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

Transcription

1 RADIO SCIENCE, VOL. 39,, doi:1.129/23rs2971, 24 Analysis of wave fields by Fourier integral operators and their application for radio occultations M. E. Gorbunov 1 and K. B. Lauritsen Danish Meteorological Institute, Copenhagen, Denmark Received 8 September 23; revised 1 May 24; accepted 9 June 24; published 19 August 24. [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. [24] 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 1 3 GHz. INDEX TERMS: 326 Mathematical Geophysics: Inverse theory; 336 Meteorology and Atmospheric Dynamics: Remote sensing; 6969 Radio Science: Remote sensing; KEYWORDS: Fourier integral operator, radio occultations, forward modeling Citation: Gorbunov, M. E., and K. B. Lauritsen (24), Analysis of wave fields by Fourier integral operators and their application for radio occultations, Radio Sci., 39,, doi:1.129/23rs 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, 199]. 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 1 Also at Institute for Atmospheric Physics, Moscow, Russia. Copyright 24 by the American Geophysical Union /4/23RS2971 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., 199; 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 1of15

2 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., 2; Pavelyev et al., 22; Gorbunov, 22b]. 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 [22a, 22b] 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 h (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, h)to(p, x), where p is the ray impact parameter, and the momentum, x, 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 x, 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, 23). 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. [23]. 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-tosatellite angle. For circular occultation geometry, the derivative of the phase, or the Doppler frequency w, of the wave field u(t) is proportional to ray impact parameter p. We can introduce the (multivalued) dependence w(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 w(t) is multivalued, it cannot be found by the differentiation of the phase of the wave field u(t). Unlike w(t), the inverse dependence t(w) is single-valued, if we assume that each impact parameter and therefore frequency w 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 ~u(w) equals t(w). 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 [22] 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. [24] obtained the exact expression for the phase function of the FIO introduced by Gorbunov and Lauritsen [22]. Jensen et al. [24] also considered the amplitude function and below we shall use the synthesis of the approaches used by Jensen et al. [24] and by Gorbunov [22a, 22b]. [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 2of15

3 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 [23]. We construct a fast new algorithm for direct modeling, which can be used for moving GPS and LEO satellites (or foeo-leo occultations [Kursinski et al., 22]). 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 y (r, t) = exp [ i h (pr Et)], where h is Planck s constant. This dictates the form of operators of momentum and energy, ^p Ê and the Schrödinger equation, Êy = H(^p, r, t)y, 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: h p ux; ð yþ ¼ exp ik h y þ ffiffiffiffiffiffiffiffiffiffiffiffiffi i 1 h 2 x ; ð1þ pffiffiffiffiffiffiffiffiffiffiffiffiffi where ( 1 h 2, h) is the unity direction vector of the plane wave. This identifies h as the momentum conjugated with coordinate y and pthe Hamilton function for a vacuum equals H = ffiffiffiffiffiffiffiffiffiffiffiffiffi 1 h 2.Ifwe describe waves in a medium with refractive index n(x, y) = 1 +N(x, y), pthen the Hamilton function takes the form H(h, y, x) = ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi n 2 h 2. In multiple phase screen modeling [Martin, 1992; Gorbunov and Gurvich, 1998a; Sokolovskiy, 