The global implications, particularly with respect to altitude dependence, of the heuristic model of Farley Buneman waves put forward initially by Milikh and Dimant (2002) are studied. This model prescribes a relationship between the background convection electric field that excites the waves and the transverse electric fields of the waves that grow in response. It also prescribes the magnetic aspect angle of the waves, which is related to their ability to heat the auroral E region. The prescription is based on the condition of marginal stability. We reformulate the basic model, which is local, and embed it in the SAMI2 ionospheric model, which includes wave and Joule heating, heat transport, cooling, temperature-dependent collisions, and related chemistry. Within the limits of its underlying assumptions, the combined model can be used to predict the phase-speeds and magnetic-aspect widths of Farley Buneman waves in the auroral zone and the heating they can cause, all as functions of altitude. Model predictions are compared with experimental results, and the efficacy of the model assessed. This modeling exercise highlights the importance of the thickness of the layer in which Farley Buneman waves exist, the strong variations in wave characteristics across the layer, and the consequences this has for coherent scatter radar measurements of the phenomenon.
 Farley Buneman waves were discovered in the equatorial electrojet during early days of operations at Jicamarca [Bowles et al., 1963; Cohen and Bowles, 1967], and a linear, local theory for the waves emerged quickly [Farley, 1963; Buneman, 1963]. A number of features of the waves not accounted for by linear theory were immediately evident, however. Among these were the apparent propagation of the waves at phase speeds close to the ion-acoustic speed rather than the electron convection drift speed, which linear theory predicts. Such “phase-speed saturation” was addressed by Sudan  in terms of quasilinear anomalous electron collisions associated with small-scale Farley Buneman wave turbulence. Other paradigms better classified as weak turbulence, where the propagation of primary waves is partially arrested by nonlinearly-pumped secondary wave effects, have also been advanced [e.g., Oppenheim et al., 1996; Otani and Oppenheim, 1998, 2006]. St.-Maurice and Hamza  also discuss a deterministic saturation mechanism rooted not in wave mechanics but in the motion of polarized plasma corpuscles acting under the Farley Buneman mechanism, while a wave scattering theory for saturation rooted in the formalism of stochastic differential equations has also been examined (Hysell, D. L., H. C. Aveiro, and J. L. Chau, Ionospheric Irregularities: Frontiers, AGU Monograph Series, Chapman Conference on Modeling the Ionosphere/Thermosphere System, in press., 2013.)
 The most intense Farley Buneman waves exist in the auroral electrojet, where the electron convection can be much more rapid than in the equatorial zone (see Eckersley  and Bowles  for descriptions of the earliest experimental work and Haldoupis , Sahr and Fejer , and Bahcivan et al.  for more recent reviews.) An important aspect of the auroral electrojet was found to be substantial electron heating in regions occupied by Farley Buneman waves, to temperatures of 1500 K or more during major geomagnetic storms [Schlegel and St.-Maurice, 1981; St.-Maurice et al., 1981; Stauning and Olesen, 1989; Williams et al., 1992; Foster and Erickson, 2000]. Robinson  attempted to account for the electron heating in terms of the electric fields associated with the waves themselves. However, those calculations neglected the contributions to heating from wave electric fields parallel to the background magnetic field. St.-Maurice and Laher  showed that parallel wave electric fields were essential to account for the heating actually observed [see also St.-Maurice, 1990; Hamza and Imamura, 2001]. Wave coupling to off-perpendicular modes is also believed to be an important mechanism for wave-induced anomalous damping and wave saturation [Hamza and St-Maurice, 1995].
 The finite wave number spread of field-aligned plasma density irregularities in the direction of the background magnetic field is reflected in the magnetic aspect sensitivity of the radar echoes. At the magnetic equator, the aspect width of Farley Buneman waves has been measured at VHF and found to be very limited of the order of a few tenths of a degree root-mean-square (RMS) or less [Kudeki and Farley, 1989; Lu et al., 2008]. However, Farley Buneman waves in the equatorial electrojet are seldom driven very far past the threshold for instability, and wave heating is generally thought to be insignificant there. Using a UHF radar, Foster et al.  estimated an aspect sensitivity of between −10 dB/° and −15 dB/° for Farley Buneman waves in the auroral electrojet, depending on the intensity of the event. Different estimates at other wavelengths have been made subsequently [e.g., Koustov et al., 2001].
