D ANISH METEOROLOGICAL INSTITUTED ANISH METEOROLOGICAL INSTITUTE SCIENTIFIC REPORT 01-02 Radiohol...

DANISH METEOROLOGICAL INSTITUTE

SCIENTIFIC REPORT

01-02

Radioholographic methods for processingradio occultation data in multipath regions

M. E. Gorbunov

DMI

COPENHAGEN 2001

ISSN 0905-3263

ISSN 1399-1949 (online)

ISBN 87-7478-433-1

Radioholographic methods for processing radio

occultation data in multipath regions

M. E. Gorbunov

Atmosphere Ionosphere Research Division

Danish Meteorological Institute, 2100 Copenhagen, Denmark

18 December 2000

Abstract

Multipath propagation in the lower troposphere plays a very importantrole in forming the structure of radio signals used for radio occultations. Pro-cessing radio occultation data in multipath regions requires the solution of theproblem of the reconstruction of the ray structure of a wave �eld. Two meth-ods were formerly suggested for processing multipath data: back-propagationand radio-optic (radio-holographic or sliding-spectral) method. In the �rstsection of this report we analyzed the two methods and their restrictions.Back-propagation method does not work in the presense of complicated causticstructures. The radio-optic method results in signi�cant errors in sub-causticzones. In the second section we suggest a combination of these methods. Ituses the spectral analysis of both measured and back-propagated �eld andthe computation of the averaged spatial spectra. It is shown that the e�ectsof caustic are suppressed in the averaged spectra, which allows for the recon-struction of refraction angles. In the third section we suggest a new methodof the reconstruction of the ray structure of a wave �eld. The method is basedon the theory of canonical transforms and Fourier integral operators. It allowsfor a very accurate reconstruction of unique pro�les of refraction angles. Allthe methods are validated in numerical simulations based on global �eld fromanalyses of European Centre for Medium-Range Weather Forecast.

1 Comparative analysis of radioholographic meth-

ods of processing radio occultation data

1.1 Introduction

Radio occultation sounding of the Earth's atmosphere using the radio signals of theGlobal Positioning System (GPS) can be a valuable source of information about keyatmospheric variables [Ware et al., 1996; Rocken et al., 1997; Kursinski et al., 1997].The experience of processing data from the Microlab-1 satellite as well as theoret-ical considerations show that the most complicated problems of the interpretation

1

of radio occultation data arise in the lower troposphere [Gorbunov et al., 1996b;Gorbunov and Gurvich, 1998a, b]. On the other hand, lower tropospheric data areof the greatest importance for numerical weather prediction (NWP). The di�cultiesof data processing can be classi�ed as follows:

1. They can be classi�ed as algorithmic problems. Processing the phase andamplitude measurements in multipath areas requires a special technique for theseparation of interfering rays.

2. They can be classi�ed as technical problems. The receiver used in theMicrolab-1 satellite was not capable of tracking the signal with the strong am-plitude and phase scintillations resulting from lower tropospheric inhomogeneitiesand multipath propagation, on the background of signi�cant refractive attenuationof the signal. This resulted in the lack of reliable lower tropospheric data.

Two di�erent approaches to processing radio occultation data in multipath areashave been discussed in the literature: (1) the analysis of spatial spectra, which wewill refer to as the radio-optic method [Lindal et al., 1987; Pavel'ev , 1998], and (2)the back propagation technique, which is also referred to as di�raction correction[Marouf et al., 1986; Gorbunov et al., 1996b; Karayel and Hinson, 1997; Gorbunovand Gurvich, 1998a, b]. Both methods can be termed radioholographic, because theyare based on the analysis of records of complex radio signals, or radio holograms. Inthis section, we analyze their capabilities in processing data of GPS radio occultationsounding of the Earth's atmosphere.

Because of the lack of reliable lower tropospheric data our analysis can only bebased on simulated radio occultations. The simulations were performed with global�elds of pressure, temperature, and humidity from analyses of the European Centrefor Medium-Range Weather Forecasts (ECMWF).

The results presented in this chapter are based on Ref. [Gorbunov et al., 2000].

1.2 Radio Occultation Technique for Sounding Planetary

Atmospheres

GPS

Earth pp

u

r

r

VLEO

VGPS

uGPS

LEO

uLEO

Figure 1: Radio occultation geometry.

2

The radio occultation technique is based on the transillumination of a planetaryatmosphere with radio waves emitted by a spaceborne transmitter and received bya spaceborne receiver. In the sounding of the Earth's atmosphere the signal is trans-mitted by a GPS satellite and received by a low-Earth orbiter (LEO) equipped witha GPS receiver. The typical observation geometry is shown in Figure 1. Duringa (setting) occultation the satellites are moving in such a way that the radio rayconnecting them immerses into the atmosphere until the receiver reaches the solidEarth's shadow or loses the signal earlier. The phase (or Doppler frequency) andamplitude of the received signal are recorded for their subsequent use for the re-construction of atmospheric refractive index, which can be interpreted in terms ofatmospheric variables [Gorbunov et al., 1996a; Ware et al., 1996; Rocken et al., 1997;Kursinski et al., 1997; Steiner et al., 1999].

Basic approximations for the interpretation of radio occultation data are ge-ometric optics and local spherical symmetry, which allows for the formulation ofthe inverse problem in terms of measured dependence of refraction (bending) angle" versus impact parameter p. The vertical pro�le of atmospheric refractive indexn(r) must be reconstructed. In a spherically symmetrical medium, rays are planecurves, and impact parameter is a ray invariant such that at each point of the ray,p = n(r)r sin = n(r) j r� uj, where is the angle between local ray direction uand radius vector r. This statement is referred to as Snell's law for a sphericallysymmetric medium or the formula of Bouger [Born and Wolf , 1964]. Using therelation between the measured Doppler frequency !LEO and the frequency of thetransmitted signal !GPS, complemented with the vector form of Snell's law, we canwrite equations for the determination of ray directions uGPS and uLEO [Fjeldbo et al.,1971; Vorob'ev and Krasil'nikova, 1994]:

c�VLEO � uLEOc�VGPS � uGPS

sc2 � V 2

GPS

c2 � V 2LEO

=!LEO!GPS

;

(1)

rLEO � uLEO = rGPS � uGPS;

where c is the light velocity in a vacuum, VPS and VEO are the GPS and LEOvelocities and rGPS and rLEO are the GPS and LEO positions. It is assumed thatboth satellites are in a vacuum.

The refraction angle can be calculated as angle " between uGPS and uLEO. Im-pact parameter p equals jrLEO � uLEOj = j rGPS � uGPSj. Dependence "(p) can beinverted using the Abel integral transform [Phinney and Anderson, 1968; Fjeldboet al., 1971], which is the exact solution for a spherically symmetrical �eld n(r) butis also used as an approximation for real atmospheres with horizontal gradients ofn.

Because Snell's law (the formula of Bouger) is violated in the real atmospherewith horizontal gradients (jrLEO � uLEOj 6= j rGPS � uGPSj), " and p determinedfrom (1) will not coincide with the real bending angle and impact parameter (thelatter will be di�erent at GPS and LEO) [Ahmad and Tyler , 1999]. Nevertheless,the dependence "(p) can be looked at as a known functional of the refractive in-

3

dex �eld and as such assimilated into global atmospheric circulation models usingthree-dimensional or four-dimensional variational assimilation techniques (3DVar or4DVar) [Eyre, 1994; Gorbunov et al., 1996a]. In this approach, (1) is looked at asthe de�nition of the functional that can be applied without making the assumptionof the spherical symmetry of the refracting medium. This approach avoids errorsdue to the approximation of the local spherical symmetry.

