Analysis of wave fields by Fourier integral operators and their application for radio occultations

Authors


Abstract

[1] Fourier integral operators (FIOs) are used for constructing asymptotic solutions of wave problems and for the generalization of the geometrical optics. Geometric optical rays are described by the canonical Hamilton system, which can be written in different canonical coordinates in the phase space. The theory of FIOs generalizes the formalism of canonical transforms for solving wave problems. The FIO associated with a canonical transform maps the wave field to a different representation. Mapping to the representation of ray impact parameter was used in the formulation of the canonical transform (CT) method for processing radio occultation data. The full-spectrum inversion (FSI) method can also be looked at as an FIO associated with a canonical transform of a different type. We discuss the general principles of the theory of FIOs and formulate a generalization of the CT and FSI techniques. We derive the FIO that maps radio occultation data measured along the low Earth orbiter orbit without first applying back propagation. This operator is used for the retrieval of refraction angles and atmospheric absorption. We give a closed derivation of the exact phase function of the FIO obtained in the “phase matching” approach by Jensen et al. [2004] We derive a novel FIO algorithm denoted CT2, which is a modification and improvement of FSI. We discuss the use of FIOs for asymptotic direct modeling of radio occultation data. This direct model is numerically much faster then the multiple phase screen technique. This is especially useful for simulating LEO-LEO occultations at frequencies of 10–30 GHz.

1. Introduction

[2] In this paper we discuss the application of Fourier integral operators (FIOs) for processing radio occultation data. The FIO is a technique for constructing short-wave asymptotic solutions of wave problems. The simplest short-wave asymptotic solution is geometric optics (GO) [Kravtsov and Orlov, 1990]. It has the following limitations of the applicability: (1) It cannot describe details of wave fields with small characteristic scales below the Fresnel zone size, (2) it does not work on caustics, where the GO amplitude goes to infinity. A generalization of asymptotic GO solution for quantum mechanics and theory of waves in inhomogeneous media was introduced in two equivalent forms: (1) the technique of Maslov operators and (2) FIO [Maslov and Fedoriuk, 1981; Mishchenko et al., 1990; Hörmander, 1985a, 1985b]. This technique significantly reduces the limitation due to Fresnel diffraction. In particular, the FIO asymptotic solution in a vacuum coincides with the exact solution.

[3] Processing radio occultation data posed the inverse problem: reconstruction of the GO rays from measurements of wave fields, especially in multipath zones. The simplest GO approximation, which can only be applied in single-ray areas, uses the connection between ray direction and Doppler frequency [Vorob'ev and Krasil'nikova, 1994]. The resolution of this approximation is limited by the Fresnel zone. Analysis of local spatial spectra is a simple technique that can separate multiple rays. It was applied for processing planetary occultations [Lindal et al., 1987], and it was also used for the analysis of GPS radio occultation data for the Earth's atmosphere, obtained by Microlab-1 and CHAMP satellites [e.g., Pavelyev, 1998; Igarashi et al., 2000; Pavelyev et al., 2002; Gorbunov, 2002b]. Another direction took origin from the early work by Marouf et al. [1986], who suggested the back propagation (BP) technique for the reconstruction of Saturn rings. This technique reduced the effects of diffraction at a large propagation distance. A very similar Fresnel inversion, based on the thin screen approximation, was introduced by Melbourne et al. [1994] and applied by Mortensen and Høeg [1998]. In addition, the combination of BP preprocessing with the standard GO inversion, which does not use the thin screen model, was suggested by Gorbunov and Gurvich [1998a, 1998b].

[4] The application of Fourier integral operators for processing radio occultations was suggested by Gorbunov [2002a, 2002b] in the framework of the canonical transform (CT) method. The CT method uses back propagation as the preprocessing tool, which reduces the radio occultation geometry to a vertical observation line. The CT method uses the phase space with coordinate y along the vertical line and conjugated momentum η (vertical projection of ray direction vector). In single-ray areas the momentum is equal to the derivative of the eikonal (optical path) of the wave field. The back-propagated complex wave field u as a function of vertical coordinate y is subjected to the Fourier integral operator associated with a canonical transform from (y, η) to (p, ξ), where p is the ray impact parameter, and the momentum, ξ, conjugated with it is equal to the ray direction angle. Under the assumption that each value of the impact parameter occurs not more than once, this transform allows for disentangling multiple rays. The phase of the transformed wave field can then be differentiated with respect to the impact parameter p giving the ray direction angle ξ, which is linked to refraction (bending) angle by simple geometrical relationships. The CT method provides high accuracy and resolution in the reconstruction of refraction angle profiles. The FIO used in its formulation allows for a fast numerical implementation on the basis of FFT. The disadvantage of this CT method is the necessity of BP, which is the most time-consuming part of the numerical algorithm. However, a very fast implementation of BP can be designed using a simple geometric optical approximation (S. Sokolovskiy, private communication, 2003). Also, the best implementations of BP using numerical computation of diffractive integral are reasonably fast.

[5] The full spectrum inversion (FSI) method was recently introduced by Jensen et al. [2003]. The simplest formulation of this method applies to a radio occultation with a circular geometry (i.e., spherical satellite orbits in the same vertical plane, spherical Earth and spherically symmetrical atmosphere). The Fourier transform is applied to the complete record of the complex field u(t) as function of observation time t or another parameterization of the observation trajectory such as satellite-to-satellite angle. For circular occultation geometry, the derivative of the phase, or the Doppler frequency ω, of the wave field u(t) is proportional to ray impact parameter p. We can introduce the (multivalued) dependence ω(t), which is by definition equal to Doppler frequency (or frequencies) of the ray(s) received at time moment t. In multipath zones, where the dependence ω(t) is multivalued, it cannot be found by the differentiation of the phase of the wave field u(t). Unlike ω(t), the inverse dependence t(ω) is single-valued, if we assume that each impact parameter and therefore frequency ω occurs not more than once. Using a stationary phase derivation, it can be easily shown that the derivative of the phase of the Fourier spectrum equation image(ω) equals −t(ω). Then, for each impact parameter p we can find the time t and therefore the positions of the GPS satellite and low-Earth orbiter (LEO) for which this impact parameter occurred. This allows for the computation of the corresponding refraction angle. The advantage of this method is its numerical simplicity. Its disadvantage is that it is formulated for circular radio occultation geometry and its generalization for realistic radio occultations required some approximation.

