### Abstract

- Top of page
- Abstract
- 1. Introduction
- 2. Helmholtz Equation, Fourier Integral Operators, and Canonical Transforms
- 3. Canonical Transform to Ray Coordinates
- 4. Numerical Simulations
- 5. Conclusions
- Acknowledgments
- References

[1] The methods of processing radio occultation data in multipath zones which were used up to now have very strong restrictions of the applicability. In this paper, we introduce a new approach to the problem of deciphering the ray structure of wave fields in multipath zones using the short-wave asymptotic solution of the wave problem. In geometric optics a canonical transform resolves multipath by introducing new coordinate and momentum in such a way that different rays are distinguished by their coordinates. The wave field is processed by a Fourier integral operator associated with the canonical transform. The transformed wave function can then be written in the single-ray approximation, which allows for the determination of refraction angles from the derivative of the eikonal. The new method retains all the advantages of the back propagation such as the removal of effects of diffraction in free space and the enhancement of the vertical resolution in retrieved profiles, but it has much wider applicability limits. The method is convenient for operational applications. We discuss a fast numerical implementation of the method and present the results of numerical simulations confirming the applicability of the method.

### 1. Introduction

- Top of page
- Abstract
- 1. Introduction
- 2. Helmholtz Equation, Fourier Integral Operators, and Canonical Transforms
- 3. Canonical Transform to Ray Coordinates
- 4. Numerical Simulations
- 5. Conclusions
- Acknowledgments
- References

[2] The interpretation of GPS radio occultation data in the lower troposphere is of a great importance for the assimilation of this type of data into numerical weather prediction models. The standard algorithm of processing radio occultation data involves the derivation of the dependence of refraction (bending) angle versus impact parameter, which is used for the formulation of the inverse problem in the geometric optical approximation [*Ware et al.*, 1996; *Rocken et al.*, 1997; *Kursinski et al.*, 1997]. The computation of refraction angles is straightforward in single-ray areas, where the derivative of the phase, or the Doppler frequency, can be related to the satellite velocities and ray directions at GPS and Low Earth Orbiter (LEO) satellites [*Vorob'ev and Krasil'nikova*, 1994]. Complementing this relation with the formula of Bouger, one can derive the refraction angle and impact parameter from the Doppler frequency.

[3] Radio signals propagated through the lower troposphere can have a very complicated structure due to the effects of multipath propagation and diffraction [*Gorbunov et al.*, 1996b; *Gorbunov and Gurvich*, 1998a, 1998b; *Sokolovskiy*, 2001a]. Refraction angle cannot be directly derived from the phase of wave field in multipath zones, because at each point the phase is defined by amplitudes and phases of multiple interfering rays.

[5] In the first method, the wave field measured along the LEO trajectory is used as the boundary condition for the back-propagating solution of the Helmholtz equation in a vacuum. The underlying ray structure of the back-propagated field may have two types of caustics: real and imaginary. There may be a single-ray area between them, where the back-propagated field can be processed in the standard way. However, the position of the single-ray area is unknown a priori, because the caustic structure depends on the profile of refraction angle, which must yet be determined. Numerical simulations with global field of atmospheric parameters from analyses of European Centre for Medium-Range Weather Forecasts (ECMWF) [*Gorbunov et al.*, 2000] show that complicated caustic structures with overlapping real and imaginary caustics may occur, which causes the back propagation method to fail. The real atmosphere contains small-scale inhomogeneities, which are not represented in numerical weather prediction models and which will result in still more complicated structures of wave fields [*Sokolovskiy*, 2001].

[6] The radio-optic method utilizes the analysis of spatial spectra of the wave field in small sliding apertures. Rays are locally associated with plane waves and are visualized as the maxima of local spatial spectra. This method has the following disadvantages: (1) it cannot be applied in sub-caustic zones, where the wave field cannot be interpreted in terms of rays and (2) it has limited resolution [*Gorbunov et al.*, 2000].

[7] In this paper, we suggest a new approach to the problem, based on the theory of Fourier integral operators. A wave problem can be associated with a canonical Hamilton system, which describes the geometric optical ray structure of the wave field [*Kravtsov and Orlov*, 1990; *Mishchenko et al.*, 1990]. The Hamilton system is written in terms of spatial coordinates and momentum related to the ray direction. We can consider the manifold of all rays in the phase space. Multipath propagation arises in areas where multiple points of this manifold have the same spatial coordinate. A canonical transform in the phase space can be used in order to introduce new spatial coordinate and momentum in such a way that the rays have unique spatial coordinates. Using Egorov's theorem [*Egorov*, 1985; *Treves*, 1982], it is possible to write a Fourier integral operator associated with the canonical transform. The operator transforms the wave function to the new representation with single-ray propagation. The ray structure can then be reconstructed from the derivative of the phase of the transformed wave function in the standard way. We show that the canonical transform to the ray coordinates (impact parameter, ray direction angle) can be used over a wide range of conditions.