 Much of what is known about Farley Buneman waves was learned from experiments with sounding rockets. During the 1998 E region Rocket-Radar Instability Study campaign in Sweden, Farley Buneman waves were observed over a broad range of altitudes between 95 and 120 km, peaking in intensity at the center of that range [Pfaff et al., 1992; Pfaff, 1995]. The data confirmed phase propagation speeds well below the convection speed predicted by linear theory at all altitudes. The dominant wavelengths observed in the auroral zone were of the order of a few meters. Finally, most of the energy was found to be in primary waves propagating in nearly the E×B Hall drift direction. Secondary waves, propagating normally to the primaries, were much weaker. We associate the primary waves with the waves observed by radars at small flow angles (angles between the electron convection drift direction and radar line of sight). It is the primary waves in which this paper is concerned. Iranpour et al.  found similar results using rocket data from the Rocket and Scatter Experiment (ROSE) campaign and additionally discerned dispersion in the wave-phase speeds, with longer-wavelength waves propagating more slowly than shorter-wavelength waves. While they asserted that this argues against an ion-acoustic interpretation for the wave-phase speed, it actually supports it in view of the fact that the acoustic speed becomes a function of wave frequency when electron thermal effects are considered (see below and also Farley and Providakes  and St.-Maurice et al. ).
 Significant theoretical progress on the problem was made recently with the advent of massively-parallel three-dimensional particle-in-cell simulations [Oppenheim and Dimant, 2013]. The simulations confirm phase propagation at the ion-acoustic speed, with wave heating and thermal effects considered self consistently, the phase-speed saturation evidently being associated with coupling to linearly stable modes, including modes with finite parallel wavelengths. Such modes are critical for wave heating, which also increases diffusivity and damping. An important finding of the new simulations is that the phase speeds of the waves constituting the turbulence do not vary significantly with magnetic aspect angle, as linear theory might predict.
 Reconciling theory with experiment and obtaining quantitative wave heating and aspect sensitivity predictions is challenging, analytically and computationally, particularly when nonlinear and nonlocal effects are to be considered. Consequently, heuristic methods have dominated the treatment of the problem. A useful heuristic model of Farley Buneman wave heating was suggested by Milikh and Dimant , Dimant and Milikh , and Milikh and Dimant  and rests upon the following assumptions:
 The RMS transverse electric field for Farley Buneman waves is equal to the background convection electric field in the ion frame of reference, less the threshold field for instability.
 The magnetic aspect angle of Farley Buneman waves, k∥/k⟂, is as large as is necessary to maintain the condition for marginal growth, i.e., a linear growth rate near zero.
 Here k∥ and k⟂ refer to the wave number components parallel and perpendicular to the geomagnetic field, respectively. The model can be extended to apply to a spectrum of waves in a packet, with k∥/k⟂ replaced by 〈k∥/k⟂〉RMS in that case, assuming that all the waves in the packet propagate with similar phase speeds (an assumption motivated by the findings of Oppenheim and Dimant ). Large magnetic aspect widths are required to guarantee marginal growth when the electron convection is rapid and also give rise to significant wave heating, elevated electron temperatures, and elevated ion-acoustic speeds. Consequently, there is a link between the convection speed and the wave frequency, albeit a circuitous one.
Milikh and Dimant  used this model to predict electron temperature enhancements observed by incoherent scatter radars as a function of the magnitude of the background convection electric field [see Foster and Erickson, 2000 and references therein]. While suprathermal electrons were produced by convection faster than about 1000 m/s, the problem remained essentially thermal at slower speeds.
 Additional support for the heuristic model comes from the particle-in-cell simulations of Farley Buneman waves described by Oppenheim et al. , Dimant and Oppenheim , and Oppenheim and Dimant . The electric fields associated with the simulated waves were found to be comparable to the background driving electric field under conditions of strong forcing. The simulated waves furthermore exhibited marginal-stability and phase-speed saturation at the ion-acoustic speed, provided the correct ratios of specific heat for electrons and ions are taken into account. Finally, the 3-D simulations showed that turbulent modes with finite parallel wave numbers play a crucial role in the wave heating and saturation.