21], the following approximations are used: p (1) pseudo-differential approximation, H(h, y, x) ffiffiffiffiffiffiffiffiffiffiffiffiffi 1 h 2 N(x, y), and (2) parabolic approximation, H(h, y, x) n( y, x) +h 2 /2. The momentum is then ^h ¼ 1 and the pseudo-differential equation is written as follows: u ¼ Hð^h; y; xþu: ð2þ 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 (ik (x, y)) [Kravtsov and Orlov, 199]. If we substitute this into equation (2) and expand it with respect to powers of k 1, then the high-order terms (k ) produce the ; y; x ; ð3þ whose characteristic lines (geometric optical rays) are described by the following canonical Hamilton system: dy ; dh : ð4þ The phase along a ray is described by the differential equation d = h dy H dx. The geometric optical amplitude A(x, y) palong ffiffiffiffiffiffiffiffiffiffiffiffiffi a ray with initial condition y is equal to j(x, y)/ Jðx; yþ, where J(x, y) =dy(x, y )/dy, and the function j is described by the transport differential equation integrated along the ray [Mishchenko et al., 199]. For our approximate Hamilton functions, j = const along each ray, which means that energy defined as E(x) = R ju(x, y)j 2 dy is conserved. The functions j and J are defined on the ray manifold, and in multipath areas j (x, y) and J(x, y) are multivalued. On caustics, where J(x, y) =, the geometric optical amplitude goes to infinity and this solution does not work. Each ray is characterized by the coordinates (y, h) 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 = we show different types of its projection to the coordinate axes. At point y 2 there are three rays, and for each of them J(x, y) 6¼, therefore the wave field can be found as a superposition of three wave fields in the GO approximation. At point y 1 there is a caustic ( y 1, h 1 ( y 1 )), where two rays are degenerated into one and J(x, y) =, and there is also a ray (y 1, h 2 (y 1 )), where J(x, y) 6¼. 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): rffiffiffiffiffiffiffiffi ik ~uðx; hþ ¼ F y!h u ¼ ux; ð yþexpð ikyhþdy: ð5þ 3of15

4 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 = ), we show different types of its projection to coordinate axes. We can also represent the transformed wave field in the form ~u(x, h) =A (x, h)exp (ik (x, h)). It can be found asymptotically using the stationary phase method [Born and Wolf, 1964]. If y s (x, h) is the stationary phase point of the Fourier integral, then we have the following relationships: ðx; hþ ¼ ðx; y s ðx; hþþ y s ðx; hþh þ g k ðx; hþ ys ðx;hþ ðx; ¼ h; y¼ys ðx;hþ ð7þ! h y s ðx; hþ ¼ y s ðx; hþ; ð8þ A ðx; hþ ¼ Ax; ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ð 2 y¼y s ðx;hþ ¼ Ax; ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ð yþ y¼y s ðx;hþ s ffiffiffiffiffiffiffiffiffiffiffiffiffiffi ¼ Ax; ð y s ðx; ð9þ 4of15 where g = ±p/4, and h s (x, y) is the solution of the equation y = y s (x, h). The term g/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 y s (x, h) belongs to the ray manifold. Equation (8) indicates that if we use momentum as the new coordinate y = h and transform the wave field to the momentum representation, then the conjugated momentum is equal to h = y. We can also introduce the new Hamilton function H (h, y )=H( y, h ) and rewrite the canonical Hamilton system for rays in exactly the same form: dy ; dh : From equation (6) we can also derive the equation for the eikonal d = y dh H dx= h dy H dx, and therefore we can also write the Hamilton-Jacobi equation for in the coordinates ( y, h ) with Hamilton function H. From equation (9) we infer energy conservation when transforming to the momentum representation: A 2 dh = A 2 dy (this also explains the choice of the normalizing factor of ( ik/) 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., ~u ¼ H ð^h ; y ; xþ~u; ð1þ and the asymptotic solution pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi will have the form ~u(x, h) = exp (ik (x, h)) j (x, h)/ J ðx; hþ, where J (x, h) =dh (x, h )/dh and j (x, h) =j (x, y(x, h)), where h = y is the initial momentum (or new coordinate) of the ray. [1] 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 j defined on the ray manifold the asymptotic solution Kj is equal to the following expression: 8 expðik ðx; yþþjðx; yþ >< p ffiffiffiffiffiffiffiffiffiffiffiffiffi ; if Jðx; yþ 6¼ ; Jðx; yþ Kj ¼ Fh!y 1 expðik ðx; hþþj ðx; hþ >: p ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ; if J ðx; hþ 6¼ ; J ðx; hþ ð11þ 1 where F h!y is the inverse Fourier transform. [11] Consider the ray manifold in Figure 1. At the point y 1 the solution will be the sum of two components, which we denote by the points of the ray manifold: (1) component from ( y 1, h 1 ( y 1 )) belongs to a caustic and

5 it must be computed in the h representation and Fouriertransformed to the y representation; and (2) component from (y 1, h 2 (y 1 )) can only be computed in the y representation, because it belongs to a caustic in the h representation. At the point y 2 there are three components, and the first one (y 2, h 1 (y 2 )) must be computed in the y representation, while the other two allow for both y and h representations 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., 199]. 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 u = u(, 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) = R pffiffiffiffiffiffiffiffiffiffiffi u h ( y)dh, where u h ( y) = ik=~u (, h) exp (ikyh) are plane waves. For each plane wave the asymptotic solution can be found in the form [Kj] (x, z, h). Then the solution can be written in the following form: ux; ð zþ ¼ rffiffiffiffiffi ik ½Kj Šðx; z; hþ~u ð; hþdh: ð12þ This is the general definition of the FIO [Mishchenko et al., 199]. If the propagation of each plane wave can be described in coordinate representation, then the corresponding FIO can be written in the following form: ux; ð zþ ¼ ^F 1 u rffiffiffiffiffi ik ¼ a 1 ðx; z; h ÞexpðikS 1 ðx; z; hþ Þ~u ð; hþdh: ð13þ Consider the situation, where the propagation of each plane wave can be described in momentum representation. Denote the momentum h. The momentum form of each plane wave is F y!h u h ( y) =d(h h)~u(, h). Its propagation is described by multiplication by the factor of a 2(x, h, h ) exp (iks 2 (x, z s (x), y s (h )) ik(z s (x)x y s (h )h )), where S 2 (x, z s (x), y s (h )) is the optical path in the coordinate representation, between points (, y s (h )) and (x, z s (x)). This is derived from the expression (6) for the phase of the wave field in momentum representation. Momentum x in the observation plane is a function of the momentum h in the source plane, x = x (h). The factor a 2 will be included into the amplitude function, which will 5of15 be determined below. The solution can then be represented as follows: rffiffiffiffiffiffiffiffi ik ux; ð zþ ¼ a 2 x; h; h ð Þexp ð iks 2ðx; z s ðþ; x y s ðhþþ ikðz s ðþx x y s ðhþhþþexpðikzxþ~u ð; hþdh ð14þ The main input into this integral comes from the vicinity of the stationary phase point, where z = z s (x (h)). This allows for canceling exp( ik(z s (x)x) exp(ikzx). We can also replace exp(iky s (h)h)~u(,h) with u(, y s (h)) multiplied by the amplitude factor defined by (9). The integration over h can then be replaced by the integration over y (the coordinate transform is defined as y = y s (h)). For simplicity s sake we collect all the amplitude factors in the function a 2 (x, z, y), which will be determined below from energy conservation. The corresponding FIO takes the following form: ux; ð zþ ¼ ^F 2 u rffiffiffiffiffiffiffiffi ik ¼ a 2 ðx; z; y ÞexpðikS 2 ðx; z; yþ Þuð; yþdy: ð15þ Here a 1,2 and S 1,2 are termed amplitude and phase functions of the FIOs, respectively. In the following we will refer to ^F 1 and ^F 2 as to FIOs of type 1 (FIO1) and type 2 (FIO2), respectively. We can introduce momenta h and x for waves in the source and observation planes, respectively. The equations for geometric optical rays can be written in the form z = z( y, h), x = x( y, h). For the phase functions we can write the following geometric