Equation (1) assumes single-ray propagation. A typical situation for the lowertroposphere is, however, multipath propagation. Doppler frequency can be com-puted as the derivative of the phase of the wave �eld, also in a multipath area.Formal application of (1) results then in a characteristic oscillating pattern in the("; p) plane, which re ects the interference oscillations of the phase [Gorbunov andGurvich, 1998b]. The resulting dependence "(p) proves ambiguous and cannot beused in inversion algorithms (inversion is usually preceded by some heuristic pro-cedure of the elimination of ambiguities). We come thus to the problem of thedetermination of the ray structure of a wave �eld measured by a spaceborne GPSreceiver.

1.3 Radio-optic Method

The radio-optic method was used for processing soundings of planetary atmospheres[Lindal et al., 1987], and recently its application to GPS radio occultation data wassuggested [Pavel'ev , 1998]. The method is based on the analysis of the local spatialspectra of the measured complex wave �eld. Local spectra are computed by Fourieranalysis of the record of the complex �eld in small sliding apertures. Rays mustcorrespond to maxima in the Fourier spectra. The limitations of this method willnow be analyzed.

We shall use Cartesian coordinates (x; z) with the x axis pointing in the directionof the initial wave propagation and the z axis transverse to it. The (0,0) point ofthe coordinate frame is located in the Earth's local curvature center.

The wave �eld is observed at distance x from the Earth's limb, which is esti-mated as

pr2LEO � r2E, where rLEO is the LEO orbit radius and rE is the Earth's

local curvature radius. The spatial spectrum of the wave �eld is analyzed insidevertical apertures with size A. The angular resolution of the spatial spectrum canbe estimated as �=A. Assuming that the wave �eld can be described in the geomet-ric optical approximation, we shall interpret this relationship in terms of refractionangles and impact parameters.

Each ray can be associated with a plane wave exp�ipk2 � k2zx+ ikzz

�. Us-

ing the approximation of small refraction angle and in�nitely remote GPS, we canwrite the following relationships for the refraction angle and impact parameter atobservation point (x; z):

" =kzk;

(2)

p(x; z; ") = z + x":

4

The second relationship can be rewritten as follows:

"G(x; z; p) =p � z

x; (3)

which speci�es the geometric relationship for the refraction angle and impact pa-rameter of a ray coming to the observation point. In order to �nd all such rays wemust solve the following equation:

"G(x; z; p) = "(p): (4)

The spectral resolution, �kz = 2�=A, determines the resolution of the refractionangle:

�" =�kzk

=�

A: (5)

On the other hand, the aperture size determines the resolution of the impactparameter:

�p =

��dpdz��A =

A

�;

(6)

� =

��1 � xd"

dp

�� :Because refraction angle and impact parameter must satisfy geometric relation-

ship (3), their uncertainties can be estimated as follows:

�p2 = �p2 + x2�"2 =A2

�2+x2�2

A2;

(7)

�"2 = �"2 +�p2

x2=�2

A2+

A2

x2�2:

These equations allow for the estimation of the optimal aperture size that minimizesboth �" and �p:

A =p�x�: (8)

The uncertainties of the refraction angle and impact parameter can then be esti-mated by their order of magnitude:

5

�" �s

�

�x;

(9)

�p �sx�

�:

We have thus the uncertainty relation

�" �p � �

�: (10)

Because d" dp is the natural measure in the space of (straight-line) rays [Santalo,1976], we can understand this relationship as the estimation of the volume of aquantum cell in this space. Two rays can be distinguished if they cannot be coveredwith one cell.

For the GPS radio occultations performed by the Microlab 1 satellite the obser-vation distance was � 3000 km, and the wavelength � 20 cm. For weak refraction(� � 1) the uncertainties of refraction angle and impact parameter are then esti-mated as 2:6 � 10�4 rad and 800 m, respectively. For the purposes of NWP wemust reconstruct pressure with an accuracy of 1 mbar (0.1%), temperature with anaccuracy of 1 K (0.3%), and geopotential heights with an accuracy of 10m. Theaccuracy of refraction angle in the lower troposphere is then estimated as 0.1%, or� 3 � 10�5 rad. The required accuracy of impact parameter must be � 10 m. Al-though the above estimations of the uncertainties exceed these values, we must notforget that multipath propagation is only signi�cant in the lower troposphere, where� can be fairly large (30-300), and the uncertainties will be reduced by a factor ofp�. It must also be noted that the amplitude of the signal in the geometric optical

approximation is also inversely proportional top� .

In the lower troposphere, where � � 1, it can be approximately written � �jx d"=dpj. Parameter � is then proportional to the observation distance x, and from(9) it follows that the uncertainty of the refraction angle can be reduced by meansof the forward propagation of the observed wave �eld to a plane located at a largerdistance from the atmosphere. This equation, however, shows that this will notimprove the uncertainty of the impact parameter, and thus the limitation of thevertical resolution will still persist. This shows that the overall uncertainty volume�� p is not so important as its shape.

The uncertainty relation is not the only restriction of the radio-optic method.Because it is based on the representation of the wave �eld in the form of a sum ofwaves corresponding to geometric optical rays, this method is incapable of takingdi�raction e�ects into account.

The relation between the geometric optical approximation and the exact solutionof the di�ractive problem was already discussed in literature [Kravtsov and Orlov ,1990], and it was shown that these solutions are very close to each other everywhere

6

in a free space except subcaustic zones and the size of a subcaustic zone was esti-mated. The approach based on the theory of canonic operators [Mishchenko et al.,1990] results in the same conclusion. This is also corroborated by the numerical sim-ulations described in Section 5. Because the wave �eld in subcaustic zones cannotbe interpreted in terms of geometric optical rays, the radio-optic method cannot beapplied there.

As we shall see in Section 5, subcaustic zones (for GPS/MET observation ge-ometry) are narrow enough to be excluded without resulting in a signi�cant loss ofdata. On the other hand, the caustic structure is a priori unknown (knowledge ofthe caustic structure is equivalent to knowledge of "(p)). At the moment we cannotformulate the algorithm of the identi�cation of subcaustic zones for their exclusion.

1.4 Back Propagation Method

The back propagation method was �rst suggested for the pro�ling of Saturn's ringsfrom di�racted images Marouf et al. [1986]. The method was initially designedto correct for the di�raction e�ects, which were signi�cant in the interpretation ofimages of Saturn's rings recorded from a large observation distance.

Back-propagation plane

y

nyry

S

pEarth

r-y

z

LEO

Real focusing

x

(p)

Imaginary focusing

r

Figure 2: Multipath propagation geometry and back propagation. Nonmonotonousvertical pro�le �(p) results in real focusing of rays and multipath propagations. Backpropagation of the wave �eld (back-propagated rays are shown as dashed lines)results in imaginary focusings. Back-propagated �eld at point r is calculated viadi�ractive integral as a superposition of ininitesimal �elds from the points of curve S.Each point y can be represented as a transmitter with intensity u0(y) and directiondiagram cos'

ry.

Recently the method was modi�ed for processing radio occultation data in mul-tipath areas [Gorbunov et al., 1996b; Karayel and Hinson, 1997; Hinson et al.,1997, 1998; Gorbunov and Gurvich, 1998a, a]. The method is based on the solutionof the boundary problem for the Helmholtz equation in a vacuum, the boundaryconditions being delivered by the measurements of the complex �eld along the LEOorbit. From the geometric optical viewpoint, this solution corresponds to the back

7

continuation of the rays as straight lines. Such a continuation conserves both raydirection (and therefore the refraction angle) and impact parameter. The �eld isback-propagated to some plane positioned, if possible, in a single-ray area (Figure 2).