 However, other recent theoretical and experimental developments would seem to be contrary to the heuristics and assumptions outlined above. Summarizing the results from a number of auroral sounding rocket flights, Bahcivan and Cosgrove  argued that the relative amplitudes of the published AC electric field perturbations in the auroral E region seem to be much less than the corresponding background (DC) field amplitudes in nature. This creates a “missing energy” problem because large wave-driven electric fields are required to account for the electron heating observed by the incoherent scatter radars according to existing theory. A resolution to the problem was offered by Bahcivan and Cosgrove  who proposed that the finite vertical background density gradient scale length in the auroral ionosphere essentially modifies the linear dispersion relation for Farley Buneman waves such that long wavelengths (tens of meters rather than a few meters) with significant, finite parallel wave numbers dominate. Such waves would be more efficient heat sources than waves in a homogeneous background ionosphere.
 Additionally, bistatic radar observations from the second RAX satellite calling into question the premise of aspect angle broadening have recently been presented (Bahcivan et al., Magnetic aspect sensitivity of high-latitude E region irregularities measured by the RAX Cubesat, submitted to Journal of Geophysical Research, 2013). The authors reported an observation of UHF coherent scatter from auroral zone Farley Buneman waves during a period of intense convection with magnetic aspect widths essentially comparable to what has been observed in the equatorial ionosphere at VHF. Aspect angles were calculated using an incisive procedure that accounts for the apparent broadening of the magnetic aspect width due to the finite thickness of the scattering layer. This observation suggests that Farley Buneman waves with short wavelengths seen by UHF radars play no significant role in wave heating. Is wave heating therefore to be attributed to the longer-wavelength waves seen by VHF radars, to even longer-wavelength waves, or to something else entirely?
 In order to begin resolving these questions, we evaluate the implications of and adaptation of the heuristic model of Dimant and Milikh  and Milikh and Dimant  outlined above, following the basic strategy pursued by Michhue . We quantify the electron and ion heating rates, Farley Buneman wave-phase speeds, and aspect angle widths implied by the model, including the effects of chemistry, temperature-dependent collisions and recombination rates, electron and ion thermal effects, and parallel transport. Most importantly, the altitude profiles implied by the model are calculated. This is essential for realistic model/data comparison. The goal is not to argue the philosophical merits of the model but to spell out its implications, which are testable, and to compare with experimental results, where possible. We discuss how this model, or models like it including whole profile predictions, is essential for interpreting any radar observations of auroral Farley Buneman waves generally.
 The linear, local dispersion relation for Farley Buneman waves can be written in the form [Farley and Providakes, 1989]
where σi,e are the ion and electron conductivities, respectively, and ω=ωr+iγ is the complex wave frequency. Following Farley and Providakes , we adopt a kinetic treatment for the ions and include ion-neutral collisions using a Bhatnagar-Gross-Krook collisional operator, yielding
where Z is the plasma dispersion function, ξ≡(νin/k)/vti is the ion-neutral collision frequency normalized to the ion thermal velocity, Θ≡(ω/k)/vti is the wave frequency normalized to the ion thermal velocity, and x≡−Θ+iξ.
 An expression for the electron conductivity based on fluid theory and incorporating the effects of frictional heating and cooling through inelastic collisions was derived by Hysell et al. . We generalize that expression hereby also allowing for waves with finite magnetic aspect angles [Michhue, 2010]
Here is the wave frequency in the neutral frame of reference through which the electrons are streaming at velocity Vd. The auxiliary variable is related to the electron ratio of specific heats, is related to frictional heating, both are related to collisional cooling, and δen is the fractional electron energy loss rate through inelastic electron neutral collisions. Adapting Gurevich , we use δen=4.8×10−3(185/Te)3/2.
 The parameter represents dispersion and damping associated with finite wave parallel wave numbers. Dispersion associated with phase-speed variations with magnetic aspect angle is absent in the three-dimensional particle-in-cell simulations of Oppenheim and Dimant , which depict Farley Buneman waves propagating in coherent packets with correlation times very long compared to their periods. Evidently, mode coupling maintains phase lock between wave packet components with different parallel wave numbers. We can therefore replace the term in α with the average for a wave packet with the assumption that this term governs the behavior of the entire packet. Assuming that the parallel wave numbers within the packets are small, we can take . Note that α is the ratio of the anisotropy factor ψto the anisotropy factor for purely perpendicular propagation ψ⟂=νenνin/ΩeΩi.