optical equations: S 1 (x, z, h) = yh + S 1 (x, z, h) and S 2 (x, z, y)=s 2 (x, z, y), where S 1,2 is the phase path along the ray between the points (, y) and (x, z), and yh is the phase of the incident plane wave. Because wave fronts are normal to geometric optical rays, we can write the differential equation ds 1,2 = x dz h dy. Thus we have the following differential equations for the phase functions (note, x is fixed): ds 1 ¼ x dz þ ydh; ds 2 ¼ x dz h dy: ð16þ ð17þ The derivation of the amplitude functions a 1,2 uses the energy conservation in geometric optics. For the first type of FIO (13) we consider the stationary phase approximation: ^F 1 u ¼ a 1ðx; z; hþa ð; hþexp½ikðs 1 ðx; z; hþþ ð; hþþš sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi S 1 ðx; z; ð; 2 h¼hðx;zþ ð18þ

6 where A (, h) and (, h) are, respectively, the amplitude and eikonal of the wave field ~u(, h) inthe momentum representation, h (x, z) is the stationary phase point determined by the 1 ðx; z; hþ ¼ : ð19þ h¼hðx;zþ Energy conservation dictates that ja 1 ðx; z; hþa ð; hþj S 1 ðx; z; ð; 2 h¼hðx;zþ dz ¼ ja ð; hðx; zþþj 2 jdhðx; zþj: ð2þ Here we also used the conservation of energy in the Fourier transform A 2 dh = A 2 dy, which follows from equation (9). This indicates that the amplitude function must be equal to the following expression: ja 1 ðx; z; hþj 2 S 1 ðx; z; ð; 2 h¼hðx;z dhðx; zþ Þ dz : ð21þ Differentiating equation (19) with respect to z results in the following 2 S 1 ðx; z; 2 ð; 2 h¼hðx;zþ dh x; z ð Þ S 1 ðx; z; : ð22þ h¼hðx;zþ Replacing h with y, we can repeat the same derivation for the second type of FIO. Finally, we have the following expressions for the amplitude functions: s a 1 ðx; z; hþ ¼ 2 S 1 ðx; ; ð23þ a 2 ðx; z; yþ ¼ 2 S 2 ðx; : ð24þ These expressions can also be derived from the requirement that ^F* = ^F 1, where ^F* is the conjugated operator [Egorov, 1985], which is equivalent to energy conservation Canonical Transforms [13] Now we will discuss the connection between FIOs and canonical transforms. Consider the phase space with coordinate y and momentum h and assume that the dynamics is described by the Hamilton function H(h, y). We can parameterize the same phase space by another coordinate z and momentum x and define a new Hamilton function H (x, z) in such a way that [Arnold, 1978] ds 1 ¼ x dz þ ydh ðh HÞdx; ð25þ ds 2 ¼ x dz h dy ðh HÞdx: ð26þ 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 S 1 (x, z, h) and S 2 (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(^h, y), which is defined as 1 follows P(^h, y)u = F h!y [P(h, y)~u(h)], where P(h, y) is termed the symbol of the operator. We can write ^F 2 P(^h, y) = Q(^x, z) ^F2, where Q(^x, z) is the representation of operator P(^h, y) in the new coordinates and ^x = is the new momentum operator. First, we compute ^F 2 P(^h, y)u as follows: rffiffiffiffiffi ik ^F 2 Pð^h; yþu ¼ a 2 ðx; z; yþexpðiks 2 ðx; z; yþþ k Pðh; yþ expðikðy y ÞhÞuðÞdydh dy rffiffiffiffiffi ik k ¼ Pðh; yþ exp ikðy y Þ h a 2 ðx; z; y ÞexpðikS 2 ðx; z; y Þ rffiffiffiffiffi ik ¼ ; y expðiks 2 ðx; z; y ÞÞuðy Þdy: dy dh a 2 ðx; z; y Þ ÞuðÞdy ð27þ Here we used the asymptotic formula: (k/) RR f ( y, h) exp(ikyh) dy dh = f (, ) [Egorov, 1985; Egorov et al., 1999], which can be easily derived by Tailor s expansion 6of15

7 of f ( y, h). Next, the operator Q(^x, z) ^F 2 can be evaluated as follows: rffiffiffiffiffi Q ^x; ik z ^F2 u ¼ ; z a 2 ðx; z; yþ expðiks 2 ðx; z; yþþuy ð Þdy: ð28þ From this we obtain that P y) =Q (@S z). Our consideration is similar to that given by Egorov [1985], who proved that for FIOs of the first type: z) = ). = = = y, we see that in both cases it follows that Q(x, z) =P(h (x, z), y(x, z)). If we redefine (pseudo-) differential operators in the representation of the new coordinate z and momentum x (this transform mixes coordinates and conjugated differential operators), then the wave function in this representation equals ^F 1,2 u. The wave equation can then be rewritten in the new coordinates as follows: H ^x; z; x ^F1;2 u ¼ ^F 1;2 u ^F 1;2 u þ ^F 1;2 u ^F 1;2 u þ ^F 1;2 Hð^h; y; xþu ^F 1;2 u þ H h ^x; z ; y ^x; z ; x ^F1;2 u: ð29þ From this we obtain that, in agreement with equations (25) and (26): H 1,2 /@x + H. Therefore the technique of FIO generalizes the technique of Maslov operators. Maslov operators use a special canonical transform (z = h, x = y; p/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 Processing Radio Occultations 3.1. Canonical Transform Without Back Propagation [14] We will now consider the complex field u( y) = A( y)exp (ik ( y)) recorded along the observation trajectory parameterized with some coordinate y, which can be, e.g., time, arc length, or satellite-to-satellite angle q Figure 2. Radio occultation geometry. Definition of refraction angle, impact parameters p L and p G at LEO and GPS, respectively, and satellite-to-satellite angular separation q. Here p L and p G are functions of impact parameter p computed from Doppler frequency shift, for a spherically symmetric atmosphere p G,L = p. LEO trajectory is shown as a dashed line. A virtual sphere is used for the computation of the amplitude. dq is the size of the intersection of the sphere with a ray tube. dq and d are full variations of q and along the element of the satellite trajectory inside the ray tube (shown as a bold dashed line). [Jensen et al., 23] (Figure 2). Here and in the following the dot denotes a full derivative with respect to the trajectory parameter y, e.g., _ = d /dy. For circular satellite orbits, and choosing y = q, we obtain _ = p q _ = p, where p is the ray impact parameter. In the momentum representation, the wave function is ~u( p)=a ( p) exp(ik ( p)), and the derivative of its eikonal d ( p)/dp is equal to satellite-to-satellite angle q s ( p) of the trajectory point, where the ray with impact parameter p was observed [Jensen et al., 23]. [15] For generic, noncircular orbits, Jensen et al. [23] introduced a phase correction, which allows for using the Fourier transform in the processing. Gorbunov and Lauritsen [22] 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 a 2 ( p, y) and phase function S 2 ( p, y). The expression for the phase function was first obtained by Jensen et al. [24]. From equation (26) it follows that 2 ðp; ¼ hðp; yþ; ð3þ where h ( 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. Ify is chosen to be equal to the time t, then h = Dw/k, where Dw is the Doppler frequency shift (we assume the time dependence of the wave field in the form A( y) exp(ik ( y) iwt)).

8 The function h ( p, y) is computed using the geometrical optics. This equation will be used for the definition of the phase function S 2 ( p, y). [16] The derivative of the eikonal of the transformed wave field ^F 2 u( p) is evaluated as follows: d ðpþ ¼ ds ð 2ðp; y s Þþ ðy s ÞÞ dp dp 2ðp; yþ þ dy ðs 2 ðp; yþþ y¼y s ðpþ 2ðp; yþ ¼ xðp; y s ðpþþ: y¼ys ðpþ [17] In this derivation we used the fact that at the stationary phase point, y s ( p), the derivative of the phase of the 2 ( p, y) + ( y))/@y, equals Phase Function [18] The phase function can be derived using equation (3). We will use the following expression for the derivative of the phase path: _ ¼ hðp; y Þ ¼ p q _ þ _r qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi G p2 r G r 2 G þ _ qffiffiffiffiffiffiffiffiffiffiffiffiffiffi p2; ð32þ where r G and are GPS and LEO (or, generally, transmitter and receiver) satellite radii (Figure 2). Here and in the following, r G,, and q are functions of the trajectory parameter y. The expression for the phase function [Jensen et al., 24] can then be found by integrating h( p, y) over y for a fixed value of impact parameter p: S 2 ðp; yþ ¼ p q _ þ _r qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi G rg 2 p2 þ _r qffiffiffiffiffiffiffiffiffiffiffiffiffiffi L rl 2 p2 dy r G ¼ pdq þ dr qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi G rg 2 p2 þ dr qffiffiffiffiffiffiffiffiffiffiffiffiffiffi L rl 2 p2 r G qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ¼ pq rg 2 p2 þ p arccos p qffiffiffiffiffiffiffiffiffiffiffiffiffiffi rl 2 p2 r G þ p arccos p ; r 2 L ð33þ where one can add an arbitrary function f ( p), which we take to equal. 