## 1. Introduction

[2] The subject of this paper is parameterizations of transionospheric ray-tracing results giving group delay and RF carrier phase advance that, unlike simple formulas for straight-line propagation [see, e.g., *Klobuchar*, 1985], take into account refraction effects, which we categorize as ray bending. Ray bending leads to significant increases in these quantities compared with the straight-line propagation approximation at frequencies such as 30 MHz, at low elevation angles, and in the presence of significant ionization (e.g., under daytime conditions). The transmitter-receiver configuration of interest is ground-to-space. The range of elevation angles being considered in the reported work is from 5° to 90°, with emphasis on the lower values, due to greater deviations from the straight-line approximation and due also to making the parameterizations available for analyzing signals from satellite receivers whose coverage extends to the horizon. The frequency range of interest is from 20 to 100 MHz. We have not presently extended the range to lower frequencies due to the increasing likelihood of reflection that is beyond the scope of the work being reported (although results will be shown that exhibit cutoffs due to reflection).

[3] The quantities of interest from ray-tracing runs are group path lengths, phase path lengths, and straight-line distances from receiver to transmitter. The group and phase path lengths are recast as differences compared with straight-line distance and are designated by Δ*l*_{g} and Δ*l*_{p}. The effects of interest can be extracted from these variables, and by working with differences, the parameterizations need not deal with absolute distances to the receiver. The parameterizations are in the form of lookup tables of Δ*l*_{g} and Δ*l*_{p} that are presently three-dimensional (3-D) in the independent variables frequency (*f*), elevation angle (θ), and straight-line total electron content *TEC*_{SL} (in units of 10^{16} electrons m^{−2}). The motivation for using *TEC*_{SL} comes from a straight-line propagation expression in terms of this variable (see section 2). A single ionospheric parameter (in this case, *TEC*_{SL}) is not sufficient for accurately specifying the table quantities in the presence of significant ray bending, for which details in the distribution of ionization along the ray path become important. Work is under way to extend the tables to four dimensions using slab thickness (*z*_{slab}) as the fourth variable. Section 4 addresses this variable and illustrates the improvement in the specification of Δ*l*_{g} and Δ*l*_{p} by its addition.

[4] The ray trace/ICED, Bent, Gallagher (RIBG) model [*Reilly*, 1993, 1991], with improvements by Reilly up to 1996, has been used to obtain the ray-tracing results for constructing the tables. It is based on the ionospheric conductivity and electron density (ICED) profile model [*Tascione et al.*, 1988] up to 1000 km, the Bent model [*Bent et al.*, 1976] above this height, and the Gallagher plasmaspheric model [*Gallagher et al.*, 1988] at still greater heights. Upgrades of RIBG later than 1996 are described and referenced by *Reilly and Singh* [2001]. The option is available to perform ray tracing either in the absence of a magnetic field or in its presence (in either the ordinary [O] or extraordinary [X] mode). The tables being reported do not include magnetic field effects. Results will follow, however, that illustrate the differences one obtains in Δ*l*_{g} and Δ*l*_{p} by including field effects.

[5] A second ray-tracing model at our disposal is that of *Jones and Stephenson* [1975]. This particular model inputs model ionospheres in tabulated form and is thus not restricted to one particular ionospheric model. Examples of ionospheric models that can be interfaced to such a ray-tracing model are the international reference ionosphere (IRI) model [see, e.g., *Bilitza*, 1997] and the parameterized ionospheric model (PIM) [*Daniell et al.*, 1995] in addition to that used by RIBG. Experience has shown that more numerical noise is present in the Jones-Stephenson model compared with RIBG, and this is the reason for using the latter model. Three-dimensional interpolations are used in the former model to specify the plasma frequency along the ray path, whereas localized analytical techniques are employed in the latter model to ensure smoothness in this variable. Were numerical noise comparable between Jones-Stephenson and RIBG, we would not expect the derived tables to be substantially different between these models using different ionospheric models. This leads to a key point with regard to the tables: They are expected to be effectively independent of the choice of ionospheric model used to generate them. This is, in fact, precisely true with increasing frequency and elevation angle where a straight-line propagation formula can be used to specify Δ*l*_{g} and Δ*l*_{p} and is a function of only *TEC*_{SL}with regard to ionospheric effects (discussed in more detail in section 2). The only basic requirement of the ionospheric model of choice is that it approximately describes the range of behavior (here excluding turbulence leading to scintillation) associated with solar activity, location (e.g., midlatitudes), and local time. In doing so, a wide range of Δ*l*_{g} and Δ*l*_{p} behavior can be sampled as is necessary for construction of the tables.

[6] The presentation of results is as follows. The concept behind the chosen parameterization is discussed in section 2 including a formula for straight-line transionospheric propagation. Section 3 contains the bulk of our results with their characteristic scatter that arises from the current limitation to a 3-D parameterization. Our choice of a fourth independent variable, namely, slab thickness, is discussed in section 4. Magnetic field effects on Δ*l*_{g} and Δ*l*_{p} are addressed in section 5 followed by a summary in section 6.