Improving the Abel transform inversion using bending angles from FORMOSAT-3/COSMIC

Abstract

The FORMOSAT-3/COSMIC satellite constellation has become an important tool toward providing global remote sensing data for sounding of the atmosphere of the earth and the ionosphere in particular. In this study, the electron density profiles are derived using the Abel transform inversion. Some drawbacks of this transform in LEO GPS sounding can be overcome by considering the separability concept: horizontal gradients of vertical total electron content (VTEC) information are incorporated by the inversion method, providing more accurate electron density determinations. The novelty presented in this paper with respect to previous works is the use of the phase change between the GPS transmitter and the LEO receiver as the main observable instead of the ionospheric combination of carrier phase observables for the implementation of separability in the inversion process. Some of the characteristics of the method when applied to the excess phase are discussed. The results obtained show the equivalence of both approaches but the method exposed in this work has the potentiality to be applied to the neutral atmosphere. Recent FORMOSAT-3/COSMIC data have been processed with both the classical Abel inversion and the separability approach and evaluated versus colocated ionosonde data.

Appendix

In order to implement separability to bending angles, it would be required to have a proportional relationship between bending angles and electron densities that would allow, for two consecutive concentric spherical layers, to write the increment of bending angles as the corresponding increment of electron densities within the two layers. Unfortunately, this is not the case and some approximations are needed to derive such proportionality. Details are given later.

We start from the definition of the ionosphere’s refractive index n derived by Appleton and Hartree, and accept that it can be approximated by first-order form with an accuracy better than 0.1% (Davies 1990),

$$ n^{2} = 1 - {\frac{{f_{p}^{2} }}{{f^{2} }}} $$

(12)

where f is the system operating frequency and f _p , the plasma frequency that is defined by

$$ f_{p} = \sqrt {{\frac{{N_{e} \cdot e^{2} }}{{\varepsilon_{0} m(2\pi )^{2} }}}} $$

(13)

where e stands for the electron charge, $\varepsilon_{0}$ the permittivity of free space and, m the rest mass of an electron. Hence, the expression in (12) now becomes

$$ n^{2} = 1 - {\frac{{N_{e} \cdot e^{2} }}{{\varepsilon_{0} m(2\pi f)^{2} }}} $$

(14)

Differentiating the latter equation gives

$$ 2ndn = - {\frac{{e^{2} }}{{\varepsilon_{0} m(2\pi f)^{2} }}}dN_{e} $$

(15)

$$ {\frac{dn}{n}} = - {\frac{1}{2}}{\frac{{e^{2} }}{{\varepsilon_{0} m(2\pi )^{2} (f^{2} - f_{p}^{2} )^{{}} }}}dN_{e} $$

(16)

Considering the nominal value of GPS frequency L ₁ $(f_{1} = 1575.42\,{\text{MHz}})$ and the plasma frequency $(f_{p} = 20\,{\text{MHz}})$, the denominator in (16) can be approximated by

$$ f^{2} - f_{p}^{2} \approx f^{2} $$

(17)

Substituting (17) into (16) leads to

$$ {\frac{dn}{n}} \approx - {\frac{1}{2}}{\frac{{e^{2} }}{{\varepsilon_{0} m(2\pi )^{2} f^{2} }}}dN_{e} $$

(18)

The latter equation provides the key to separability implementation when using the L ₁ bending angle α ₁ as input data for the inversion since it will give the proportionality relationship between bending angle and electron density.

Using Bouguer’s formula, which is equivalent to Snell’s law in a spherically symmetric medium, we can establish the relationship of bending angle α and the refractive index n

$$ nr\sin \theta = a $$

(19)

The symbol r stands for the geocentric distance, θ the zenith angle of the LOS vector and a the impact parameter. Considering a layered ionosphere, the change between different layers is obtained by differentiating (19)

$$ \Updelta nr\sin \theta + n\Updelta r\sin \theta + nr\cos \theta \Updelta \theta = 0 $$

(20)

Rearranging the terms in previous equation gives

$$ \Updelta \theta = - {\frac{\Updelta n}{n}}\tan \theta - {\frac{\Updelta r}{r}}\tan \theta $$

(21)

The first term in (21) takes into account the change in the ray path due to the changes in the refractive index while the second depends on the geometric variations. Actually, the first term provides the definition of the bending angle change in terms of n,

$$ \Updelta \alpha = - {\frac{\Updelta n}{n}}\tan \theta $$

(22)

Recalling the expression in (18), it can be derived that the increment of bending angle between consecutive layers with different refractive index is

$$ \Updelta \alpha = - {\frac{\Updelta n}{n}}\tan \theta = {\frac{1}{2}}{\frac{{e^{2} }}{{\varepsilon_{0}^{{}} m(2\pi f)^{2} }}}\Updelta N_{e} \tan \theta $$

(23)

Therefore, the total bending angle for one ray path would be obtained by adding all bending angle contributions

$$ \alpha = \sum {\Updelta \alpha_{i} } = {\frac{1}{2}}{\frac{{e^{2} }}{{\varepsilon_{0} m(2\pi f)^{2} }}}\sum {\Updelta N_{ei} \tan \phi_{i} } $$

(24)