 Given a specification of the background state parameters, including the electron convection speed, and for a given wavelength, (1) can be solved for the frequency and aspect width giving marginally unstable (γ=0) waves. We solve locally at every altitude using a modified Powell's hybrid method.
 We can now estimate electron and ion heating rate profiles, respectively, using the following heuristic formulas:
where σ∘e and σpi are the electron direct and ion Pedersen conductivities, respectively, and B is the magnetic induction. Also, vde′ is the electron E×B drift speed in the frame of reference of (partially) E×B-drifting ions and v∘ is the minimum value of vde′ required for instability. We regard (vde′−v∘)Bas an estimate of the transverse, wave-driven electric field, following the Dimant and Milikh  prescription. Multiplying by the RMS aspect angle width yields our estimate of the wave-driven parallel electric field, as in Dimant and Milikh .
 The Ecn term in (5) is the convection electric field in the neutral frame of reference. It is well known that the Joule heating rate in the auroral zone depends crucially on the wind speed and direction and that the winds can both increase and decrease the heating [e.g., Thayer, 1998]. We are ill-prepared to fully account for the effects of varying wind speeds on the Joule heating rate here. However, as Joule heating occurs mainly above the altitude where Farley Buneman waves propagate, we can to some degree neglect this consideration for the time being. For the purposes of this study, we somewhat arbitrarily take Ecn to be half the field associated with the convection speed. We will eventually argue that the properties of auroral Farley Buneman waves turn out to be largely independent of the Joule heating-induced ion temperature variations.
 On the basis of the electron continuity equation, the quasineutrality condition, and the amplitude of the transverse wave electric field assumed above, we can estimate the amplitude of the electron density perturbation associated with Farley Buneman waves
From this, we can postulate an altitude averaging kernel K∝|δn|2. Such a kernel is necessary for estimating the characteristics of radar signals scattered from Farley Buneman waves occupying a range of altitudes. It is therefore essential for comparing model output with some radar-derived data sets. Note that this kernel depends on the background density profile, which varies significantly in nature. Modeling nighttime E region density profiles is a notoriously difficult problem, even under quiet geomagnetic conditions in the absence of particle precipitation. The results presented below are consequently only representative. Other factors may influence those characteristics as well, notably the finite magnetic aspect width of the waves (see below) and also refraction, which can complicate matters tremendously and is beyond the scope of the present study.
 Lastly, the electron and ion heating rates are introduced to a modified version of the NRL SAMI2 model [Huba et al., 2000]. We use SAMI2 to solve for equilibrium electron and ion temperature, ion composition, and electron density profiles, incorporating the effects of heating, cooling, chemistry, temperature-dependent collision frequencies and rate constants, and parallel transport. Seven ion species are considered—, NO+, , O+, N+, He+, and H+. Of these, the first two dominate in the E region, accompanied by small amounts of O+. We run a one-dimensional version of the model, seeking solutions along a single magnetic flux tube at auroral latitudes during winter conditions at night. Our version has been modified and now includes fine grinding in the E region, the aforementioned provisions for auroral electron and ion heating rates, and complete heat equations of all the ion species. Equilibrium profiles are found for different electron convection speeds ranging between 400 and 1000 m/s.
3 Model Results
 Figure 1 shows the results of the heuristic model computed for Farley Buneman waves with 5 m wavelength. Model runs for 16 different electron convection speeds ranging from 400 to 1000 m/s are shown. Profiles for the lowest convection speeds reflect background conditions. Figure 1 (left) shows electron density, electron temperature, and ion temperature profiles. The elevated electron and ion temperatures result from wave and Joule heating, respectively. For these runs, as mentioned above, we assume, somewhat arbitrarily, that the neutral wind speed is half the ion drift speed. Different choices lead to different families of curves for Ti, naturally (see below). Where the electron temperatures are elevated, the electron density is increased, a consequence of the temperature dependence of the recombination rate.