The new feature of the method is that although the back propagation is per-formed as in a vacuum, the back propagation plane can (and must) be positionedinside the atmosphere, and the back-propagated �eld is thus not the real wave �eldat the location of the back propagation plane.

Assuming that the source of the wave �eld (GPS) is stationary and that the LEOorbit is located in a vertical plane, which we will refer to as the occultation plane,the back-propagated �eld is calculated using the di�ractive integral [Vladimirov ,1971; Gorbunov et al., 1996b]:

u(r) =

�k

2�

�1=2 ZS

u0(y)�

� cos'ry

exp (�ikjr� yj+ i�=4)

jr� yj1=2 dSy; (11)

where 'ry

is the angle between vector r � y and normal ny to curve S at currentintegration point y, and u0(y) is the boundary condition. Curve S is the LEOtrajectory. Although real occultations are not vertical, it is possible to reduce theproblem to the case where the GPS satellite is stationary, and the LEO satellite ismoving vertically, by means of an appropriate coordinate transform Gorbunov et al.[1996b].

As discussed in Section 3, di�ractive e�ects in single-ray areas are negligible.Therefore, if the back propagation plane is positioned in a single-ray area, then theback-propagated �eld can be interpreted in terms of geometrical optics. The Dopplerfrequency is calculated by means of the numerical di�erentiation of its phase. Therefraction angles and impact parameters are then found by solving (1).

The calculation of Doppler frequency by means of the numerical di�erentiationof the phase can be performed with a much smaller e�ective aperture than thatrequired for the radio-optic method. The wavenumber-aperture uncertainty relationdoes not apply here because of the use of a very strong assumption of single-raypropagation. This can be explained as follows: Given a complex wave �eld u(z), theuncertainty relation states that �kz �z � 2�, where �z is the aperture size and �kzis the spectral resolution. If the spatial spectrum is grouped around some frequencyk(0)z , then for an aperture size approximately equal to the oscillation period of thewave �eld, 2�=k(0)z , the spatial spectrum cannot be determined, because �kz � k

(0)z .

The situation is di�erent if it is a priori known that the �eld is measured in a single-ray area. It can then be represented in the form u(z) = a(z) exp

�iRkz(z)dz

�,

where amplitude a(z) and spatial frequency kz(z) are very slowly changing insidean oscillation period. In this case the frequency can be accurately calculated from

the oscillation period, and wave vector k(z) =hp

k2 � k2z (z); kz(z)ipoints in the

direction of the single ray at a given point.The basic restriction of the back propagation method is related to the con�gura-

tion of rays and caustics of the back-propagated �eld in the occultation plane, which

8

will now be investigated. The straight line associated with a ray characterized byrefraction angle " and impact parameter p is described by the following equation:

x sin "(p) + z cos "(p) = p: (12)

A caustic is the envelope of the family of rays. In the simplest cases each causticis the boundary between multipath and single-ray propagation areas. The equationfor caustics is derived by complementing (12) with its derivative with respect toparameter p Korn and Korn [1968]. Solving the resulting system, we arrive at theequations for caustics:

x = p sin "(p) +cos "(p)

"0p(p);

(13)

z = p cos "(p)� sin "(p)

"0p(p):

Assuming that j"0p(p)j <1, we see that the complete envelope can be representedas a set of continuous caustics corresponding to the monotonous fragments of thedependence "(p). They can be subdivided into two classes: real caustics with "0p(p) >0 arising because of focusing of rays in the observation plane, and imaginary causticswith "0p(p) < 0 arising because of defocusing (or imaginary focusing).

Equation (13) indicates that each ray with a p such that "0p(p) 6= 0 touches onlyone caustic (real or imaginary according to the sign of "0p(p)). The perigee points ofthe rays de�ned by the equations

x = p sin "(p);

(14)

z = p cos "(p)

are located between real and imaginary caustics, and thus they are in a single-rayarea. This suggests back-propagating the measured wave �eld to this curve. Thissolution is, however, associated with algorithmic di�culties, because the dependence"(p) itself must yet be determined, and iterations are thus necessary. Additionaldi�culties will also result from the complicated shape of such a curve. Currently,we back-propagate the measured �eld to a vertical straight line, whose position xcan be estimated from (14) as x = rE"max where the maximum refraction angle canbe estimated from the processing of the measurements without back propagation.In most cases this estimation is su�cient for disentangling multipath ray structures.Other practical criteria for the right choice of the back propagation plane positioncan be the following.

1. There are minimum amplitude uctuations of back-propagated �eld Karayeland Hinson [1997]. This criterion is based on the fact that the strongest amplitude uctuation arise due to interference in multipath areas.

9

2. There is a minimum di�erence of optical path in L1 and L2 frequency chan-nels. This criterion uses the strong frequency dependence of interference e�ects inmultipath areas, while in one ray areas the frequency dependence of optical pathcan only be due to di�raction e�ects, which are very weak in single-ray areas forGPS/MET observations.

The situation becomes more complicated in the presence of a superrefractionlayer. In this case the geometric optical dependence "(p) is characterized by a verysharp peak, where formally j"0p(p)j = 1. As seen from (13), the caustic structurewill then contain real and imaginary caustics very close to each other. In such casesthe back propagation plane will always have intersections with caustics. Refractionangle pro�les with sharp spikes caused by superrefraction or conditions approachingsuperrefraction constitute the basic limitation of the back propagation method.

As we have already seen, however, strong refraction resulting in large values of� is most favorable for the radio-optic method. This suggests combining the twomethods for processing lower tropospheric data.

1.5 Numerical Simulations

(p)

Refraction angle

Impa

ct p

aram

eter

(x,z ,p)G 0

|u(x,z ,k )|~0 z

Figure 3: Geometry of the calculation of spatial spectra. Each ray is representedas a point in the (�; p) plane. Each observation point (x; z0) is represented as curve�G(x; z0; p). Each spatial frequency kz corresponds to point (�(x; z0; kz); p(x; z0; kz))of this curve. Rays coming to given point are de�ned by the equation �G(x; z0; p) =�(p). Absolute values of the spatial spectra of wave �eld j~u(x; z0; kz)j must havemaxima corresponding to rays.

In the numerical simulations we compared the results of processing arti�cial occul-tation data using the back propagation and radio-optic methods. The arti�cial datawere generated using the split-step numerical solution of the di�raction problem inthe atmosphere [Martin, 1992; Karayel and Hinson, 1997; Gorbunov and Gurvich,1998b]. The model of the atmospheric refractive index �eld was based on the global

10

0.01 0.015 0.02 0.025 0.03 0.035Refraction angle, rad

2

3

4

5

6

7

Ray

heig

ht,

km

b Spatial spectra

0.01 0.015 0.02 0.025 0.03 0.0352

3

4

5

6

7


2

3

4

5

6

7R

ayhe

igh

t,km

GOBPRO mainRO secondary

a

-500 0 500 1000X, km

6330

6340

6350

6360

6370

Y,k

m

c Caustic structure

Figure 4: Simulated occultation event 0478 (February 2, 1997, 1637 UTC) at 29.3�S173.0�E: a) refraction angle pro�les (1) geometric optical (GO), (2) from back-propagated �eld (BP), (3) from phase measured at LEO, and (4) spectral max-ima from radio optical analysis (RO); b) local spatial spectra in ray coordinates inpseudo-colour; c) caustic structure.

�elds of temperature, humidity, and pressure from analyses of ECMWF. The oc-cultation geometry was taken from real GPS/MET occultations and transformed tovertical occultation planes. The analysis �elds are given with a time step of 6 hours;the analysis nearest to the time of the GPS/MET occultation event was chosen.