[6] Gorbunov and Lauritsen [2002] suggested a synthesis of the CT and FSI methods. They derived a general formula for the FIO applied directly to radio occultation data measured along a generic LEO trajectory without the BP procedure. The phase function of the FIO is described by a differential equation. The equation was approximately solved for a generic occultation geometry, to the first order in a small parameter measuring the deviation from a circular orbit. It was shown that for a circular occultation geometry this operator reduces to a Fourier transform. This method was thus a generalization of the FSI technique for generic observation geometry. Recently, Jensen et al. [2004] obtained the exact expression for the phase function of the FIO introduced by Gorbunov and Lauritsen [2002]. Jensen et al. [2004] also considered the amplitude function and below we shall use the synthesis of the approaches used by Jensen et al. [2004] and by Gorbunov [2002a, 2002b].

[7] The paper is organized as follows: In section 2 we give a survey of the FIO technique. We discuss the method of the Maslov operator and show that its application for the computation of the Green function results in the FIO. We also discuss the similarity between canonical transforms in the geometric optical approximation and FIOs in asymptotic approximation in wave optics. We introduce a new class of operators, FIOs of the second type (FIO2). In section 3 we discuss the application of the new FIO2 technique for processing radio occultation data. We show that the FSI method can also be classified as a specialization of the FIO2 technique, which explains the link between the FSI and CT methods. We describe a generalization of FSI for generic observation geometry and derive a new simple inversion algorithm, which can be implemented using FFT. The algorithm combines the numerical efficiency of the FSI and improved accuracy of the computation of the impact parameter. In section 4 we make a further development of the FIO technique for direct modeling of radio occultation data, which was discussed by Gorbunov [2003]. We construct a fast new algorithm for direct modeling, which can be used for moving GPS and LEO satellites (or for LEO-LEO occultations [Kursinski et al., 2002]). Section 5 presents the results of numerical modeling. In section 6 we make our conclusions.

2. Asymptotic Solutions of Wave Problems

2.1. Maslov Operator

[8] The technique of Maslov operators [Maslov, 1965] was developed for solving the Schrödinger equation in quantum mechanics and pseudo-differential or parabolic equation in the theory of waves in inhomogeneous medium. In quantum mechanics, a particle with momentum p and energy E is described by the plane wave ψ (r, t) = exp [equation image(prEt)], where h is Planck's constant. This dictates the form of operators of momentum and energy, equation image = − ihequation image, Ê = ihequation image, and the Schrödinger equation, Êψ = H(equation image, r, t)ψ, where H is the Hamilton function, which expresses the energy of a particle as function of momentum and time-spatial coordinates.

[9] Consider now wave propagation in a 2-D plane with Cartesian coordinates x, y, where the axis x is the preferable direction of wave propagation. If backscattering can be neglected then the wave problem can be reduced to a pseudo-differential or parabolic equation, where x plays the role of time. Plane waves have the following form:

equation image

where (equation image, η) is the unity direction vector of the plane wave. This identifies η as the momentum conjugated with coordinate y and the Hamilton function for a vacuum equals H = − equation image. If we describe waves in a medium with refractive index n(x, y) = 1 + N(x, y), then the Hamilton function takes the form H(η, y, x) = −equation image. In multiple phase screen modeling [Martin, 1992; Gorbunov and Gurvich, 1998a; Sokolovskiy, 2001], the following approximations are used: (1) pseudo-differential approximation, H(η, y, x) ≈ −equation imageN(x, y), and (2) parabolic approximation, H(η, y, x) ≈ −n(y, x) + η2/2. The momentum operator is then equation image, and the pseudo-differential equation is written as follows:

equation image

This equation describes direct waves propagating in the nearly x direction. It can also be derived by the operator factorization of the Helmholtz equation, which describes both direct and back scattered waves [Martin, 1992]. In order to construct a short-wave asymptotic solution of this equation, the wave field is represented in the form u(x, y) = A(x, y) exp (ikΨ (x, y)) [Kravtsov and Orlov, 1990]. If we substitute this into equation (2) and expand it with respect to powers of k−1, then the high-order terms (k0) produce the Hamilton-Jacobi equation

equation image

whose characteristic lines (geometric optical rays) are described by the following canonical Hamilton system:

equation image

The phase along a ray is described by the differential equation dΨ = η dyHdx. The geometric optical amplitude A(x, y) along a ray with initial condition y0 is equal to ϕ(x, y)/equation image, where J(x, y) = dy(x, y0)/dy0, and the function ϕ is described by the transport differential equation integrated along the ray [Mishchenko et al., 1990]. For our approximate Hamilton functions, ϕ = const along each ray, which means that energy defined as E(x) = ∫ ∣u(x, y)∣2dy is conserved. The functions ϕ and J are defined on the ray manifold, and in multipath areas ϕ (x, y) and J(x, y) are multivalued. On caustics, where J(x, y) = 0, the geometric optical amplitude goes to infinity and this solution does not work. Each ray is characterized by the coordinates (y, η) in the cross section of the phase space related to some x. Figure 1 shows an example of a ray manifold in the phase space. For the cross section of the ray manifold for x = 0 we show different types of its projection to the coordinate axes. At point y2 there are three rays, and for each of them J(x, y) ≠ 0, therefore the wave field can be found as a superposition of three wave fields in the GO approximation. At point y1 there is a caustic (y1, η1(y1)), where two rays are degenerated into one and J(x, y) = 0, and there is also a ray (y1, η2(y1)), where J(x, y) ≠ 0. In order to find the form of the asymptotic solution that works on caustics, we consider the Fourier transform of the wave field (momentum representation):