 Figure 1 (middle) shows phase velocity and RMS aspect width profiles consistent with marginal wave growth. Both quantities increase with increasing convection speed. Figure 1 (right) shows electron and ion heating rates for different convection speeds. Also shown is the relative intensity (amplitude squared) of the density fluctuations of the Farley Buneman waves versus altitude (labeled as K for averaging kernel).
 Finally, Figure 1 (bottom) shows kernel-averaged wave-phase speeds and RMS aspect angle widths. These quantities are expected to be related to, but not identical to, radar-derived measurements of the same quantities. The vertical lines drawn through the phase-speed markers span the range of possible ion-acoustic speeds (kernel averaged). The bottom of each line corresponds to the case of isothermal electrons, and the top to adiabatic electrons with 1 degree of freedom. (The ions are taken to be adiabatic with 3 degrees of freedom.) The markers evidently move from the bottom of the lines upward as phase speed and frequency of the waves increase.
 Notice first that the electron and ion temperatures show the most elevation above the altitudes where heating is concentrated. This is because the electron and ion cooling rates also decrease with altitude. The ion-acoustic speed increases sharply with altitude, particularly above about 110 km where the electron and ion temperatures are the most elevated. Since the diffusion rate is proportional to the ion-acoustic speed, more diffusive damping consequently occurs at altitudes above 110 km than below. This means that more damping due to aspect width broadening is required at the lower altitudes to maintain the condition for marginal wave growth. Hence, the aspect widths are greater at lower E region altitudes.
 Since the waves are largely excluded from altitudes above 110 km by the enhanced diffusion there, the averaging kernel peaks below this altitude. In fact, the peak height of K descends with increasing convection speed. This tends to suppress the growth of 〈Vφ〉 with increasing convection speed while enhancing the growth of 〈θRMS〉. The curve for 〈Vφ〉 is consequently somewhat flat for convection speeds below about 700 m/s and then rises more quickly as the convection speed continues to increase. The curve for 〈θRMS〉 starts to approach a value of about 1 degree asymptotically at fast convection speeds.
 Different choices of neutral wind speeds give rise to different ion temperature curves, with smaller neutral winds resulting in larger ion temperatures. Farley Buneman waves tend to be excluded from regions with high ion temperatures, however, and the averaging kernel K consequently excludes high-altitude portions of the E region where Joule heating is made to be strong. Consequently, kernel-averaged quantities like those plotted in Figure 1 (bottom) tend to be surprisingly resilient and insensitive to ion temperature changes associated with Joule heating. The modeling results would seem to be fairly robust even in the absence of accurate neutral wind specification.
 We do not necessarily expect the markers in Figure 1 (bottom) to reflect experimental results from radar experiments, however. This is because the finite aspect width of the irregularities will weight the echoes from different altitudes somewhat differently in experiment, in concert with K, depending on the experimental geometry. Echoes arising closest to the locus of perpendicularity (the altitude where the condition for field-aligned radar backscatter is exactly satisfied) will be weighted the most heavily. As the aspect widths are generally greater below 110 km than above it, the weighting will tend to favor lower altitudes, since those will be attenuated the least when the condition for field-aligned backscatter is not satisfied exactly. The kernel-averaged values shown in Figure 1 (bottom) are consequently only shown for purposes of comparison.
 The aspect sensitivity of the irregularities will also affect aspect sensitivity measurements in an even more complicated way (see discussion below), and so the values shown in Figure 1 (bottom) are presented just for the sake of completeness. However, the information in the curves in Figure 1 (middle) will influence both Doppler frequency and aspect sensitivity measurements.
 Figure 2 shows modeling results calculated for Farley Buneman waves with 1 m wavelengths. Such waves are subject to more Landau damping than 5-m waves and so achieve marginal stability at lower temperatures and narrower magnetic aspect angle widths. The waves tend to be excluded from high altitude, high temperature regions, and so the averaging kernel has a maximum near or below 105 km. The electrons undergo less heating for a given convection speed, and the associated photochemical effects are more subtle. The kernel-averaged-phase speeds and magnetic-aspect widths follow similar trends but have smaller values than the 5 m waves. In accordance with their higher frequency, these waves are generally closer to being adiabatic than the 5 m waves.