[8] We designed a fast numerical implementation of the method based on the FFT. For the validation of the method we process simulated radio occultations, where the application of the back propagation method results in big errors. The simulations show that the method has a much wider applicability than the back propagation.

### 2. Helmholtz Equation, Fourier Integral Operators, and Canonical Transforms

- Top of page
- Abstract
- 1. Introduction
- 2. Helmholtz Equation, Fourier Integral Operators, and Canonical Transforms
- 3. Canonical Transform to Ray Coordinates
- 4. Numerical Simulations
- 5. Conclusions
- Acknowledgments
- References

[9] In a radio occultation experiment, the wave field propagated through the atmosphere is recorded along the LEO trajectory *S* (Figure 1). We shall use Cartesian coordinates (*x*, *y*) in the occultation plane defined in Figure 1. The recorded complex field can be back-propagated to some location *x*. From the geometric optical point of view, this procedure is equivalent to the continuation of rays backwards as straight lines. In the previous papers [*Gorbunov et al.*, 1996b; *Karayel and Hinson*, 1997; *Hinson et al.*, 1997, 1998; *Gorbunov and Gurvich*, 1998a, 1998b] this method was applied in the assumption that there is a single-ray area between real and imaginary caustics. Here we shall always analyze the back-propagated field, because this procedure conserves impact parameters and refraction angles of rays.

[10] We will use the direct and inverse 1D *k*-Fourier transform in the following definition:

where *k* is the wave vector. We shall also use differential operators like .

[11] The back-propagated field *u*(*x*, *y*) is described by the Helmholtz equation:

The propagation coordinate *x* can be looked at as time, because we consider the propagation of very short waves, and the backscattering can be neglected. This equation can be rewritten using the operator factorization [*Martin*, 1992]:

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

where *H*(*y*, *D*_{y}) is the Hamilton operator, and −*D*_{x} can be understood as the operator of energy. The Hamilton operator is understood as multiplication by in the Fourier space. We shall use notation *u*_{x}(*y*) for *u*(*x*, *y*).

[12] The wave equation (5) can be solved asymptotically for short waves by means of expansion with respect to the small parameter *k*^{−1} (similar to *h* in quantum mechanics) [*Kravtsov and Orlov*, 1990; *Mishchenko et al.*, 1990]. For single-ray propagation the asymptotic geometric optical (GO) solution is written in the form *A*_{x}(*y*) exp(*ik*Ψ_{x}(*y*)), where amplitude *A* and eikonal Ψ are smooth functions of the spatial coordinates [*Kravtsov and Orlov*, 1990]. The GO rays are described by the canonical system [*Arnold*, 1978] with the Hamilton function *H*(*y*, η), where η is momentum, associated with operator *D*_{y}.

[13] Retaining the main order terms (*k*^{0}) and neglecting the derivatives of the amplitude *D*_{x,y}*A*, which contribute to the next order (*k*^{−1}), we derive the equation for the momentum and the Hamilton-Jacobi equation:

From these equations we derive , which clarifies the geometrical meaning of the momentum. ∇Ψ is the unit ray direction vector. This relationship between the derivatives of the eikonal and the ray direction (the normal to the wave front) is used for the determination of refraction angles in single-ray areas [*Vorob'ev and Krasil'nikova*, 1994].

[14] Using the notation *z* and ξ for the spatial coordinate and momentum in the observation line at *x*_{1}, we write the solution of the Hamilton system as function of the initial conditions in the source line at *x*_{0}, *z* = *z*(*y*, η), ξ = ξ(*y*, η):

where Δ *x* = *x*_{1} − *x*_{0} is the propagation distance. In order to simplify the notation we use indexes 0 and 1 instead of *x*_{0} and *x*_{1}, respectively. We consider now the Cauchy problem with the boundary condition *u*_{0}(*y*) = *A*_{0}(*y*) exp(*ik*Ψ_{0}(*y*)). For finding its solution we need to know the ray structure of field *u*_{0}(*y*). For single-ray field *u*_{0}(*y*), the momentum, or the vertical component of ray direction vector, η(*y*) at each point *y* is computed using (6) as ∂Ψ_{0}(*y*)/∂*y*. Given a ray starting point *y* and momentum η(*y*), we can find the ray's destination *z*(*y*, η(*y*)). The GO amplitude is then found in the standard way [*Kravtsov and Orlov*, 1990]:

[15] Single-ray propagation is defined by the condition that for each *z* not more than one *y* satisfies the equation *z* = *z*(*y*, η(*y*)). In this case the GO wave field *u*_{1}(*z*) equals *A*_{1}(*z*)exp(*ik*Ψ_{1}(*z*)) for points *z* covered by the rays and 0 in the shadow. For multiple interfering rays the wave field can be written as the sum of GO fields of all the rays.