equation image

We can also represent the transformed wave field in the form equation image(x, η) = A′ (x, η)exp (ikΨ′(x, η)). It can be found asymptotically using the stationary phase method [Born and Wolf, 1964]. If ys(x, η) is the stationary phase point of the Fourier integral, then we have the following relationships:

equation image
equation image
equation image
equation image

where γ = ±π/4, and ηs(x, y) is the solution of the equation y = ys(x, η). The term γ/k in the eikonal asymptotically vanishes and it will be neglected. Equation (7) follows from the fact that at a stationary point the derivative of the phase of the integrand equals zero, and it shows that the stationary point ys(x, η) belongs to the ray manifold. Equation (8) indicates that if we use momentum as the new coordinate y′ = η and transform the wave field to the momentum representation, then the conjugated momentum is equal to η′ = −y. We can also introduce the new Hamilton function H′ (η′, y′) = H(y′, −η′) and rewrite the canonical Hamilton system for rays in exactly the same form:

equation image

From equation (6) we can also derive the equation for the eikonal dΨ′ = −ydη −Hdx = η′ dy′ − Hdx, and therefore we can also write the Hamilton-Jacobi equation for Ψ′ in the coordinates (y′, η′) with Hamilton function H′. From equation (9) we infer energy conservation when transforming to the momentum representation: A2dη = A2dy (this also explains the choice of the normalizing factor of (−ik/2π)1/2 in the Fourier transform). Using this fact and conservation of energy along the rays mentioned above, we can infer that the wave problem can be formulated in the same way in the momentum representation, i.e.,

equation image

and the asymptotic solution will have the form equation image(x, η) = exp (ikΨ′(x, η)) ϕ′ (x, η)/ equation image, where J′(x, η) = dη (x, η0)/dη0 and ϕ′ (x, η) = ϕ (x, y(x, η)), where η0 = y0 is the initial momentum (or new coordinate) of the ray.

Figure 1.

Schematic ray manifold in the phase space. The ray manifold evolves along the coordinate x. For the cross section of the ray manifold in the source plane (x = 0), we show different types of its projection to coordinate axes.

[10] This results in the following construction of the Maslov operator K (in Russian literature also referred to as the canonical operator). Given ray manifold and a normalized amplitude ϕ defined on the ray manifold the asymptotic solution Kϕ is equal to the following expression:

equation image

where Fη→y−1 is the inverse Fourier transform.

[11] Consider the ray manifold in Figure 1. At the point y1 the solution will be the sum of two components, which we denote by the points of the ray manifold: (1) component from (y1, η1(y1)) belongs to a caustic and it must be computed in the η representation and Fourier-transformed to the y representation; and (2) component from (y1, η2(y1)) can only be computed in the y representation, because it belongs to a caustic in the η representation. At the point y2 there are three components, and the first one (y2, η1(y2)) must be computed in the y representation, while the other two allow for both y and η representations.

2.2. Fourier Integral Operators

[12] Fourier integral operators arise when the Maslov operator is used for the derivation of the Green function of a wave propagation problem [Mishchenko et al., 1990]. The asymptotic solution in FIO form is based on the GO ray configuration, which defines the kernel of the FIO transforming the GO solution to a more accurate asymptotic solution. Given initial condition u0 = u(0, y) in the source plane, we will find the solution u(x, z), where we introduce another notation z for the vertical coordinate for the propagated wave in the observation plane (Figure 1). Here z is a duplicate of the y coordinate. In the further consideration, it will be necessary to consider z a different coordinate in a different space. We use the expansion u(y) = ∫ uη(y)dη, where uη(y) = equation image (0, η) exp (ikyη) are plane waves. For each plane wave the asymptotic solution can be found in the form [Kϕ] (x, z, η). Then the solution can be written in the following form:

equation image

This is the general definition of the FIO [Mishchenko et al., 1990]. If the propagation of each plane wave can be described in coordinate representation, then the corresponding FIO can be written in the following form:

equation image

Consider the situation, where the propagation of each plane wave can be described in momentum representation. Denote the momentum η′. The momentum form of each plane wave is Fy→η′uη(y) = δ(η′ − η) equation image (0, η). Its propagation is described by multiplication by the factor of a2(x, η, η′) exp (ikΣ2(x, zs(ξ), ys(η′)) − ik(zs(ξ)ξ − ys(η′)η′)), where Σ2(x, zs(ξ), ys(η′)) is the optical path in the coordinate representation, between points (0, ys(η′)) and (x, zs(ξ)). This is derived from the expression (6) for the phase of the wave field in momentum representation. Momentum ξ in the observation plane is a function of the momentum η in the source plane, ξ = ξ (η). The factor a2 will be included into the amplitude function, which will be determined below. The solution can then be represented as follows:

equation image

The main input into this integral comes from the vicinity of the stationary phase point, where z = zs(ξ (η)). This allows for canceling exp (−ik(zs(ξ)ξ)exp (ikzξ). We can also replace exp (ikys(η)η) equation image(0,η) with u(0, ys(η)) multiplied by the amplitude factor defined by (9). The integration over η can then be replaced by the integration over y (the coordinate transform is defined as y = ys(η)). For simplicity's sake we collect all the amplitude factors in the function a2(x, z, y), which will be determined below from energy conservation. The corresponding FIO takes the following form:

equation image

Here a1,2 and S1,2 are termed amplitude and phase functions of the FIOs, respectively. In the following we will refer to equation image1 and equation image2 as to FIOs of type 1 (FIO1) and type 2 (FIO2), respectively. We can introduce momenta η and ξ for waves in the source and observation planes, respectively. The equations for geometric optical rays can be written in the form z = z(y, η), ξ = ξ(y, η). For the phase functions we can write the following geometric optical equations: S1(x, z, η) = yη + Σ1(x, z, η) and S2(x, z, y) = Σ2(x, z, y), where Σ1,2 is the phase path along the ray between the points (0, y) and (x, z), and yη is the phase of the incident plane wave. Because wave fronts are normal to geometric optical rays, we can write the differential equation dΣ1,2 = ξ dz − η dy. Thus we have the following differential equations for the phase functions (note, x is fixed):

equation image
equation image

The derivation of the amplitude functions a1,2 uses the energy conservation in geometric optics. For the first type of FIO (13) we consider the stationary phase approximation:

equation image

where A′(0, η) and Ψ′(0, η) are, respectively, the amplitude and eikonal of the wave field equation image(0, η) in the momentum representation, η (x, z) is the stationary phase point determined by the equation:

equation image

Energy conservation dictates that

equation image

Here we also used the conservation of energy in the Fourier transform A2dη = A2dy, which follows from equation (9). This indicates that the amplitude function must be equal to the following expression:

equation image

Differentiating equation (19) with respect to z results in the following equation:

equation image

Replacing η with y, we can repeat the same derivation for the second type of FIO. Finally, we have the following expressions for the amplitude functions:

equation image
equation image

These expressions can also be derived from the requirement that equation image* = equation image−1, where equation image* is the conjugated operator [Egorov, 1985], which is equivalent to energy conservation.

2.3. Canonical Transforms

[13] Now we will discuss the connection between FIOs and canonical transforms. Consider the phase space with coordinate y and momentum η and assume that the dynamics is described by the Hamilton function H(η, y). We can parameterize the same phase space by another coordinate z and momentum ξ and define a new Hamilton function H′ (ξ, z) in such a way that [Arnold, 1978]

equation image
equation image

Then the same dynamics can also be described in the new coordinates with the new Hamilton function. This type of coordinate transform in the phase space is termed a canonical transform, with S1(x, z, η) and S2(x, z, y) being different types of its generating functions [Arnold, 1978]. The geometric optical solution of a wave problem can be written in different canonical coordinates. We can also ask: is it possible to reformulate the wave problem in the new phase space with the new Hamilton operator? This question was first studied by Egorov [1985] and Egorov et al. [1999], who showed that the FIO of the first type maps the wave field to the new representation. Below we show that the FIO of the second type can also be used for the same purpose. We will derive the formula for the commutation of a FIO of the second type and a pseudo-differential operator P(equation image, y), which is defined as follows P(equation image, y)u = Fη→y−1[P(η, y)equation image(η)], where P(η, y) is termed the symbol of the operator. We can write equation image2P(equation image, y) = Q(equation image, z) equation image2, where Q(equation image, z) is the representation of operator P(equation image, y) in the new coordinates and equation image = equation image is the new momentum operator. First, we compute equation image2P(equation image, y)u as follows:

equation image

Here we used the asymptotic formula: (k/2π) ∫ ∫ f(y, η)exp (ikyη) dydη = f(0, 0) [Egorov, 1985; Egorov et al., 1999], which can be easily derived by Tailor's expansion of f(y, η). Next, the operator Q(equation image, z) equation image2 can be evaluated as follows:

equation image

From this we obtain that P (−equation image, y) = Q (equation image, z). Our consideration is similar to that given by Egorov [1985], who proved that for FIOs of the first type: Q(equation image, z) = P(η, equation image). Because −equation image = η, equation image = equation image = ξ, equation image = y, we see that in both cases it follows that Q(ξ, z) = P(η (ξ, z), y(ξ, z)). If we redefine (pseudo-) differential operators in the representation of the new coordinate z and momentum ξ (this transform mixes coordinates and conjugated differential operators), then the wave function in this representation equals equation image1,2u. The wave equation can then be rewritten in the new coordinates as follows:

equation image

From this we obtain that, in agreement with equations (25) and (26): H′ = −∂S1,2/∂x + H. Therefore the technique of FIO generalizes the technique of Maslov operators. Maslov operators use a special canonical transform (z = η, ξ = −y; π/2 rotation of phase plane), while FIOs are associated with a wide class of canonical transforms. This has a big advantage, because it may allow for writing the global solution of a wave problem in a situation where the Maslov operator would require sewing together local solutions in order to obtain a global solution. Another important advantage is that FIOs can be used for solving inverse problem of reconstruction of the ray structure of wave fields, as will be discussed in section 3.

3. Processing Radio Occultations

3.1. Canonical Transform Without Back Propagation

[14] We will now consider the complex field u(y) = A(y)exp (ikΨ (y)) recorded along the observation trajectory parameterized with some coordinate y, which can be, e.g., time, arc length, or satellite-to-satellite angle θ [Jensen et al., 2003] (Figure 2). Here and in the following the dot denotes a full derivative with respect to the trajectory parameter y, e.g., equation image = dΨ/dy. For circular satellite orbits, and choosing y = θ, we obtain equation image = pequation image = p, where p is the ray impact parameter. In the momentum representation, the wave function is equation image(p) = A′ (p)exp(ikΨ′ (p)), and the derivative of its eikonal dΨ′ (p)/dp is equal to satellite-to-satellite angle θs(p) of the trajectory point, where the ray with impact parameter p was observed [Jensen et al., 2003].

Figure 2.

