Completion of Band-Limited Data Sets on the Sphere

Abstract

In this study we propose the complementation of satellite-only gravity field models by additional a priori information to obtain a complete model. While the accepted gravity field models are restricted to a sub-domain of the frequency space, the complete models form a complete basis in the entire space, which can be represented in the frequency domain (spherical harmonics) as well as in the space domain (data grids). The additional information is obtained by the smoothness of the potential field. Using this a priori knowledge, a stochastic process on the sphere is established as a background model. The measurements of satellite-only models are assimilated to this background model by a subdivision into the commission, transition and omission sub-domain. Complete models can be used for a rigorous fusion of complementary data sets in a multi-mission approach and guarantee also, as stand-alone gravity-field models, full-rank variance/covariance matrices for all vector-valued, linearly independent functionals.

Appendix

1.1 Expectation and Variance of a Stochastic Process in Amplitude/Phase Notation on the Sphere

The stochastic process $\mathcal{U}(\vartheta,\lambda )$ on the sphere is defined by

$$\displaystyle{ \mathcal{U}(\vartheta,\lambda ) =\sum _{ \ell=0}^{\infty }\sum _{ m=0}^{\ell}A_{\ell m}\bar{P}_{\ell m}(\cos \vartheta )\cos \left (m\lambda + \mathcal{P}_{\ell m}\right ) }$$

(22)

where the phases constitute random variables. The distribution is defined by

$$\displaystyle\begin{array}{rcl} & & f_{\boldsymbol{\mathcal{P}}}(\mathbf{p}) =\prod \limits _{ \ell=0}^{\infty }\prod \limits _{ m=0}^{\ell}f_{ \mathcal{P}_{\ell m}}(p_{\ell m}) \\ & & \qquad \quad \mbox{ with}f_{\mathcal{P}_{\ell m}}(p_{\ell m}) = \left \{\begin{array}{crcccr} 0 & & &p_{\ell m}& \leq &-\pi \\ \frac{1} {2\pi } & \ \ -\pi &\leq &p_{\ell m}& \leq & \pi \\ 0 & \pi & \leq &p_{ \ell m}&. \end{array} \right.{}\end{array}$$

(23)

This means we have uniformly distributed phases, for each degree ℓ and order m, and phases for different degrees/orders are mutually independent.

We are interested in the expectation

$$\displaystyle\begin{array}{rcl} & & E\left \{\mathcal{U}(\vartheta,\lambda )\right \} =\int \limits _{ -\infty }^{\infty }u(\vartheta,\lambda )f_{\boldsymbol{ \mathcal{P}}}(\mathbf{p})d\mathbf{P} {}\\ & & =\!\int \limits _{ -\infty }^{\infty }\sum _{ \ell=0}^{\infty }\sum _{ m=0}^{\ell}\!\!A_{\ell m}\bar{P}_{\ell m}(\cos \vartheta )\cos \left (m\lambda + \mathcal{P}_{\ell m}\right )f_{\boldsymbol{\mathcal{P}}}(\mathbf{p})\;d\mathbf{P}\ . {}\\ \end{array}$$

Due to the independence of the phases the integral can be rewritten as

$$\displaystyle{ \int \limits _{-\infty }^{\infty }\sum _{ \ell=0}^{\infty }\sum _{ m=0}^{\ell}\!\!A_{\ell m}\bar{P}_{\ell m}(\cos \vartheta )\cos \left (m\lambda \,+\,\mathcal{P}_{\ell m}\right )f_{\mathcal{P}_{\ell m}}(p_{\ell m})\mathit{dp}_{\ell m}. }$$

If we now interchange integration and summation and introduce the individual distribution (23) we get

$$\displaystyle\begin{array}{rcl} & & E\left \{\mathcal{U}(\vartheta,\lambda )\right \} =\sum _{ \ell=0}^{\infty }\sum _{ m=0}^{\ell}A_{\ell m}\bar{P}_{\ell m}(\cos \vartheta ) {}\\ & & \qquad \qquad \qquad \qquad \times \ \frac{1} {2\pi }\int _{-\pi }^{\pi }\cos \left (m\lambda + p_{\ell m}\right )\mathit{dp}_{\ell m} {}\\ & & =\sum _{ \ell=0}^{\infty }\sum _{ m=0}^{\ell}A_{\ell m}\bar{P}_{\ell m}(\cos \vartheta )\;\frac{1} {2\pi }\big(-\sin m\lambda +\sin m\lambda \big)\ . {}\\ \end{array}$$

Finally we see that

$$\displaystyle{ E\left \{\mathcal{U}(\vartheta,\lambda )\right \} = 0\ . }$$

(24)

The variance of this process is given by

$$\displaystyle{ \mbox{ Cov}\left \{\mathcal{U}(\vartheta,\lambda ),\mathcal{U}(\vartheta ',\lambda ')\right \} =\int _{ -\infty }^{\infty }u(\vartheta,\lambda )\;u(\vartheta ',\lambda ')\;f_{\boldsymbol{ \mathcal{P}}}(\mathbf{p})d\mathbf{P}\ . }$$

(25)

Because of the independence of the random variables this can be written as

$$\displaystyle\begin{array}{rcl} & & \mbox{ Cov}\left \{\mathcal{U}(\vartheta,\lambda ),\mathcal{U}(\vartheta ',\lambda ')\right \} \\ & & \qquad =\sum _{ \ell=0}^{\infty }\sum _{ m=0}^{\ell}A_{\ell m}^{2}\bar{P}_{\ell m}(\cos \vartheta )\bar{P}_{\ell m}(\cos \vartheta ') \\ & & \qquad \times \ \int _{-\infty }^{\infty }\cos \left (m\lambda + p_{\ell m}\right )\cos \left (m\lambda ' + p_{\ell m}\right ) \\ & & \qquad \times f_{\mathcal{P}_{\ell m}}(p_{\ell m})\;dp_{\ell m}. {}\end{array}$$

