## 1. Introduction

[2] Simulations of propagation of high-frequency waves through inhomogeneous media play pivotal roles in as diverse fields as medical tomography, seismics and geophysics, atmospheric science, microscopy, remote sensing and telecommunications, meteorology, astronomy, quantum mechanics and optics amongst many others; the instance of the problem we consider in this text, evaluation of electromagnetic-wave propagation through the atmosphere, is a centerpiece in the field of remote sensing. Much attention has centered over the last century around the high-frequency volumetric-propagation problem, and a wide range of methodologies have been developed for its treatment, focusing mainly around four general approaches: (1) geometrical optics and ray tracing [*Keller*, 1958; *Lewis and Keller*, 1995], (2) parabolic approximations [*Levy*, 2000; *Bamberger et al.*, 1988], (3) approximations based on arrays of particles [*van de Hulst*, 1981] and (4) finite difference/finite element simulations [*Sei and Symes*, 1994]. While significant understanding in many areas of science and engineering has arisen from such mathematical treatments, there still remain many important scientific problems, like the problem we consider in this paper, propagation of GPS (Global Positioning System) signals in fully three-dimensional atmospheres, for which the methodologies 1 through 4 are either inadequate or exceedingly costly. For example, methods based on the parabolic approximation [*Coles et al.*, 1995; *Martin and Flatté*, 1988; *Reilly*, 1991; *Rubio et al.*, 1999; *Shkarofsky and Nickerson*, 1982] turn out to be extremely expensive for large fully three-dimensional atmospheric configurations, and, thus, studies based on such methodologies assume two-dimensional atmospheres, i.e., atmospheres for which the refractive index is constant in the cross-range direction [*Levy*, 2000]. Similar considerations apply to the multiple phase screen (MPS) method [*Karayel and Hinson*, 1997; *Sokolovskiy*, 2001; *Ao et al.*, 2003] that is used often in GPS applications.

[3] In this contribution we focus on the problem of propagation of electromagnetic waves through the atmosphere, with specific examples drawn from GPS occultation configurations [*Karayel and Hinson*, 1997; *Sokolovskiy*, 2001; *Ao et al.*, 2003], and we present a new methodology that extends significantly the range of volumetric propagation problems that can be adequately treated by computational algorithms. This methodology, which is based on localization of Rytov integration domains to small tubes around geometrical optics paths, can accurately solve problems of propagation through realistic atmospheres in computing times that are orders-of-magnitude shorter than those required by other available algorithms. For example, the proposed algorithm can produce solutions for, say, propagation of ≈20 cm waves across hundreds of kilometers of realistic, three-dimensional atmospheres (including cross range refractivity variations) in computing times on the order of 1 hour in a present-day single-processor workstation, a task for which other algorithms would require, in such single-processor computers, computing times on the order of months or even years.

[4] To emphasize this point we provide some comparisons. We first consider [*Levy*, 2000, p. 175] where, with reference to application of the parabolic approximation to two dimensional atmospheres, we read “Some radar applications involve calculations of the electromagnetic field in very large domains, up to several hundreds of km in range and several km in height. Even larger sizes may be required for the modeling of refractive effects on Earth/space paths. As PE integration times depend on frequency, propagation angles and domain size, calculations become prohibitively expensive for such large domains.” In view of these comments we conclude that, certainly, the parabolic approximation cannot be expected to produce accurate results for the significantly more challenging fully three-dimensional atmospheres in reasonable computing times. The MPS approach [*Karayel and Hinson*, 1997; *Sokolovskiy*, 2001], in turn, has successfully been applied to two-dimensional atmospheres. In an application of this method to fully three-dimensional atmospheres, however, the computational times required by this approach would be increased by a factor equal to the number of FFT sampling points required for cross-range sampling: of the order of hundreds of thousands of points for accurate resolution of the wavelength over distances of the order of tens of kilometers [*Ao et al.*, 2003]. Clearly, an application of MPS to three-dimensional geometries would require inordinately long computing times, even in very large present-day parallel computers; in reference [*Sokolovskiy*, 2003, p. 24] we read in these regards: “Modeling of the propagation through 3-D tropospheric irregularities in RO, computationally, is very difficult”. In view of the exceedingly large computational cost required even by these specialized approaches, and since, certainly, use of standard solvers such as those based on finite difference or finite element approaches would be even more costly, one could turn to use of the classical approach based on the ray tracing (geometrical optics) approximation, which can be very fast indeed. Unfortunately, as shown in Figure 8, for example, a direct use of a ray-tracing geometrical optics can give rise to incorrect solutions for measured terrestrial atmospheric refractive index distributions. This difficulty arises as the refractive index can have sub-Fresnel scale structures that the geometrical optics approach cannot handle. As Figure 8 shows, in contrast, the effects of these structures are captured by our method without difficulty. A rationale for the failure of the geometrical optics methodology for realistic, experimentally measured atmospheres is provided in section 4, with reference to Figure 12. In all, we suggest that the methodology proposed in this paper is the first one that can successfully evaluate propagation over hundreds of kilometers of the fully three dimensional upper troposphere.

