Ray trace calculation of ionospheric propagation at lower frequencies
 The Raytrace/Ionospheric Conductivity and Electron Density–Bent-Gallagher model has been revised to make it applicable to ionospheric propagation at low radio frequencies (0.5–5.0 MHz), where the ionosphere and magnetic anisotropy drastically alter propagation paths and provide a severe test of propagation model algorithms. The necessary revisions are discussed, and the model is applied to the problem of ionospheric penetration from a source below the ionosphere to a receiver above the ionosphere. It is necessary to include the electron collision frequency in the Appleton-Hartree index of refraction in order to permit ionospheric penetration for radio frequencies below the maximum plasma frequency (e.g., whistler modes). The associated reformulation of the ray trace equations for a complex index of refraction is straightforward. Difficulties with numerical methods are cited for the lowest frequencies, and future improvements are indicated.
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 The Raytrace/Ionospheric Conductivity and Electron Density–Bent-Gallagher (RIBG) ionospheric propagation model [Reilly, 1993] adapts a ray trace propagation model [Reilly, 1991] to a global climatological ionospheric model, whose components are the Ionospheric Conductivity and Electron Density (ICED) model and the Bent and Gallagher ionospheric models. RIBG starts with the ICED model [Tascione et al., 1988] for heights 0–1000 km. ICED is based on monthly median global maps of the F2 layer maximum plasma frequency foF2 [Rush et al., 1989] and the maximum usable frequency factor M(3000)F2, which is used to determine the height at maximum density hmF2. The notation used here is the same as that of Davies , who describes the use of these maps. For heights 1000–3000 km, the fourth and fifth exponential layers of the Bent model [Bent et al., 1975] are joined continuously with the ICED model at 1000 km. At height 3000 km, the Gallagher plasmaspheric model [Gallagher et al., 1988], the update of which [Gallagher and Craven, 2000] we have not yet incorporated, is joined continuously onto the Bent model and is extended upward to a cutoff height, defined by the condition ln(ne) = 1 for electron density ne (cm–3).
 We have modified RIBG since the 1993 paper by (1) adapting it to the WGS-84 oblate Earth geometry; (2) incorporating updates of the geomagnetic reference field; (3) changing the bottomside (below hmF2) model of ICED so that superposition of Chapman height profiles for E, F1, and F2 layers does not alter the values determined for the critical densities and heights of these layers, (4) improving the method by which the Chapman topside F2 Chapman profile in ICED joins with the Elkins-Rush topside profile, and (5) joining the Gallagher plasmaspheric model continuously to the top of the Bent profile at height 3000 km. The Elkins-Rush profile has the form exp[−A tan−1 (αh)], where h is the height and A and α are constants that depend on latitude and longitude (but not height). The height derivative of the exponent defines a scale height, and joining with the F2 topside Chapman layer occurs where the two scale heights match. Otherwise, height profile segment joining is made continuous by adjusting the amplitude factor for the upper segment.
 The RIBG electron density height profile is driven by effective sunspot numbers “SF2” for foF2, “SM3” for M(3000)F2, and “SWDTH” for the width parameter of the F2 layer. The other driving parameters are “CFAC,” a constant in the denominator of the expression for parameter A in RIBG, and the usual planetary 3-hour geomagnetic activity index Kp, which primarily affects the high-latitude model in ICED. The quoted driving parameters are the parameter names we use in the RIBG program. Ideally, they can be determined from external ionospheric data in the space-time vicinity of the region of interest, whereas the index Kp is obtained from the Internet (e.g., see www.sec.noaa.gov/ftpmenu/indices/old_indices.html). For simulations of ionospheric penetration in this paper, however, we select typical driving parameters.
 RIBG obtains results for propagation parameters in either point-to-point or area coverage calculations. In the latter, rays are sent out from a point in a fan of azimuth and elevation angles to a specified height. Rays are traced through a user-specified grid of latitude-longitude points. Each grid point is associated with a set of parameters that specify the height versus electron density profile, which consists of a series of analytic height profile segments. As the ray path calculation traverses the ionospheric model grid, the electron density variation is continuous. The ray path necessarily crosses boundaries between adjacent height profile segments, but care is taken to ensure that the ray is not in any way scattered by these artificial model boundaries. Details are found in work by Reilly . The program works well for frequencies at HF or above, but we are applying it now to the low-frequency regime 0.5–4.0 MHz and ionospheric penetration from a radio source on the ground to an elevated receiver. In the process, numerical procedures are strained, and the need for modifications becomes apparent. We find we need to include the effect of collisions in the index of refraction in the low-frequency regime. Otherwise, ionospheric penetration is limited to frequencies in excess of the maximum plasma frequency along the ray path (assuming this is larger than the electron gyrofrequency). However, ionospheric penetration of low-frequency whistler modes of radio propagation is a well-known phenomenon, and we find that inclusion of the electron collision frequency gives them to us.