Using the geometric optical solution, we calculated the caustic structure, whichallowed for the determination of the optimal position of the back propagation plane.The radio-optic method was applied to the complex �eld forward-propagated toa plane located at x = 5000km, which allowed for a slight improvement of theresolution. The forward-propagation plane was covered by sliding apertures with asize of

p�x�. Factor � was evaluated from (6), where "(p) was computed from the

observed phase with a very strong smoothing. In our simulation the accuracy of thisestimation was su�cient. The basic sliding step of the sliding apertures was � 1:5km.

Inside an aperture centered at z0 the (forward propagated) complex �eld u(z)was represented in the form

11

0.02 0.025 0.03 0.035 0.04 0.045 0.05Refraction angle, rad

2

2.5

3

3.5

4

Ray

heig

ht,

km

b Spatial spectra

0.02 0.025 0.03 0.035 0.04 0.045 0.052

2.5

3

3.5

4


2

2.5

3

3.5

4R

ayhe

igh

t,km


a

-500 0 500 1000X, km

6360

6370

6380

6390

6400

6410

Y,k

m

c Caustic structure

Figure 5: Simulated occultation event 0681 (February 2, 1997, 2342 UTC) at 29.1�S171.4�W: a) refraction angle pro�les (1) geometric optical (GO), (2) from back-propagated �eld (BP), (3) from phase measured at LEO, and (4) spectral max-ima from radio optical analysis (RO); b) local spatial spectra in ray coordinates inpseudo-colour.

u(z) = a(z) exp (i'(z))

(15)

'(z) = �0 + �1(z � z0) ++ �2(z � z0)2 +�'(z)

where a(z) is the amplitude, '(z) is the accumulated phase, and �i are the regres-sion coe�cients. The complex �eld a(z) exp [i�'(z)] with downconverted spatialfrequency was subjected to the Fourier analysis. This procedure approximately cor-responds to the representation of the wave �eld u(z) as a sum of spherical waves,and the square phase term in (15) approximately corrects for the wave front curva-ture. The discretization step �z was chosen to be equal to �=�" � 200 m, where�" = 0:001 rad is the a priori estimation of the width of the angular spectrum.

Downconverted frequencies �z correspond to spatial frequencies kz = �1 + �z ofthe initial �eld u(z) in the center of the aperture. The wave vector is then equal to

k = (kx; kz) =�p

k2 � k2z ; kz�. Using Snell's law, we can compute refraction angles

"(x; z0; kz) and impact parameters p(x; z0; kz) corresponding to the ray direction ofwave vector k. This de�nes a mapping of (z0; kz) to ("; p). For �xed observation point(x; z0), dependencies "(x; z0; kz) and p(x; z0; kz) are the parametric representation

12

0.015 0.02 0.025 0.03 0.035 0.04 0.045 0.05Refraction angle, rad

2

2.5

3

3.5

4

Ray

heig

ht,

km

b Spatial spectra

0.015 0.02 0.025 0.03 0.035 0.04 0.045 0.05Refraction angle, rad

2

2.5

3

3.5

4R

ayhe

igh

t,km


a

0.015 0.02 0.025 0.03 0.035 0.04 0.045 0.052

2.5

3

3.5

4

0 100 200 300 400 500X, km

6345

6350

6355

6360

6365

Y,k

m

c Caustic structure

Figure 6: Simulated occultation event 0353 (February 7, 1997, 1236 UTC) at 18.0�N177.3�W: a) refraction angle pro�les (1) geometric optical (GO), (2) from back-propagated �eld (BP), (3) from phase measured at LEO, and (4) spectral max-ima from radio optical analysis (RO); b) local spatial spectra in ray coordinates inpseudo-colour.

of curve "G(x; z0; p) in the ("; p) plane, where kz is the curve parameter. Eachlocal spatial spectrum eu(x; z0; kz) can then be plotted along this curve in ("; p)coordinates. For all possible z0, curves "G(x; z0; p) cover the complete ("; p) plane,and the maxima of the spectra must thus trace the dependence "(p).

This is schematically shown in Figure 3, where dependencies "G(x; z0; p) and "(p)have three intersections, i.e. three rays interfere at a given observation point, andthree maximamust be expected for absolute values of spatial spectrum jeu(x; z0; kz)j.

An example of processing arti�cial occultation data using the back propagationand radio-optic methods is shown in Figure 4. We computed the geometric opticalrefraction angles "GO(p) by means of the standard ray-tracing technique Gorbunovet al. [1996a]. Using them as the exact solution, we validate the back propagationand the radio-optic methods. Panel (a) of Figure 4 shows refraction angle pro�les:(1) geometric optical "GO(p), (2) "BP(p) computed from back-propagated �eld, (3)"LEO(p) computed from phase measured at LEO, and (4) spectral maxima from radiooptical analysis. Instead of impact parameter we use the ray leveling height de�nedas p � rE. Panel (b) shows the local spatial spectra in ray coordinates in pseudo-colour. Panel (c) shows the con�guration of caustics of the back-propagated wave�eld. The back propagation plane can be located between the real and imaginarycaustics at x = 200 km. For the radio-optic analysis, factor � was estimated at 60,which corresponds to aperture size A = 8 km.

13

0.005 0.01 0.015 0.02 0.025 0.03 0.035 0.04 0.045 0.05Refraction angle, rad

2

2.5

3

3.5

4

4.5

5

Ray

heig

ht,

km

b Spatial spectra

0.005 0.01 0.015 0.02 0.025 0.03 0.035 0.04 0.045 0.052

2.5

3

3.5

4

4.5

5

0.005 0.01 0.015 0.02 0.025 0.03 0.035 0.04 0.045 0.05Refraction angle, rad

2

2.5

3

3.5

4

4.5

5R

ayhe

igh

t,km


a

0 100 200 300 400 500X, km

6350

6355

6360

6365

6370

Y,k

m

c Caustic structure

Figure 7: Simulated occultation event 0504 (February 7, 1997, 1754 UTC) at 39.4�S156.6�E: a) refraction angle pro�les (1) geometric optical (GO), (2) from back-propagated �eld (BP), (3) from phase measured at LEO, and (4) spectral max-ima from radio optical analysis (RO); b) local spatial spectra in ray coordinates inpseudo-colour.

The multipath regions are marked with strong oscillations of the dependence"LEO(p), which becomes ambiguous there. The oscillations are concentrated nearthe branch of the dependence "GO(p) corresponding to the strongest of the interfer-ing rays. Spectral maxima computed in the radio-optic method show a good agree-ment with "GO(p) everywhere, except in relatively narrow vicinities of the caustics,where di�raction e�ects are signi�cant and spectral maxima cannot be interpretedin terms of rays. The best agreement with the exact solution is indicated by theback propagation method.

Figure 5 show another example of an arti�cial radio occultation, where the backpropagation method is not so successful in the lowest troposphere. The causticstructure shown in Figure 5 is very characteristic for a superrefraction layer. Nopositioning of the back propagation plane can get rid of multipath propagation. Forthe radio-optic method, the value of � was estimated to be 215, which correspondsto aperture size A = 15 km. The vertical pro�le of refraction angles is characterizedby a sharp spike at height of � 3 km. This spike is responsible for the real andimaginary caustic with overlapping projections to the x axis. For ray heights below3.5 km the back propagation method proves incapable of reconstructing the true raystructure. The dependence "(p) is ambiguous here. This ambiguity, however, canserve as the indication that the back propagation plane intersects caustics and theback propagation method is inapplicable. Such a situation can also be diagnosed

14

by the amplitude uctuations [Karayel and Hinson, 1997]. For this occultationthe radio-optic method yields a much better agreement of the reconstructed raystructure with the exact geometric optical solution. However, the problem of extraspectral maxima in subcaustic areas remains unsolved. Figures 6 and 7 show furtherexamples of occultations with complicated caustic structures.

