On the integral inversion of satellite-to-satellite velocity differences for local gravity field recovery: a theoretical study

Abstract

The gravity field can be recovered locally from the satellite-to-satellite velocity differences (VDs) between twin-satellites moving in the same orbit. To do so, three different integral formulae are derived in this paper to recover geoid height, radial component of gravity anomaly and gravity disturbance at sea level. Their kernel functions contain the product of two Legendre polynomials with different arguments. Such kernels are relatively complicated and it may be impossible to find their closed-forms. However, we could find the one related to recovering the geoid height from the VD data. The use of spectral forms of the kernels is possible and one does not have to generate them to very high degrees. The kernel functions are well-behaving meaning that they reduce the contribution of far-zone data and for example a cap margin of $7^{\circ }$ is enough for recovering gravity anomalies. This means that the inversion area should be larger by $7^{\circ }$ from all directions than the desired area to reduce the effect of spatial truncation error of the integral formula. Numerical studies using simulated data over Fennoscandia showed that when the distance between the twin-satellites is small, higher frequencies of the anomalies can be recovered from the VD data. In the ideal case of having short distance between the satellites flying at 250 km level, recovering radial component of gravity anomaly with an accuracy of 7 mGal is possible over Fennoscandia, if the VD data is contaminated only with the spatial truncation error, which is an ideal assumption. However, the problem is that the power of VD signal is very low when the satellites are close and it is very difficult to recognise the signal amongst the noise of the VD data. We also show that for a successful determination of gravity anomalies at sea level from an altitude of 250 km mean VDs with better accuracy than 0.01 mm/s are required. When coloured noise at this level is used for the VDs at 250 km with separation of 300 km, the accuracy of recovery will be about 11 mGal over Fennoscandia. In the case of using the real velocities of the satellites, the main problems are downward/upward continuation of the VDs on the mean orbital sphere and taking the azimuthal integration of them.

Appendix

The starting point for the mathematical derivation of the kernel (3e) in the closed-form is the following expression, see, e.g. (Anli and Gungor 2007):

$$\begin{aligned}&\sum _{n=0}^\infty {t^{n} P_n (x) P_n (y)} =\frac{2 K(k)}{\pi \sqrt{a-b}}, k=\sqrt{\frac{2b}{b-a}}, a=1-2xyt+t^{2},\nonumber \\&\quad b=-2t\sqrt{1-x^{2}}\sqrt{1-y^{2}}, \end{aligned}$$

(6)

where $-1 \le x \le 1$ and $-1 \le y \le 1$ and K(k) is the complete elliptic integral of the first kind:

$$\begin{aligned} K(k)=\int \limits _0^1 {\frac{\hbox {d}u}{\sqrt{1-u^{2}}\sqrt{1-k^{2}u^{2}}}}. \end{aligned}$$

(7)

Alternatively, Eq. (6) may be rewritten in terms of the Gauss hypergeometric function $_{2}F_{1}$:

$$\begin{aligned} \sum _{n=0}^\infty {t^{n} P_n (x) P_n (y)} =\frac{_2 F_1 (1/2,1/2,1,k^{2})}{\sqrt{a-b}}, \end{aligned}$$

(8)

where

$$\begin{aligned} {}_2 F_1 \left( {e,f,g,h} \right) =\sum _{n=0}^\infty {\frac{\left( e \right) _n \left( f \right) _n }{\left( g \right) _n }} \frac{h^{n}}{n!} \end{aligned}$$

(9)

is the Gauss hypergeometric function (Abramowitz and Stegun 1972). The last equation results from the relationship between the complete elliptic integral of the first kind and the Gauss hypergeometric function, i.e., $K(k)=\uppi /2_{2}F_{1}(1/2, 1/2, 1, k^{2})$, (see e.g. ibid. Eq. 17.3.9). Differentiating Eq. (8) with respect to the variable t we get:

$$\begin{aligned} \sum _{n=0}^\infty {n t^{n-1} P_n (x) P_n (y)}= & {} \frac{1}{\left( {a-b} \right) ^{3/2}}\left[ {(xy-t-\sqrt{1-x^{2}}\sqrt{1-y^{2}}) _2F_1 (1/2,1/2,1,k^{2})} \right. \nonumber \\&\left. {-\frac{b(1-t^{2})}{2t(a-b)^{2}} F_1(3/2,3/2,2,k^{2})} \right] , \end{aligned}$$

(10)

where the rule for differentiating the Gauss hypergeometric function has been applied, i.e.,:

$$\begin{aligned} \frac{\partial _2 F_1 (e,f,g,h)}{\partial h}=\frac{ef}{g}F_1(e+1,f+1,g+1,h). \end{aligned}$$

(11)

Having Eqs. (9) and (10) in hand, we may now provide the final closed-form formula for the kernel (3e). After some algebraic manipulations we get:

$$\begin{aligned}&\sum _{n=0}^\infty {(2n+1) t^{n+1} P_n (x) P_n (y)}\nonumber \\&\quad =\frac{t(1-t^{2})}{\left( {a-b} \right) ^{3/2}}\left[ {_2 F_1 (1/2,1/2,1,k^{2})-\frac{b}{a-b^{2}} F_1 (3/2,3/2,2,k^{2})} \right] . \end{aligned}$$

(12)