Full-wave reflection of lightning long-wave radio pulses from the ionospheric D region: Numerical model



[1] A model is developed for calculating ionospheric reflection of electromagnetic pulses emitted by lightning, with most energy in the long-wave spectral region (f ∼ 3–100 kHz). The building block of the calculation is a differential equation full-wave solution of Maxwell's equations for the complex reflection of individual plane waves incident from below, by the anisotropic, dissipative, diffuse dielectric profile of the lower ionosphere. This full-wave solution is then put into a summation over plane waves in an angular direct Fourier transform to obtain the reflection properties of curved wavefronts. This step models also the diffraction effects of long-wave ionospheric reflections observed at short or medium range (∼200–500 km). The calculation can be done with any arbitrary but smooth dielectric profile versus altitude. For an initial test, this article uses the classic D region exponential profiles of electron density and collision rate given by Volland. With even these simple profiles, our model of full-wave reflection of curved wavefronts captures some of the basic attributes of observed reflected waveforms recorded with the Los Alamos Sferic Array. A follow-on article will present a detailed comparison with data in order to retrieve ionospheric parameters.

1. Introduction

[2] In the low-frequency (LF; 30–300 kHz) and especially in the very low frequency (VLF; 3–30 kHz) spectrum, the relatively long wavelengths dictate a full-wave approach to the calculation of many problems in ionospheric radio reflection. Various full-wave numerical methods were developed in the 1950s, and with the arrival of more capable computers early in the 1960s, several workers attempted to calculate full-wave reflection of LF/VLF plane waves from the underside of the lower ionosphere (D region). Their aim was to compare theory with observations for single-frequency, continuous-wave (CW) radio links between fixed stations, such as between Rugby and Cambridge UK. A particularly elegant and successful technical approach was that of Pitteway [1965] (hereinafter referred to as P65), and it was applied to explaining a variety of LF/VLF ionospheric data [Piggott et al., 1965].

[3] VLF propagation at very long ranges (thousands of km) has been treated elsewhere by a sophisticated waveguide-mode long-wave propagation model [see Pappert and Ferguson, 1986, and references therein]. The waveguide approach has been successful for modeling global-scale propagation of narrow-band beacon signals [McRae and Thomson, 2000; Thomson, 1993; Thomson and Clilverd, 2001; Thomson et al., 2004, 2005]. For intermediate ranges, authors have used the waveguide-propagation model with a truncated order of modes to predict the spectrum of the propagated field, when the radio source is the broadband emissions from lightning [Cheng and Cummer, 2005; Cheng et al., 2006; Cummer, 1997; Cummer et al., 1998]. This approach using broadband emissions is interpreted using a waveguide-propagation model framework. By contrast, the single-reflection model we are applying here is useful only for sufficiently short ranges (hundreds of km) where the sole ionospheric effect is a single discrete reflection. This simpler framework will aid in elucidating the physical origin of various reflection effects, and will be more accurate for short-range reflections.

[4] A careful and extremely exact approach to calculating ionospheric reflection has been incrementally developed over a few decades by Isamu Nagano and his colleagues [Nagano et al., 1975, 2003; Xiang-Yang et al., 1996]. They divide the D region into a stack of thin layers, within which the ionospheric parameters are constant, and invoke the Fresnel reflection conditions at each interface between adjacent layers. The calculation of the wavefield everywhere, for a plane wave incident from below, is provided by a matrix method rather than a differential equation solver. This offers advantages such as being amenable to parallel-computing methods [Nagano et al., 2003]. During the last two decades the Nagano team has incorporated their matrix-based plane wave solution into a solution for spherical waves, using an exact expansion for spherical waves in terms of a sum over plane waves [Stratton, 1941].

[5] Altogether different from the linear, spectral methods just mentioned, is the time domain finite difference approach. In this latter method the electromagnetic wave-propagation problem is solved as differential equations in both time and space, not just space as in our approach [Cho and Rycroft, 1998; Hu and Cummer, 2006]. Moreover, in the time domain approach, nonlinearities can be included, by modifying the medium's collision rate and electron density in time as the electric fields heat electrons and ionize further neutrals. For modeling the fast responses of the D region to electromagnetic heating in the lightning VLF radiation, it is clear that only the nonlinear time domain approach is suitable. Adapting linear spectral methods to this nonlinear problem inevitably requires ad hoc steps in modifying the medium [see Nagano et al., 2003, section 3 and Figure 5].

[6] Our single-reflection model is motivated by the need to model a wealth of recent VLF-LF reflection findings [Jacobson et al., 2007b, 2008] from the Los Alamos Sferic Array, or LASA [Shao et al., 2006; Smith et al., 2002]. Our technical approach for studying ionospheric reflections has been based on the recording of short-pulse VLF-LF LASA signals emitted by the unique lightning stroke known as an “NBE,” or Narrow Bipolar Event [Jacobson, 2003a; Jacobson and Light, 2003; Smith et al., 1999]. The NBE main pulse is fast (∼10-μs width) and powerful (peak effective radiated power in range 109–1011 W). The NBE emission provides a transmitter of opportunity for high-time-resolution time-delayed reflectometry (TDR) from the ionosphere. NBEs are a common (but not ubiquitous) intracloud discharge in many electrified storms [Jacobson et al., 2007a; Jacobson and Heavner, 2005; Suszcynsky and Heavner, 2003; Wiens et al., 2008]. The NBE emission rate from an active thunderstorm can be as high as several per minute, although such a high rate is unusual [Wiens et al., 2008]. Recording of the NBE ground wave, followed by both the ionospheric and the ground-then-ionospheric echoes, allows retrieval of the source height and “ionospheric reflector” height [Smith et al., 2004], for lightning-to-receiver ranges 200–800 km.

2. Background Profiles

[7] The ionospheric D region is characterized by diffuse profiles of electron-collision rate and of electron density, the two variables most relevant to modeling VLF/LF propagation. The diffuse rather than sharp profile renders artificial any sharp-boundary, Fresnel-reflection approach. Moreover, the profile's gradient lengths are not long compared to a VLF wave's radian wavelength. Thus, a full-wave reflection model is required, because the ray optics assumption is not satisfied. For initial presentation of the model and its main features, this article confines itself to a simple D region profile, variations of which have been used over several decades by the VLF-propagation community. A subsequent article will examine departures from the standard profile, in a process of ionospheric parameter retrieval based on detailed comparison of modeled and observed reflected waveforms.

[8] In our plane wave calculation which serves as the computational building block, the ionospheric variables (electron density ne, electron-neutral collision rate νen) vary versus only the vertical coordinate z. We ignore ion mobility and ion inertia, as the studied frequency range (f ≥ 2 kHz) lies above both the ion gyrofrequency and the ion plasma frequency. The only particles that contribute significantly to the complex dielectric are the free electrons. The neutrals play a secondary role as a momentum and energy sink, via electron-neutral collisions. Model profiles for both electron density ne and electron-neutral collision rate νen are each exponential in z [Wait and Spies, 1964] and have become standard “unperturbed” profile parametrizations for D region studies [Cheng and Cummer, 2005; Cheng et al., 2006; Cummer, 1997; Cummer et al., 1998; McRae and Thomson, 2000; Thomson, 1993; Thomson and Clilverd, 2001; Thomson et al., 2004, 2005]. We will use these standard profiles with slight modification. In Volland's notation [see Volland, 1995, equation (3.23), section 3.2.3],

equation image
equation image

with n0 = 3 × 108 m−3, p = 0.15 km−1 (daytime) or 0.35 km−1 (nighttime), hp = 70 km (daytime) or 85 km (nighttime), ν0 = 5 × 106 s−1, hq = 70 km, and q = 0.15 km−1. Thus the electron-neutral collision rate is the same both night and day. By contrast, the electron density profile differs markedly between day and night, extending lower in altitude and being shallower in slope during the day than during night. The difference between the Volland profiles and the more commonly used Wait and Spies profile is that Volland uses a reference density n0 = 3 × 108 m−3, while Wait and Spies use n0 ∼ 4 × 108 m−3. We immerse the reflection region in a constant vector magnetic field lying in the y-z plane, pointing obliquely below horizontal (the y axis) by a dip angle I.