1.6 Conclusions

The presented analysis allows for the following comparative characterization of theback propagation and radio-optic methods. The radio-optic method has two basiclimitations.

1. One limitation is the uncertainty relation of refraction angle and impactparameter. The uncertainties, however, decrease in the lowest troposphere underthe conditions of strong refraction, where the limitation of the resolution becomesless signi�cant and thus the biggest impact of this method is expected.

2. Di�ractive e�ects in subcaustic zones result in \false" maxima of the spatialspectra of the wave �eld, i.e., maxima that cannot be interpreted in terms of rays.Sorting out these maxima can become an important problem, especially with com-plicated pro�les of refraction angles which can be expected in the lower troposphere.

The basic limitation of the back propagation method is related to complicatedcaustic structures, which can arise in the presence of strong refraction or super-refraction. Because strong refraction reduces the uncertainties of refraction angleand impact parameter in the radio-optic analysis, these two methods can be com-bined.

The basic advantage of the back propagation method is its high accuracy in thereconstruction of the pro�le of refraction angle when it is possible to position theback propagation plane in a single-ray area. However, even when it is impossible,this situation can at least be diagnosed by ambiguities in the dependence "(p) or bysevere amplitude uctuations in the back propagation plane.

2 Radioholographic analysis of radio occultation

data in multipath zones

2.1 Introduction

Numerical simulations of GPS radio occultation experiments show that the e�ectsof multipath propagation must play a signi�cant role in the lower troposphere Gor-bunov and Gurvich [1998b]; Gorbunov et al. [2000]. Two techniques for processingradio occultation data in multipath regions were discussed in the literature: (1) theback propagation method Gorbunov et al. [1996b]; Hinson et al. [1997]; Gorbunovand Gurvich [1998a, a]; Hinson et al. [1998] and (2) the radio optic (or radioholo-graphic) method Lindal et al. [1987]; Pavel'ev [1998].

The former method consists in the numerical back propagation of the wave �eldrecorded during a radio occultation experiment to a signle-ray region. The latter

15

method is based on the analysis of local spatial spectra of the wave �eld, where thespectral maxima are interpreted as rays.

The analysis of both mehtods Gorbunov et al. [2000] indicates that they havetheir restrictions. The back propagation method does not work for complicatedcaustic structures. The radio optic method cannot locate rays in the sub-causticzones, where the wave �eld cannot be described in terms of geometric optics. Theradio optic method has also a limited resolution. This restriction, however, is notvery signi�cant in the lower troposphere.

In this section we suggest a combination of both methods, which can be usedfor processing radio occultation data in multipath regions in the situations withcomplicated caustic structures. The combined method is based on the radio opticanalysis of both the wave �eld recorded at the LEO orbit and the back-propagated�eld. The two local spatial spectra are computed in the ray coordinates (refractionangle and impact parameter), and the averaged spectrum is the computed. Thenwe localize the main maxima of the averaged spectra for each impact parameter.The refraction angle coordinates of the maxima trace the dependence of refractionangle versus impact parameter.

The method is validated on the material of global �elds of atmospheric vari-ables from analyses of the European Centre for Medium Range Weather Forecast(ECMWF).

The results presented in this chapter are based on an article submitted to RadioScience.

2.2 Data Analysis

The back propagation method is used in order to correct for the e�ects of multipathpropagation [Gorbunov et al., 1996b; Hinson et al., 1997; Gorbunov and Gurvich,1998a, b; Hinson et al., 1998]. It uses the solution of the boundary problem of theHelmholtz equation in a vacuum. In this technique the complex wave �eld recordedalong the LEO orbit during a radio occultation experiment is numerically back-propagated to a virtual observation plane located inside the atmosphere. From thegeometric optical viewpoint this corresponds to the back continuation of the rays asstraight lines, which often allows for disentangling the multipath structure.

The dependence of refraction angle � versus impact parameter p de�nes a con-�guration of straight back-propagated rays and corresponding caustics. There aretwo classes of caustics: real and imaginary ones [Gorbunov et al., 2000], and theback-propagation plane must be positioned in the single-ray area between them. Ifthis is possible, then the back-propagation method provides a high accuracy in thereconstruction of refraction angle pro�les. In the presense of strong refraction orsuperrefraction, however, structures with overlapping real and imaginary causticscan arise. In this case none positioning of the back-propagation plane can get rid ofmultipath propagation.

Another problem with this method is the choise of the back propagation planeposition. This position depends on the caustic structure, which is a priori unknown.

An alternative technique of processing radio occultation data in multipath areasis the radio-optic method [Lindal et al., 1987; Pavel'ev , 1998]. It is based on the

16

LEO

LEO

BP

BP

(p)

Refraction angle

Impa

ct p

aram

eter

Figure 8: Geometry of calculation of spatial spectra. Thick dashed lines (LEO) rep-resent two intersections of LEO orbit with caustics; thick solid lines (BP) representtwo intersections of the back-propagation plane with caustics.

analysis of local spatial spectra of the recorded complex �eld in small sliding aper-tures. Rays correspond to the maxima of the spectra. This method, however, doesnot work in sub-caustic zones, where the wave �eld cannot be described in terms ofgeometric optics.

In this section we describe an improved approach to the radio-optic analysisbased on combining it with the back-propagation.

We assume a vertical occultation with an immovable GPS. In the occultationplane we introduce cartesian coordinates x; z. Given an observation point at theLEO orbit, the radio-optic method locates the rays at this point using the Fourieranalysis of the complex �eld in a surrounding aperture. Rays coming to a �xedobservation point (x; z) must satisfy the geometric relationship:

x sin �+ z cos � = p: (16)

In the ray coordinates �; p this equation speci�es a curve representing the ob-servation point. Each spatial frequency of the complex �eld corresponds to a raydirection at the observation point. Thus it speci�es a ray with "; p coordinates sat-isfying relationship (16). This means that each spatial frequency can be representedas a point of this curve. The spatial spectra of the recorded complex �eld can thusbe plotted in the ray coordinates [Gorbunov et al., 2000]. The maxima of the spectramust then trace the dependence �(p).

For an observation point located on a caustic or in a sub-caustic zone, however,the spectral maxima cannot be interpreted in terms of rays. The spectrum will then

17

contain \false" maxima. If we represent the observation point with a correspondingcurve in the ray coordinates, then the false maxima will be located along this curve.

The slope of curve (16) can be estimated as the observation distance x. Thismeans that when the back-propagated wave �eld is subjected to the radio-opticanalysis, then the locations of \false" maxima will be di�erent as compared to theanalysis of the �eld recorded at the LEO orbit. The situation is schematically shownin Figure 8

We use the following data analysis algorithm. We compute the logarithmicspatial spectra ~uLEO(�; p) and ~uBP(�; p) of the wave �eld at the LEO orbit andin the back propagation plane, respectively. For this purpose, the spatial spectra~uLEO(�; p) and ~uBP(�; p) computed along di�erent curves (16) are interpolated to astandard rectangular p-" grid. Then we compute the averaged spatial spectrum:

~u(�; p) =1

2(~uLEO(�; p) + ~uBP(�; p)) : (17)

For each �xed impact parameter p, we �nd the location �(p) the main maximumof the cross section of the spectrum ~u(�; p). This speci�es the dependence �(p).


