2nd GRAS SAF User Workshop 1

Radio Occultation and Multipath Behavior

Kent Bækgaard Lauritsen

Danish Meteorological Institute (DMI), Denmark

2nd GRAS SAF User Workshop, 11-13 June 2003

2nd 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

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Radio Occultation Geometry

Impact parameter

Bending angle

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Radio Occultation Signal

Physical signal: E, B

Measured signal: u(t)

u(t) = uEM + 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

2

,))(exp()()( kikAu EMEMEM rrr

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Wave Optics Simulation Example

Standard atmosphere

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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

1km157 dr

dN

Horizontal gradients:

- normally, one assumes spherical symmetry in order to obtain

the refractivity N(r) from (p) using the Abel transform

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Multipath Example

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Schematic Ray Manifold

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Inversion of 1-Ray Signal

Measured signal:

Doppler shift (‘wave vector’ along the t coordinate):

dt

tdt

)()(

Bending angle, (p), obtainable from (t) (using geometry)

Refractivity, N(r), using the Abel transform (& spherical symmetry)

Atmospheric quantities: P, T, q, …

))(exp()()( titAtu

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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:

uz(z): z-representation with 1 ray at any given value of the ‘coordinate’ z

))(exp()()( zkizAzu zzz

Wave vector along the z-coordinate:

dz

zdkz z )(

)(

Bending angle, (p), obtainable from (z)

Phase space: (z, ) are new coordinates, replacing (y,ky)

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Back-Propagation Method

Back-propagation maps the measured field u(t) to a new field with x xB:

Wave vector along the yB-coordinate:

B

BBB dy

ydkk

)(

Bending angle, (p), obtainable from kB:

)())(exp()()( BBBBBBBB yuykiyAyu

)/arcsin( kkB

Does yB 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: (yB,kB) are new coordinates in the (y,ky) phase space

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Back-Propagation Plane at xB

yB

xB

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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: up(p)

Mathematical physics provides the recipe for calculating up(p):

)()( pupu 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

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Schematic Drawing of the p-Representation

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Canonical Transform Method

Map to the 1-ray p-representation:

)())(exp()()( pupkipApu ppp

Wave vector along the p-coordinate:

dp

pdkp p )(

)(

Bending angle, (p), obtainable from (p): (p) = (p)

(plus a correction when the GPS satellite is at a finite position)

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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 (yB, kB) to (p, ) coordinates

Canonical transform (of type 2):

- CT method based on directly mapping the measured field u(t) to the

p-representation, up(p) [FSI]

- FIO, 2, based on a canonical transform from (t, ) to (p, ) coordinates

- up(p) can be chosen to be identical to the one obtained by a CT of type 1

- GPS satellite is not assumed stationary

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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