[9] In this study we slightly modify the standard Volland profile to provide additional calculational convenience. First, at E region altitudes, we force the electron density to plateau at a capping value 1011 m−3, in cases where the Volland exponential profile would exceed that value. The transition to the capping value is smoothed with a tanh function in the vicinity of the transition, and thus remains analytic. This has no sensible effect on our numerical solutions but does make it easier to impose a radiation boundary condition above the D region. Second, below 50 km, we transition to a steeper falloff of ne(z), in order to minimize model conductivity at stratospheric heights. For practical purposes the propagation is indistinguishable from vacuum-like at heights near and below 50 km anyway, so this modification, also, has no sensible effect on the solutions.

[10] Figure 1 shows the modified-Volland background profiles as functions of height z, with a logarithmic ordinate scale. For Figure 1, we choose the cyclical wave frequency f = ω/(2π) = 10 kHz. We choose the cyclical electron cyclotron frequency fce = ωce/(2π) = 1300 kHz (as is correct for Florida circa 1999–2002). The thick solid curve is (ωpe/ω)2 for midday, while the thin solid line is (ωpe/ω)2 for midnight, where ωpe is the electron plasma radian frequency. The altitude domain is z = 25 to 105 km. The bottom dashed horizontal line is the level where (ωpe/ω)2 = 1. The left and center asterisks on this bottom dashed line indicate the daytime and nighttime altitudes at which (ωpe/ω)2 = 1. The diagonal dashed line shows the scaled collisionality νen/ω, and its intercept with the bottom horizontal line, marked by the right asterisk, occurs where the wave radian frequency equals the collision rate. The top dashed horizontal line shows ωce/ω, which we approximate (with negligible error) to be constant throughout the reflection region.

Figure 1.

Modified profiles of Volland for D region electron density and electron-neutral collision rate, with logarithmic vertical scale. Descending dashed line is the collision rate divided by the wave frequency, same at all local times. Solid lines are proportional to electron density for midday (thick line) and midnight (light line). Top horizontal dashed line is the electron gyrofrequency divided by wave frequency. Bottom horizontal dashed line lies at unity. Left and center asterisks show where electron density passes critical level for wave frequency f = 10 kHz. Right asterisk shows where collision rate falls below ω = 2πf.

[11] Figure 1 shows that the collision rate exceeds the typical VLF wave radian frequency in much of the D region. Thus, electron collisions tend to dominate over electron inertia in the electron equation of motion, so that electrostatic plasma oscillations are deeply suppressed. It does not follow, however, that the dielectric anisotropy is suppressed by the collisionality. The diagonal dashed curve in Figure 1 indicates that the radian gyrofrequency exceeds the collision rate above z ∼ 65 km. Thus for much of the reflection region, the electrons gyrate faster than they collide with neutrals, and the dielectric tensor will therefore be somewhat anisotropic. This anisotropy survives despite the electron collisionality, at least with Volland's canonical parameter values for νen in equation (1b).

3. Geometrical Approach

[12] Figure 2 shows the schematic geometry in the plane of incidence. A source transmitter dipole is located at P1 and is oriented along the local vertical. The source dipole is elevated by a height Hs above the local ground. It radiates spherical wavefronts, whose constant-phase surfaces are shown as concentric circles. A receiver dipole is located on the ground at P2 and is aligned along the local vertical. The transmitter-to-receiver distance along the Earth's surface is ρ. The Earth is shown in gray shading.

Figure 2.

Geometry of propagation. Source P1 is an elevated dipole at height Hs above the Earth. Spherical wavefronts depart from the dipole, with dipole amplitude pattern whose null is at local zenith. Receiver P2 is a dipole at ground level, similarly with a dipole gain pattern whose null is at local zenith. The great circle path between P1 and P2 is ρ. We define a rectilinear coordinate system (s, p, z) with z along the local vertical at the midpoint C of the great circle path, p along the local horizontal at C, and s out of the page at C. A volume of D region roughly above C is responsible for the wave reflection. The ionosphere varies with geocentric radius but here will be approximated in the calculation as varying only versus z in the entire reflection volume. In the (s, p, z) system, the source is at z = zS, while the receiver is at z = −zR.

[13] If the propagation could be described by ray optics, which it cannot, there would be a single “specular” ray that propagates (via ionospheric reflection) to the receiver. If the reflection were sharp boundary, which it is not, the reflection would occur entirely at a discrete altitude Hi. Since we can rely neither on ray optics nor on a sharp-boundary reflection, we must deal with the implications of curved wavefronts within a diffuse dielectric profile. Without any approximations, this would lead to a very complicated, computationally burdensome, nonsymmetric calculation. The radius of curvature of the spherical wavelets encountering the ionosphere is only on the order of a few times Hi, which lies in the range 70–90 km, while the ionospheric radius of curvature is on the order of the geocentric radius Re + Hi, or ∼ 6450 km. These two radii are widely mismatched, breaking a potential spherical symmetry that would allow treatment by a spherical-harmonic expansion of the transmitted field.

[14] We reiterate that the matrix method of Isamu Nagano and colleagues [Nagano et al., 1975, 2003; Xiang-Yang et al., 1996] solves the spherical-wave problem exactly, using a stack of ionospheric layers within each of which the plasma density and collision rate are held constant. In contrast, our work retains the Pitteway approach of a differential equation solution and then, to represent wavefront curvature, uses a more approximate, and computationally less burdensome, approach than that of Nagano.

[15] To regain some simplicity while retaining the key physics, we now impose two geometrical approximations:

[16] 1. Planar ionosphere: The large radius of curvature of the ionosphere, compared to the radius of curvature of the wavefronts anywhere in our problem, suggests a convenient approximation: At the center “C” of the P1-to-P2 arc (see Figure 2), we define a right-hand Cartesian coordinate system (z, p, s), with z along the vertical at C, p in the plane of incidence and tangent to the Earth at C, and s normal to the plane of incidence and tangent to the Earth at C. The ionosphere will be approximated as varying only along the z direction. This will lead to a slight underestimation of the reflected-field magnitude at the receiver, because the planar geometry lacks the slight focusing effect of the spherical ionosphere. In the (z, p, s) system, the transmitter is at zs = Hsρ2/(8Re), and the receiver is at zr = −ρ2/(8Re), where the adjustment −ρ2/(8Re) is due to the curvature of the Earth's surface.

[17] 2. Cylindrical wavefronts: We will confine our modeling to nonnormal incidence, i.e., to ranges ρHi, and we will avoid any short ranges near normal incidence (ρ = 0). This ensures that (1) the plane of incidence is well defined and (2) the refractive and reflective effects on the wavefronts will be mainly within the plane of incidence. That is, the Poynting flux will be approximately within the plane of incidence. We will then be allowed to replace the spherical geometry with cylindrical geometry. We will confine attention to that component of Earth curvature in the plane of incidence. Likewise we will represent the emitted curved wavefronts as cylindrical, with curvature only in the plane of incidence. That is, the cylinder axis is in the s direction, normal to the plane of incidence. This approximation renders moot any small effect of refraction out of the plane of incidence. At the cost of this small loss of realism, we will make far less burdensome the plane wave expansion of the curved wavefronts (see Appendix A).

