frac{1}{2}v_{+1}\bar{C}_{n,2} \\ \bar{S}_{n+1,1}^{x}&\!\!= +\frac{1}{2} v_{+1}\bar{S}_{n,2}\\ \end{array}\right.} \\&{\left\{ \begin{array}{ll} \bar{C}_{n+1,m}^{x}&\!\!= -\frac{1}{2} v_{-1}\bar{C}_{n,m-1} + \frac{1}{2}v_{+1} \bar{C}_{n,m+1}\\ \bar{S}_{n+1,m}^{x}&\!\!= -\frac{1}{2} v_{-1}\bar{S}_{n,m-1} + \frac{1}{2}v_{+1} \bar{S}_{n,m+1}\\ \end{array}\right.}\\&\quad V_y: \\&{\left\{ \begin{array}{ll} \bar{C}_{n+1,0}^{y}&\!\!= \frac{1}{2} v_{+1}\bar{S}_{n,1}\\ \bar{S}_{n+1,0}^{y}&\!\!= 0 \\ \end{array}\right.} \\&{\left\{ \begin{array}{ll} \bar{C}_{n+1,1}^{y}&\!\!= +\frac{1}{2} v_{+1}\bar{S}_{n,2} \\ \bar{S}_{n+1,1}^{y}&\!\! = -v_{-1}\bar{C}_{n,0} - \frac{1}{2}v_{+1}\bar{C}_{n,2}\\ \end{array}\right.} \\&{\left\{ \begin{array}{ll} \bar{C}_{n+1,m}^{y}&\!\! = \frac{1}{2} v_{-1}\bar{S}_{n,m-1} + \frac{1}{2}v_{+1} \bar{S}_{n,m+1}\\ \bar{S}_{n+1,m}^{y}&\!\!= -\frac{1}{2} v_{-1}\bar{C}_{n,m-1} - \frac{1}{2}v_{+1} \bar{C}_{n,m+1}\\ \end{array}\right.}\\&\quad V_z: \\&{\left\{ \begin{array}{ll} \bar{C}_{n+1,0}^{z}&\!\!= -v_0 \bar{C}_{n,0}\\ \bar{S}_{n+1,0}^{z}&\!\!= 0 \\ \end{array}\right.} \\&{\left\{ \begin{array}{ll} \bar{C}_{n+1,1}^{z}&\!\!= -v_0 \bar{C}_{n,1} \\ \bar{S}_{n+1,1}^{z}&\!\!= -v_0 \bar{S}_{n,1} \\ \end{array}\right.} \\&{\left\{ \begin{array}{ll} \bar{C}_{n+1,m}^{z}&\!\!= -v_0 \bar{C}_{n,m} \\ \bar{S}_{n+1,m}^{z}&\!\!= -v_0 \bar{S}_{n,m} \\ \end{array}\right.} \\&\quad V_{xx}:\\&{\left\{ \begin{array}{ll} \bar{C}_{n+2,0}^{xx}&\!\!= -\frac{1}{2} t_0\bar{C}_{n,0} + \frac{1}{4}t_{+2} \bar{C}_{n,2} \\ \bar{S}_{n+2,0}^{xx}&\!\!= 0 \\ \end{array}\right.} \\&{\left\{ \begin{array}{ll} \bar{C}_{n+2,1}^{xx}&\!\!= -\frac{3}{4} t_0\bar{C}_{n,1} + \frac{1}{4}t_{+2} \bar{C}_{n,3} \\ \bar{S}_{n+2,1}^{xx}&\!\!= -\frac{1}{4} t_0\bar{S}_{n,1} + \frac{1}{4}t_{+2} \bar{S}_{n,3} \\ \end{array}\right.} \\&{\left\{ \begin{array}{ll} \bar{C}_{n+2,2}^{xx}&\!\! = \frac{1}{4} t_{-2}\bar{C}_{n,0} - \frac{1}{2}t_0 \bar{C}_{n,2} + \frac{1}{4}t_{+2} \bar{C}_{n,4} \\ \bar{S}_{n+2,2}^{xx}&\!\!