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: ðp; yþ ¼ q arccos p arccos p : ð34þ r G Then we can express the derivative of the eikonal of the transformed wave field as follows [Jensen et al., 24]: xðp; yþ ¼ q þ arccos p þ arccos p ¼ ðp; yþ: ð35þ r G 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 ^F 2 u(p) =A ( p) exp(ik ( p)) of impact parameter p. The derivative of its eikonal ( p) with a negative sign is then equal to the refraction angle = ( p, y s ( p)), where y s ( p)is the coordinate of the trajectory point, where the ray with impact parameter p was observed ( y s ( p) is the stationary point of oscillating integral (15)). [19] For a circular occultation geometry (r G = const, = const, q _ = const) the phase function has the form S 2 (p, y) = pq + F(p), and the FIO is reduced to the Fourier transform. In the FSI method, for perfectly circular occultation the phase function S 2 (p, y) is equal to pq. The derivative of the phase is then equal to q, and refraction angle can be found as function of q and p using equation (34) ( = q + F (p)). 8of Amplitude Function [2] The amplitude function a 2 ( p, y) of the FIO does not play any role in the computation of refraction angles. For example, we could set a 2 ( p, y) 1. However, the correct definition of the amplitude of the transformed wave field is necessary for the retrieval of atmospheric absorption [Gorbunov, 22a; Jensen et al., 24; M. S. Lohmann et al., manuscript in preparation, 23]. Gorbunov [22a] 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 (dy G /dp) 1/2, where y G 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. FoEO-LEO occultations, the accuracy of this approximation is worse. Jensen et al. [24] 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. [24] 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 q. [21] For the derivation of the amplitude function a 2 ( p, y), we use energy conservation. We can write an equation similar to equation (2) for the coordinate y, amplitude A, eikonal, and amplitude function a 2. For Cartesian coordinate y, we defined infinitesimal element

9 of energy as ju(x, y)j 2 dy. For a generic coordinate y, we must introduce a measure m and replace dy with mdy. The measure will be used in the local formulation of energy conservation and will be understood as a function m( p, y). Using the expression (33) for the phase function derived above we arrive at the following expression for the amplitude function: sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi a 2 ðp; yþ ¼ S 2 ¼ m q _ _r G p pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi _ p 1=2 pffiffiffiffiffiffiffiffiffiffiffiffiffiffi : r G rg 2 p2 rl 2 p2 ð36þ [22] For the definition of the measure m 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 de R A 2 cos y ds = de T, where de R is the energy received in the aperture ds, y is the angle between the ray tube and the normal to the aperture, de T is the transmitted energy inside the ray tube. Because the GO amplitude A only depends on r G,, and q, rather than on satellite velocities, we can consider any virtual receiving aperture. If it is chosen to be an infinitesimal element of the sphere (,G = const), then we can write the distribution of transmitted and received energy, E T and E R, respectively [Eshleman et al., 198; Sokolovskiy, 2; Jensen et al., 23, 24; S. Leroy, unpublished manuscript, 21]. de T ¼ P dy 1 G 2 sin y Gdf ¼ P 1 1 p G pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi rg 2 df p2 2 r G G de R ¼ A 2 1 cos y L 2 sin qdf qffiffiffiffiffiffiffiffiffiffiffiffiffiffi ¼ A 2 rl 2 1 p2 L 2 sin qdf dp G dp dp; dq dq dy dy; ð37þ ð38þ where terms is curly brackets {...