[16] Howsoever simple and even trivial these considerations are, they indicate a significant problem in the application of the geometric optics for the solution of the Cauchy problem: There is no any general way of associating a ray structure with the wave field *u*_{0}(*y*) without any additional assumptions. This is straightforward for plane or spherical waves, where we can use (6), but it probably cannot be done in general for any arbitrary wave fields. This problem arises in the interpretation of radio occultation data. These data are interpreted in terms of geometric optics, which requires the determination of the ray structure of the registered wave field. The standard technique of the computation of refraction angles from the derivative of the phase, or Doppler frequency, cannot be applied in multipath zones.

[17] There are two other problems with the application of the GO solution: (1) this solution does not work in the vicinity of caustics, where GO amplitude (9) is singular and (2) diffraction effects can become significant for big propagation distances *x*. These difficulties can, however, be overcome by using Maslov operators or Fourier integral operators [*Mishchenko et al.*, 1990]. It is important that the theory of Fourier integral operators also gives a clue for the solution of the problem deciphering the ray structure of a wave field diffracted by the Earth's atmosphere.

[18] We shall now discuss the solution of the Cauchy problem using Fourier integral operators. We consider a general approach, which can be used for an inhomogeneous medium. The initial condition *u*_{0}(*y*) is represented as the superposition of incident plane waves _{0}(η)exp(*iky*η). Each wave creates a range of parallel rays with initial momentum η and infinitesimal amplitude (*k*/2π)_{0}(η) *d*η. The rays are propagated using the geometric optics. Assuming that not more than one ray from each incident plane wave with momentum η arrives at each observation point *z*, we can write the solution of the Cauchy problem as follows:

where *y* = *y*(*z*, η) is the initial point of the single ray starting in the direction η and arriving at point *z*. Functions *a*(*z*, η) and Σ(*z*, η) describe the variation of the amplitude and the phase delay of the ray. In a vicinity of the initial ray momentum η, phase delay Σ can be expressed as a function of ray boundary points, *z* and *y*. From the fact that rays are normal to phase fronts, and the gradient of the phase is the ray direction vector, or directly using (6), we infer the following differential equation:

where ξ is the momentum of the ray at the point *z*. This results in following equation for the phase function *S*(*z*, η) = Σ(*z*, η) + *y*η of operator (10):

[19] The amplitude *a*(*z*, η) of the operator is derived using equation (9) and the relationship *y* = ∂ *S*/∂ η, which follows from (12):

[20] For a fixed propagation distance Δ*x*, the solution (*z*, ξ) of the Hamilton system, as a function of initial conditions (*y*, η) specifies a canonical transform [*Arnold*, 1978, chapter 9]. A canonical transform replaces the old coordinates (*y*, η) in the phase space with the new ones (*z*, ξ) in such a special way that the canonical form of the Hamilton system is preserved. For a transform to be canonical, the form η *dy* − ξ *dz* must be equal to a full differential, which we denote *d*Σ. Redefining Σ = *S* − *y*η we arrive at equation (12). Σ and *S* are different types of the generating function of the canonical transform [*Arnold*, 1978, chapter 9].

[21] Formulas (10), (12), and (13) provide the asymptotic solution of the Cauchy problem in the form of the Fourier integral operator associated with the canonical transform from (*y*, η) to (*z*, ξ) [*Egorov*, 1985; *Mishchenko et al.*, 1990; *Treves*, 1982]. The theory of Fourier integral operators is employed here in the simplest, so-called “naive”, variant [*Treves*, 1982]. As a simple example, consider the vacuum propagation. For canonical transform (8) we have generating function and the amplitude *a*(*z*, η) = 1. The solution for *x*_{0} = 0, *x*_{1} = *x* takes the following form:

This is equivalent to the following relationship:

which can be derived directly from (5) using the Fourier transform with respect to *y* [*Zverev*, 1975]. An important fact is that this asymptotic solution for a vacuum is exact. This is due to the fact that the GO solution for plane waves in a vacuum is exact.

