[1] The Radon Transform plays a central role in the image reconstruction technique known as computed tomography, used commonly in radio astronomy and medical imaging. Although usually formulated as a projection of a spatial density function along straight ray paths, the Radon Transform kernel also permits curved path projections, providing the path can be defined. Reformulation of the Radon Transform as a path integral for the case of a radio ray refracting in a spherically symmetric atmosphere leads directly to the Abel Transform formulation commonly used in atmospheric radio occultation.

[2] Radio occultation is a technique which, among other applications, is employed to probe planetary atmospheres in search of their physical properties. Considering, for convenience, only the neutral atmosphere, radio waves propagating within the gas are refracted and simultaneously retarded in phase. The degree of bending that occurs depends on the gas refractivity, which is controlled by gas composition and density, and reflects the atmospheric structure that the wave encounters along the propagation path. The refractivity profile ν(r) of the atmosphere under study is retrieved by processing of radio occultation data, subject to the assumption of local conformance to spherical symmetry. For a known or assumed composition, plus the additional strong assumption of hydrostatic equilibrium, atmospheric pressure and temperature profiles local to the experimental ray path are deduced from the refractivity data. The refractivity profile normally is retrieved from preprocessed occultation data through the use of the Abel Transform. The details of this procedure may be found in Fjeldbo et al. [1971], Eshleman [1973], Tyler [1987], Karayel and Hinson [1997], and Ahmad and Tyler [1998].

[3] The Radon Transform (RT), broadly employed in computed tomography (CT), is a more general reduction tool than the Abel Transform (AT) since spherical symmetry is not required. The RT is formulated for straight ray paths. In radio occultation, however, the functional argument to the AT is modified to accommodate curved ray paths. When the RT is reformulated for curved ray paths in a spherically symmetric system, it reduces to the form of the AT familiar in radio occultation studies.

2. Atmospheric Radio Occultation

[4] The geometry of an atmospheric radio occultation measurement is shown in Figure 1. Radio waves from the transmitter, T, which is moving in the frame of the planet with a velocity _{T}(t), propagate along an initial path defined by _{T}. The waves arrive along unit vector −_{R} at the receiver, R, moving with velocity _{R}(t). Refraction by the neutral atmosphere perturbs the path of radio rays, bending them towards the center of the refractivity field, as shown in Figure 1. In actual atmospheres, both positive and negative refractivity can occur, corresponding to the neutral atmospheric gas and the ionosphere, respectively, but the same analysis applies to both. In the ionosphere, the refractivity gradient can be either positive or negative.

[5] The basic observable of radio occultation is the perturbation of signal frequency associated with the Doppler shift that occurs as a result of the refractive bending of a ray linking a moving transmitter and receiver as the ray passes through an intervening atmosphere. Doppler shift f_{D}(t) is induced in the received signal by the combination of the relative motion of T and R with respect to the center of the planet O, and by the bending of the rays in the planet's atmosphere [Ahmad and Tyler, 1999]. For the classical result,

where the time dependence of the quantities in (1) is omitted to simplify the notation. The geometric solution of this system is unique under the assumption of spherical symmetry, which constrains the ray to lie in the plane defined by the positions of the transmitter, receiver, and the center of refractive symmetry, which is notionally the center of the planet. With this assumption and knowledge of the positions and velocities of the transmitter and receiver, observation of f_{D} is sufficient to determine the vectors _{T} and _{R} uniquely, and the total bending angle α and ray asymptote closest approach distance a, or impact parameter, found.