Radio occultation geometry. Definition of refraction angle ε, impact parameters pL and pG at LEO and GPS, respectively, and satellite-to-satellite angular separation θ. Here pL and pG are functions of impact parameter p computed from Doppler frequency shift, for a spherically symmetric atmosphere pG,L = p. LEO trajectory is shown as a dashed line. A virtual sphere is used for the computation of the amplitude. δθ is the size of the intersection of the sphere with a ray tube. dθ and drL are full variations of θ and rL along the element of the satellite trajectory inside the ray tube (shown as a bold dashed line).

[15] For generic, noncircular orbits, Jensen et al. [2003] introduced a phase correction, which allows for using the Fourier transform in the processing. Gorbunov and Lauritsen [2002] suggested processing radio occultation data by a generic Fourier integral operator (15) of the second type. Below we present the derivation of its amplitude function a2(p, y) and phase function S2(p, y). The expression for the phase function was first obtained by Jensen et al. [2004]. From equation (26) it follows that

equation image

where η (p, y) is equal to the derivative of the eikonal of the geometric optical ray with given impact parameter p observed at the trajectory point y. If y is chosen to be equal to the time t, then η = −Δω/k, where Δω is the Doppler frequency shift (we assume the time dependence of the wave field in the form A(y)exp (ikΨ (y) − iωt)). The function η (p, y) is computed using the geometrical optics. This equation will be used for the definition of the phase function S2(p, y).

[16] The derivative of the eikonal of the transformed wave field equation image2u(p) is evaluated as follows:

equation image

[17] In this derivation we used the fact that at the stationary phase point, ys(p), the derivative of the phase of the integrand, ∂(S2(p, y) + Ψ(y))/∂y, equals 0.

3.2. Phase Function

[18] The phase function can be derived using equation (30). We will use the following expression for the derivative of the phase path:

equation image

where rG and rL are GPS and LEO (or, generally, transmitter and receiver) satellite radii (Figure 2). Here and in the following, rG, rL, and θ are functions of the trajectory parameter y. The expression for the phase function [Jensen et al., 2004] can then be found by integrating η (p, y) over y for a fixed value of impact parameter p:

equation image

where one can add an arbitrary function f(p), which we take to equal 0. Refraction angle can be expressed as a function of trajectory parameter y, which defines the satellite positions, and impact parameter p, which defines the directions of emerging and incident rays:

equation image

Then we can express the derivative of the eikonal of the transformed wave field as follows [Jensen et al., 2004]:

equation image

For the reconstruction of the refraction angle profile from the measurements of the wave field along the orbit we apply the FIO2 operator (15), which produces a function equation image2u(p) = A′(p) exp(ikΨ′ (p)) of impact parameter p. The derivative of its eikonal Ψ′ (p) with a negative sign is then equal to the refraction angle ε = ε (p, ys(p)), where ys(p) is the coordinate of the trajectory point, where the ray with impact parameter p was observed (ys(p) is the stationary point of oscillating integral (15)).

[19] For a circular occultation geometry (rG = const, rL = const, equation image = const) the phase function has the form S2(p, y) = −pθ + F(p), and the FIO is reduced to the Fourier transform. In the FSI method, for perfectly circular occultation the phase function S2(p, y) is equal to −pθ. The derivative of the phase is then equal to −θ, and refraction angle can be found as function of θ and p using equation (34) (ε = θ + F′ (p)).

3.3. Amplitude Function

[20] The amplitude function a2(p, y) of the FIO does not play any role in the computation of refraction angles. For example, we could set a2(p, y) ≡ 1. However, the correct definition of the amplitude of the transformed wave field is necessary for the retrieval of atmospheric absorption [Gorbunov, 2002a; Jensen et al., 2004; M. S. Lohmann et al., manuscript in preparation, 2003]. Gorbunov [2002a] discussed the possibility of retrieving the atmospheric absorption from the amplitude of the transformed wave field in the framework of the standard CT method and it was shown that in the absence of absorption the CT amplitude is very close to a constant. Indeed, the CT amplitude describes the distribution of energy with respect to impact parameters. The transmitted energy is homogeneously distributed over spatial transmission angle. For an immovable transmitter and 2-D case the CT amplitude is proportional to (dψG/dp) 1/2, where ψG is the angle between outgoing ray and GPS radius (cf. the discussion of direct modeling in section 4). For GPS-LEO occultations, where the distance from the transmitter to the planet limb is big, it is a good approximation to assume a homogeneous distribution of transmitted energy with respect to impact parameters. For LEO-LEO occultations, the accuracy of this approximation is worse. Jensen et al. [2004] discussed normalizing the amplitude to the distribution of the transmitted energy with respect to impact parameters. Our consideration follows that given by Jensen et al. [2004] with some generalizations and refinements: we consider both 2-D and 3-D cases, use more accurate derivation of the amplitude and parameterize trajectory with coordinate y instead of θ.

[21] For the derivation of the amplitude function a2(p, y), we use energy conservation. We can write an equation similar to equation (20) for the coordinate y, amplitude A, eikonal Ψ, and amplitude function a2. For Cartesian coordinate y, we defined infinitesimal element of energy as ∣u(x, y)∣2dy. For a generic coordinate y, we must introduce a measure μ and replace dy with μdy. The measure will be used in the local formulation of energy conservation and will be understood as a function μ(p, y). Using the expression (33) for the phase function derived above we arrive at the following expression for the amplitude function:

equation image

[22] For the definition of the measure μ we consider the energy conservation in geometrical optics. We consider an infinitesimal ray tube for given satellite positions. The amplitude A is defined by the requirement that dERA2 cos ψ dS = dET, where dER is the energy received in the aperture dS, ψ is the angle between the ray tube and the normal to the aperture, dET is the transmitted energy inside the ray tube. Because the GO amplitude A only depends on rG, rL, and θ, rather than on satellite velocities, we can consider any virtual receiving aperture. If it is chosen to be an infinitesimal element of the sphere (rL,G = const), then we can write the distribution of transmitted and received energy, ET and ER, respectively [Eshleman et al., 1980; Sokolovskiy, 2000; Jensen et al., 2003, 2004; S. Leroy, unpublished manuscript, 2001].

