1. 2 nd GRAS SAF User Workshop 1 Radio Occultation and Multipath Behavior Kent Bækgaard Lauritsen Danish Meteorological Institute (DMI), Denmark 2 nd GRAS SAF User Workshop, 11-13 June 2003

2. 2 nd GRAS SAF User Workshop 2 Outline of the Talk Introduction Multipath behavior Inversion of 1-ray and multipath signals Back-propagation Canonical transform methods Conclusions and outlook

3. 2 nd GRAS SAF User Workshop 3 Radio Occultation Geometry Impact parameter Bending angle

4. 2 nd GRAS SAF User Workshop 4 Radio Occultation Signal Physical signal : E, B  Measured signal: u(t) u(t) = u EM + Receiver noise & tracking errors Receiver: - small noise will not cause problems - tracking errors: need to be known in order to be able to correct for them Two tracking modes: - closed loop: phase-locked loop ( PLL ) - open loop: raw signal

5. 2 nd GRAS SAF User Workshop 5 Wave Optics Simulation Example Standard atmosphere

6. 2 nd GRAS SAF User Workshop 6 Tropics: dense water vapor layers will in general give rise to multipath propagation of radio signals Critical refraction condition: - ducting of rays Water Vapor and Multipath Horizontal gradients: - normally, one assumes spherical symmetry in order to obtain the refractivity N(r) from  (p) using the Abel transform

7. 2 nd GRAS SAF User Workshop 7 Multipath Example

8. 2 nd GRAS SAF User Workshop 8 Schematic Ray Manifold

9. 2 nd GRAS SAF User Workshop 9 Inversion of 1-Ray Signal Measured signal : Doppler shift (‘wave vector’ along the t coordinate): Bending angle,  (p), obtainable from  (t) (using geometry) Refractivity, N(r), using the Abel transform (& spherical symmetry) Atmospheric quantities: P, T, q, …

10. 2 nd GRAS SAF User Workshop 10 Inversion of Multipath Signal u(t): t-representation, with caustic with 3 rays at a given time, t Map to a 1-ray representation : Measured, multi-ray representation: u z (z): z-representation with 1 ray at any given value of the ‘coordinate’ z Wave vector along the z-coordinate : Bending angle,  (p), obtainable from  (z) Phase space: (z,  ) are new coordinates, replacing (y,k y )

11. 2 nd GRAS SAF User Workshop 11 Back-Propagation Method Back-propagation maps the measured field u(t) to a new field with x  x B : Wave vector along the y B -coordinate : Bending angle,  (p), obtainable from k B : Does y B uniquely define the rays? - no, real and imaginary caustics may overlap - multipath tend to be reduced, thus results are slightly improved  B : known from the Green’s function for the Helmholtz equation Phase space: (y B,k B ) are new coordinates in the (y,k y ) phase space

12. 2 nd GRAS SAF User Workshop 12 Back-Propagation Plane at x B yByB xBxB

13. 2 nd GRAS SAF User Workshop 13 Impact Parameter Representation Physical insight: for a spherical symmetric atmosphere, the impact parameter, p, uniquely defines a ray [Gorbunov]; with horizontal gradients the assumption will be fulfilled to a good approximation Thus, choose z = p and map the measured field to the p-representation: u p (p) Mathematical physics provides the recipe for calculating u p (p): where  is a Fourier integral operator (FIO) with phase function being equal to the generating function for the canonical transform from the old to the new (p,  ) coordinates; note, there are infinitely many  ’ s that map to the p-representation

14. 2 nd GRAS SAF User Workshop 14 Schematic Drawing of the p-Representation

15. 2 nd GRAS SAF User Workshop 15 Canonical Transform Method Map to the 1-ray p-representation : Wave vector along the p-coordinate : Bending angle,  (p), obtainable from  (p):  (p) =  (p) ( plus a correction when the GPS satellite is at a finite position )

16. 2 nd GRAS SAF User Workshop 16 Canonical Transform Method of ‘‘Type 2’’ Canonical transform (of type 1): - Gorbunov’s original CT method which involves first doing back-propagation - FIO, , based on a canonical transform from (y B, k B ) to (p,  ) coordinates Canonical transform (of type 2): - CT method based on directly mapping the measured field u(t) to the p-representation, u p (p) [FSI] - FIO,  2, based on a canonical transform from (t,  ) to (p,  ) coordinates - u p (p) can be chosen to be identical to the one obtained by a CT of type 1 - GPS satellite is not assumed stationary

17. 2 nd GRAS SAF User Workshop 17 Conclusions and Outlook Radio occultations and multipath behavior  Water vapor, critical refraction, receiver tracking errors Mapping from multi-ray to 1-ray representation  Multi-ray: caustics  1-ray: Impact parameter representation Inversion methods  Standard methods: handle 1-ray signals  Back-propagation: can reduce multi-ray behavior  Canonical transform methods: handle multi-ray behavior  Gorbunov’s original CT & CT without back-propagation (CT of type 2)  Increased vertical resolution (about 50 m)  Improved product accuracy