Theory of GNSS Radio Occultation - GNSS Remote Sensing: Theory, Methods and Applications - page 114

of the bending angle retrieval, it is necessary to use the radio-holographic (RH)
methods, which is based on the analysis of the records of complex radio signals, or
radio holograms.
There are many RH methods proposed for processing radio occultation signals
in multipath regions: (1) back-propagation (BP) (also referred to as diffraction
correction), which propagates signals backward in the vacuum toward a plane
located in a single-ray area (Hinson et al. 1997 ; Gorbunov and Gurvich 1998 ); (2)
radio-optics, which analyzes the local spatial spectra of the measured complex wave
field through Fourier analysis (Lindal et al. 1987 ; Pavelyev 1998 ; Hocke et al. 1999 ;
Sokolovskiy 2001 ); (3) Fresnel diffraction theory (Marouf et al. 1986 ; Mortensen
and Høeg 1998 ); (4) canonical transform (CT) (Gorbunov 2002 ); (5) full-spectrum-
inversion (FSI) (Jensen et al. 2003 ) and (6) phase matching method (Jensen et al.
2004 ).
The widely used CT method consists of using Fourier Integral Operators (FIO)
to find directly the dependence of the bending angle on the impact parameter
for each physical ray in multi-path conditions. The CT method requires that the
electromagnetic field is back-propagated to a straight line before the canonical
transformation can be applied to transform the coordinate to impact parameter rep-
resentation (Gorbunov 2002 ). The back-propagation from the observation trajectory
to the auxiliary straight-line trajectory results in the most computational time in CT
processing. The Fourier transform method has theoretical limitation that requires
circular satellite orbits. The approximation developed by Jensen et al. ( 2003 )in
FSI method expands the Fourier method to be applied for near-circular/realistic
satellite orbits. The FSI method utilizes the relation between the derivative of the
phase of a physical ray on the instantaneous frequency in the full Fourier spectrum
of the RO signal and the time of intersection of the physical ray with LEO satellite
orbit. Similar to FSI method, the phase-matching method is based on the synthetic
aperture concept and the method of the stationary phase and thus preserves the high
vertical resolution properties. Moreover, it is a more general method that is directly
applicable for noncircular/realistic orbits and eliminates the intermediate step of
propagation of complex electromagnetic field to circular orbits (e.g., for FSI) or
straight line (e.g., for CT). However the phase-matching method comes with higher
computational cost, as it can't apply Fast-Fourier-Transform (FFT).
All the RH methods are based on stationary electromagnetic theory. Other than
BP method, the other RH methods provide the bending angle as a single-valued
function of impact parameter in the multi-path region. The BP method, however,
may result in muli-valued function in particular in the presence of ducting or super-
refraction (Gorbunov 2002 ).
In terms of resolution and ability to handle multipath, the most efficient radio-
holographic methods are currently the Fourier operator based methods such as
canonical transform and FSI. Both methods are widely adapted by many GNSS
occultation data processing centers. For GPS and close to circular LEO orbit, the FSI
method is an optimal method (most accurate and fast) for bending angle retrieval in
the mult-path condition.