[18] 3. Plane wave expansion of cylindrical wave: We will expand the cylindrical wave as a direct Fourier transform (DFT) over plane waves, as explained in Appendix A. For our regime, the far-field criterion ≫ 1 is robustly satisfied, allowing us to ignore the curvature terms in spatial derivatives of the wavefields. Each constituent plane wave will be identified by an angle of incidence θi, as shown in Figure 3. (The Earth's surface has been removed in Figure 3. Later, the conducting Earth's surface can be reimposed as a boundary condition.) The upgoing (solid lines) and reflected (dashed lines) wavefronts are shown, for the “vacuum” below the ionosphere. The angle of incidence can be any angle, not just the “specular” angle of ray optics. All angles will contribute to the angular Fourier summation, albeit with effectiveness that is confined to a zone roughly centered on the specular angle. Appendix A contains full details.

Figure 3.

Plane wave geometry below the ionosphere for a single frequency and angle of incidence, in the (s, p, z) coordinate system. Within the ionosphere, the wavefronts have more complicated z dependence. Both the upgoing (incident) and downgoing (reflected) plane waves have the same horizontal wave vector, by Snell's law, while their vertical wave numbers have the same magnitude but opposite sign.

4. Plane Wave Propagation Solution

4.1. Numerical Approach

[19] Following P65, we solve Maxwell's equations for an anisotropic, collisional, cold-electron dielectric tensor appropriate to the lower ionosphere. The magnetic field B lies in the y-z plane and dips downward to the magnetic north (positive y) by the “dip angle” relative to horizontal. Thus a positive dip angle is for the Northern magnetic latitudes. We take B as a constant vector within the reflection region. We solve Maxwell's equations as four simultaneous complex ordinary differential equations (ODEs), in the complex horizontal components of the electric and magnetic fields: Ex, −Ey, Z0Hx, Z0Hy. Here, Z0 is the impedance of free space, used to render E and Z0H of same units. The y axis is horizontal in the magnetic meridian, the x axis is horizontal normal to the magnetic meridian, and the z axis is upward vertical. The detailed method of solution is to treat eight simultaneous ODEs, four for the real and four for the imaginary components of the four field variables. We use the DDEABM solver [Shampine and Gordon, 1973] based on the Adams-Bashford-Moulton method for ODEs, ported into Interactive Data Language (IDL) and kindly made available by Craig Markwardt (NASA/Goddard Space Flight Center, Code 662, Greenbelt, Maryland 20770, USA).

4.2. Coordinate System

[20] The ODEs are solved assuming plane wave incidence at a single angular frequency ω. Below the region of the conductivity's practical effects on the numerical solution, the waves will be of the form

equation image

where Ex0 is complex and contains the polarization information. (Similar equations apply for y and z in vacuum.) This form of the solution is assumed at the lowermost grid point (z = 25 km) where we impose a lower boundary condition. Here, the wave number in vacuum is given by

equation image

where c is the speed of light in vacuum. In the computational grid above its base (at z = 25 km), the solution has the same x and y dependence, as required by Snell's law, but the z dependence is computed numerically and is not prescribed by equations (2) and (3). The x axis is horizontal and normal eastward to the magnetic meridian, the y axis is horizontal and within the magnetic meridian, and the z axis is vertical. The only background variations are along z; both x and y are ignorable coordinates. The wave's magnetic azimuth is defined as the angle clockwise from the y axis, and the wave's angle of incidence is the tilt of the wavefronts from horizontal in the vacuum region where equation (2) applies. (The angle of incidence is meaningful only below the refractive ionosphere.)

[21] Figure 3 shows the simple geometry of the plane wave solver, below the region of plasma/conductivity effects, i.e., in “vacuum.” The x,y,z coordinate system's origin is at C. The p axis is horizontal and in the plane of incidence. Were the propagation azimuth 0, then p would coincide with y (and s with x); were the azimuth 90 deg, then p would coincide with x (and s with −y). The upgoing waves have been launched from below the z = 0 level by an ideal plane wave generator. The downgoing (reflected) waves in the subionospheric vacuum have the opposite sign of kz but the same kx and ky, as the upgoing waves. The angle of incidence is θi. We stress that this plane wave z dependence breaks down within the ionospheric medium, where the ODE solver must provide the z axis variations.

[22] In the anisotropic permittivity tensor that describes the dielectric/conductive response of the medium, the background parameters appear as characteristic frequencies scaled by ω. The electron-neutral collision rate appears as νen/ω, the electron density appears via the plasma radian frequency (ωpe), as (ωpe/ω)2, and the magnetic field magnitude appears via the electron cyclotron radian frequency, as ωce/ω; see Figure 1.

4.3. Ordinary Differential Equation Solution

[23] Unlike in ray optics, the full-wave solution for arbitrary z variations does not in general have identifiable eigenmodes, that is, solutions of a Helmholtz equation. However, there are still two independent and quasi-distinct major solutions in the ODE results, and for each independent major solution, there is a pair of subsolutions, differing only trivially, by the sign of the vertical logarithmic derivative. The major solutions have different identifiable asymptotic behavior at the upper end of the domain (z = 105 km), in terms of well-known ray optic propagation modes. At 105 km, we impose a radiative boundary condition, and select only the subsolutions whose propagation (or evanescence) is upward. For our solutions at 105 km, (ωpe/ω) > 1 and (ωce/ω) > 1 (see Figure 1), and thus the two major solutions are semicollisional versions of either (1) a propagating electron-whistler wave or (2) an evanescent cutoff wave, respectively. These two solutions differ dramatically in their downward reflection from the D region. Because of the high transmission upward at 105 km, the whistler-associated wave lets more of its energy penetrate the ionosphere and escape into the magnetosphere. For this reason we follow P65 in calling this solution the “penetrating” solution. Similarly, the cutoff solution, which is evanescent at 105 km, cannot radiate upward, and hence tends to have a higher reflectivity downward from the D region, and we call this the “nonpenetrating” solution. The difference in reflectivity is mostly due to the difference in transionospheric transmission, not to any difference in dissipation within the ionosphere.

[24] At z = 105 km we impose the radiation boundary condition and iteratively solve for the WKB-like eigenpolarizations (eigenmodes) and their complex wave numbers, to launch the numerical ODE solver. This initialization is obviously a ray optics (WKB) approximation, invoked only for launching the numerical calculation, which then develops naturally (going downward in z) into full-wave. Later, we will see that at the top end, z = 105 km, the WKB approximation is quite good; the inadequacies of WKB occur lower in the D region. There are four solutions for the eigenpolarizations and their wave numbers, but two are trivial (upward) variations of the downward solutions. We choose the upward propagating and upward evanescent solutions to supply starting polarizations and starting logarithmic derivatives, for the launch of ODE numerics.

4.4. Numerical Swamping

[25] Looking downward into the medium from the starting point (z = 105 km), one numerical solution (the penetrating solution) is oscillatory versus z, while the other solution (nonpenetrating) is exponentially growing downward. In lieu of any intervention, the numerical solver will eventually allow the penetrating solution to lose its identity as it is “swamped” by the exponentially growing solution growing out of errors. To avoid this, we employ the method of P65, and periodically (in z) retransform the penetrating mode to be orthogonal to the nonpenetrating mode's (“non”) complex conjugate, in the sense that the transformed penetrating solution (“pen”) is repeatedly forced to obey

equation image

This transformation is implemented several places in the range of z where the nonpenetrating mode exhibits exponential growth downward. After the z integration is completed, we then subtract the corrections from the penetrating mode, and recover the penetrating solution that would have been obtained if the numerics had infinite precision and hence if the numerics had not been vulnerable to numerical swamping. This artifice is due to P65.