= - \frac{1}{2}t_0 \bar{S}_{n,2} + \frac{1}{4}t_{+2} \bar{S}_{n,4} \end{array}\right.} \\&{\left\{ \begin{array}{ll} \bar{C}_{n+2,m}^{xx}&\!\!= \frac{1}{4} t_{-2}\bar{C}_{n,m-2} - \frac{1}{2}t_0 \bar{C}_{n,m} + \frac{1}{4}t_{+2} \bar{C}_{n,m+2} \\ \bar{S}_{n+2,m}^{xx}&\!\!= \frac{1}{4} t_{-2}\bar{S}_{n,m-2} - \frac{1}{2}t_0 \bar{S}_{n,m} + \frac{1}{4}t_{+2} \bar{S}_{n,m+2} \\ \end{array}\right.}\\&\quad V_{yy}:\\&{\left\{ \begin{array}{ll} \bar{C}_{n+2,0}^{yy}&\!\!= -\frac{1}{2} t_0\bar{C}_{n,0} - \frac{1}{4}t_{+2} \bar{C}_{n,2} \\ \bar{S}_{n+2,0}^{yy}&\!\!= 0 \\ \end{array}\right.} \\&{\left\{ \begin{array}{ll} \bar{C}_{n+2,1}^{yy}&\!\!= -\frac{1}{4} t_0\bar{C}_{n,1} - \frac{1}{4}t_{+2} \bar{C}_{n,3} \\ \bar{S}_{n+2,1}^{yy}&\!\!= -\frac{3}{4} t_0\bar{S}_{n,1} - \frac{1}{4}t_{+2} \bar{S}_{n,3} \\ \end{array}\right.} \\&{\left\{ \begin{array}{ll} \bar{C}_{n+2,2}^{yy}&\!\!= -\frac{1}{4} t_{-2}\bar{C}_{n,0} - \frac{1}{2}t_0 \bar{C}_{n,2} - \frac{1}{4}t_{+2} \bar{C}_{n,4} \\ \bar{S}_{n+2,2}^{yy}&\!\!= - \frac{1}{2}t_0 \bar{S}_{n,2} - \frac{1}{4}t_{+2} \bar{S}_{n,4} \end{array}\right.} \\&{\left\{ \begin{array}{ll} \bar{C}_{n+2,m}^{xx}&\!\!= \frac{1}{4} t_{-2}\bar{C}_{n,m-2} - \frac{1}{2}t_0 \bar{C}_{n,m} + \frac{1}{4}t_{+2} \bar{C}_{n,m+2} \\ \bar{S}_{n+2,m}^{xx}&\!\!= \frac{1}{4} t_{-2}\bar{S}_{n,m-2} - \frac{1}{2}t_0 \bar{S}_{n,m} + \frac{1}{4}t_{+2} \bar{S}_{n,m+2} \\ \end{array}\right.}\\&\quad V_{zz}:\\&{\left\{ \begin{array}{ll} \bar{C}_{n+2,0}^{zz}&\!\!= t_0\bar{C}_{n,0} \\ \bar{S}_{n+2,0}^{zz}&\!\!= 0 \\ \end{array}\right.} \\&{\left\{ \begin{array}{ll} \bar{C}_{n+2,1}^{zz}&\!\!= t_0\bar{C}_{n,1} \\ \bar{S}_{n+2,1}^{zz}&\!\!= t_0\bar{S}_{n,1} \\ \end{array}\right.} \\&{\left\{ \begin{array}{ll} \bar{C}_{n+2,2}^{zz}&\!\!= t_0\bar{C}_{n,2} \\ \bar{S}_{n+2,2}^{zz}&\!\!= t_0\bar{S}_{n,2} \end{array}\right.} \\&{\left\{ \begin{array}{ll} \bar{C}_{n+2,m}^{zz}&\!\!= t_0\bar{C}_{n,m}\\ \bar{S}_{n+2,m}^{zz}&\!\!= t_0\bar{S}_{n,m} \\ \end{array}\right.}\\&\quad V_{xy}:\\&{\left\{ \begin{array}{ll} \bar{C}_{n+2,0}^{xy}&\!