 Section 2 shows the derivation of the ray trace equations for the complex index of refraction. We include the derivation because it is unusual and elegant and leads directly to the results of interest. The derivation in Appendix A of Reilly  is similar to, but falls short of, the present derivation. Also included is the expression for the electron collision frequency versus height, which fits graphs given by Budden  and Davies . Section 3 discusses what the Appleton index tells us about low-frequency ionospheric penetration and then gives results of the calculations of ionospheric penetration. Section 4 discusses modifications of numerical procedures that were necessary for the calculations and will be necessary for future work. Section 5 discusses results and gives conclusions and suggestions for future work.
2. Ray Trace Equations for a Complex Index of Refraction
 Budden  shows the derivation of the Appleton formula. For electromagnetic field components with time behavior exp(iωt), we conclude from the derivation that a field component solution of the coupled Maxwell and Newton's second law equations for inhomogeneous electron plasma in an external magnetic field has the form
where we recast the exponent in terms of time as the independent variable. We use the summation convention for repeated subscripts. The wave vector magnitude satisfies k = ηω/c, where c is the speed of light in a vacuum and η = η′ − iη″ is the complex index of refraction, given by the Appleton formula as
where fN is the electron plasma frequency, f is the wave frequency, ν is the electron collision frequency, e is the electron charge (<0), m is the electron mass, Bj is a component of the Earth's magnetic field vector, and fH is the associated electron gyrofrequency. The contribution of ions to the index of refraction is ignored, which is justified for frequencies of interest here (f > 0.5 MHz). The plus or minus signs in (2a) are associated with ordinary (O) and extraordinary (X) modes of propagation. The O mode is the one that minimizes the magnetic effect for transverse magnetic field geometry (YL = 0). Consequently, the O mode is associated with the plus sign for X < 1 and with the minus sign for X > 1. The alternate sign choices are made for the X mode. For these sign choices, the index is continuous at X = 1 for each mode [cf. Budden, 1985]. This choice of signs is easy enough to see for no collision frequency and real index, but when we include the collision frequency, the method for computing the complex index can affect the choice of sign for O and X modes. This is worth checking. In the calculation of D in (2a), we first let ρ1 exp(iϕ1) = YT2/2(U − X) and compute ϕ1 = a tan 2((ν/ω), 1 − X), where the intrinsic function atan2(y, x) computes the actangent of y/x and returns the result in the interval (−π, π] in standard Fortran or C++ implementations. In this case, as X increases from 0 to 1, ϕ1 increases from a small positive value to π/2. Further increase of X causes ϕ1 to increase from π/2 toward π. Next, we compute the radicand in D by letting ρ22 exp(2iϕ2) = YL2 + ρ12 exp(2iϕ1). Now we see that as 2ϕ1 increases in the interval (0, 2π] with X, the computed value of 2ϕ2 = atan2(ρ12 sin(2ϕ1), YL2 + ρ12 cos(2ϕ1)) ranges from a small positive value when X is small up to π when X = 1. As X increases further, 2ϕ2 switches discontinuously from π to −π and then increases toward zero. Hence, because the calculation behaves this way, the result for DisD = U − ρ1 exp(iϕ1) ± ρ2 exp(iϕ2), where the phase angles increase from a small value up to π/2 as X increases from 0 to 1. However, as X increases further, ϕ1 increases from π/2 toward π, whereas ϕ2 increases from −π/2 toward 0. Consequently, the above sign switch for O and X modes is appropriate with or without the collision frequency in our computation scheme.