4.5. Lower Boundary Condition

[26] The numerical plane wave solution is integrated downward and finally is halted at z = 25 km. At this altitude, the solutions' behavior is that of transverse electromagnetic (TEM) waves in vacuum. The solutions are, in general, elliptically polarized. Each solution consists of two components, which can be identified as upgoing and reflected waves (see Figure 3). The penetrating and nonpenetrating upgoing waves are of opposite rotation sense from each other, and the same holds for the reflected waves.

[27] Our practical application involves radiation from a dipole along the local vertical. The dipole lies within the plane of incidence. Thus we must linearly combine the penetrating and nonpenetrating solutions, to ensure that for z ≤ 25 km the upgoing E is linearly polarized purely within the plane of incidence. We shall call this linear combination the “total” solution, and for z ≤ 25 km it has a linearly polarized upgoing component and a (generally) elliptically polarized downgoing component. Each major solution's total upgoing field is set to unity amplitude below the ionosphere, and this serves to renormalize the fields throughout the calculational domain.

4.6. Example of Plane Wave Solution

[28] We illustrate the plane wave numerical solution's behavior with a specific example: midday modified-Volland profile, angle of incidence θi = 60 deg, azimuth = 90 deg, fce = 1300 kHz, dip angle = 59 deg, and wave frequency in the low VLF, f = 10 kHz. We show Z0Hy for the daytime modified-Volland D region, for the nonpenetrating (Figure 4a), the penetrating (Figure 4b), and total (Figure 4c) solutions. The real and imaginary parts of the solutions are shown as solid and dashed curves, respectively. The upward (“incident”) electric field below the ionosphere has unit amplitude (not shown), so the ordinate scales are relative to unity incident electric field vector amplitude from below. The vertical lines indicate the three characteristic transitions along the profile from ray optics modes. The nonpenetrating solution evanesces above ∼65-70 km, while the penetrating solution transitions into an oblique electron-whistler mode. The shortening of the wavelength and growth in magnitude are due to the whistler's reduced wave impedance (compared to Z0) and to the increase in electron density with increasing altitude. The penetrating mode's wavelength is sufficiently short at the top end (z = 105 km) that its propagation there has become practically WKB-like. We reiterate that the increase in the Hy magnitude for z > 80 km is also due to the decrease of wave impedance; the wave-magnetic field increase does not imply an upward growing Poynting flux. The real component of the whistler is 90 deg delayed from the imaginary component, so the whistler seen at higher altitudes in Figures 4b and 4c is propagating upward.

Figure 4.

Vertical profile at t = 0 of the y component of a single plane wave solution's complex magnetic field, multiplied by impedance of free space Z0, for angle of incidence θi = 60 deg, azimuth = 90 deg, fce = 1300 kHz, dip angle = 59 deg, and wave frequency in the low VLF, f = 10 kHz. Real (solid curves) and imaginary (dashed curves) portions are shown separately. (a) Nonpenetrating solution, (b) penetrating solution, and (c) total solution satisfying boundary condition that upgoing E lie in the plane of incidence at the source.

[29] The appearance of an electron whistler for altitudes z > 70 km in Figure 4 is a sign of dielectric anisotropy. A whistler-like wave requires magnetic field-controlled anisotropy, and could propagate in neither an isotropic plasma nor an isotropic conductor. Although νen/ω ≫ 1 below 90 km, a collisional precursor of the whistler can nonetheless develop, because νen/fce < 1 (i.e., the electrons are “magnetized”) above 65 km (see Figure 1).

5. Summation Over Plane Waves to Construct Curved-Wavefront Solution

5.1. Phase of Plane Wave Angular Spectrum

[30] The summation over plane waves to construct the curved-wavefront solution at a field point (see Appendix A) is effectively controlled by the plane wave phase evaluated at the field point. Figure 5 examines the plane wave phase at the field point (with the plane wave phase zeroed at the source point), for the particular example of midday modified-Volland profile, azimuth = 90 deg, fce = 1300 kHz, dip angle = 59 deg, source height Hs = 12 km, and f = 10 kHz. The angle of incidence θi is varied in 353 discrete 0.25-deg steps, from 1.0 deg to 89.0 deg. The phase has been unwrapped to avoid the ±2π jumps from the arctan function. The range is stepped from ρ = 0 to ρ = 450 km in discrete 50-km steps, with a separate curve for each range. Hence Figure 5 comprises results from 3530 different plane wave calculations. Figure 5a shows the entire phase, with each curve vertically offset so as to be zeroed at its minimum. Figure 5b shows the same but with a finer vertical scale (capped at +3 radians), to highlight the parabola-like minima. The solution for each of the 3530 plane wave calculations in Figure 5 is the total solution (the linear combination of nonpenetrating and penetrating solutions), which alone obeys the lower boundary condition that upgoing E lies in the plane of incidence.

Figure 5.

(a) Phase variation for midday model plane wave calculation, for Ez at receiver versus angle of incidence, for azimuth = 90 deg, fce = 1300 kHz, dip angle = 59 deg, Hs = 12 km, and wave frequency in the low VLF, f = 10 kHz. Each curve is for a separate range, ρ = 0, 50, 100, 150, 200, 250, 300, 350, 400, and 450 km. (b) Same, but for vertical scale of only −1 to +3 vertical radians to show width of Fresnel zones (see text). Note: Each curve has been vertically offset to place its minimum at zero radians.

[31] For each range's curve in Figure 5, the location of the minimum (“stationary-phase” point) may be interpreted as a quasi-specular angle of incidence. The parabolic width of the curve controls the angular width of the plane wave expansion that effectively contributes to the constructed quasi-cylindrical-wave solution at the field point. Within δθi = ±1 rad about the point of stationary phase, the plane waves tend to add in phase to the solution, while on the steeper flanks at ∣ δθi ∣ > 1 rad the phase varies versus θi with increasing rapidity, so that neighboring plane waves add out of phase with each other and thus to contribute less to the summation over plane waves (see Appendix A).

[32] We show the 10-kHz, ρ = 250 km plane wave solution's total electric field components along the z (Figure 6a), x (Figure 6b), and y (Figure 6c) axes, for the conditions of Figure 5. The heavy curves are the real and imaginary (solid and dashed, respectively) parts of the downward propagating portion Edown of the total solution, evaluated at the receiver. The exp{iωt} dependence is not included; the solutions shown are at t = 0. The light curves are the corresponding upward propagating portion Eup, evaluated at the source. The upward propagating components have zero imaginary part, because the upward phase is set to zero at the source, as the phase reference. The upward propagating component lies in the x-z plane, which is the plane of incidence for this case of az = 90 deg; the upward propagating Ey is by definition zero for this example. The z and x components of Eup vary as sin(θi) and cos(θi), respectively. The vector Eup has unit amplitude, lies within the plane of incidence, and is normal to the upgoing wave vector at the source. The thin solid curve labeled δϕ in Figure 6 is the phase of the downward propagating Ez at the receiver, vertically offset to place the minimum of that phase at the baseline. The vertical span of phase for this curve is 33 rad. The phase curve climbs 1 rad above its minimum in a half-angle of ∼10 deg; this is therefore the half-angle subtended by the Fresnel zone. Thus the 10-kHz angular spectrum will have effective contributions from a fan of plane waves on the order of 20 deg wide. This is a full-wave diffractive effect, missed by WKB approaches. The stationary-phase angle (∼65 deg) corresponds to a quasi-specular reflection to which WKB would be limited.

Figure 6.