\!= \frac{1}{4}t_{+2} \bar{S}_{n,2} \\ \bar{S}_{n+2,0}^{xy}&\!\!= 0 \\ \end{array}\right.} \\&{\left\{ \begin{array}{ll} \bar{C}_{n+2,1}^{xy}&\!\!= -\frac{1}{4} t_0\bar{S}_{n,1} + \frac{1}{4}t_{+2} \bar{S}_{n,3} \\ \bar{S}_{n+2,1}^{xy}&\!\!= -\frac{1}{4} t_0\bar{C}_{n,1} - \frac{1}{4}t_{+2} \bar{C}_{n,3} \\ \end{array}\right.} \\&{\left\{ \begin{array}{ll} \bar{C}_{n+2,2}^{xy}&\!\!= \frac{1}{4} t_{+2}\bar{S}_{n,4} \\ \bar{S}_{n+2,2}^{xy}&\!\!= \frac{1}{2}t_{-2} \bar{C}_{n,0} - \frac{1}{4}t_{+2} \bar{C}_{n,4} \end{array}\right.} \\&{\left\{ \begin{array}{ll} \bar{C}_{n+2,m}^{xy}&\!\!= -\frac{1}{4}t_{-2} \bar{S}_{n,m-2} + \frac{1}{4}t_{+2} \bar{S}_{n,m+2} \\ \bar{S}_{n+2,m}^{xy}&\!\!= \frac{1}{4}t_{-2} \bar{C}_{n,m-2} - \frac{1}{4}t_{+2} \bar{C}_{n,m+2} \\ \end{array}\right.}\\&\quad V_{xz}:\\&{\left\{ \begin{array}{ll} \bar{C}_{n+2,0}^{xz}&\!\!= -\frac{1}{2}t_{+1} \bar{C}_{n,1} \\ \bar{S}_{n+2,0}^{xz}&\!\!= 0 \\ \end{array}\right.} \\&{\left\{ \begin{array}{ll} \bar{C}_{n+2,1}^{xz}&\!\!= t_{-1}\bar{C}_{n,0} - \frac{1}{2}t_{+1} \bar{C}_{n,2}\\ \bar{S}_{n+2,1}^{xz}&\!\!= -\frac{1}{2} t_{+1}\bar{S}_{n,2}\\ \end{array}\right.} \\&{\left\{ \begin{array}{ll} \bar{C}_{n+2,m}^{xz}&\!\!= \frac{1}{2} t_{-1}\bar{C}_{n,m-1} -\frac{1}{2} t_{+1}\bar{C}_{n,m+1} \\ \bar{S}_{n+2,m}^{xz}&\!\!= \frac{1}{2} t_{-1}\bar{S}_{n,m-1} -\frac{1}{2} t_{+1}\bar{S}_{n,m+1} \\ \end{array}\right.}\\&\quad V_{yz}:\\&{\left\{ \begin{array}{ll} \bar{C}_{n+2,0}^{yz}&\!\!= -\frac{1}{2}t_{+1} \bar{S}_{n,1} \\ \bar{S}_{n+2,0}^{yz}&\!\! = 0 \\ \end{array}\right.} \\&{\left\{ \begin{array}{ll} \bar{C}_{n+2,1}^{yz}&\!\!= -\frac{1}{2} t_{+1}\bar{S}_{n,2} \\ \bar{S}_{n+2,1}^{yz}&\!\!= t_{-1}\bar{C}_{n,0} + \frac{1}{2}t_{+1} \bar{C}_{n,2} \\ \end{array}\right.} \\&{\left\{ \begin{array}{ll} \bar{C}_{n+2,m}^{yz}&\!\! = -\frac{1}{2} t_{-1}\bar{S}_{n,m-1} -\frac{1}{2} t_{+1}\bar{S}_{n,m+1} \\ \bar{S}_{n+2,m}^{yz}&\!\!= \frac{1}{2} t_{-1}\bar{C}_{n,m-1} +\frac{1}{2}t_{+1}\bar{C}_{n,m+1} \\ \end{array}\right.} \end{aligned}$$