[6] An occultation experiment proceeds by measuring the ray parameters α and a at many points throughout the planet or moon's atmosphere—from the top of the sensible atmosphere, where the first bending of the ray is detected, to the point where the ray path intersects the surface of the occulting planet/moon, or is otherwise undetectable due to atmospheric refraction, absorption, or defocusing. A sequence of such measurements yields a profile of the bending angle α(a) spanning the measurable limits of the atmosphere. Temperature and pressure profiles are derived from the measurement of α(a) according to [e.g., Fjeldbo et al., 1971],

where T(r) is temperature, p(r) is atmospheric pressure, is the mean molecular mass, k is Boltzmann's constant, g(r) is the acceleration due to gravity, n_{t}(r) is the total number density of neutral atoms and molecules, ν(r) is the refractivity of the atmosphere, and r_{o} is the closest approach of the ray to the planet's surface, as shown in Figures 1 and 2. The refractivity is related to the refractive index μ(r) according to ν(r) = μ(r) − 1. The refractive index is retrieved directly from the measured bending angle α via Abelian transformation, as discussed below.

3. Applicability of the Abel Transform to the Radio Occultation Technique

[7]Fjeldbo et al. [1971] showed that determination of the refractive index profile μ(r) from α(a) is a special case of Abelian inversion. Consider the geometry of a radio ray, refracting as it propagates, as depicted in Figure 2, where the view is in the plane of the ray. The center of the coordinate system is located at the center of the planet under study, which is taken to be the center of symmetry.

[8]Appendix A shows that the relationship of the bending angle α as a function of the ray asymptote a is given by,

Fjeldbo et al. [1971] showed that since = , (7) can be solved by Abel inversion to find a solution for the refractive index as a function of the impact parameter,

4. Radon Transform

[9] The Radon Transform can be thought of as a generalization of the Abel Transform, in instances where the requirement of spherical symmetry is removed. The Radon Transform, f(R, θ), is defined as the set of all straight line integrals through a spatial function h(x, y) at some perpendicular distance R from the origin of the coordinate system [Bracewell, 2000],

The argument of the Dirac delta function in the integrand defines a line as illustrated in Figure 3. The distance from the origin to the intercept P is calculated using the Pythagorean theorem,

The variables x and y can also be expressed in terms of sines and cosines,

With a slight modification:

we can substitute (12) into (10) to obtain the equation of the line,

A zero of the argument to δ(·) in (9) occurs when a point (x, y) lies on the line,

[10] For fixed values of θ and R (θ_{1} and R_{1}, say), then the integral f(R_{1}, θ_{1}) represents the projection of the density function h(x, y) along the line L = L_{θ}. The equation of the scan line L_{θ} is sometimes expressed in a more standard y = mx + b form,

Note that in Figure 3, the line L has been drawn with a negative slope (i.e. m → −m).

5. Radon and Abel Transform Equivalence for Radio Occultation

[11] In the RT, (9) and (15) above, the argument of the Dirac delta defines a straight line. Physically, a straight line scan is a mathematical approximation to a thin pencil beam in the CT technique used in imaging, and used by Bracewell to image radio galaxies [Bracewell, 1956].

[12] In medical CT, individual pencil beams are arranged to form a fan of beams that sweeps around a patient. Detectors are placed opposite the fan such that each pencil beam is aligned with an individual detector. The function h(x,y) is a measure of the spatial absorption and/or scattering of the pencil beam energy at the frequency of transmission. Images are reconstructed by applying an inverse RT—most commonly one of several approximations to this inverse—to recover h(x, y) from the data, yielding f(R, θ).

[13] For the case of ray propagation through a refractive atmosphere, there is some deviation between the actual ray path and a straight line path. In this instance, the argument of the Dirac delta must be reformulated to define the curved path of a ray refracting as it propagates in the atmosphere.

[14] Let us modify the Radon Transform (9)—using imprecise language for the moment—to allow for a curved path,

where the argument p of the Dirac delta defines a curved path in (x, y). The variable R is a ray path-related parameter that remains undefined for the moment. Reformulation in polar coordinates gives,

Assuming that the field h(r, θ) is spherically symmetric, h(r, θ) → h(r). With this,

The projection operation performed by the Dirac delta in the integral (18) is equivalent to integration along the path defined by p(r, θ). Thus, in instances for which a differential ray path element dl can be defined, we can rewrite (18) as follows,

where dl is a small element along the path.