Complex E at receiver versus angle of incidence, for the ρ = 250-km plane wave curve in the example of Figure 5. Real and imaginary portions of downgoing field are shown as solid and dashed heavy curves, respectively. Overall parabola-like curve repeated in each plot is the phase curve for ρ = 250 km. Real component of upgoing field is shown as light dashed curve. (a) Ez, (b) Ex, and (c) Ey.

5.2. Spectral Transfer Function

[33] In a practical field observation, one measures the vertical component of the VLF/LF wavefield at a point. One cannot separate plane waves; rather, the measured field is the entire field of the curved wavefront impinging on the receiver. Thus we must take the analysis of the previous section one step further, to predict the measured field at the receiver in a manner that incorporates the summation over plane waves to construct a quasi-cylindrical wave. Toward this end, we calculate a spectral transfer function based on summing the plane wave solutions over angle, weighted by the apodizing functions shown in Appendix A. This is done separately for each of 80 frequencies from 2 to 160 kHz, in 2-kHz steps. The summation is performed for a given pair of length parameters: range ρ and source height Hs. This pair of length parameters can be changed without repeating the burdensome step, which is the differential equation numerical calculation for each of the 353 discrete plane waves. Thus for each choice of ionospheric profile, dip angle, azimuth, electron cyclotron frequency, and wave frequency, we obtain and archive all 353 plane wave solutions. We then expand the plane wave-solution archive to include all 80 wave frequencies. Thus to construct a spectral transfer function for any given pair of range ρ and source height Hs, we need to have already calculated and stored a matrix of 353 × 80 = 28240 plane wave solutions of Maxwell's equations. On the other hand, that 28240 plane wave-solution archived matrix is useable for any new pair of range ρ and source height Hs, as long as the following remain the same: ionospheric profile, dip angle, azimuth, and electron cyclotron frequency. This archived matrix we call the “spectral transfer function.”

5.3. Construction of Model Time Domain Ionospheric Reflection Signals

[34] Armed with the spectral transfer function, a choice of distance parameters, and a (contrived or recorded) transmitted waveform, we construct a model time domain reflection signal as follows:

[35] 1. Direct Fourier transform the transmitted waveform in time, providing the transmitted signal's complex Fourier coefficients. We use a 500-μs time window, giving 2-kHz steps in the direct Fourier transform. We apodize the high-frequency response with a constant-phase tanh function to provide complete low-pass filtering slightly below 160 kHz, eliminating any frequency-edge effects.

[36] 2. Multiply the Fourier transform by the spectral transfer function from section 5.2.

[37] 3. Perform an inverse Fourier transform to obtain the time domain reflection signal at the receiver.

[38] We illustrate this procedure with the case of midday modified-Volland profile, azimuth = 90 deg, fce = 1300 kHz, dip angle = 59 deg, range ρ = 250 km, and source height Hs = 12 km. We show the steps to calculate the time domain reflection from the spectral transfer function: low-pass filter to apodize input spectrum (Figure 7a); contrived input waveform (imitating a typical positive-polarity Narrow Bipolar Event lightning waveform) (Figure 7b); spectral amplitude of filtered input waveform (Figure 7c); time domain direct wave evaluated at the receiver (Figure 7d); spectral amplitude of reflected field (Figure 7e); and time domain reflected waveform (Figure 7f). The direct wave is the “ground wave,” but since we have lifted the ground out of our problem (see Figure 3), it is just labeled direct. Both the direct-wave and reflected-wave time domain waveforms have been renormalized so that the direct wave has unit peak value. Thus, for this example, the model predicts that the reflection waveform would have a maximum value of ∼0.08, relative to the direct wave's peak. The negative frequencies in the Fourier domain are ignored, as they would correspond to downward propagating incident waves, which are contrary to our radiation boundary condition (that above z = 105 km, the only waves are upward propagating or upward evanescent).

Figure 7.

Illustration of complex transfer function and time domain solutions for the case of midday modified-Volland profile, azimuth = 90 deg, fce = 1300 kHz, dip angle = 59 deg, range ρ = 250 km, and source height Hs = 12 km. (a) Constant-phase low-pass filter to apodize input spectrum, (b) contrived input waveform (imitating a typical Narrow Bipolar Event lightning waveform), (c) spectral amplitude of filtered input waveform, (d) time domain direct wave evaluated at the receiver, (e) spectral amplitude of reflected signal, and (f) time domain reflected waveform. Note 1: The reason why the direct wave in Figure 7d is not exactly identical to the input waveform in Figure 7b is that the direct wave has undergone constant-phase low-pass filtering (see Figure 7a), which distorts the waveform. Note 2: The input pulse timing in Figure 7a is located at 250 μs to render it most visible in the center of the window. The direct wave in Figure 7d is arbitrarily placed at 50 μs although its actual delay from the input pulse would be ∼833 μs (the speed of light transit time for ρ = 250 km). The timing of the ionospheric reflection in Figure 7f is correct in terms of the timing of the direct wave in Figure 7d. This offset of Figures 7d and 7f together, relative to Figure 7b, is done to keep the time window short for visibility of waveform shapes.

[39] The low-pass filter (Figure 7a) is constant phase, hence noncausal. This creates the slight leading ramp on the direct-wave pulse prior to the main positive peak. The reflection Fourier amplitude (Figure 7e) drops off sharply versus frequency, leading to the dissimilarity of the reflection waveform (Figure 7f) to the direct waveform (Figure 7d). Most of the higher-frequency (f > 40 kHz) content is lost to dissipation in the midday D region, and the reflected waveform is built of the minor spectral content at the input spectrum's lower frequencies.

6. General Properties of Time Domain Constructed Solutions

[40] We now use the modified-Volland D region profiles to illustrate two basic features of the modeled time domain waveforms of reflected pulses: (1) noticeably different reflection amplitudes for different propagation azimuths and (2) radically different waveform results in midday versus midnight conditions. We use the example case of dip angle = 59 deg, range ρ = 250 km, and source height Hs = 12 km, but we let the propagation azimuth take two values, either 90 deg (toward the east) or 270 deg (toward the west).

[41] Figure 8a shows the midday total solution's Fourier amplitude versus frequency for the direct wave (light dotted curve), the 90-deg azimuth's reflection (heavy solid curve), and the 270-deg azimuth's reflection (heavy dotted curve). The reduced reflection amplitude of the westward versus eastward propagation is consistent with observations [Bickel et al., 1970; Jacobson et al., 2008]. The amplitude difference is more pronounced near f ∼ 10 kHz and disappears for f > 30 kHz. Figure 8b shows the midday total solution's time domain waveforms corresponding to the Fourier amplitudes of Figure 8a. Both propagation azimuths' reflection waveforms show the same strong low-pass filtering by the ionosphere. The time-stretched appearance of the reflected pulses, compared to the direct pulse, indicates a basic problem with the sharp-boundary model used in retrieval of the “reflection height.” The main transition occurs at the center of the reflected pulse, but the reflected energy extends tens of microseconds both prior to, and after, this transition. Thus even in daytime, the “best behaved” of the D region conditions, the reflected waveform may be dramatically dissimilar to the incident (direct) pulse waveform. The reflection-height retrieval is based upon the timing of the reflection with respect to the direct pulse and is implemented by correlating the direct waveform with the reflected waveform, as a function of time delay [Jacobson et al., 2007b; Smith et al., 1999, 2002, 2004]. Owing to the narrowness of the direct waveform compared to the reflected waveform, this height retrieval is strongly biased toward the steep transition midway between the leading negative and trailing positive halves of the reflected pulse. Figure 8b shows, however, that the reflection is distributed over a range of “heights” (if that language is even appropriate in a full-wave process), and that substantial energy arrives over a duration that is long compared to either the duration of the transition or the width of the direct pulse. In particular, the reflection precursor prior to the main transition typically is seen for several tens of microseconds before the transition. This precursor represents distributed energy reflection from D region heights lower than the single, discrete “height” usually mentioned for LASA data [Jacobson et al., 2007b].