Normalized Hotine’s factors for the acceleration vector read (use of \(v\) implicates they refer to a vector):

$$\begin{aligned} v_0&= \sqrt{\frac{(2n+1)(n+m+1)(n-m+1)}{2n+3}}, \\ v_{-1}&= a \sqrt{\frac{(2n+1)(n+m+1)(n+m)}{2n+3}}\\&\text{ with}\,\, a =\frac{1}{\sqrt{2}}\,\, {\text{ for}}\,\, m=1, a=1 \,\,\text{ otherwise} \\ v_{+1}&= a \sqrt{\frac{(2n+1)(n-m+1)(n-m)}{2n+3}}\\&\text{ with}\,\, a=\sqrt{2} \,\,\text{ for}\,\, m=0,a=1\,\,\text{ otherwise} \end{aligned}$$

and similarly for the components of the Eötvös tensor (use of \(t\) implicates they refer to a tensor):

$$\begin{aligned} t_0 \!&= \!\sqrt{\frac{(2n\!+\!1)(n\!+\!m\!+\!2)(n\!+\!m\!+\!1)(n\!-\!m\!+\!2)(n\!-\!m\!+\!1)}{2n\!+\!5}}, \\ t_{\!-\!1} \!&= \!a\sqrt{\frac{(2n\!+\!1)(n\!+\!m\!+\!2)(n\!+\!m\!+\!1)(n\!+\!m)(n\!-\!m\!+\!2)}{2n\!+\!5}} \\&\text{ with}\,\,a=\frac{1}{\sqrt{2}} \,\,\text{ for}\,\, m=1,a=1 \,\,\text{ otherwise}\\ t_{\!+\!1} \!&= \!a\sqrt{\frac{(2n\!+\!1)(n\!+\!m\!+\!2)(n\!-\!m\!+\!1)(n\!-\!m)(n\!-\!m\!+\!2)}{2n\!+\!5}}\\&\text{ with}\,\, a=\sqrt{2} \,\,\text{ for}\,\, m=0,a=1\,\, \text{ otherwise} \\ t_{\!+\!2} \!&= \!a \sqrt{\frac{(2n\!+\!1)(n\!+\!m\!+\!2)(n\!+\!m\!+\!1)(n\!+\!m)(n\!-\!m\!+\!2)}{2n\!+\!5}}\\&\text{ with}\,\, a=\sqrt{2} \,\,\text{ for}\,\, m=0,a=1 \,\,\text{ otherwise} \\ t_{\!-\!2} \!&= \!a\sqrt{\frac{(2n\!+\!1)(n\!+\!m\!+\!2)(n\!+\!m\!+\!1)(n\!+\!m)(n\!-\!m\!-\!1)}{2n\!+\!5}}\\&\text{ with}\,\, a=\sqrt{2} \,\,\text{ for}\,\, m=2,a=1 \,\,\text{ otherwise}. \end{aligned}$$

The better overview of the exceptions defined by multiplier \(a\) is shown in Fig. 5, where the exceptions are red points. The exceptions hold for the whole order.

Fig. 5
figure 5

Use of the multiplier \(a\) in \(v_{+1},v_{-1},t_{+1},t_{-1},t_{+2},t_{-2}\) (in red) for degree 4

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Appendix B: Rotational matrices

With the ordinary rotational matrix \(R\)

$$\begin{aligned} R= \begin{pmatrix} r_{11}&\quad r_{12}&\quad r_{13} \\ r_{21}&\quad r_{22}&\quad r_{23} \\ r_{31}&\quad r_{32}&\quad r_{33} \end{pmatrix}, \end{aligned}$$

and the transformation law for tensors \(V_{ij}=R V_{ij}^{^{\prime }} R^\mathrm{T} \) we get the matrix \(S\)