[15] An element dl along a curved radio ray path in a refracting atmosphere can be calculated by analysis similar to that of section 3. Referring to Figure 2, the path element dl for the case of a spherically-symmetric refracting medium is given by,

Substituting dθ as previously defined in (A11) into (20) and simplifying yields,

Thus, the θ-dependance of the ray path is removed from the path integral. Incorporating the assumption of a symmetric ray path and the result (21) allows (19) to be rewritten,

[16] In section 3, the specific form of g(x) defined in (6), combined with the substitution x = μr transforms the general form of the AT, (5), into specific forms used in atmospheric radio occultation, (4), (7). Using (6) in (22) by applying the substitution h(r) = μ^{−1}g(r), and simplifying yields the following,

[17] A comparison of Figures 1 and 3 shows that the variable R, which has been defined to this point as a ray-path parameter, is actually the impact parameter. Accordingly, R has been replaced by a in (23). Equation (23), found by exchanging the Dirac delta kernel of the Radon Transform for the path integral of a refracting ray, is identical to the Abel integral (4) which resulted from first principles analysis of refraction in a spherically symmetric medium, where f(a) = α(a).

6. Discussion and Conclusion

[18]Ahmad and Tyler [1998] identified the substitution x = μr as a transformation between straight and curved ray paths, for the purpose of computing α, μ, and a. Fjeldbo et al. [1971] employed this substitution to invert (4) via standard Abelian inversion, yielding a closed-form expression for refractive index as a function of impact parameter, (8). No approximations are introduced into the procedure as a result of these substitutions. For both the AT and the path integral derived from the RT, a connection to the bending angle formula (4) exists via the functions g(r) and μ^{−1}g(r), respectively, which transform the Abel and Radon Transform kernels from straight ray to curved ray path formulations.

[19] Normally, the ray path in the RT kernel is a straight line, but use of a refractive path is not precluded by the mathematics. The Dirac delta kernel of the RT performs a sampling operation on the spatial density function h(r, θ), effectively projecting h(r, θ) along a path for which the argument of the Dirac delta is zero. In computed tomography, this is the path of a ray propagating through the medium under study. Formulating the RT as a path integral defines a curved ray path in the Radon kernel, with the path element dl prescribed by the geometry of a radio ray propagating in a spherically symmetric refractive medium. The resulting equation, (23), is the same as the specific form of the AT, (4)x, commonly used in atmospheric radio occultation.

Appendix A

[20] Following Fjeldbo et al. [1971], suppose that the distance r to a ray changes by the distance dr as the position vector to the ray sweeps through the angle d. The local inclination of the ray, ξ, can be expressed in terms of,

The relationship between a, μ, and r is obtained from Bouguer's rule [Born and Wolf, 1999],

where a is the impact parameter discussed above, and depicted in Figure 1. It is implicit in (A2) that μ = μ(r) and ξ = ξ(r). By inspection, the three angles θ, ξ, and ψ sum to ,

therefore taking the derivative yields,

or,

We would like to develop expressions for dθ and dξ to use in (A5), thus deriving an expression for dψ that we can integrate to yield the total bending angle α. We begin this task by differentiating Bouguer's rule (A2) with respect to r and solving for dξ,

Armed with expressions for dθ and dξ, we combine (A9) and (A11) in (A5) to obtain an expression for dψ,

A final simplification of terms gives an expression for dψ in terms of a, μ, and r,

Integration of the quantity dψ along the entire ray path yields the total bending angle α as shown in (4). α is conventionally defined as positive for bending towards the center of the planet/moon, as shown in Figure 1. The impact parameter a is related to the closest approach distance r_{o} by Bouguer's rule, (A2), with ξ = π/2, thus a = μr_{o} (also see Figure 2).