(26)

If we extend the first cosine term in the integral to $\left (m(\lambda -\lambda ') + m\lambda ' + p_{\ell m}\right )$, use the relation

$$\displaystyle{ cos(x + y)\cos (x - y) = \frac{1} {2}(\cos 2x +\cos 2y) }$$

and substitute

$$\displaystyle\begin{array}{rcl} x& =& \frac{1} {2}m(\lambda -\lambda ') + m\lambda ' + p_{\ell m} {}\\ y& =& \frac{1} {2}m(\lambda -\lambda ') {}\\ \end{array}$$

the integral can be solved and yields

$$\displaystyle\begin{array}{rcl} & & \frac{1} {2}\sin \left (m(\lambda -\lambda ') + 2m\lambda ' + 2p_{\ell m}\right )\Big\vert _{p_{\ell m}=-\pi }^{p_{\ell m}=\pi } + {}\\ & & \qquad \qquad \qquad \qquad \qquad \qquad + \frac{1} {2}\cos \left (m(\lambda -\lambda ')\right )p_{\ell m}\Big\vert _{p_{\ell m}=-\pi }^{p_{\ell m}=\pi }\ \ . {}\\ \end{array}$$

The first term vanishes because of the skew symmetry of the sine and only the cosine term is relevant. Substituting this result into (26) yields

$$\displaystyle\begin{array}{rcl} & & \quad \mbox{ Cov}\left \{\mathcal{U}(\vartheta,\lambda ),\mathcal{U}(\vartheta ',\lambda ')\right \} {}\\ & & =\sum _{ \ell=0}^{\infty }\sum _{ m=0}^{\ell}\frac{1} {2}A_{\ell m}^{2}\bar{P}_{\ell m}(\cos \vartheta )\bar{P}_{\ell m}(\cos \vartheta ')\cos \left (m\lambda - m\lambda '\right )\ \ . {}\\ \end{array}$$

Applying the addition theorem

$$\displaystyle{ \cos \left (m\lambda - m\lambda '\right ) =\cos m\lambda \cos m\lambda ' +\sin m\lambda \sin m\lambda ' }$$

and recalling the definition of Laplace’s surface spherical harmonics (9) the right hand side can be reformulated as

$$\displaystyle\begin{array}{rcl} & & \sum _{\ell=0}^{\infty }\sum _{ m=0}^{\ell}\frac{1} {2}A_{\ell m}^{2}\Big(\bar{C}_{\ell m}(\vartheta,\lambda )\bar{C}_{\ell m}(\vartheta ',\lambda ') {}\\ & & \quad +\bar{ S}_{\ell m}(\vartheta,\lambda )\bar{S}_{\ell m}(\vartheta ',\lambda ')\Big)\ \ . {}\\ \end{array}$$

Introducing now the amplitudes defined in (12) we get

$$\displaystyle\begin{array}{rcl} & & \sum _{\ell=0}^{\infty }\sum _{ m=0}^{\ell} \frac{1} {2\ell + 1}\sigma _{\ell}^{2}\Big(\bar{C}_{\ell m}(\vartheta,\lambda )\bar{C}_{\ell m}(\vartheta ',\lambda ') {}\\ & & \quad +\bar{ S}_{\ell m}(\vartheta,\lambda )\bar{S}_{\ell m}(\vartheta ',\lambda ')\Big)\ \ . {}\\ \end{array}$$

The decomposition formula or addition theorem for spherical harmonics (cf. e.g. Moritz 1980, p. 23 (3–30))

$$\displaystyle\begin{array}{rcl} & & \quad P_{\ell}\big((\vartheta,\lambda );\;(\vartheta ',\lambda ')\big) = {}\\ & & \frac{1} {2\ell + 1}\sum _{m=0}^{\ell}\left (\bar{C}_{\ell m}(\vartheta,\lambda )\bar{C}_{\ell m}(\vartheta ',\lambda ') +\bar{ S}_{\ell m}(\vartheta,\lambda )\bar{S}_{\ell m}(\vartheta ',\lambda ')\right ) {}\\ \end{array}$$

allows the for further simplification

$$\displaystyle{ \mbox{ Cov}\left \{\mathcal{U}(\vartheta,\lambda ),\mathcal{U}(\vartheta ',\lambda ')\right \} =\sum _{ \ell=0}^{\infty }\sigma _{ \ell}^{2}P_{\ell}\big((\vartheta,\lambda );\;(\vartheta ',\lambda ')\big) }$$

where the function value of the Legendre polynomial $P_{\ell}\big((\vartheta,\lambda );\;(\vartheta ',\lambda ')\big)$ depends only on the spherical distance cosψ between $(\vartheta,\lambda )$ and $(\vartheta ',\lambda ')$

$$\displaystyle{ \cos \psi =\cos \vartheta \cos \vartheta ' +\sin \vartheta \sin \vartheta '\cos (\lambda -\lambda ')\ . }$$

(27)

Finally this results in

$$\displaystyle\begin{array}{rcl} \mbox{ Cov}\left \{\mathcal{U}(\vartheta,\lambda ),\mathcal{U}(\vartheta ',\lambda ')\right \}& =& \sum _{\ell=0}^{\infty }\sigma _{ \ell}^{2}\;P_{\ell}(\cos \psi ) = \mathrm{cov}(\psi,\sigma _{\ell}^{2}). {}\\ \end{array}$$