equation image
equation image

where terms is curly brackets {…}3-D relate to the 3-D case, P is the transmitter power, ϕ is the angle of rotation around the axis from the curvature center to the transmitter, ψG,L = arcsin (p/rG,L) are the angles between the ray and satellite radii at the transmitter and receiver, pG,L = rG,L sin ψG,L are impact parameters at transmitter and receiver, A is the geometric optical amplitude of a received ray, and δθ is the virtual variation of the satellite-to-satellite angle along the sphere (rL,G = const) (Figure 2). For a spherically symmetrical atmosphere pG,L = p, for a nonspherical atmosphere p is computed from the Doppler frequency shift [Vorob'ev and Krasil'nikova, 1994], and therefore p is a function of pG and pL. Because ψL is the angle between a ray and the normal to the virtual circular orbit (rather than the real satellite orbit) it should be used in combination with the virtual infinitesimal element of the sphere δθ (cf. the use of dθ computed along the real trajectory by Jensen et al. [2003]). This allows for the definition of the measure such that (P/2π)dp = A2μdy (under the assumption of spherical symmetry):

equation image

For the definition of the virtual variation δ θ we can write:

equation image

where we used equation (34). Finally, we can write the following expression for the amplitude function:

equation image

3.4. Representation of Approximate Impact Parameter

[23] The FIO defined by the phase and amplitude functions (33) and (41), respectively, solves the problem of the extraction of refraction angles from measurements of the complex field along a satellite trajectory, directly without back propagation. However, this operator cannot be implemented as a Fourier transform and therefore its numerical implementation, especially for high frequencies such as 10–30 GHz, is slow. Here we shall describe an approximation that allows for writing the FIO2 operator in the form of a Fourier transform.

[24] Consider the measured complex field u(t) = A(t) exp (ikΨ(t)) and corresponding momentum σ = dΨ/dt. We use a FIO associated with the canonical transform from the (t, σ) to the (p, ξ) representation. The impact parameter p is a function of t, σ: p = p(t, σ). Instead of exact impact parameter we introduce its approximation equation image:

equation image
equation image

where σ0(t) is a smooth model of Doppler frequency, p0(t) = p(t, σ0(t)), and ∂p0/∂σ = equation image. We compute σ0(t) by differentiation of the eikonal with a strong smoothing over approximately 2 s time interval. We now parameterize the trajectory with the coordinate Y = Y(t), where we use the notation Y in order to distinguish between this specific choice of the trajectory coordinate and the generic coordinate y. For brevity we use the notation u(Y) instead of u(t(Y)). For the coordinate Y and the corresponding momentum η we use the following definitions:

equation image

Then, we can write the following linear canonical transform:

equation image

where we use the notation f(Y) instead of f(t(Y)). The generating function of this canonical transform is easily computed from the differential equation dS2 = ξdequation image − ηdY = −Ydequation image − (equation imagef(Y)) dY:

equation image

Using equation (32) we can find ∂σ/∂p and therefore dY:

equation image

So we can approximately write δθ/dY ≈ 1. This explains the geometrical interpretation of coordinate Y (Figure 2). Given an infinitesimal element of trajectory (shown as bold dashed line in Figure 2) inside a ray tube corresponding to the smoothed Doppler model σ0(t), dY along the element of trajectory equals δθ for the intersection of the ray tube with the sphere. An accurate expression can be easily derived using equation (40), but we do not use it, because the accuracy of the above approximation is sufficient. Because ∣∂2S2/∂equation imageY∣ = 1, the amplitude function (36) equals equation image and it can be written as follows:

equation image

The amplitude function a2(equation image, Y) in the FIO can be replaced with a2(equation image, Ys(equation image)) and factored out from within the integral, where Ys(equation image) is the stationary phase point of the FIO. The resulting operator can be written as the composition of adding a model, ikf(Y)dY, to the phase, the Fourier transform, and an amplitude factor:

equation image

The function Ys(equation image) equals −ξ, where the momentum ξ is the derivative of the eikonal of the integral term in (49). This operator maps the wave field to the representation of the approximate impact parameter equation image. Using equation (42), the exact value of σ and therefore impact parameter p(σ, t) can be found from equation image, as a function p(equation image). Practically, however, the difference between exact and approximate impact parameters, p and equation image, is negligibly small: typically it is below 1 m, and for very strong multipath conditions it may reach 5 m.

[25] The FSI method also uses an approximation of impact parameter [Jensen et al., 2004]. Then we have the following relation between the FSI approximation of impact parameter, c, and its accurate value p:

equation image

This equation signifies the identity of the Doppler frequencies expressed through the approximate impact parameter c and through the accurate impact parameter p. For the estimation of the impact parameter error, we neglected the slow movement of the GPS satellite. pm = pm(θ) is a model of impact parameter variation [Jensen et al., 2003]. In the linear approximation we can write cp = −γ (ppm), where γ = (drL/dθ)(c0/rL)/equation image. Therefore the impact parameter error is proportional to the first-order term describing the deviation of the satellite trajectory from a circle. For GPS/MET occultation orbits the characteristic values of the factor γ are 0.004–0.01. In regions with strong multipath, pc0 may reach 4 km. The worst-case impact parameter error will then be 40 m. Above we discussed the possibility of evaluating the accurate impact parameter from the approximate one. This can also be applied to the FSI method. The accurate impact parameter p can be found as a function p(c, t), provided that the approximate impact parameter c is a unique coordinate for the ray manifold. Because c is a function of both p and t, this restriction may be broken under strong multipath conditions.

