Estimation of source altitude through a linear fit of the time-of-flight (TOF) slope of energy–time dispersions is a method widely used in the literature e.g.. It assumes that electrons of different energies are released simultaneously from a single altitude, so that differences in arrival time arise only from the differences in time-of-flight between electrons with different velocities. This assumption does not accurately represent the expected spatial and temporal characteristics of a wave-particle interaction, which is thought to happen over a wide range of altitudes as DAWs propagate downward toward the ionosphere . These limitations are known e.g., but the method remains useful as it provides a single characteristic distance that can be compared across events and studies. Using the Akaike Information Criterion (AIC), showed that a 2nd order polynomial (quadratic curve) is frequently a better fit to the dispersion curves of the delay in arrival time Δt over the inverse of the electron velocity (1/v) than the classical linear fit. This suggests that the relationship between arrival time and inverse velocity is not strictly linear, which is expected if acceleration occurs over a range of altitudes rather than at a single point . The fitted curve can then be differentiated to obtain a range of source altitudes varying with electron energy, though this still implies a simultaneous release at all energies over the range.

As shown in Fig. , numerous distinct time–energy dispersed structures are visible in the VISIONS-2 data, to which the TOF fitting methods described above can be applied. Figure  presents an illustrative example using one event identified at around 338 s (Fig. b). Figure a shows the energy–time dispersed structure. Measurements on either side of the structure (displayed with higher transparency) are excluded from the analysis, as we attribute them to adjacent precipitation. For each energy slice, the electron arrival time is determined by calculating the flux-weighted mean time. These mean values are marked with white dots in Fig. a, and the white horizontal bars indicate the width in time of the structure as given by the square root of the weighted second moment calculated around each mean (standard deviation). Figure b shows the same mean arrival time, Δt, and estimated widths, plotted in Δt over 1/v space. The blue curve shows the best linear fit of the form Δt=a+b(1/v), while the red curve shows the best quadratic fit of the form Δt=a+b(1/v)+c(1/v)2 where a, b and c are the best fit parameters, following the approach proposed by . In addition, we fit a logarithmic curve of the form Δt=a-bln⁡(1/v), shown in green in Fig. . This curve corresponds to a straight line in an energy-time space plotted on a semi-log scale such as in Fig. a, which seems to visually correspond well with several events in our dataset. Arrival times for energies under 70 eV are excluded from the analysis. Measurements at these lower energies are generally noisier, making the identification of well-defined peaks in arrival time more challenging. Tests made by varying the cut-off value from 50 to 100 eV did not change the overall trend of the results significantly, indicating that the analysis is not strongly sensitive to the exact value of this threshold. A more detailed discussion of this choice is provided in Sect. .

Following and , a range of source altitudes can be estimated using the function F relating Δt and 1v through Δt=F1v. Taking the derivative dFd1v provides an estimate of the source distance at any given 1/v via a local linear TOF approximation of the form Δt=d1v, where d corresponds to the slope of the tangent of F at that point. For the linear fit, this slope is constant, yielding the same source altitude for all energies. In contrast, the slope varies with 1/v for the non-linear fits, resulting in different altitudes for different energies. Both methods, however, still rely on the assumption of an instantaneous electron release over the inferred altitude range.

Although the energy–time linear TOF method as described in the previous section is frequently used, fewer studies have exploited pitch-angle–time dispersions for source altitude estimation. The advantage of such a method would be that, as long as the dispersions are resolved across multiple energy channels, the source altitude can be estimated independently for each electron energy. This is a relaxation of the assumption of simultaneous release across energies that underlies the energy–time methods described in the previous section. For instance, attempted to estimate source altitudes from the different arrival times of electrons at 0° and 30° pitch-angle, but the resulting source altitude estimates carried large uncertainties (several thousand kilometers), which were attributed to insufficient time and angular resolution. They concluded that measurements with higher resolution were necessary. The time resolution of our measurements (∼ 50 ms) is similar to theirs (∼ 41 ms), but the angular resolution of the EESA onboard VISIONS-2 (see Table ) allows us to resolve clear pitch-angle–time dispersions across a higher number of pitch-angle channels (from -110 to 110°) as seen in Fig. .

Here, we test and further expand on the idea of using pitch-angle–time dispersions by developing and using a simple forward model, with two main components.

The first component simulates the propagation of electrons from a source altitude zsource to a measurement altitude zmeas. The electrons, all with the same kinetic energy, are launched simultaneously from zsource with a range of pitch-angles. The magnetic mirroring force is accounted for by conservation of the magnetic moment, using the IGRF-14 model for the magnetic field strength, which is close to a dipole field model over the heights of interest. As such, the pitch-angle is allowed to evolve along the particle trajectory, which can change the total path-length.