[5] The numerical method introduced in this paper is applicable to nonspherically symmetric atmospheric refractive index distributions that amount to sufficiently small perturbations from spherically symmetric smooth distributions; as shown through a variety of examples drawn from actual atmospheric data, the departures from spherical symmetry that actually occur in the Earth's atmosphere fall within the domain of applicability of the proposed methodology. One of the main elements of our algorithm is the well known Rytov approximation: we express a given atmospheric refractive index distribution *n*(**r**) as a sum *n*_{0}(*r*) + *n*_{1}(**r**) of an “unperturbed” spherically symmetric and smooth refractive index *n*_{0}(*r*) and a small “perturbation” *n*_{1}(**r**) which contains the three-dimensional variations of *n*(**r**). The character of the problem under consideration, however, is such that even its solution on the basis of Rytov's method gives rise to extremely high computational costs. We thus resort to (and extend) a high-frequency localization methodology introduced recently [*Bruno et al.*, 2004; *Bruno and Geuzaine*, 2009], which reduces computational costs in the high frequency regime through localization around the sets of points of stationary phase (which, as shown in section 2.4, actually coincide with light rays). In conjunction with this strategy, we evaluate the necessary spherically symmetric Green's function by means of geometrical optics, which is permissible in view of our assumption of smoothness of the underlying unperturbed atmosphere. The solutions of the geometrical optics problems arising from both localization and evaluation of the spherically symmetric Green's function are produced through high-order evaluation of integrals and differential operators, and, in particular, do not require use of numerical ODE solvers. Interestingly, unlike all other numerical methods applicable to this problem, the accuracy and computational costs of the proposed approach do not change as frequencies are increased. Thus, the proposed methodology will become even more attractive as it is applied to future mission designs, that propose use of much higher frequency (microwave) signals to probe the atmosphere [*Kursinski et al.*, 2002; *Gorbunov and Kirchengast*, 2007].

[6] The validity of the Rytov approximation itself as a solver for problems of wave propagation within the atmosphere has been the subject of several discussions [*Brown*, 1966, 1967; *Fried*, 1967; *Keller*, 1969]; roughly speaking, an application of the theory of *Brown* [1966] indicates an agreement with our contention of validity for a tropospheric region of the order of 1000 km in horizontal dimensions. While Brown's theory [*Brown*, 1966] is based on statistical assumptions that may be difficult to verify, the results presented in this paper show unequivocally that under the types of refractive index variations present in the upper troposphere, the Rytov approximation produces very accurate solutions for domains of several hundred kilometers in range.

[7] This paper is organized as follows: After introducing the fast localized Rytov strategy in section 2, in section 3 we present a detailed account of our computational implementation of that methodology. A variety of numerical results presented in section 4 demonstrate the applicability of the proposed approach to realistic three-dimensional configurations, as well as its accuracy and efficiency. These results clearly indicate, in particular, the appropriateness of the use of Rytov's approximation in the context of the atmospheric propagation problems we consider. Appendix A, finally, presents details of implementation for the underlying spherically symmetric geometrical optics calculations. Concluding remarks are presented in section 5.