 Figure 2 (bottom) shows the empirical relationship between convection speed and phase speed for Scandinavian Twin Auroral Radar Experiment (STARE) radar echoes during eastward flows found by Nielsen and Schlegel . The radar echoes examined by Nielsen and Schlegel  were dominated by contributions from about 105 km altitude. The agreement between the theoretical and empirical models is rather good, although the influence of the scattering geometry on the radar echo characteristics has not been explicitly accounted for here.
 Similar results can be obtained from the model for primary waves with 35 cm wavelengths. Such waves are linearly unstable only for convection speeds above about 550 m/s under our assumed conditions. The waves exhibit relatively small amplitudes and aspect widths and cause relatively little electron heating compared to their longer-wavelength relatives. The waves are essentially adiabatic and have phase speeds of about 550 m/s for all the unstable convection speeds considered. Since electron heating is minimal, the phase speed varies little with convection speed. Magnetic aspect angle widths approach about 0.5° for Vd = 1000 m/s in our model.
 We do not show the model results for the 35 cm waves because they are irrelevant. The elevated electron temperatures caused by the dominant meter-scale Farley Buneman waves that would inevitably accompany the 35 cm waves would cause them to be linearly stable. The only way to excite such waves would be through nonlinear mode coupling to the linearly unstable waves. This process is entirely outside the scope of this modeling exercise. The simple heuristic model being explored here has no implications for coherent echoes observed by UHF radar.
 The main findings of this modeling study are the following:
 The span of altitudes over which the model predicts meter-scale Farley Buneman waves to be excited is consistent with rocket experiments.
 The electron temperature enhancements for meter-scale primary Farley Buneman waves at 112 km (where Te is a maximum) are consistent with the measurements of Schlegel and St.-Maurice , which is still the most extensive record of wave heating in the domain of low convection speeds. The modeled electron temperature profile shapes are also consistent with measurements [Schlegel and St.-Maurice, 1981; St.-Maurice et al., 1981; Igarashi and Schlegel, 1987; Stauning and Olesen, 1989]. There is considerable scatter in the experimental record, however, and a more exhaustive experimental study needs to be conducted for the model to be validated.
 Similar remarks hold for the radar Farley Buneman wave-phase-speed measurements reported by Nielsen and Schlegel  and the predictions are shown in Figure 2, although the exceptional agreement is partially fortuitous, since the experimental effects of magnetic aspect sensitivity have not been folded into our analysis. Furthermore, since a broadband spectrum of Farley Buneman waves is thought to be responsible for auroral heating, neither Figure 1 nor Figure 2 can be expected to fully capture the phenomenon. A model of broadband heating needs to be considered.
 The Dimant and Milikh  model equates the transverse RMS electric fields of Farley Buneman waves with the background convection field, less the threshold field required for instability. This prescription is not inconsistent with rocket data showing wave fields smaller than background fields, particularly when the instrument function of double-probe field sensors (which implies strong attenuation for wavelengths comparable to the boom length) is taken into account [Kelley and Mozer, 1973].
 The heuristic model only applies to linearly unstable waves. It makes no predictions for waves observed with UHF radars, their phase-speeds or magnetic-aspect widths, since those waves should be linearly stabilized by wave heating by linearly unstable waves.
 The ion-acoustic speed depends on electron and ion temperature which can be altered, respectively, by wave and Joule heating and heat transport, effects which must be considered self consistently. The ratio of specific heats of the electrons moreover depends on wave frequency and, consequently, on the forcing and also on the probe wavelength, in the case of radar measurements. Assessing whether Farley Buneman waves actually propagate at the ion-acoustic speed necessitates a forward modeling effort like this one.
 The analysis above highlights the importance of the finite thickness of the layer inhabited by Farley Buneman waves when testing the model against observations. All of the important wave characteristics vary with altitude, and according to the model, radar measurements should reflect averages over broad profiles, weighted by a number of factors. Some of these factors depend on experimental geometry and conditions and cannot be accounted for here. This is particularly true of magnetic aspect angle and aspect sensitivity measurements, which are at once incisive tools for testing the model and potentially misleading outside of a modeling context. This issue is discussed in more detail in Appendix A.
