Alternative validation method of satellite gradiometric data by integral transform of satellite altimetry data

Abstract

Integral transforms of the disturbing gravitational potential derived from satellite altimetry onto satellite gradiometric data are formulated, investigated and applied in this article. First, corresponding differential operators, that relate the disturbing gravitational potential to the six components of the disturbing gradiometric tensor in the spherical local north-oriented frame, are applied to the spherical Abel-Poisson integral equation. This yields six new integral equations for which respective kernel functions are given in both spectral and spatial forms. Second, truncation error formulas for each of the integral transforms are provided in the spectral form. Also expressions for the corresponding truncation error coefficients are derived. Third, practical estimators for evaluation of the disturbing gravitational gradients are formulated, and their correctness and expected accuracy are investigated. Finally, the practical estimators are applied for validation of a sample of the gradiometric data provided by the GOCE satellite mission. Obtained results demonstrate applicability of the new apparatus as an alternative validation method of the satellite gravitational gradients.

Appendices

Appendix A: Global integrals of the disturbing gravitational potential

To obtain the spectral form of truncation errors for the integral transforms of Eq. (13), some global integrals of the disturbing gravitational potential are needed. The global integrals are summarized as follows, see, e.g. Thalhammer (1994) and Eshagh (2009a):

$$\begin{aligned} \frac{2 n + 1}{4 \pi } \int \limits _{\Omega '} T(R, \Omega ') \,P_{n,0}(u)\,{\mathrm {d}}\Omega ' = T_{n}(R, \Omega ), \end{aligned}$$

(44)

$$\begin{aligned} \frac{2 n + 1}{4 \pi } \int \limits _{\Omega '} T(R, \Omega ') \,P_{n,1}(u)\,\cos \alpha \,{\mathrm {d}}\Omega ' = \frac{\partial T_{n}(R, \Omega )}{\partial \varphi }, \end{aligned}$$

(45)

$$\begin{aligned} \frac{2 n + 1}{4 \pi } \int \limits _{\Omega '} T(R, \Omega ') \,P_{n,1}(u)\,\sin \alpha \,{\mathrm {d}}\Omega '{=}\frac{1}{\cos \varphi } \frac{\partial T_{n}(R, \Omega )}{\partial \lambda },\nonumber \\ \end{aligned}$$

(46)

$$\begin{aligned}&\frac{2 n + 1}{4 \pi } \int \limits _{\Omega '} T(R, \Omega ') \,P_{n,2}(u)\,\cos 2 \alpha \,{\mathrm {d}}\Omega ' \nonumber \\&\quad = \left[ n (n + 1) T_{n}(R, \Omega ) + 2 \frac{\partial ^2 T_{n}(R, \Omega )}{\partial \varphi ^2}\right] , \end{aligned}$$

(47)

$$\begin{aligned}&\frac{2 n + 1}{4 \pi } \int \limits _{\Omega '} T(R, \Omega ') \,P_{n,2}(u)\,\sin 2 \alpha \,{\mathrm {d}}\Omega ' \nonumber \\&\quad = \frac{2}{\cos \varphi } \left[ \tan \varphi \frac{\partial T_{n}(R, \Omega )}{\partial \lambda } + \frac{\partial ^2 T_{n}(R, \Omega )}{\partial \lambda \partial \varphi }\right] . \end{aligned}$$

(48)

Appendix B: Spectral and recurrence representation of the modified truncation error coefficients

The modified truncation error coefficients $Q_{n}^{\bullet }$, $Q_{n}^{\bullet t}$ and $Q_{n}^{\bullet tt}$ in the spectral form are defined in the following proposition:

Proposition 8

The spectral representations of the modified truncation error coefficients are:

$$\begin{aligned}&Q_{n}^{\bullet }(t, u_0) = \sum \limits _{k=0}^{\infty } t^{k+2} (2k + 1)\ e_{n,k} (u_0), \end{aligned}$$

(49)

$$\begin{aligned}&Q_{n}^{\bullet t}(t, u_0) = \sum \limits _{k=0}^{\infty } t^{k+1} (2k + 1)(k + 2)\ e_{n,k} (u_0),\end{aligned}$$

(50)

$$\begin{aligned}&Q_{n}^{\bullet tt}(t, u_0) = \sum \limits _{k=0}^{\infty } t^{k} (2k + 1)(k + 1)(k + 2)\ e_{n,k} (u_0), \end{aligned}$$

(51)

where:

$$\begin{aligned} e_{n,k} (u_0) = \int _{-1}^{u_0} P_{n,0} (u)\ P_{k,0} (u)\, {\mathrm {d}}u. \end{aligned}$$

(52)

Proposition 8 follows from Eqs. (40)–(42) and from the spectral form of the modified isotropic kernel $K^{\bullet }$ given by Eq. (37). Note that Eq. (52) defines the so-called Paul coefficients, which can be evaluated by recurrence formulas, see, e.g. Paul (1973).