Figure 8.

Midday/midnight and eastward/westward behavior of solutions for fce = 1300 kHz, dip angle = 59 deg, range ρ = 250 km, and source height Hs = 12 km. Eastward (azimuth = 90 deg) reflected-wave solution in heavy solid curves, westward (azimuth = 270 deg) reflected-wave solution in heavy dotted curves, and direct-wave solution in light dotted curves. (a) Midday Fourier amplitude versus frequency, (b) midday time domain waveform versus time, (c) midnight Fourier amplitude versus frequency, and (d) midnight time domain waveform versus time.

[42] Figures 8c and 8d repeat Figures 8a and 8b but for midnight conditions. Midnight reflections are qualitatively different from those of midday, in two ways: First, the Fourier amplitude extends to higher frequencies at midnight (Figure 8c) than at midday (Figure 8a). Second, the time domain waveform (Figure 8d) exhibits spectral dispersion at midnight (but not at midday), with the higher frequencies arriving in a late coda.

[43] There are two other items of comparison between midday and midnight modeled reflections. First, the azimuth difference seen at midday (Figure 8a) is also seen at midnight (Figure 8c) with the exception that at higher frequencies (f > 30 kHz) the westward propagation's reflection Fourier amplitude is enhanced with respect to the eastward propagation's. However, since most of the reflective energy is at lower frequencies, the overall energy reflection still is favored in eastward over westward propagation. Second, the reflected pulse shape is radically dissimilar to the incident (direct) pulse shape, so that determination of “reflection height” for nighttime reflections shares with daytime reflections the basic conceptual inadequacy of a sharp-boundary description. Moreover, the reflection waveform shape varies sensibly with azimuth, in a manner that can further confuse the already-problematic sharp-boundary reflection concept.

7. Comparison of Model With General Features of LASA Reflection Waveforms

[44] The modified-Volland profiles of D region electron density and electron-collision rate (see Figure 1) are not optimized for agreement with LASA data but suffice, nonetheless, to illustrate a broad comparison between model and observations.

[45] We restrict this comparison to daytime propagation conditions, which are the simplest and most stable. Again, we use the example case of dip angle = 59 deg, range ρ = 250 km, electron gyrofrequency fce = 1300 kHz, and source height Hs = 12 km, but we let the propagation azimuth take two values, either 90 deg (toward the east) or 270 deg (toward the west). For observations, we use the LASA archive of ∼ 65,000 NBE recordings recently described for retrievals of reflection “height” and energy reflectivity [Jacobson et al., 2007b, 2008]. From this pool of recordings, we now select small subsets for a data cube given by midday (solar zenith angle < 60 deg), range ρ = 220–280 km, and either azimuth = 60–120 deg (for eastward propagation) or azimuth = 240–300 deg (for westward propagation). These small subsets are then further reduced by requiring an exceptionally high threshold of reflected-pulse signal-to-noise ratio. Specifically, we select for only those NBE recordings in which the ionospheric-echo amplitude exceeds 3% of the direct-wave amplitude, the direct-wave amplitude exceeds 5 V/meter, the fitted ionospheric height is in the range 60–90 km, and the chi-square of the event location is less than 0.001. This selection process results in 94 example recordings for eastward and 55 example recordings for westward propagation. Next, we combine each subset's recordings in order to illustrate the broadest daytime reflection features, by superimposing the reflected-pulse waveforms and summing the results within each subset of recordings. Within each subset, the source heights and overall “reflection heights” can vary widely [Jacobson et al., 2007b, 2008], so we must realign the reflection curves in time prior to superposition and summation. We do this by a cross-correlation technique, comparing the reflection features and aligning the traces horizontally to maximize the cross correlation. The result is an aligned “superposed epoch” average waveform in each subset of NBE recordings. The realignment is implemented only for the reflection-pulse portions of the recording, not for the direct-wave portion.

[46] Figure 9a shows the averaged direct-wave recording as a dashed curve, which is quite similar to the input-waveform curve for the model (solid curve). Figure 9b shows the modeled reflected-pulse waveforms for both eastward (heavy solid curve) and westward (light solid curve) propagation. Figure 9c shows the average observations for eastward (heavy dashed curve) and westward (light dashed curve) propagation. The NBE lightning discharge is always substantially elevated above Earth, by typically 8–18 km [Jacobson, 2003b; Smith et al., 1999]. Thus in the observations there are two reflected paths, the first from the ionosphere directly, and the second from the ground then ionosphere [Jacobson et al., 2007b; Smith et al., 2004]. Thus the superposed-epoch reflection waveforms in Figure 9c are more complicated than is the model, which considers only one ionospheric reflection. To avoid that complication, we truncate the average-data curves at the onset of the second reflected pulse.

Figure 9.

Comparison of the model for fce = 1300 kHz, dip angle = 59 deg, range ρ = 250 km, and source height Hs = 12 km, with superposed-epoch averages of observed data from LASA (see text). (a) Input wave, (b) model echo waveforms, and (c) superposed-epoch observations. Note 1: The observed reflection waveforms in the data would include a second pulse due to the ground-ionosphere reflection, which should be overlooked in comparison of observations with model. For this reason, the curves are truncated before the second pulse in Figure 9c. Note 2: The ionospheric reflections in Figures 9b and 9c are advanced (together) by 833 μs with respect to the input pulse of Figure 9a, in order to keep the time duration of Figure 9 short and the waveform details more visible.

[47] The comparison of model (Figure 9b) and average observations (Figure 9c) indicates the following:

[48] 1. The midday model is consistent with the daytime observation that eastward propagation has larger reflection amplitude than does westward propagation.

[49] 2. The midday model is consistent with the daytime observation that the eastward propagating arrives later than does the westward propagating reflected pulse.

[50] 3. The midday model predicts a longer pulse than the observations, by about 50%. Otherwise the model is consistent with the overall low-pass-filtering effect of daytime reflections.

[51] 4. The model predicts less timing difference between eastward and westward reflected pulses than is seen in the data. This discrepancy will be explored in the parameter estimation done in the follow-on paper. Here we simply note that the Volland daytime profile does not adequately account for the magnitude of the observed timing differences.

[52] More systematic comparisons of the full-wave model with data, with a view to retrieving ionospheric-structure parameters, will be described in forthcoming publications.

8. Summary and Conclusions

[53] 1. The present full-wave model borrows a prior conceptual plane wave-solution approach [Piggott et al., 1965; Pitteway, 1965] as its innermost calculational step.

[54] 2. We embed this plane wave solution within an angular direct Fourier transform to construct curved wavefronts from a summation over plane waves at a range of angles of incidence. In this step, the wavefronts are approximated as cylindrical, with the cylinder axis normal to the plane of incidence. In this two-dimensional approximation of a three-dimensional problem, we retain the essential full-wave diffraction effects within the plane of incidence, while reducing the computational burden.

[55] 3. We calculate and archive a complex spectral transfer function, relating Fourier components of the received electric field at a receiver to corresponding Fourier components of the emitted pulse at a transmitter. A standard D region model is used. Time domain received waveforms are constructed by a second direct Fourier transform, back to the time domain.

[56] 4. The modeled results for daytime reflections predict some basic features of the recorded waveforms from the LASA system. These include the extreme low-pass filtering of the daytime reflection, as well as the observed systematic difference between propagation toward the East versus toward the west.