 A wave packet signal associated with a transmitter is associated with an integral of (1) over frequency, to proscribe the wave packet's temporal extent, and wave vector components, to proscribe its spatial extent. The phase of a wave packet component is seen to be
where we keep only the real part of the wave vector, associated with η′. The ray trace equations follow from the requirement that phase variation, subject to the Appleton formula constraint, within the wave packet be stationary in order for there to be appreciable field intensity. The stationary phase requirement δP = 0 is of a standard type treated in the calculus of variations, that is, variation subject to constraints [see, e.g., Arfken, 1985]. The result is that arbitrary dependent variable displacements between fixed end points in an integral of the form P = L(yi, dyi/dτ, τ)dτ, subject to a constraint G(yi, t) = 0, result in stationary P if and only if the integrand satisfies
where λ is an undetermined multiplier. Students of classical mechanics will recognize this as the derivation of Lagrange's equations with constraints. These are the ray trace equations we need if we make the following substitutions in the preceding formula: L = ω − k′i (dxi/dt) and yi = (ω, k′1, k′2, k′3, x1, x2, x3). Hence
The first equation solution for λ is substituted into the remaining six equations, which are then the six basic ray trace equations that are integrated in progressive ray path increment solutions from the transmitter to the receiver position in point-to-point homing calculations or to the receiver height in area coverage calculations, as explained by Reilly . One chooses the mode for the calculation (O or X) and assumes that the phase expression is preserved along the entire path. It is obvious from the way it was derived that ct is the group path length. It is straightforward to add rate equations for phase path length, deviative absorption, and other parameters of interest. We find it convenient and sufficient to choose G as
The collision frequency model we use provides a good analytic fit to the graphs given by Budden  and Davies :
where h is the height variable. Units of the ai are Hz and units of the αi are km–1.
3. Ionospheric Penetration Results
 We start with a source on the ground and an elevated receiver height, which we later set to be 1000 km in our calculations. We also assume initially that there is no electron collision frequency, that is, U = 1 in equations (2a) and (2b). First, we discuss O-mode propagation. The ray from the source enters the ionosphere, and the index decreases from 1 toward 0 in its upward progression. The ray will be refracted downward when the index reaches some value that depends on its angle of incidence at the bottom of the ionosphere. Smaller angles (more vertical) correspond to smaller values of index for ray turnabout. For frequencies less than the electron gyrofrequency (about 1.4 MHz), Y > 1, and the effect of D in (2a) can be quite large, especially for nearly longitudinal propagation, where the component YL dominates. In this case, D slows the descent of the index toward zero. Nevertheless, as X increases toward 1 (i.e., the wave works its way upward from the bottom of the ionosphere), the YT term in D eventually dominates, and in the near vicinity of X = 1, D has an X dependence given by
Hence the index necessarily passes through zero at X = 1, and there is virtually no possibility of ionospheric penetration, unless X < 1 along the entire path to the receiver. For YL > 1, (2a) gives D = 0 at X − 1 = YT2/(YL2 − 1) so that this is a singular point in η, beyond which η is large and real (the whistler region) and propagation to the receiver would be unimpeded. The ray might reach this point if the collision frequency terms could make Rl(D) > 1 at X = 1, thus enabling a ray at nearly vertical incidence to reach into the region X > 1, where it would encounter the singularity, soon if YL ≫ YT. Collisions move the singularity away from the real x axis and possibly enable whistler mode penetration of the ionosphere. Hence we would need near-vertical incidence and relatively large values of YL near X = 1. Thereafter, the whistler mode ray would bend toward maximum values of the index, which are associated with relatively large values of YL.
 Next, we assume X-mode propagation without the electron collision frequency in the low-frequency regime (f < fH). On its upward path into the ionosphere, the ray propagates freely up to the vicinity of X = 1, since D < 0 and η > 1. However, when it enters the region X > 1, the sign change makes D > 1, and then the index steadily decreases toward zero as X increases further, until ray turnabout occurs. Near to vertical incidence, rays will be refracted downward near η = 0, which occurs when X = 1 + Y. If the maximum plasma frequency along the path to the receiver is small enough to keep X to the left of this singularity, then X-mode penetration can occur. However, this limit on plasma frequency is given by fN2 < f2 + ffH ≤ 2fH2. The maximum plasma frequency is typically greater. Also, if YL < 1 and is larger than YT, the ray's upward path may first be arrested by a singularity in the X-mode index at 1 − X = YT2/(1 − YL2). The singularity attracts the ray on its high-index side. Unlike the case for the O mode, including the electron collision frequency apparently cannot help to enable X-mode penetration.