0.02 0.025 0.03 0.035 0.04 0.045 0.05 0.055 0.06Refraction angle, rad

2

2.5

3

3.5

4

Ray

heig

ht,

km

c Spatial spectra

0.02 0.025 0.03 0.035 0.04 0.045 0.05 0.055 0.062

2.5

3

3.5

4

220 230 240 250 260 270 280Dry temperature, K

0

0.5

1

1.5

2

2.5

3

Alti

tud

e,km

GOBPBP+RO

d

-500 0 500 1000X, km

6320

6330

6340

6350

6360

6370

Y,k

m

b Caustic structure


2

2.5

3

3.5

4

Ray

heig

ht,

km

GOBPBP+RO

a

Figure 9: Occultation 0308, February 02, 1997: a) refraction angles: (GO) geomet-ric optical, (BP) back propagation, and (BP+RO) combined algorithm; b) causticstructure; c) averaged spatial spectra; d) reconstrcuted temperatures.

In the numerical simulations we processed a few arti�cial occultation data sets.The arti�cial data were generated for the locations of real GPS/MET occultation

18


2

2.5

3

3.5

4

Ray

heig

ht,

km

c Spatial spectra

0.02 0.025 0.03 0.035 0.04 0.045 0.05 0.055 0.062

2.5

3

3.5

4

-500 0 500 1000X, km

6360

6365

6370

6375

6380

6385

6390

6395

6400

6405

6410

Y,k

m

b Caustic structure


0

0.5

1

1.5

2

2.5

3

Alti

tud

e,km

GOBPBP+RO

d


2

2.5

3

3.5

4R

ayhe

igh

t,km

GOBPBP+RO

a


events using global �elds of atmospheric variables from analyses of the ECMWF.We used the wave optics model [Gorbunov and Gurvich, 1998b] for generating thearti�cial data. We also computed the refraction angles using the geometric opticalray equations, which were treated as the exact solution. We chose the occultationevents with complicated caustic structures, where the back propagation method isinapplicable.

Figures 9, 10, and 11, show refraction angles, caustic structures, averaged lo-cal spatial spectra, and reconstructed temperature. We compare refraction anglescomputed using the geometric optics (GO), refraction angles computed using theback propagation method (BP), and refraction angles computed using the combinedalgorithm described above (BP+RO). For each refraction angle pro�le we computecorresponding dry temperature pro�le.

For the simulations we chose three occultations where the application of the backpropagation method results in big errors in the reconstruction of the refraction an-gle pro�les. This is the case if the refraction angle pro�les have sharp spikes, whichresults in structures with overlapping real and imaginary caustics [Gorbunov et al.,2000]. The combined radioholographic analysis, however, gives much better results.The error of reconstruction of the dry temperature using the radioholographic anal-ysis is 0.5-1 K in all the examples. This error, however, reaches 10-20 K, when usingthe back propagation method.

19


2

2.5

3

3.5

4

Ray

heig

ht,

km

c Spatial spectra

-500 0 500 1000X, km

6355

6360

6365

6370

6375

6380

6385

Y,k

m

b Caustic structure


2

2.5

3

3.5

4R

ayhe

igh

t,km

GOBPBP+RO

a


0

0.5

1

1.5

2

2.5

3

Alti

tud

e,km

GOBPBP+RO

d

0.01 0.02 0.03 0.04 0.05 0.062

2.5

3

3.5

4


2.4 Conclusions

Two methods of processing radio occultation data in multipath regions were previ-ously suggested, which are (1) the back propagation method and (2) the radio opticmethod. It was also shown that both methods have their restrictions. The radiooptic method cannot be applied in sub-caustic zones, where the di�raction e�ectsare signi�cant, and the wave �eld cannot be interpreted in terms of rays. On theother hand, the back propagation method cannot be applied for complicated causticstructures, which can arise if the refraction angle pro�le has sharp spikes.

We suggested the combination of the two methods. For this purpose, we computethe local spatial spectra of the wave �eld at the LEO orbit (observation plane) andof the back-propagated wave �eld. Both spectra are computed in the ray coordinates"; p. For the averaged spectrum, we �nd its main maximum for each given impactparameter p. The positions of the main maximuma with respect to the " coordinatestraces the dependence "(p)

The simulations showed that the combined algorithm can signi�cantly improvethe accuracy of the reconstruction of the refraction angles for the cases where theback propagation method does not work.

20

3 Canonical transform method for processing GPS

radio occultation data in lower troposphere

3.1 Introduction

The interpretation of GPS radio occultation data in the lower troposphere is of agreat importance for the incorporation of this type of data into numerical weatherprediction models. The standard algorithm of processing radio occultation data in-volves the derivation of the dependence of refraction (bending) angle versus impactparameter, which is used for the formulation of the inverse problem in the geomet-ric optical approximation [Ware et al., 1996; Rocken et al., 1997; Kursinski et al.,1997]. The computation of refraction angles is straightforward in one-ray areas,where the derivative of the phase, or the Doppler frequency, can be related to thesatellite velocities and ray directions at GPS and Low Earth Orbiter (LEO) satel-lites [Vorob'ev and Krasil'nikova, 1994]. Complementing this relation with Snell'slaw, one can derive the refraction angle and impact parameter from the Dopplerfrequency.

However, radio signals propagated through the lower troposphere can have a verycomplicated structure due to the e�ects of multipath propagation and di�raction[Gorbunov et al., 1996b; Gorbunov and Gurvich, 1998b, b]. The refraction anglecannot be directly derived from the phase of wave �eld in multipath zones, becauseat each point the phase is de�ned by amplitudes and phases of multiple interferingrays. This results in the formulation of the problem of the determination of the raystructure of wave �eld.

Two methods were suggested for processing radio occultation signals in multipathzones: (1) back-propagation method [Gorbunov et al., 1996b; Karayel and Hinson,1997; Hinson et al., 1997, 1998; Gorbunov and Gurvich, 1998a, a] and (2) radio-optic (or radioholographic) method [Lindal et al., 1987; Pavel'ev , 1998; Hocke et al.,1999]. Both of them are based on simple physical considerations. The comparativeanalysis of the two methods performed by Gorbunov et al. [2000] indicates that bothmethods have very strong restrictions of their applicability.

In the �rst method, the wave �eld measured along the LEO trajectory is used asthe boundary condition for the back-propagating solution of the Helmholtz equationin a vacuum. The underlying ray structure of this back-propagated �eld may havetwo types of caustics: real and imaginary. There may be a one-ray area betweenthem, where the back-propagated �eld can already be processed in the standard way.The di�culty with the application of this method is, however, that the position of theone-ray area is unknown a priori, because the caustic structure depends on the pro-�le of refraction angle, which must yet be determined. Numerical simulations, whichwere performed on the material of global �eld of atmospheric parameters from anal-yses of European Medium-Range Weather Forecast Center (ECMWF) [Gorbunovet al., 2000], show that complicated caustic structures with overlapping real andimaginary caustics may occur, which causes the back-propagation method to fail.It is also clear that the real atmosphere must contain small-scale inhomogeneities,which are not represented in numerical weather prediction models and which willresult in still more complicated structures of wave �eld propagating through the

21

atmosphere.The radio-optic method utilizes the analysis of spatial spectra of the wave �eld

in small sliding apertures. It is assumed that each ray can be locally associatedwith a basic wave form such as a plane or a spherical wave. Then the maxima oflocal spatial spectra identify interfering rays. The most important disadvantage ofthis method is that it cannot be applied for the analysis of wave �eld in sub-causticzones, where the wave �eld has a complicated structure and cannot be interpretedin terms of rays [Gorbunov et al., 2000].