The second component models the detector response. A Gaussian with width σt is centered on the computed time-of-arrival of each electron, representing an injection at source altitude extended in time. Electrons arriving within a certain pitch-angle and time window are then integrated to produce a simulated total count for each detector pixel. The time windows are 1 ms wide and repeated every 50 ms to reproduce the behaviour of the instrumental energy sweep. The pitch-angle window widths match the angular resolution of the instrument channels. Since the absolute time of injection is unknown, a free offset parameter toffset is applied uniformly to all arrival times.

For each energy channel, electron trajectories are computed and the three model parameters zsource, σt and toffset are varied. The advantage of this approach is that, unlike with the energy-time dispersion fits, each energy bin will have an independently fit release height and time. The best fitting values are found by minimizing the sum of squared errors (SSE) between the normalized observed and modelled pitch-angle–time dispersions. Because the intensity distribution across pitch-angle in the source population is not constrained by the model, each pitch-angle channel is normalized by its total flux count before computing the SSE. This normalization also mitigates potential differences in sensitivity between pitch-angle channels in the observed data. As a result, the optimization compares only the morphology of the pitch-angle–time dispersions, rather than their absolute amplitudes.

In this work, we restrict the forward model to the down-going pitch-angle channels (-70, -45, -20, -5, 0, 5, 20, 45, 70°). As mentioned above, the dispersed structures are typically visible in a higher number of channels, from -110 to 110°, which also includes up-going channels (mirroring electrons). As a general rule, more channels lead to better constraints on the forward model. However, because the Alfvénic precipitation is strongly field-aligned, the differential flux decreases with increasing pitch-angle. Additionally, the channels of this specific EESA are narrower in angular width close to -90 and 90° (Table ). The resulting lower signal-to-noise ratio (SNR) of these channels could potentially degrade the quality of the fits rather than improve them, and it was decided not to include them. We note that future instruments with higher sensitivity might be able to exploit the information contained in these high pitch-angle and up-going channels and further improve the quality of the results.

An example of the application of the forward model is shown in Fig. . The event is the same as the one whose energy–time dispersion was presented in Fig. , and the energy slice considered here is taken at 358 eV. Figure a shows the observed pitch-angle–time dispersion after normalization, while Fig. b presents the best-fit modelled dispersion. Figure c–e illustrate the variation of the SSE as a function of each of the three model parameters zsource, σt and toffset, with the remaining two parameters held fixed at their best-fit values. The best-fit value of each parameter is indicated by a vertical dashed line and an annotation. Finally, Fig. f shows the mapping between pitch-angles at the source altitude and those at the measurement altitude for the best-fit solution. In this example, electrons measured with a pitch-angle of 60° at the rocket altitude started with a pitch-angle of approximately 40° at the inferred source altitude.

Once the best-fitting parameters are found, the uncertainty in the estimated source altitude is obtained through a linearised uncertainty estimation . The parameter covariance matrix Σp is calculated as Σp=σr2(JTJ)-1 where σr is the root mean square difference between the observed and best-fit modelled dispersions, and J is the Jacobian matrix of the model evaluated around the best-fit parameters. The standard error on the source altitude is then the square root of the first diagonal element of Σp.

To infer at which altitude the acceleration of the electrons took place, we have used two types of methods, one based on energy–time dispersions and one on pitch-angle–time dispersions. Both approaches are subject to uncertainties, which we discuss below.

The first source of uncertainty relates to the finite resolution of measurements. The energy–time dispersions methods rely on fitting the arrival times of electrons at different energies. However, as can be seen in Fig. , the observed structures are often only a few pixels wide in time at each energy. Although we mitigate this by calculating a weighted mean, the limited time resolution still introduces uncertainty on the precise arrival time of each energy. We expect a higher temporal resolution to better constrain the dispersion fits and the inferred source altitudes. Similarly, the pitch-angle–time dispersion method is also affected by measurement resolution. This is visible in Fig. , where the uncertainties generally increase with source altitude (e.g. the large error bars in the 500 s event in Fig. f). This reflects the reduced sensitivity of the pitch-angle–time dispersion shape to source altitude at large distances. Higher resolution measurements in both time and pitch-angle would better constrain the forward model and reduce these uncertainties.

A second source of uncertainty concerns the interpretation made of the dispersed structures themselves: do they reflect temporal variations in the precipitation, or spatial structures sampled by the moving spacecraft. This distinction is a long-standing question in in-situ observational studies of auroral precipitation e.g.. During the measurements, the rocket had a horizontal velocity of 1–2 km s⁻¹, implying that it covered spatial distances of roughly 0.5 to 1 km over the typical duration of these structures (< 0.5 s). Previous multi-point measurement studies have reported simultaneous measurement of similar precipitation signatures across payloads separated by ∼ 0.9–3 km , suggesting that these structures extend over larger spatial scale than the displacement of the rocket and therefore supporting a temporal interpretation. In addition, the coincident observation of dispersions in both energy and pitch-angle (Fig. ) further strengthens a temporal origin , as a purely spatial interpretation would imply simultaneous change across all pitch-angles.