[57] 5. The modeled results for nighttime reflections predict the observed enhancement of higher frequencies, the strong spectral dispersion, and the basic azimuth dependence of these features. However, the apparent temporal irreproducibility of nighttime reflections, at least near thunderstorms, might require consideration of electric field-generated variabilities [Cummer et al., 1998; Inan et al., 1996; Rodger et al., 2001; Taranenko et al., 1993] as well as effects of particle precipitation [see, e.g., Cheng et al., 2006].

Appendix A

A1. Plane Wave Expansion of Curved-Wavefront Solutions

[58] The approximated z-dependent ionosphere makes a plane wave solution most convenient, with the required subsequent step to construct the curved-wavefront solution from a summation over plane waves. A waveguide-scattering model [MacPhie and Wu, 1995, 1999] has developed just such a series expansion of a vector electromagnetic cylindrical wavefield in terms of plane waves. The MacPhie-Wu approach is essentially an angular direct Fourier transform (DFT) of the cylindrical field. In the following, we will implement an angular DFT by straightforward numerical means, rather than the MacPhie-Wu expansion. Our ad hoc approach is simpler but more limited in its application (see below).

[59] A Fourier transform summation should use a weighting function to minimize edge effects. Figure A1 illustrates the angular weighting functions employed for a specific case, ρ = 250 km and Hs = 12 km. The wider dashed curve shows the overall radiated amplitude-response lobe of the source dipole. The source dipole axis is tilted from the z axis (see Figure 2), toward negative p, by an angle (125 km)/Re radians, or about 1.1 deg. That accounts for the nonzero value of the dipole lobe at θi = 0 deg in Figure A1. The two narrower curves in Figure A1 show the weighting function for the ionospheric reflection (solid curve) and for the direct wave (narrow dashed curve). These narrow curves are 90-deg-wide portions of the dipole function, centered, respectively, on 45 deg and 90 deg. Each weighting function is apodized by two tanh functions to provide a taper at either of its own boundaries (i.e., at 0 and 90 deg for the ionospheric reflection, and at 45 and 135 deg for the direct wave). The reflected-wave angular window is further from the emitter dipole's lobe center, compared to the direct wave's window.

Figure A1.

Source antenna lobe (outer, light dashed curve) and artificial weighting functions for angular direct Fourier transforms (inner heavier curves, solid for reflected wave, and dashed for direct wave). The source antenna lobe's zero is offset from the z axis owing to Earth curvature over the half-range ρ/2, with source height Hs = 12 km. Each of the artificial weighting functions is a portion of the source lobe formed by tapering with two tanh functions.

[60] Since our plane wave expansion of the vector electromagnetic cylindrical wave is ad hoc, its validity for our parameter range must be demonstrated. This requires showing that the essential features of a vector cylindrical wavefield are reproduced in the expansion. We now illustrate this, taking as an example the direct wave, whose constituent plane waves propagate as if through vacuum. We tie the phase of all the constituent plane waves together at the transmitter, by setting their phase to zero at that location. We then examine the phase of the various plane wave contributions at a distant field point on the periphery of a circle centered on the transmitter. At a fixed point on this circle of radius ρ, two different plane waves differing in angle of incidence by ɛ will deviate in phase by δϕ:

equation image

The angular scale ɛcrit, related to the “Fresnel radius,” is the angular separation between two plane waves whose phase differs (at the field point) by 1 rad:

equation image

For example, for ρ = 250 km and f = 10 kHz, ɛcrit = 0.2 rad = 11 deg. Thus for 10 kHz, the contribution to the wavefield at a range of 250 km is expected to involve a fan of plane waves of width at least ±11 deg. This is the unavoidable diffractive width of the contribution, dictated by diffraction. It is important that our weighting functions be wider than the intrinsic diffractive width ±ɛcrit, and this will be satisfied for all but our lowest frequency (2 kHz).

[61] The angle of incidence with an extremum (stationary point) of phase corresponds to the specular angle in ray optics. Equations (A1) and (A2) indicate that the phase will vary quadratically versus θi around the stationary-phase angle. We launch a regular grid of unit-amplitude plane waves phased together at the source, and the number of plane waves that will effectively contribute to the sum should scale as 2ɛcrit X (number of angular grid points per radian of θi). This effective number will therefore scale as ρ−1/2, which assures that correct cylindrical-wave amplitude scaling is inherent to our expansion approach.

[62] Figure A2 illustrates the performance of the angular DFT in approximating a cylindrical wave. Figure A2a shows the product of the apodization function for the direct wave. This example is for f = 30 kHz, about the midpoint of the VLF spectral region. Figure A2b shows the amplitude of the summation of plane waves versus polar angle η on a circle of radius ρ. The amplitude closely approximates a flat plateau for field points near η = 90 deg, which is the site of the receiver in our problem (see Figures 2 and 3). Another requirement of a proper cylindrical wave is that the vector electric field be normal to the radius from the source. This amounts to requiring

equation image

Figure A2c shows that the angular DFT solution has the correct polarization over a large angular region around η = 90 deg. The diagonal dashed line is the equality of equation (A3).

Figure A2.

Numerical test of ability of angular direct Fourier transform to construct cylindrical waveform for the direct vacuum wave, for ρ = 250 km, Hs = 12 km, and f = 30 kHz. A curved surface is placed at radius 250 km from the source dipole. The polar angle η indicates angular position on this surface. The receiver is at η = 90 deg, on the p axis. (a) The direct-wave weighting function from Figure A1. (b) Complex amplitude of direct-wave vector electric field, as a function of polar angle. The receiver is in the center of a well-behaved angular zone, ∼40-deg wide in η, in which the amplitude is constant, as in a cylindrical wave. (c) Angle of vector electric field versus η, based on arctan of ratio of horizontal to vertical magnitudes. The receiver is in the center of a well-behaved angular zone, ∼40-deg wide in η, in which the electric field is tangent to the curved surface of 250-km radius, as in a cylindrical wave.

[63] In addition to amplitude and polarization as shown in Figures A2b and A2c, there is the matter of derivatives: does the angular DFT summation provide derivatives that approximate those of a true cylindrical wave? The Maxwell's equations involve the curl operator, and for short radius (or long wavelength), the curvature terms can compete with the partial-derivative terms. The DFT in terms of plane waves lacks the curvature terms in the curl. Fortunately, in our case, the curvature terms are inevitably small, as we note that kρ ≫ 1, for all the frequencies important to our analysis. For example, for Figure A2's example of ρ = 250 km and f = 30 kHz, we have kρ = 157. As long as ≫ 1 we are assured that the derivatives, and hence the curl terms in Maxwell's equations, are adequately approximated by our plane wave summation, even though our plane wave summation cannot replicate the near-field curvature terms varying as ρ−1.

A2. Benchmarks With Analytic Results

[64] The ordinary differential equation (ODE) solver, the coding of the susceptibility tensor, and the method for computing the reflectivity used in this article can be tested by comparison to solutions that can be obtained analytically for certain fortuitous cases. We will choose four complementary tests that would reveal errors (if there are any) in the calculation's handling of (1) magnetic anisotropy, (2) background gradients, and (3) the imposition of boundary conditions to calculate a reflection coefficient.