The modified truncation error coefficients $Q_{n}^{\bullet }$, $Q_{n}^{\bullet t}$ and $Q_{n}^{\bullet tt}$ may be defined by the following set of recurrence formulas:

Proposition 9

The recurrence representations of the modified truncation error coefficients are:

$\underline{{\mathbf {(A) for the coefficient}}\ Q_{0}^{\bullet }}$:

$$\begin{aligned} Q_{0}^{\bullet }(t, u_0)&= t (1 - t) \left( \frac{1 + t}{g_0} - 1\right) ,\nonumber \\ Q_{1}^{\bullet }(t, u_0)&= (t - 1) \left[ 1 - (t + 1) \left( g_0 - t + \frac{t u_0}{g_0}\right) \right] ,\nonumber \\ Q_{n}^{\bullet }(t, u_0)&= \frac{1 + t^2}{t}\ Q_{n-1}^{\bullet }(t, u_0)\nonumber \\&\quad - Q_{n-2}^{\bullet }(t, u_0) - \frac{t (1 - t^2)}{g_0}\ I_{n-1}(u_0), \end{aligned}$$

(53)

$\underline{{\mathbf {(B) for the coefficient}}\ Q_{0}^{\bullet t}}$:

$$\begin{aligned} Q_{0}^{\bullet t}(t, u_0)&= 2 t - 1 \nonumber \\&\quad + \frac{1}{g_0} \left[ 1 - 3 t^2 + \frac{t (1 - t^2) (u_0 - t)}{g_0^2}\right] ,\nonumber \\ Q_{1}^{\bullet t}(t, u_0)&= t \Bigg \{3 t - g_0 + \frac{1}{g_0} \Bigg [(t + u_0) (u_0 - 2 t)\nonumber \\&\quad + \frac{2 t u_0 (u_0 - t)^2}{g_0^2}\Bigg ]\Bigg \},\nonumber \\ Q_{n}^{\bullet t}(t, u_0)&= \frac{1 + t^2}{t}\ Q_{n-1}^{\bullet t}(t, u_0)\nonumber \\&\quad - Q_{n-2}^{\bullet t}(t, u_0) - \frac{1 - t^2}{t^2}\ Q_{n-1}^{\bullet }(t, u_0)\nonumber \\&\quad - \left\{ g_0 + \frac{t}{g_0} \left[ 3 u_0 - 5 t + \frac{2 t (u_0 - t)^2}{g_0^2}\right] \right\} \nonumber \\&\qquad \times I_{n-1}(u_0), \end{aligned}$$

(54)

$\underline{{\mathbf {(C) for the coefficient}}\ Q_{0}^{\bullet tt}}$:

$$\begin{aligned} Q_{0}^{\bullet tt}(t, u_0)&= 2 - \frac{7 t}{g_0} + \frac{u_0 - t}{g_0^3} \Bigg [2 (1 - 4 t^2)\nonumber \\&\quad + \frac{3 t (1 - t^2) (u_0 - t)}{g_0^2}\Bigg ],\nonumber \\ Q_{1}^{\bullet tt}(t, u_0)&= 6 t - g_0 + \frac{(2 t + u_0) (u_0 - 3 t) - t^2}{g_0}\nonumber \\&\quad + \frac{t (u_0 - t)}{g_0^3}\left[ (u_0 - 2 t) (5 u_0 + t) \right. \nonumber \\&\quad +\left. \frac{6 t u_0 (u_0 - t)^2}{g_0^2}\right] ,\nonumber \\ Q_{n}^{\bullet tt}(t, u_0)&= \frac{1 + t^2}{t}\ Q_{n-1}^{\bullet tt}(t, u_0) - Q_{n-2}^{\bullet tt}(t, u_0)\nonumber \\&\quad - \frac{2(1 - t^2)}{t^2}\ Q_{n-1}^{\bullet t}(t, u_0) + \frac{2}{t^3}\ Q_{n-1}^{\bullet }(t, u_0)\nonumber \\&\quad - \frac{1}{g_0} \left\{ 2 u_0 - 9 t + \frac{t (u_0 - t)}{g_0^2}\right. \nonumber \\&\quad \left. \times \left[ 7 u_0 - 13 t + \frac{6 t (u_0 - t)^2}{g_0^2}\right] \right\} \, I_{n-1}(u_0), \end{aligned}$$

(55)

where:

$$\begin{aligned} g_0 = g(t, u_0) = \sqrt{1 - 2 t u_0 + t^2}, \end{aligned}$$

(56)

and:

$$\begin{aligned} I_{0}(u_0)&= 1 + u_0, \quad I_{1}(u_0) = \frac{1}{2} (u_0^2 - 1),\nonumber \\ I_{n}(u_0)&= \frac{1}{n + 1} [(2 n - 1)\ u_0\ I_{n-1}(u_0) {-} (n - 2)\ I_{n-2}(u_0)]. \end{aligned}$$

(57)

To prove Proposition 9, we start with the modified truncation error coefficient $Q_{n}^{\bullet }$. First, the spatial form of the modified isotropic kernel $K^{\bullet }$, see Eq. (37), is inserted into Eq. (40). Second, we make use of the analytical solution for the resulting integral given by Pavlis (1991, Eq. A. 29a–b) and Eq. (53) immediately follow. The modified truncation error coefficients $Q_{n}^{\bullet t}$ and $Q_{n}^{\bullet tt}$ are the first and second derivatives of $Q_{n}^{\bullet }$ with respect to the variable $t$, see Eqs. (41), (42). Performing the differentiations of Eq. (53), we get the recurrence formulas of Eqs. (54) and (55).