} 3-D relate to the 3-D case, P is the transmitter power, j is the angle of rotation around the axis from the curvature center to the transmitter, y G,L = arcsin (p/r G,L ) are the angles between the ray and satellite radii at the transmitter and receiver, p G,L = r G,L sin y G,L are impact parameters at transmitter and receiver, A is the geometric optical amplitude of a received ray, and dq is the virtual variation of the satellite-to-satellite angle along the sphere (,G = const) (Figure 2). For a spherically symmetrical atmosphere 9of15 p G,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 p G and p L. Because y L 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 dq (cf. the use of dq computed along the real trajectory by Jensen et al. [23]). This allows for the definition of the measure such that (P/)dp = A 2 mdy (under the assumption of spherical symmetry): qffiffiffiffiffiffiffiffiffiffiffiffiffiffiqffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi m ¼ p2 p2 r 2 L r 2 G r G p sin q dq dy : ð39þ For the definition of the virtual variation dqwe can write: dq ¼ dr G p ¼ dq dr G p pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi d p p ffiffiffiffiffiffiffiffiffiffiffiffiffiffi ; ð4þ r G rg 2 p2 rl 2 p2 where we used equation (34). Finally, we can write the following expression for the amplitude function: qffiffiffiffiffiffiffiffiffiffiffiffiffiffiqffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi a 2 ðp; yþ ¼ rl 2 p2 rg 2 r 1=2 G p2 p sin q q _ _r G p pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi _ p pffiffiffiffiffiffiffiffiffiffiffiffiffiffi : ð41þ r G rg 2 p2 rl 2 p 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 1 3 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 (ik (t)) and corresponding momentum s = d /dt. We use a FIO associated with the canonical transform from the (t, s) tothe(p, x) representation. The impact parameter p is a function of t, s: p = p(t, s). Instead of exact impact parameter we introduce its approximation ~p: ~pðt; sþ ¼ p ðþþ s s ðþ t Þ ¼ fðþþ s;

10 fðþ¼p t s ðþ t ¼ p dq dt dr G dt r G p p ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi r 2 G p2 d dt! 1 p pffiffiffiffiffiffiffiffiffiffiffiffiffiffi rl 2 s ; p2 ð43þ where s (t) is a smooth model of Doppler frequency, p (t) = p(t, s (t)), /@s s=s (t). We compute s (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 h we use the following definitions: dy 1 dt dt; h s: Then, we can write the following linear canonical transform: ~p ¼ fðyþþh; x ¼ Y ; ð45þ 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 ds 2 = x d~p h dy = Y d~p ( ~p f ( y)) dy: S 2 ð~p; Y Y Þ ¼ ~py þ fðy ÞdY : ð46þ Using equation (32) we can and therefore dy: dy ¼ dq dr G p pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi r G rg 2 d p pffiffiffiffiffiffiffiffiffiffiffiffiffiffi p2 r L rl 2 dq: ð47þ p2 So we can approximately write dq/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 s (t), dy along the element of trajectory equals dq for the intersection of the ray tube with the sphere. An accurate expression can be easily derived using equation (4), but we do not use it, because the accuracy of the above approximation is sufficient. Because j@ 2 S 2 /@~p@yj =1,the pffiffiffi amplitude function (36) equals m and it can be written as follows: qffiffiffiffiffiffiffiffiffiffiffiffiffiffiqffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi a 2 ð~p; Y Þ ¼ rl 2 ~p2 rg 2 r G sin q 1=2 ~p2 : ~p ð48þ The amplitude function a 2 ( ~p, Y ) in the FIO can be replaced with a 2 ( ~p, Y s ( ~p)) and factored out from within the integral, where Y s ( ~p) is the stationary phase point of the FIO. The resulting operator can be written as the composition of adding a model, ik R f ( y)dy, to the phase, the Fourier transform, and an amplitude factor: rffiffiffiffiffiffiffiffi ik ^F 2 u ð~p Þ ¼ a 2 ð~p; Y s ð~p ÞÞ expð ik~py Þ 1 Y exp@ ik fðy ÞdY AuðYÞdY : ð49þ The function Y s ( ~p) equals x, where the momentum x 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 ~p. Using equation (42), the exact value of s and therefore impact parameter p(s, t) can be found from ~p, as a function p( ~p). Practically, however, the difference between exact and approximate