[22] The relation between canonical transforms in phase space and Fourier integral operators constitutes the contents of Egorov's theorem [*Egorov*, 1985; *Treves*, 1982]. Particularly, it establishes the form of the asymptotic solution of Cauchy problem for wave equations. But the meaning of this theorem is much wider. A canonical transform is the choice of a coordinate system in a phase space. If the phase space represents the geometric optical rays of a wave problem, then Egorov's theorem describes the asymptotic transformation of the wave function to a different representation: This transformation is performed by the Fourier integral operator associated with the canonical transform.

[23] Above we discussed a time-independent canonical transform of the phase space variables *y*, η. Consider a canonical transform from (*y*, η) to (*z*, ξ) with a generating function *S*_{x}(*z*, η) which depends on *x*. In this case the old Hamilton function *H* will be replaced with the new one, *H*′, defined from the differential equation [*Arnold*, 1978]:

The wave problem can be reformulated in the (*z*, η)-representation. Egorov's theorem specifies the Fourier integral operator transforming the wave function to the new representation. This may allow for simplifying the Hamilton operator by the choice of the most convenient canonical coordinates [*Treves*, 1982]. The conjugated operator associated with the inverse canonical transform performs the inverse transformation [*Egorov*, 1985]. Given the solution *u* of the wave problem with initial condition *u*_{0} in the (*y*, η)-representation, we can find the solution *u*′ with the initial condition in the (*z*, ξ)-representation. The asymptotic character of the transformation means that has the order *k*^{−1} or higher and vanishes for big *k*.

### 3. Canonical Transform to Ray Coordinates

- Top of page
- Abstract
- 1. Introduction
- 2. Helmholtz Equation, Fourier Integral Operators, and Canonical Transforms
- 3. Canonical Transform to Ray Coordinates
- 4. Numerical Simulations
- 5. Conclusions
- Acknowledgments
- References

[24] Now we can use the theory of Fourier integral operators for the solution of the problem of the determination of the ray structure of a wave field in multipath zone. Multipath propagation means that multiple rays arrive at the same spatial location. Interfering rays always have different momenta. This means that in the phase space rays never intersect. This follows from the fact that initial conditions (*y*, η) for the Hamilton system specify an unique ray.

[25] We assume that rays can be identified by their impact parameters. The new coordinate *z* can then be taken equal to ray impact parameter *p*. Using the unity normal to a straight ray , the ray impact parameter (the distance from the point (0, 0) to the ray) is expressed as .

[26] The new momentum ξ is defined by the condition of the conservation of the volume in the phase space [*Arnold*, 1978]:

If we assume that ξ = ξ (η), then we have

The canonical transform has thus the following form:

which indicates that the new impact ξ is the ray direction angle with respect to the *x*-axis. The generating function of this transform and the corresponding amplitude, can be readily derived:

This allows for writing the Fourier integral operator as follows:

In order to reduce oscillations of spectrum _{x}(η) we subtract the Earth curvature radius *a* from the arguments of the functions *u*_{x} and . Assuming that *p* = *a* + Δ*p* and *y* = *a* + Δ*y*, we introduce the functions *v*_{x}(Δ*y*) = *u*_{x}(*a* + Δ*y*), and . The operator takes then the following form:

The 3rd-order term *ika*(arcsin η−η) in the argument of the oscillating exponential function is very significant. Neglecting this term would completely destroy the operator and turn it practically into the back propagation to *x* = 0.

[27] This operator can be rewritten by using the coordinate ξ = arcsin η, which turns it into the inverse *k*-Fourier transform of the following function:

The algorithm can thus be numerically implemented using the FFT.

[28] In the coordinates (*p*, ξ) we have single-ray propagation. Given the transformed wave function in the form *A*′(Δ*p*)exp(*ik*Ψ′(Δ*p*), the momentum can be found using (6):

The refraction angle is computed as follows:

where the second term corrects for the ray direction angle at the GPS satellite located at (*x*_{GPS}, *y*_{GPS}).

[29] Although both the operator and back-propagated field *u*_{x} depend on *x*, this dependence vanishes in , because it contains the combination

which is independent of *x*, as seen from (15). This agrees with the fact that the Hamilton function *H*′ in the (*p*, ξ)-representation is equal to 0, which follows from (15) and the fact that ∂*S*_{x}/∂*x* = *H*.

[30] This shows that the operator includes back propagation to *x* = 0 and therefore it inherits the advantages of the back propagation technique such as the removal of effects of diffraction in free space and the enhancement of the vertical resolution in retrieved profiles. But the canonical transform method has much wider limits of the applicability: it can always be applied if the refraction angle is a single-valued function of the impact parameter, regardless of how complicated the caustic structure is. Besides, this method does not have such a tuning parameter as the back propagation plane position, which must be found individually for each occultation. This makes this method convenient for operational applications. In our implementation we use the back propagation to some fixed position, and the back propagated field is then subjected to the Fourier integral operator.