A2.1. Check 1: Collisionless Electron-Whistler Wave, to Test Anisotropy Handling

[65] The electron-whistler wave in a uniform, magnetized, cold, collisionless plasma has a well-known dispersion relation and polarization [see, e.g., Krall and Trivelpiece, 1973, equation (4.10.1) and section 4.10]. The whistler thus provides a convenient test for the function of our ODE solver and anisotropic susceptibility tensor. Within the upper zone of our computational domain (z = 100 − 105 km) treated by the ODE solver, the plasma density for the midnight profile is essentially constant (see Figure 1, midnight curve). Moreover, if we make the magnetic field vertical for this test, any correctly calculated dispersion would need to be azimuthally symmetric. Finally, in this region the penetrating-mode wave has a short wavelength (see Figure 4b). We therefore will test the ODE solver's calculated penetrating-mode dispersion relation by seeing if it agrees with the analytic whistler dispersion relation. To do this, we tabulate the calculated penetrating mode's −dln(Ey)/dz at 104.99 km and graph it versus frequency, for normal incidence onto the midnight profile, with a vertical magnetic field (dip angle = 90 deg). Figure A3 shows −dln(Ey)/dz (at 104.99 km) versus frequency (open squares), compared to the predicted kz for the whistler propagating parallel to B (continuous curve) [see, e.g., Krall and Trivelpiece, 1973, equation 4.10.1 and section 4.10]. The agreement of the calculated dispersion with the analytic whistler is excellent. To test for correct treatment of propagation azimuth, we need to calculate the dispersion for oblique incidence. In the inset graph in Figure A3 we show the calculated variation of −dln(Ey)/dz (at 104.99 km) versus azimuth, for a single, fixed frequency f = 4 kHz, and for oblique incidence of θi 70 deg. Since the system is rotationally symmetric about the magnetic field (which in this test case is vertical), an error in the coding for the susceptibility tensor, or the operation of the ODE solver, would be manifested as a spurious variation of the dispersion versus azimuth in this test case. Fortunately, the calculated dispersion does not depend on azimuth in this vertical-magnetic field case, and thus passes check 1.

Figure A3.

Dispersion relation for nighttime modified-Volland profile at z = 104.99 km, with zero collision rate and vertical magnetic field. Square symbols show the calculated logarithmic derivative of Ey for the penetrating solution for normal incidence; solid curve shows the Appleton-Hartree electron-whistler dispersion for the same context. Inset: Azimuth dependence of calculated logarithmic derivative of Ey for the penetrating solution, for a fixed frequency f = 4 kHz and angle of incidence = 70 deg. The lack of variation versus azimuth agrees with the Appleton-Hartree dispersion relation, as the susceptibility tensor is rotationally symmetric for vertical magnetic field.

A2.2. Check 2: Airy Linear-Density Solution, to Test Gradient Handling

[66] The special case of wave reflection from a linear electron-density gradient in a cold, collisionless, unmagnetized plasma has an analytic solution in terms of Airy functions [see, e.g., Budden, 1985, section 8.7 and Figure 8.5]. This provides a test for whether the ODE solver can correctly integrate through a gradient in the susceptibility tensor. Figure A4 shows the adopted profile (dashed curve) of the diagonal term, 1−(fpe/f)2, in the susceptibility. The classic turning point for normal incidence is thus at z = 70 km. The solid curve in Figure A4 is the calculated transverse electric field versus z; it agrees exactly with the Airy solution [see, e.g., Budden, 1985, section 8.7 and Figure 8.5]. The wave phase is constant versus z, because its reflection coefficient is unity and the calculated solution is a perfect standing wave. Any error in the ODE solver's handling of a gradient would be manifested as a spurious departure from the Airy solution.

Figure A4.

Standing-field pattern for 100-kHz normal incidence on a nonmagnetized, collisionless, linear-density-gradient test problem. Dashed curve: Profile of susceptibility, with critical level at z = 80 km. The calculated wavefield agrees exactly with the analytic Airy solution [see, e.g., Budden, 1985, Figure 8.5, section 8.7].

A2.3. Check 3: Nearly Sharp-Boundary Solution, to Test Boundary-Condition Handling

[67] The sharp-boundary reflection coefficient is given by the simple Fresnel reflection coefficient [see, e.g., Volland, 1995, section 3.2.3, chap. 3]. Although an ODE solver cannot integrate across a perfectly sharp boundary, we can integrate across an analytic (hyperbolic tangent) function of short (∼1-km) gradient scale, which should approximate a sharp boundary for wavelengths greatly exceeding the gradient length. Figure A5a shows the quasi-sharp-boundary profile of plasma frequency versus z used for this test: fpe = 2.5*(1 + tanh(z-80 km)), centered at 80 km, and asymptotic to 5 kHz at higher altitudes. We employ the ODE solver on this profile with zero magnetic field, zero collisionality, and normal incidence. Figure A5b shows the calculated amplitude reflection coefficient's magnitude versus frequency as a solid curve. The dashed curve is the analytic Fresnel-reflection coefficient's magnitude [see, e.g., Volland, 1995, section 3.2.3, chap. 3]. The close agreement at lower frequencies is due to the radian wavelength's being large compared to the profile gradient length. For example, at f = 10 kHz, the radian wavelength is kz−1 ∼ 5 km, much longer than the gradient length (∼ 1 km) of the profile (Figure A5a). On the other hand, at f = 25 kHz, we have kz−1 ∼ 1 km, so it is not surprising that the sharp-boundary model slightly overstates the true reflectivity at the higher frequencies. The tanh profile in Figure A5a behaves like a sharp boundary for the lower frequencies but slightly nonsharp for the highest frequencies in Figure A5b. We conclude that the reflection accounting of this code correctly reproduces the sharp-boundary analytic result where applicable.

Figure A5.

Test of calculated reflectivity in the case of a nearly sharp boundary. (a) Analytic test profile for the plasma frequency, approximating a sharp boundary for kz−1 ≫ 1 km. Below the transition the plasma frequency is 0; above, it is 5 kHz. The gradient length within the transition is ∼1 km. (b) Amplitude reflectivity magnitude versus frequency from the analytic test fpe profile, for normal incidence, nonmagnetized, collisionless conditions. Solid curve, calculated solution; dashed curve, Fresnel reflection formula.

A2.4. Check 4: Constant-Collision-Rate, Exponential-Electron-Density Analytic Solution

[68] For a strong magnetic field (fcef) there exists an approximate analytic solution for full-wave propagation at normal incidence, with B vertical, constant electron collision rate versus z, and an exponential (e.g., Volland as used in this paper) electron-density profile [Stanley, 1950], as reported by Budden [see Budden, 1985, equation (19.20), section 19]. This analytic solution gives the amplitude reflection coefficient for an upgoing linearly polarized wave, with reflected-wave components both parallel (coefficient = R11) and perpendicular (coefficient = R21) to the upgoing wave. We tabulate these two amplitude reflection coefficients for electron collision rate 104 s−1, fce = 5000 kHz, and the Volland midnight electron-density profile (equation (1a)). Figure A6 shows these reflection coefficients as continuous curves versus frequency, from 0.2 to 16 kHz, in steps of 0.2 kHz. Also plotted are discrete symbols showing our numerical code's corresponding plane wave-solution reflection coefficients. The agreement is quite good. The 0.2–16 kHz frequency range highlights the interesting transition zone where the wave radian frequency 2πf first is less than, then is equal to, and finally exceeds the electron collision frequency 104 s−1.

Figure A6.

Test of calculated reflectivity in the case of Stanley's analytic solution for constant electron collisionality, Volland midnight electron density, vertical magnetic field, and normal incidence; see text. Continuous curves are Stanley's solution, while discrete symbols are the parallel (squares) and perpendicular (diamonds) amplitude-reflection coefficients.


[69] We are indebted to Patrick Colestock for insightful suggestions on the calculation of curved-wavefront solutions. Two of the authors (A.J. and R.H.) have been supported in this work by the National Science Foundation, under grant NSF 0809988, “Using Powerful, Low-Frequency Radio Waves from Lightning to Diagnose the D-region Ionosphere.”

[70] Zuyin Pu thanks the reviewers for their assistance in evaluating this paper.