[26] The introduced FIO2 mapping (49) generalizes the FSI method [Jensen et al., 2003]: FSI uses a composition of a phase model with a Fourier transform with respect to angle θ. Our definition of the coordinate Y takes into account the generic occultation geometry. This coordinate can be used for an arbitrary observation curve, and the impact parameter error is defined by the second-order terms not accounted for by the linear approximation (42). For circular orbits Y = θ. In this case, p is a linear function of Doppler frequency. Therefore, for circular orbits, the linear approximation of impact parameter, equation image, equals the accurate impact parameter p. We also use a more accurate derivation of the amplitude function a2, as discussed in section 3.3. We shall refer to this CT inversion technique based on the FIO of the second type as the CT2 algorithm.

4. Direct Modeling

[27] Fourier integral operators can also be used for asymptotic direct modeling [Gorbunov, 2003]. The FIOs equation image1,2 with amplitude functions (23) and (24) can be easily inverted: equation image1,2−1 = equation image1,2*. For example, equation image2−1 is a FIO2 with phase function S*2 (y, p) = −S2(p, y) and amplitude function a2* (y, p) = a2(p, y). If the amplitude function equals a2 = ∣μ∂2S2/∂equation imagey1/2, then the inverse operator can be approximately written with amplitude function a*2 = ∣μ−12S2/∂equation imagey1/2, because μ is a slowly changing function (the amplitude function at the stationary point can be factored out and the μ and μ−1 in the composition of the direct and inverse transform will cancel).

[28] If we use the representation of approximate impact parameter equation image, then the direct model is especially efficient. Given a 3-D atmospheric model, we first perform geometric optical modeling, and iteratively find the trajectory point Ys(equation image), where the ray with the impact parameter p(equation image) is observed. The wave function in the equation image representation is then equal to w(equation image) = A′ (equation image)exp (−ikYs(equation image)dequation image), where the amplitude A′ (equation image) equals a normalizing constant in the light zone and 0 in the geometric optical shadow. This function is then mapped into the Y representation by the inverse FIO2:

equation image
equation image

where pL and pG are functions of equation image computed for given atmosphere and satellite trajectories. For modeling atmospheric absorption, the amplitude A (equation image) must also be multiplied by a factor of exp (−knds), where n″ is the imaginary part of refractive index, and the integral is taken along the ray with the impact parameter p(equation image). A similar direct modeling algorithm can be constructed on the basis of inverting the operator used in the FSI method.

[29] This technique of direct modeling has the following limitation of applicability: the approximate impact parameter equation image must be an unique coordinate of the ray manifold. Another restriction of the asymptotic technique (for both inverse and direct modeling) is that diffraction inside the atmosphere is neglected (M. E. Gorbunov et al., Comparative analysis of radio occultation processing approaches based on Fourier integral operators, submitted to Radio Science, 2003, hereinafter referred to as Gorbunov et al., submitted manuscript, 2003).

[30] Gorbunov [2003] discussed the inversion of the standard BP + CT composition based on the FIO of the first type for direct modeling. The wave function in the p representation equals w(p) = A′(p) exp (ik ∫ ξ (p)dp), where ξ is the geometric optical momentum, and the amplitude for the 2-D case and immovable transmitter was taken to equal A0 (dψG/dp)1/2, so that A2dp is the infinitesimal element of transmitted energy. Then, equation image1−1w is equal to the wave field along a vertical line, and it can be forward propagated to the LEO orbit using the diffractive integral. The draw-backs of this algorithm are (1) the necessity of the forward propagation, which may be time-consuming for high frequencies and (2) the modeling of immovable transmitter only. The technique based on equation image2−1 suggested in this paper, is free from these disadvantages: it requires only one FFT, which is much faster than the computation of diffractive integrals, and it allows for modeling simultaneous movement of the transmitter and receiver, which is important for LEO-LEO occultations.

[31] Simulation of moving transmitter and receiver was also discussed by Mortensen et al. [1999], who used an approximation based on the composition of geometrical optics and the thin screen approximation. However, their technique proved extremely time-consuming (for each sample of simulated data, it requires geometric optical propagation from transmitter and receiver to points of an intermediate thin screen and computation of one diffractive integral). Besides, it only works above 4 km [Mortensen et al., 1999].

5. Numerical Simulations

[32] Here we compare the performance of the CT2 inversion technique introduced above and the standard composition of BP and CT techniques. For this purpose, we modeled a spherically symmetric atmosphere using a high-resolution tropical radio sonde profile of refractivity. We simulated radio occultation signals using multiple phase screens (MPS) for the standard GPS frequencies. The occultation model included an immovable transmitter and a receiver moving along a circular orbit. The MPS simulation followed the scheme described by, e.g., Gorbunov and Gurvich [1998a] and Sokolovskiy [2001]: the signal is propagated through the atmosphere using parallel phase screens, and from the last phase screen to the satellite orbit it is propagated using the diffractive integral. We used 2-D formulas for the amplitude. The screen-to-screen step was 5 km, and the integration step for geometric optical modeling was 2.5 km. The simulated radio occultation signals were processed by the two inversion algorithms and the results of the reconstruction of refraction angle profile were compared with the exact geometric optical solution. The comparison presented in Figure 3 indicates a very good agreement between both inversion algorithms and the geometric optical solution. All the differences are in small scales below 50 m, which cannot be effectively resolved for the GPS frequencies, due to diffraction inside the atmosphere (Gorbunov et al., submitted manuscript, 2003).

Figure 3.

Comparison of different modifications of the CT technique. Refraction angle profiles as functions of ray height (impact parameter minus Earth's curvature radius): (1) reference geometric optical solution (GO, solid line), (2) standard composition of BP and CT (BPCT, dotted line), and (3) CT2 algorithm (dashed line).