 The example we choose for a RIBG calculation of ionospheric penetration is 1 September 2004, 1900 hours (UT), SF2 = SM3 = SWDTH = 50, CFAC = 0.87, Kp = 3. Source point locations on the ground are varied on a latitude-longitude grid of points that span a latitude range of −88° to 88° in steps of 22° and a longitude range from 0° to 330° in steps of 30°. For each source point location, a fan of rays is sent out for frequencies in the range 0.5–4.0 MHz for both O modes and X modes. Ray launch angles in the fan vary in elevation from 20° to 86° in steps of 11° and in azimuth from 0° to 330° in steps of 30°, a total of 84 angles. The ray tracing is set to end at a receiver height of 1000 km, at the ground at height 0 km, or when the ground range exceeds 15,000 km. Table 1 shows penetration count for the O mode at 0.5 MHz. Numbers indicate how many of the 84 rays in each source point cell make it to the receiver height. Table 1 includes the collision frequency. Without it, no penetration was found. No entry for a source location indicates that no rays penetrated. Penetration elevation angles ranged from 42° to 86°, with heavy weighting toward higher values in this range. The corresponding O-mode results for 1.0 MHz are shown in Table 2. It is seen that only a relatively few rays penetrate at this frequency because of a smaller collision frequency influence. All but three of the rays that penetrate have the highest elevation angle, 86°. The other three have the next highest value, 75°. The O-mode results for 1.5 MHz show no penetrating rays. None of the X-mode results show penetration to the receiver height for any of the aforementioned frequencies. For 1.5 MHz and higher frequencies, it is found that ray trace programs with and without the collision frequency yield virtually identical results. There are several instances at low frequency where a ray penetrates above the E layer maximum and then is ducted between the E and F layers all the way out to the ground range cutoff limit of 15,000 km. The penetration of low-frequency O modes seems consistent with the discussion at the beginning of this section. It is surprising to see how rapidly penetration drops off between 0.5 and 1.0 MHz. Interestingly, the simulation indicates that most instances of low-frequency O-mode penetration occur in the Northern Hemisphere in the hours around midday. Penetration occurs from a complicated interplay of frequency, magnetic field, plasma frequency, propagation direction, and collision frequency. We do not yet have an adequate explanation of the hemispherical asymmetry.
Table 1. Number of Rays that Penetrate for O Mode at 0.5 MHz
|44|| || || || || || ||6||12||17||27||13||7|
|22|| || || || ||1||1||3||10||15||16||9||2|
|0|| || ||1|| || || ||2||3|| ||1||2||2|
|−22|| || || || ||1|| ||7||3||4|| ||1|| |
|−44|| || || || || || ||6||9||14||4||2||6|
|−66|| || ||3||3||2||5||8||5||6||8||6||1|
Table 2. Number of Rays That Penetrate for O Mode at 1.0 MHz
|88||2|| || || || || || || || || || || |
|66||2|| || || || || ||1||1||1||1|| || |
|44|| || || || || || || || ||2|| || || |
|22|| || || || || || || || ||1|| || || |
|0|| || || || || || || || ||1|| || || |
|−22|| || || || ||1|| || || || ||1|| || |
|−44|| || || || ||1|| || || || || || || |
|−66|| || ||1|| || || || ||1|| || || || |
|−88|| || || || ||1|| || || ||1|| || || |
 At the remaining frequencies, 2, 3, and 4 MHz, penetration shows up again at around 3 MHz, first in the nighttime and early morning Southern Hemisphere and then elsewhere as the frequency increases. These are the rays whose frequency exceeds the maximum plasma frequency along their path, and they are virtually unaffected by the collision frequency.
4. Numerical Difficulties and Modifications
 The original RIBG program traced rays through a latitude-longitude grid of height profile parameters, using triangular interpolation in between the grid points. The smallest triangle of grid points surrounding a ray trace point determined a three-term linear expansion in latitude and longitude for the interpolation of height profile parameters. The trouble with this at low frequencies was that as the definition of the grid point interpolation triangle changed, there was a small discontinuity in the plasma frequency This was particularly troublesome when the index approached zero and it would stop the calculation. We solved this by using rectangular bilinear interpolation [Press et al., 1992], involving interpolation from four surrounding grid points. Even when the definition of the grid point interpolation rectangle changes, the plasma frequency variation is continuous.