$$\begin{aligned}&S = \begin{pmatrix} s_{1}^{xx}&\quad s_{2}^{xx}&\quad s_{3}^{xx}&\quad s_{4}^{xx}&\quad s_{5}^{xx}&\quad s_{6}^{xx} \\ s_{1}^{yy}&\quad s_{2}^{yy}&\quad s_{3}^{yy}&\quad s_{4}^{yy}&\quad s_{5}^{yy}&\quad s_{6}^{yy} \\ s_{1}^{zz}&\quad s_{2}^{zz}&\quad s_{3}^{zz}&\quad s_{4}^{zz}&\quad s_{5}^{zz}&\quad s_{6}^{zz} \\ s_{1}^{xy}&\quad s_{2}^{xy}&\quad s_{3}^{xy}&\quad s_{4}^{xy}&\quad s_{5}^{xy}&\quad s_{6}^{xy} \\ s_{1}^{xz}&\quad s_{2}^{xz}&\quad s_{3}^{xz}&\quad s_{4}^{xz}&\quad s_{5}^{xz}&\quad s_{6}^{xz} \\ s_{1}^{yz}&\quad s_{2}^{yz}&\quad s_{3}^{yz}&\quad s_{4}^{yz}&\quad s_{5}^{yz}&\quad s_{6}^{yz} \end{pmatrix} =\\&\quad \begin{pmatrix} r_{13}^2&\, r_{12}^2&\, r_{11}^2&\, 2r_{11}r_{12}&\, 2r_{13}r_{11}&\, 2r_{13}r_{12} \\ r_{23}^2&\, r_{22}^2&\, r_{21}^2&\, 2r_{21}r_{22}&\, 2r_{21}r_{23}&\, 2r_{22}r_{23} \\ r_{33}^2&\, r_{32}^2&\, r_{31}^2&\, 2r_{31}r_{32}&\, 2r_{31}r_{33}&\, 2r_{32}r_{33} \\ r_{13}r_{23}&\, r_{12}r_{22}&\, r_{11}r_{21}&\, (r_{12}r_{21}+r_{11}r_{22})&\, (r_{13}r_{21} +r_{11}r_{23})&\, (r_{13}r_{22} + r_{12}r_{23}) \\ r_{13}r_{33}&\, r_{12}r_{32}&\, r_{11}r_{31}&\, (r_{12}r_{31}+r_{11}r_{32})&\, (r_{13}r_{31} +r_{11}r_{33})&\, (r_{13}r_{32} + r_{12}r_{33}) \\ r_{23}r_{33}&\, r_{22}r_{32}&\, r_{21}r_{31}&\, (r_{22}r_{31} +r_{21}r_{32})&\, (r_{23}r_{31} +r_{21}r_{33})&\, (r_{23}r_{32} + r_{22}r_{33}) \end{pmatrix}. \end{aligned}$$

For example, transfer from the EFF into the spherical LNOF is provided by rotational sequence:

$$\begin{aligned} R\!=\!R_y(-\theta )R_z(\lambda - \pi )\!=\! \begin{pmatrix} \cos \theta&\quad 0&\quad \sin \theta \\ 0&\quad 1&\quad 0 \\ -\sin \theta&\quad 0&\cos \theta \\ \end{pmatrix} \begin{pmatrix} \cos (\lambda - \pi )&\quad \sin (\lambda - \pi )&0 \\ -\sin (\lambda - \pi )&\quad \cos (\lambda - \pi )&0 \\ 0&0&1 \end{pmatrix}. \end{aligned}$$

Adding one more rotation around \(y\) axis for a difference between latitudes \(\alpha ={\varphi }_g-\varphi _s=\theta -\vartheta \) we move from the spherical into the ellipsoidal LNOF:

$$\begin{aligned} R=R_y(\alpha )R_y(-\theta )R_z(\lambda - \pi ). \end{aligned}$$

Appendix C: Matlab codes of the design matrix for \(V_i\) and \(V_{ij}\)

The input matrix with the ALFs has to keep the triangular convention from Fig. 5 (order in columns, degree in rows with \(\bar{P}_{0,0}\) at upper left corner). For the acceleration vector, this matrix must be generated up to \(N+1\), whereas for the Eötvös tensor up to \(N+2\). The routines evaluate all degree/order functions \(v_0,v_{\pm 1},t_0,t_{\pm 1},t_{\pm 2}\) each time the function is called. This is only due to the compactness of the code presented. As these d/o functions do not depend on (\(r,\theta ,\lambda \)), they can be computed before the routines “tiA”, “tijA” are called.

1.1 C.1 The design matrix for the acceleration vector

figure a1

1.2 C.2 The design matrix for the Eötvös tensor

figure a2
figure a3

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Sebera, J., Wagner, C.A., Bezděk, A. et al. Short guide to direct gravitational field modelling with Hotine’s equations. J Geod 87, 223–238 (2013). https://doi.org/10.1007/s00190-012-0591-2

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  • DOI: https://doi.org/10.1007/s00190-012-0591-2

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