In this section, we suggest a new approach to the problem, based on the theoryof Fourier integral operators. As shown in [Mishchenko et al., 1990], a wave problemcan be associated with a canonical Hamilton system, which describes the geometricoptical ray structure of the wave �eld. The Hamilton system is written in terms ofspatial coordinates and impact, the latter corresponding to the di�erential operatorin the wave problem. On the other hand, coordinate vector and impact de�ne theinitial conditions of a ray. We can thus consider the manifold of all possible raysand introduce ray coordinates, which are impact parameter and refraction angle. Wecan also write a canonical transform [Arnold , 1978] mapping spatial coordinates andimpact to ray coordinates. It can be shown that the impact parameter now playsthe role of the spatial coordinate, while the refraction angle is the new impact. AFourier integral operator can be associated with this canonical transform [Egorov ,1985]. This operator transforms the wave function to the representation of raycoordinates. The argument of the transformed wave function is now the ray impactparameter, and the derivative of its phase (the new impact) is the refraction angle.This method is applicable if an unique dependence of refraction angles versus impactparameter exists, which means that rays can be uniquely identi�ed by their impactparameters.

We designed a fast numerical implementation of the method based on the FFT.For the validation of the new method we process simulated radio occultations, wherethe application of the back-propagation method results in big errors.

The results presented in this chapter are based on an article submitted to RadioScience.

3.2 Helmholtz Equation, Fourier Integral Operators, and

Canonical Transforms

We shall use the direct and inverse 1D k-Fourier transform in the following de�nition:

ef(�) =

Ze�ikx�f(x)dx;

(18)

f(x) =k

2�

Zeikx� ef (�)d�;

where k is the wave vector. We shall also use di�erential operators likeDx =1

ik

@

@x:

22

Back-propagation plane

S

pEarth

y

LEO

Real focusing

x

(p)

Imaginary focusing

Figure 12: The geometry of multipath propagation.

We shall use Cartesian coordinates (x; y), where the x-axis points in the directionof the incident wave, and the y-axis is transversal to it. The geometry is shown inFigure 12. In a radio occultation experiment, the wave �eld propagated throughthe atmosphere is recorded along the LEO trajectory S. The recorded complex �eldcan be back-propagated to some location x. From the geometric optical point ofview, this procedure is equivalent to the continuation of rays backwards as straightlines. In the previous papers [Gorbunov et al., 1996b; Karayel and Hinson, 1997;Hinson et al., 1997, 1998; Gorbunov and Gurvich, 1998a, b] this method was appliedin order to �nd the single-ray area between real and imaginary caustics. As waspointed out above, it is not always possible. Here we shall, however, always analyzethe back-propagated �eld, because this procedure conserves impact parameters andrefraction angles of rays. The back-propagation technique is thus an immanent partof the suggested method.

The back-propagated �eld u(x; y) is described by the Helmholtz equation:

��D2x �D2

y + 1�u = 0: (19)

The variable x can be looked at as time, because we consider the propagationof very short waves, and the backscattering can be neglected. Using the operatorfactorization of this equation [Martin, 1992], we can rewrite it in the following form:

�Dx +

q1�D2

y

��Dx +

q1�D2

y

�u = 0: (20)

Assuming that the incident wave has the form exp(ikx), we can separate theequation for the forward propagating wave:

�Dxu = H(y;Dy)u � �q1�D2

yu; (21)

23

where H(y;Dy) is the Hamilton operator, and �Dx can be understood as the op-erator of energy. The Hamilton operator belongs to the class of pseudo-di�erentialoperators, which are de�ned as follows [Egorov , 1985]:

H(y;Dy)f =k

2�

ZH(y; �)eiky� ef (�)d�; (22)

where the function H(y; �) is the symbol of the operator. For big �, jH(y; �)j mustbe majorated by C j�jm, where m is called the order of the operator.

We shall use notation ux(y) for u(x; y). Using the Fourier transform with respectto y, we can write the solution of the boundary problem Zverev [1975]

eux(�) = eikxp

1��2eu0(�): (23)

The solution can thus be written in the form of a Fourier integral operator trans-forming function u0(y) to u(z), and z is here a duplicate of the vertical coordinate:

ux(z) =k

2�

Zeikhxp

1��2+z�ieu0(�)d�: (24)

A Fourier integral operator is de�ned as an operator of the following type [Egorov ,1985]:

�u(z) =k

2�

Za(z; �)eikS(z;�)eu(�)d�; (25)

where a(z; �) is the symbol of the operator, and S(z; �) is its phase function. Egorov'stheorem [Egorov , 1985] states that this operator is associated with a canonical trans-form from the old coordinates (y; �) to the new coordinates (z; �) in the phase space.A canonical transform preserves the Hamilton structure in the phase space [Arnold ,1978]. For a transform to be canonical, the form �dy��dz must be a full di�erential,which is equivalent to the following condition [Arnold , 1978]:

yd� + �dz = dS(z; �): (26)

where S(z; �) is the generating function of the canonical transform from (y; �) to(z; �). And vice versa, given a function S(z; �), we can write equations de�ning acanonical transform [Arnold , 1978]:

y =@S(z; �)

@�;

(27)

� =@S(z; �)

@z:

24

If we now consider a Fourier integral operator with the phase function S(z; �), thenEgorov's theorem states that in the main order

�Q = P�; (28)

where Q = Q(y;Dy); P = P (z;Dz) are pseudo-di�erential operators, whose symbolsare related by the canonical transform with the generating function S(z; �):

Q

�@S(z; �)

@�; �

�= P

�z;@S(z; �)

@z

�: (29)

This means that the operator � transforms the wave function from (y; �)-representationto (z; �)-representation. If we, for example, consider the function S(z; �) = z�, thenthe corresponding canonical transform is the identity: y = z; � = �, and the Fourierintegral operator with phase function S(z; �) and symbol a(z; �) = 1 is the composi-tion of the direct and inverse Fourier transforms, which is also identity. The integraloperator (24) corresponds to the canonical transform:

z = y +�p

1 � �2x;

(30)

� = �;

which describes the evolution of straight rays with direction vectors�p

1� �2; ��.

For negative x, this operator describes the back-propagation.

3.3 Canonical Transform to Ray Coordinates

We have by now introduced all the concepts necessary for the derivation of theFourier integral operator transforming the wave function to the representation ofthe ray coordinates. First, we need to �nd the necessary canonical transform. Weassume that rays can be identi�ed by their impact parameters. Thus we can chosethe new coordinate z to be equal to ray impact parameter: z = �x� + y

p1 � �2.

For the de�nition of the new impact � we have the condition [Arnold , 1978]:d� ^ dy = d� ^ dz, which results in the equation:

@�

@�

@z

@y� @�

@y

@z

@�= 1 (31)

If we assume that � = �(�), then we have

@�

@�=

�@z

@y

��1

=1p

1� �2: (32)

25

The canonical transform has thus the following form:

z = �x� + yp1 � �2;

(33)

� = arcsin �;

which indicates that the new impact � is the ray direction angle with respect to thex-axis. The generating function of this transform can be readily derived:

S(z; �) = z arcsin � � xp1 � �2 (34)

The symbol of the Fourier integral operator can be chosen in the form [Egorov ,1985]:

a(z; �) =

�� @2S@z@�

��1=2 = �1 � �2�1=4

: (35)

This allows for writing the Fourier integral operator as follows:

�ux(z) =k

2�

Z �1 � �2�1=4 eikhz arcsin��xp1��2

ieux(�)d� (36)

The transformed wave function �ux(p) has the formA(p) exp�ikR�(p)dp

�, where

A(p) is the amplitude. This allows for the computation of the impact as follows

�(p) =1

k

d

dparg �ux(p) (37)

Finally, the refraction angle is computed as follows:

"(p) = ��(p) + arcsin

jxGPS j �

px2GPS � 2a (p � a)

a

!;

where a is the Earth's radius, and the second term corrects for the ray directionangle at the GPS satellite.