impact parameters, p and ~p, 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., 24]. Then we have the following relation between the FSI approximation of impact parameter, c, and its accurate value p: c _ q þ _ q ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi r 2 L p2 m ¼ p _ q þ _ qffiffiffiffiffiffiffiffiffiffiffiffiffiffi rl 2 p2 ð5þ 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. p m = p m (q) is a model of impact parameter variation [Jensen et al., 23]. In the linear approximation we canpwrite ffiffiffiffiffiffiffiffiffiffiffiffiffiffi c p = g ( p p m ), where g =(d /dq)(c / )/ rl 2 c2. 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 g are.4.1. In regions with strong multipath, p c may reach 4 km. The worst-case impact parameter error will then be 4 m. Above we discussed the possibility of evaluating the accurate impact parameter from the approximate one. This can also be applied to the FSI 1 of 15

11 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., 23]: FSI uses a composition of a phase model with a Fourier transform with respect to angle q. 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 = q. In this case, p is a linear function of Doppler frequency. Therefore, for circular orbits, the linear approximation of impact parameter, ~p, equals the accurate impact parameter p. We also use a more accurate derivation of the amplitude function a 2, 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, 23]. The FIOs ^F 1,2 with amplitude functions (23) and (24) can be easily inverted: ^F1,2 1 = ^F 1,2 *. For example, ^F2 1 is a FIO2 with phase function S* 2 ( y, p) = S 2 ( p, y) and amplitude function a 2 *( y, p) =a 2 ( p, y). If the amplitude function equals a 2 = jm@ 2 S 2 /@~p@yj 1/2, then the inverse operator can be approximately written with amplitude function a* 2 = jm 2 S 2 /@~p@yj 1/2, because m is a slowly changing function (the amplitude function at the stationary point can be factored out and the m and m 1 in the composition of the direct and inverse transform will cancel). [28] If we use the representation of approximate impact parameter ~p, 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 Y s ( ~p), where the ray with the impact parameter p( ~p) is observed. The wave function in the ~p representation is then equal to w( ~p) =A ( ~p)exp ( ik R Y s ( ~p)d~p), where the amplitude A ( ~p) equals a normalizing constant in the light zone and in the geometric optical shadow. This function is then mapped into the Y representation by the inverse FIO2: rffiffiffiffiffi 1 ik Y uy ð Þ ¼ exp@ ik fðy ÞdY A expðik~py Þa* 2 ðy s ð~p Þ; ~p Þw ð~p Þd~p; ð51þ a* 2 ðy; ~p Þ ¼ 1 p pffiffiffiffiffiffiffiffiffiffiffiffiffiffi G rl 2 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi p2 L rg 2 p2 r G L r G sin q! 1=2 dp G ; d~p ð52þ where p L and p G are functions of ~p computed for given atmosphere and satellite trajectories. For modeling atmospheric absorption, the amplitude A (~p) must also be multiplied by a factor of exp ( k R n ds), where n is the imaginary part of refractive index, and the integral is taken along the ray with the impact parameter p( ~p). 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 ~p 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, 23, hereinafter referred to as Gorbunov et al., submitted manuscript, 23). [3] Gorbunov [23] 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 R x ( p)dp), where x is the geometric optical momentum, and the amplitude for the 2-D case and immovable transmitter was taken to equal A (d G/dp) 1/2, so that A 2 dp is the infinitesimal element of transmitted energy. Then, ^F1 1 w 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 drawbacks of this algorithm are (1) the necessity of the forward propagation, which may be time-consuming for high frequencies and (2) the modeling of immov- ^F 2 1 able transmitter only. The technique based on 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 foeo- 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 diffrac- 11 of 15