 To appreciate the universal importance of the finite layer thickness, as predicted by the model, on experimental outcomes, consider the power p(r) received by a conventional, ground-based radar observing coherent scatter from Farley Buneman waves at some range r
Here Ω is shorthand for bearing (azimuth and elevation), dΩis differential solid angle, B is the radar two-way antenna gain, I is the absolute intensity of the irregularities as a function of altitude z, and A gives the magnetic aspect angle dependence of the backscatter. The altitude z is determined by range and bearing, z = z(r,Ω). Also, Θ is the angle the radar ray path makes with the geomagnetic field, which is also a function of range and bearing. In this formulation, both I and A depend explicitly on altitude, in accordance with our model. The C1 is a system constant.
 In principle, information about the aspect sensitivity of the irregularities can be derived from measurements of power made in different volumes where the main beam of the antenna intersects the geomagnetic field at different angles. If the beam of the probing radar were exceptionally narrow, or if it were narrow in azimuth and the irregularities were confined to a very thin layer in altitude (slab model), then the product of B and I would effectively form a delta function in bearing, and the received power versus range would become a proxy for the aspect angle dependence of the irregularities, i.e., p(r)∝A(Θ(r)). In view of the model results above, the slab model in particular is poorly justified. In practice, the thickness of the irregular layer should broaden the span of ranges through which significant backscatter power will be observed, at least for radars without exceptionally narrow beams in elevation observing nearby targets.
 In fact, all aspects of the waves observable by conventional radar have to be evaluated in the context of finite layer thickness if they are to be interpreted correctly. For example, formally, the observed Doppler velocity of the echoes can be expressed as
The measured phase speed, 〈vφ(r)〉, represents the wave-phase-speed profile, vφ(z), weighted by the antenna beam shape B(Ω), the irregularity intensity profile I(z), and by magnetic aspect sensitivity considerations A(z,Θ) itself influenced by experiment geometry through Θ(r,Ω). Even if phase speed does not vary with magnetic aspect angle, as has been presumed throughout this entire modeling effort, the observed phase velocity would be expected to vary with antenna pointing (i.e., bore sight pointing with respect to the magnetic field) because contributions from different altitudes will be weighted differently. Distinguishing the various influences on experimental observations necessitates forward modeling.
 In writing (A2), we make use of the fact that the first moment of a spectrum resulting from an average is the average of the first moments of the component spectra. This is the basis for the notion of the measured phase speed representing an average of the phase speeds of waves propagating at different strata. The same relationship holds for all the moments of the spectrum. It is not true for spectral width, which is derived from a nonlinear combination of the first and second spectral moments. Spectral width predictions based on this theory are therefore not straightforward and will require considerable care to formulate.
 As first recognized by Bahcivan et al. (submitted manuscript, 2013), similar remarks hold for bistatic aspect sensitivity measurements of the kind made by the RAX satellite. In this case, the radar is bistatic, and the illuminating radar has a sufficiently narrow beam to provide the delta function in bearing. In (A3), we write the power received by the satellite as a function of its position s along its trajectory. The C2 is another system constant. As before, the layer thickness acts to extend the span of locations at which the satellite should receive significant echo power. The authors of the aforementioned study estimated and deconvolved the I(z) altitude dependence to arrive at their aspect width estimates.
 Finally, it should also be possible to estimate the magnetic aspect width using interferometry, as has been done in the equatorial zone already. Given an appropriate experimental geometry, interferometry provides an estimate of 〈Θ2〉−〈Θ〉2. These moments can be expressed formally as in (A4). At the magnetic equator, range and altitude are practically interchangeable, I(z) is essentially uniform over all bearings of interest, and (A4) gives the moments of the aspect angle given the averaging kernel determined by A. At high latitudes, different radar bearings span different altitudes, and the tendency will be for the finite span I(z) to make the averaging kernel more compact. This will reduce the apparent aspect width of the irregularities, as determined from interferometry. The effect is opposite to that described immediately above. At large ranges, discriminating the finite aspect width of the irregularities from their finite altitude extent interferometrically could prove very challenging.
 In any event, knowledge of the functions I(z) and A(z,Θ) is sufficient to predict the outcomes of the aforementioned aspect sensitivity measurements. Inferring these functions from the measurements requires the application of inverse methods. In this paper, we have proposed a means of estimating these functions based on a combination of theory and heuristics. They can serve as starting points for data inference and model/data closure.
 This work was supported by NSF award AGS-1042057 to Cornell University. The research of JDH was supported by NRL Base Funds.