An advantage of this method is that it allows for the computation of the refractionangle " for a given ray impact parameter p, unlike the other methods, where " andp are computed simultaneously for given point in the LEO trajectory or in theback-propagation plane, and noise may result in a non-unique function "(p).


For the numerical simulations we used simulated occultation data computed for sta-tistical comparisons of GPS/MET data and ECMWF analyses [Gorbunov and Ko-rnblueh, 2001]. Figures 13 and 14 show the results of the simulations. The refraction

26

-500 0 500 1000X, km

6320

6330

6340

6350

6360

6370

Y,k

m

Caustic structureb

220 230 240 250 260 270Dry temperature, K

0

0.5

1

1.5

2

Alti

tude

,km GO

BPCT

c

200 250 300 350 400Refractive index, N-units

0

0.5

1

1.5

2

Alti

tud

e,km

GOBPCT

d

0.02 0.03 0.04 0.05 0.06Refraction angle, rad

2

2.5

3

3.5

4R

ayhe

ight

GOBPCT

a

Figure 13: Simulated occultation 0308, February 02, 1997, UTC 10:41, 8.6�N62.3�E: (a) refraction angles computed using geometric optical model (GO), back-propagation method (BP) and canonic transform (CT), (b) caustic structure of theback-propagated �eld, (c) reconstructions of dry temperature, and (d) reconstruc-tions.

angles computed using the back-propagation and the canonical transform methodswere compared with the geometric optical refraction angles, which were treatedas the exact solution. In these examples, the application of the back-propagationmethod results in signi�cant errors in the reconstruction of the refraction anglepro�les in multipath zones. This is explained by the complicated caustic structurewith overlapping real and imaginary caustics. The inversions of the refraction anglepro�les computed by the back-propagation method indicate signi�cant errors in thereconstructed pro�les of refractivity and dry temperature. The canonical transformmethod, however, allows for a very accurate computation of the refraction anglepro�les, which results in a high accuracy of inversions.

Figure 15 presents a simulated occultation, where strong horizontal inhomo-geneities result in a non-unique dependence "(p). For a spherically symmetricmedium, ray impact parameter is de�ned as p = nr sin , where is the angle be-tween the ray direction and local vertical. In a horizontally inhomogeneous mediumthis value is not constant. Its evolution along a ray is described by the dynamicalequation [Gorbunov et al., 1996a; Kravtsov and Orlov , 1990]:

_p = n@n

@�; (38)

where � is the polar angle in the occultation plane. This results in errors of the de-

27

-500 0 500 1000X, km

6350

6360

6370

6380

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m

b Caustic structure

230 240 250 260 270 280 290 300Dry temperature, K

0

0.5

1

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tude

,km

GOBPCT

c


0

0.5

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2

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tud

e,km

GOBPCT

d


2

2.5

3

3.5

4R

ayhe

ight

GOBPCT

a

Figure 14: Simulated occultation 0180, February 03, 1997, UTC 06:22, 42.3�S129.7�E: (a) refraction angles computed using geometric optical model (GO), back-propagation method (BP) and canonic transform (CT), (b) caustic structure of theback-propagated �eld, (c) reconstructions of dry temperature, and (d) reconstruc-tions of refractivity.

termination of the e�ective refraction angle and impact parameter from the Dopplerfrequency in the assumption of spherical symmetry [Ahmad and Tyler , 1999]. Thismay also result in di�erent rays having the same e�ective impact parameter. In thiscase the measured dependence "(p) is non-unique.

The variation of the ray impact parameter can be estimated as a�n, wherea is the Earth radius, and �n is the variation of refractivity due to horizontalgradients along the ray. If we assume a 10 K variation of temperature and 3� 10�3

kg/kg variation of speci�c humidity across an atmospheric front, then the horizontalvariation of refractivity is about 30 N-units. The corresponding variation of theimpact parameter is then about 200 m.

In Figure 15, the length of the impact parameter interval, where "(p) is non-unique is about 300 m, which is very close to the above estimation. In this interval,the transformed wave function �ux(p) is approximately a superposition of threefunctions Aj(p) exp

�ikR�j(p)dp

�corresponding to multiple rays with the same im-

pact parameters. Like in multipath zones, the computation of the refraction angleby means of the di�erentiation of the phase results in errors. The computed pro�le"(p) is mostly in uenced by the strongest of the interfering rays. In this case thetransformed wave function �ux(p) can also be subjected to the spectral analysisin sliding apertures, like in the radio-optical method, which should reveal multiplebranches of "(p).

28

-500 0 500 1000X, km

6310

6320

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m

b Caustic structure

210 220 230 240 250 260Dry temperature, K

0

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tude

,km

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0

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tud

e,km

GOBPCT

d


2

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4R

ayhe

ight

GOBPCT

a

Figure 15: Simulated occultation 0052, February 05, 1997, UTC 01:48, 11.2�S171.1�E: (a) refraction angles computed using geometric optical model (GO), back-propagation method (BP) and canonic transform (CT), (b) caustic structure of theback-propagated �eld, (c) reconstructions of dry temperature, and (d) reconstruc-tions of refractivity.

3.5 Conclusions

In this section, we suggested a new approach to the problem of the computationof refraction angles from radio occultation data measured in multipath zones. Theapproach is based on the theory of Fourier integral operators. We consider theHelmholtz equation and the underlying Hamilton structure describing the rays ofthe geometric optical approximation for this equation. For this Hamilton structurewe found the canonical transform mapping the spatial coordinate and impact to theray coordinates. With this canonical transform we can associate a Fourier integraloperator, which transforms the wave function from the representation of the spatialcoordinates to the representation of the rays coordinates. In this representation, thewave function depends on the ray impact parameter, and the derivative of its phaseis the refraction angle. If we assume that the rays can be uniquely identi�ed by theirimpact parameters, then this method allows for disentangling the ray structure.

The limitation of the applicability of the canonical transform method is therequirement that the dependence of the refraction angle versus impact parametershould be unique. This condition can be broken in the presence of strong regularhorizontal gradients of the atmospheric refractivity, which can be caused, for exam-ple, by atmospheric fronts. Such strong gradients, however, occur rarely. A possiblesolution for this situation could be the analysis of the spectra of the transformedwave function in small sliding apertures. On the other hand, in this case it is also

29

very di�cult to formulate the inverse problem of atmospheric refraction, even if theexact refractivity pro�le is known.

The canonical transform method is an exact asymptotic solution of the problemof the reconstruction of the ray structure of a wave �eld. It allows for the recon-struction of unique dependencies of refraction angle versus impact parameter witha high accuracy. This method can be understood as a generalization of the back-propagation, which can also be written in the form of a Fourier integral operatorassociated with a canonical transform. However, the limits of the applicability ofthe canonical transform method are much wider than those of the back-propagationmethod.

4 Conclusions and Acknowledgements

This report is based on research performed prior to and during a visit to the DanishMeteorological Institute (DMI) as a visiting scientist from October to December2000. This activity was sponsored by the GRAS Meteorology Satellite ApplicationFacility (SAF) project.

The GRAS SAF is a Eumetsat development project hosted by DMI and withthe two partner institutes the UK Met O�ce and IEEC in Spain. The scope ofthe GRAS SAF is to develop and set up a processing facility to deliver atmosphericparameters in real time that can be used for numerical weather prediction. Themethod is based on GPS satellite observations using the GRAS instrument onboardthe Metop satellite.

The objective of this study was to focus on di�raction correction methods in-cluding the back propagation method and radio-optical (spectral) methods and todevelop prototype algorithms for radio occultations with multipath propagation foruse in the processing of the GRAS SAF products.

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