[31] Another advantage of this method is that it allows for the computation of the refraction angle ε for a given ray impact parameter *p*, unlike the other methods, where ε and *p* are computed simultaneously for given point in the LEO trajectory or in the back propagation plane, and noise may result in a multivalued function ε(*p*).

### 4. Numerical Simulations

- Top of page
- Abstract
- 1. Introduction
- 2. Helmholtz Equation, Fourier Integral Operators, and Canonical Transforms
- 3. Canonical Transform to Ray Coordinates
- 4. Numerical Simulations
- 5. Conclusions
- Acknowledgments
- References

[32] The numerical simulations were performed with artificial occultation data generated for statistical comparisons of GPS/MET data and ECMWF analyses [*Gorbunov and Kornblueh*, 2001]. The refraction angles computed by the back propagation (BP) and the canonical transform (CT) methods were compared with the GO refraction angles, which were computed by ray-tracing. The results are shown in Figures 2 and 3. The refraction angles are plotted in two ways: (1) as functions of the ray leveling height above the planet limb, Δ*p* = *p* − *a*, in panels (a), and (2) as functions of the ray perigee height above the Earth's surface, in panels (c). The ray perigee heights *z*_{p} were computed using the retrieved profiles of refractive index *n*(*z*) versus altitude *z* and the standard relationship (*a* + *z*_{p})*n*(*z*_{p}) = *a* + Δ *p*. In panels (b) we plot the amplitude of . Panels (d) show the Abel inversions of the GO, CT, and BP refraction angles. We plot the differential refractivities defined as the deviations from the exponential model 300 exp(−*z*/7.5 km).

[33] Because multivalued profiles ε(*p*) cannot be used for inversion, in panels (c) we always plot single-valued profiles computed by the monotonization procedure: Given gridded data {*p*_{i}, ε_{i}}, the sequence {*p*_{i}} is nonmonotonous for a multivalued profile. We replace it with the *L*_{2}-nearest monotonous sequence.

[34] In Figures 2 and 3, the BP refraction angles indicate significant errors in the multipath zones. This is explained by the complicated caustic structure with overlapping real and imaginary caustics [*Gorbunov et al.*, 2000]. This results in big retrieval errors. The CT refraction angles show a very good agreement with the GO solution, and the retrieval accuracy is here much higher than in the BP inversions.

[35] The amplitude of can be used for locating the geometric optical shadow. The border between the light and shadow zones is marked by a very sharp drop of the amplitude in the (*p*, ξ)-representation. This is explained by the fact that the Fourier integral operator contains the back propagation, which corrects for refractive attenuation and diffraction over a long propagation distance in a free space. In addition, this operator corrects for multipath propagation by transforming the wave function to a single-ray representation. The resulting amplitude is free from multipath interference, refractive attenuation, and the diffraction effects are suppressed to a significant extent. The characteristic width of the light-to-shadow transfer zone is here about 30 m, which also gives the estimation of the resolution of the method.

[36] Figure 4 presents a simulated occultation, where strong horizontal inhomogeneities result in a multivalued dependence ε (*p*). For a spherically symmetric medium, ray impact parameter is defined as *p* = *nr* sin ψ, where ψ is the angle between the ray direction and local vertical. In a horizontally inhomogeneous medium this value is not constant. Its evolution along a ray is described by the dynamical equation [*Kravtsov and Orlov*, 1990; *Gorbunov et al.*, 1996a; *Gorbunov et al.*, 2000]:

where θ is the polar angle in the occultation plane. Due to this, rays with different impact parameters at the transmitter may acquire identical impact parameters at the receiver. This results in multivalued profiles ε(*p*). (A discussion of the computation of refraction angle and impact parameter from Doppler frequency and errors due to nonsphericity can be found in the works of [*Ahmad and Tyler*, 1999; *Gorbunov et al.*, 2000]).

[37] The ray impact parameter variation is estimated as *a*Δ*n*, where Δ*n* is the horizontal variation of refractivity. For a 10 K variation of temperature and 3 g/kg variation of specific humidity across an atmospheric front, Δ*n* is about 30 N-units. The impact parameter variation is then about 200 m. This is close to a 300 m length of interval of multivalued ε(*p*) in Figure 4. In this interval multipath propagation persists also in the (*p*, ξ)-representation, and the refraction angles computed by differentiation of the phase indicate big errors. The amplitude of indicates strong interference oscillations in this area. Like in the radio-optics method, local spatial spectra of must be able to reveal multiple branches of ε(*p*).