 Of the three numerical fourth-order integration options in RIBG [Reilly, 1991], which are the Runge-Kutta, Runge-Kutta-Fehlberg, and Adams predictor-corrector (for variable interval sizes) methods [Burden and Faires, 2005], the Runge-Kutta-Fehlberg method with error control seemed to work best at low frequencies. Our use of this method requires that we specify the number of significant figures of accuracy desired in a ray path increment solution. This implies a local truncation error bound, which limits the size of the independent time increment value. We also specify a lower limit on the length of the ray path increment that we can tolerate before we stop the integration. We used lower limits in the range 10–8 km. At the lower end of the frequencies of our study (0.5, 1.0, and 1.5 MHz), the calculation was frequently terminated by the lower bound on ray path increment size. In particular, this occurred near the index singularities mentioned in section 3. As expected, the rays bent toward the associated high-index regions. As a consequence, the calculations for penetrating O modes also stopped. Apparently, the index variations for these cases cannot be accurately handled by conventional numerical integration techniques. We are investigating stiff differential equation techniques as a way to solve this problem. We did find that the ray trace programs, with or without collisions, do not experience this difficulty and do yield virtually identical results in the range of frequencies above the whistler mode regime (>2 MHz), which defines the lower limit of validity of our present ray-tracing programs with respect to numerical accuracy of results. A better numerical integration technique may enable us to extend the range of validity of our RIBG program with collision frequency down to about 0.5 MHz.
 We need to explain how we obtained the tabular penetration results in section 3. We simply forced the ray trace equations not to terminate on the condition for lower limit on the ray path increment size. When the ray path increment decreases below the lower limit, we set it equal to the lower limit and push on with the calculation. We realize that we are compromising numerical accuracy, but we weigh the significance of the results by how invariant and stable they are to variations of the lower-limit value on ray path increments. The results of section 3 pass this test. We found that we had to vary the ray path increment limit for different source locations. For example, a lower limit of 10−3 km worked well for most source locations, yielding stable results in a fairly reasonable amount of time. However, in order to get the calculation not to “hang up” at some of the lower latitudes, we had to increase the lower limit several fold for these locations. Finding the right lower limit and checking stability was a lengthy process. We estimate from this process that the numbers in Table 1 are correct with a possible count error of 1 or 2 in a few entries, and we similarly estimate that the entries in Table 2 are correct, except possibly in a relatively few cases. This is not enough to invalidate our qualitative statements about whistler O-mode penetration in the presence of the electron collision frequency.
5. Discussion and Conclusions
 RIBG without the collision frequency predicts the propagation of radiation from a ground transmitter to a receiver above the maximum electron plasma frequency fN max of the ionosphere only when
for the wave frequencies of the O mode and X mode, respectively, unless we have the unlikely case that fN max ≤ (2)1/2fH, in which case low-frequency (fx < fH) X-mode radiation may propagate to the receiver. On the other hand, inclusion of the collision frequency in RIBG moves the troublesome singularities in the index that occur for low frequency off the real X axis and permits ionospheric penetration of O modes, especially in the frequency regime f < 1 MHz. The RIBG program with collisions agrees with RIBG without collisions and both are now numerically accurate and trouble-free (no breakdowns) down to about 2 MHz, a frequency that exceeds the whistler mode regime (f < fH). In the whistler mode regime, however, RIBG with collisions would be used, except it is not numerically accurate and trouble-free, because the numerical integration routine in use is not sufficiently capable of handling the wild variations near index singularities that occur. Hopefully, an appropriate stiff differential equation solver will make if possible to extend the numerically accurate and trouble-free frequency region of RIBG with collisions down to about 0.5 MHz.
 The identification of whistlers with the O mode is certainly not new. For example, the index of refraction for whistler modes, given as equation (10.2) by Davies , is associated with the O-mode sign choice in our equations (2a) and (2b). More explicitly, whistlers are identified with the O mode in Figure 4.6 and near the beginning of section 13.8 of Budden . Both authors state that thunderstorms are a source for whistlers that propagate to the topside ionosphere. We show that the propagation from source to end point can occur in a relatively benign model ionosphere as an O mode, when collisions are included.
 The RIBG ionospheric propagation model does not contain irregularities or mode interactions, which may promote the penetration of whistler O and X modes. If the boundaries of an irregularity are sufficiently sharp, we may get conversion or scattering of the X mode in the region X > 1 to an O mode, which can subsequently propagate unencumbered up to satellite height. On the other hand, if the region of the irregularity encompasses the satellite and is substantially depleted of electrons, we may get the condition X < 1 all the way to the satellite, which would enable the detection of O-mode radiation and especially enable X-mode detection at an elevated satellite receiver.
 The plan is to extend the propagation model to include mode conversion at both low and high frequencies. Mode conversion for frequencies in excess of about 50 MHz is known to occur when a ray passes through a transverse propagation condition (YL ≈ 0) [Cohen, 1960].