[33] For the validation of the asymptotic direct modeling we performed numerical simulations with a simple spherically symmetrical phantom (refractive index field model). The phantom represents an exponential model with a quasiperiodical perturbation:

equation image

where z is the height above the Earth's surface, N0 = 300 × 10−6 is the characteristic refractivity at the Earth's surface (300 N units), H = 7.5 km is the characteristic vertical scale of refractivity field, α = 0.003 is the relative magnitude of the perturbation, h = 0.3 km is the period of the perturbation, L = 3.0 km is the characteristic height of the perturbation area. This phantom was smoothly combined with the MSIS climatological model above 20 km.

[34] We simulated radio occultation signals using MPS and the asymptotic solution described in section 4 for the frequency 9.7 GHz, which is intended to be used in LEO-LEO occultations. The results of the comparison of the amplitude of the simulated wave field for these two modeling techniques are presented in Figure 4. The peculiarity of the amplitude around 28.5 s is due to the transfer from MSIS to the test phantom. Between 40 and 47.5 s the amplitude indicates large-scale oscillations reproducing the oscillations of the refractivity profile. In this area there is no multipath propagation. After 47.5 s we notice increasing small-scale scintillations due to emergence of multipath propagation. The occultation fragment from 57 to 59 s with strong multipath scintillations is enlarged and shown separately. Figure 4 illustrates a good agreement of both these simulation techniques.

Figure 4.

Validation of asymptotic direct modeling. Amplitude of simulated radio occultation signal as function of time: (1) MPS simulation (solid line) and (2) asymptotic simulation based on the FIO2 (A, dotted line).

[35] The asymptotic modeling is significantly faster than MPS modeling. One run of the asymptotic propagator took 4 min, while the MPS modeling took 2 hours on a system based on a Pentium-III processor (1 GHz). For multiple channels, the computational time for MPS simulations is proportional to the number of channels. In the asymptotic propagator, the most time-consuming part is the geometric optical modeling, which is common for all the channels. The computation of an FFT with 221 = 2097152 points, which is required for the simulation of a 22.6 GHz channel, takes 3 s. Therefore, when simulating 3 channels, the asymptotic propagator will take approximately the same time, 4 min, while an MPS simulation will require 6 hours.

[36] Figure 5 shows the geometric optical refraction angle profile and the results of the inversion of the simulated data. We present four combinations of the two simulation techniques: (1) the FIO asymptotic solution and (2) multiple phase screens; and the two inversion techniques: (1) CT2 and (2) the standard combination of BP and CT. This figure shows good agreement between the GO solution and the retrieved refractivity profile. The strongest deviations of retrieved refraction angles from the reference GO profile are observed for processing MPS simulations in the lowest 200 m. This can be accounted for by the diffraction on the Earth's surface.

Figure 5.

Refraction angle profiles from geometric optical ray tracing and from simulated radio occultation data: (1) reference GO profile (GO, solid line), (2) asymptotic simulation processed by CT2 (A-CT2, dotted line), (3) asymptotic simulation processed by BP + CT (A-BPCT, dotted line), (4) MPS simulation processed by CT2 (MPS-CT2, dashed line), and (5) MPS simulation processed by BP + CT (MPS-BPCT, dashed line).

6. Conclusions

[37] FIOs are a very efficient means of analysis of wave fields. These operators describe short-wave asymptotic solutions of wave equations. Furthermore, these operators are linked to canonical transforms because the dynamics of geometric optical rays is also described by a canonical transform. The canonical Hamilton system describing geometric optical rays can be written in different canonical coordinates (coordinate and momentum). Because momentum is associated with a differential operator, a canonical transform can also be understood as a transform of the Hamilton operator of the wave problem. The FIO associated with the canonical transform maps the wave field to the representation of the new canonical coordinates, and the wave equation can be asymptotically rewritten in the new representation. This makes this technique valuable for the inverse problem of the reconstruction of the ray manifold of a wave problem. Using impact parameter as a new coordinate, it is possible to map the wave function into the impact parameter representation. If impact parameter is a unique coordinate in the ray space, then in this representation there is no multipath and the momentum is equal to the derivative of the eikonal. The momentum is a function of ray direction or refraction angle. Previously, the technique of FIOs was applied in the composition with back propagation. We describe a FIO applied directly to radio occultation data measured along the LEO orbit, without BP. The phase function of the FIO was first found by Jensen et al. [2004]. Using a linearized canonical transform, we design the CT2 algorithm that allows a very effective numerical implementation based on a single Fourier transform. CT2 is a modification and improvement of FSI. It provides a better accuracy of impact parameter and, unlike FSI, it can be used for an arbitrary observation geometry.

[38] Another application of FIOs is the direct modeling. The asymptotic solution of the direct problem uses the mapping of the geometric optical solution in the impact parameter representation to the standard coordinate representation. The method based on the inverse CT2 is very efficient numerically because it can be implemented as the composition of the geometric optical solution and a single Fourier transform. This is important for direct modeling and processing radio occultation data at high frequencies (10–30 GHz), where the computation of diffractive integrals may be numerically very slow.

Acknowledgments

[39] The authors are grateful to A. S. Jensen and Per Høeg (Danish Meteorological Institute, Copenhagen), V. I. Klyatskin, I. G. Yakushkin, A. S. Gurvich, S. V. Sokolovskiy (Institute for Atmospheric Physics, Russian Academy of Sciences, Moscow), Y. A. Kravtsov (Space Research Institute, Moscow), and G. Kirchengast (Institute for Geophysics, Astrophysics, and Meteorology, Graz, Austria) for useful scientific discussion. One of the authors (M. E. Gorbunov) has been supported by the Russian Foundation for Fundamental research (grants 01-05-64269 and 03-05-64366). The other author (K. B. Lauritsen) has been supported by the GRAS Meteorology SAF.

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