A third source of uncertainty arises from the assumptions underlying each method. When deriving source altitudes from the energy–time dispersion fits, it is assumed that field-aligned electrons are released simultaneously over the whole source region, which is not fully consistent with the expected picture of a moving wave-particle interaction . For the linear fit case, commonly used in the literature, this release is further assumed to occur from a single point. Similar to and , we find that the non-linear fits of the energy–time dispersions are generally preferred over the linear fits, supporting the case of a wider range of source altitude. Nevertheless, we also observe that the point source altitudes inferred from the linear fits typically fall near the middle of the altitude ranges derived from the non-linear fits (Figs. , ). As such, it seems that the linear fit can provide a convenient first-order estimate, straightforward to compute, and that can be used for a quick comparison across events and studies.

In order to relax the assumption of simultaneous release associated with the energy–time dispersion fits, we developed and applied a second method based on the pitch-angle–time dispersions. This approach allows the source altitude to be estimated independently for each electron energy, and instead assumes the instantaneous release of a broader pitch-angle distribution. We expect this assumption to be more accurate the higher the source altitudes. Indeed, at greater source distances, the injected electrons have a narrower pitch-angle distribution and are therefore more likely to originate from a similar altitude. For instance, the source altitude of the 4300 eV electrons of the 500 s event (Fig. f) is inferred to be around 6400 km, with pitch-angles at the source ranging from 0 to 25°, meaning that all the electrons had a field-aligned velocity component between cos⁡(25°)≈0.9 and 1 of their total speed.

Despite the aforementioned uncertainties, we found both approaches, each with different underlying assumptions regarding electron injection, to yield comparable ranges of source altitudes across events (Figs.  and ). This agreement between the two types of methods increases confidence in the inferred source altitudes.

Having discussed the robustness and validity of the inferred source altitudes, we now examine how these results compare with previous observational studies and theoretical expectations.

The source altitudes inferred both from the energy–time and pitch-angle–time dispersions methods show some variability across events (Figs.  and ), despite being observed during the same rocket flight. This suggests differences in plasma conditions and/or wave parameters along nearby magnetic field lines. Whether these differences arise from local temporal variations or a larger spatial structuring of the acceleration region remains unclear due to space-time ambiguities that cannot be resolved with a single sounding rocket. It is expected that conditions along field lines can evolve following earlier auroral precipitation , but also that these regions can present strong spatial structuring, such as in the ionospheric Alfvén resonator (IAR) .

Despite this variability, our observations generally indicate source altitudes of ∼ 1000–3000 km for electrons with energies of a few hundred eV, and ∼ 2000–5000 km for electrons with energies of a few keV. The general trend is for high-energy electrons to be released at higher altitudes than low-energy electrons, consistent with expected Alfvén velocity profiles along magnetic field lines . However, these values lie on the lower end of theoretical predictions e.g., especially for the dayside .

For comparison, from their observations also made on the dayside at a similar MLT (around 13.3 MLT) with a sounding rocket, inferred source altitudes of ∼ 2000–3000 km for ∼ 30 eV electrons and ∼ 5000–6000 km for ∼ 200 eV electrons using a non-linear (quadratic) fit method. These values are more consistent with the theoretical estimations of . However, the maximum electron energies in their dispersed structures were generally lower (a few hundred eV) than in our observations (up to several keV), suggesting different plasma and wave conditions in the acceleration-region between our studies.

When compared with source altitudes reported in previous studies (Table ) from linear fits of energy–time dispersions, our results also fall on the lower end of the observed range. Using the same linear fitting approach, our estimates typically lie between 1000 and 3000 km (blue crosses in Fig. ). However, similar source altitudes have been reported in as well as and for events measured on the nightside and reaching several keV. For comparison with the other results reported in Table , the VISIONS-2 rocket was at altitudes between 480 and 600 km and around a MLT of 13.5 during the observations of Fig. .

The results of were obtained using data from the Acute Precipitating Electron Spectrometer (APES) instrument onboard the same rocket as in our study. By design, their instrument resolves primarily the highest energy part of the dispersions . The source altitudes of the 8 events they report range from approximately 1700 to 6300 km, which is consistent with the higher-energy portion of the source altitudes we inferred from the non-linear energy fits and pitch-angle approaches presented here.

Overall, the wide range of source altitudes reported across studies (Table ), from ∼ 500 km to several RE, does not appear to clearly correlate with the spacecraft type (rocket vs. satellite), the spacecraft altitude, or the magnetic local time (MLT) of observations. Similar to the event-to-event variations observed in our data, these differences suggest different plasma and wave conditions along the magnetic field lines.

To further interpret our inferred source altitudes in terms of the plasma environment along the magnetic field lines, we compare our observations with theoretical expectations derived from different plasma density profiles in the acceleration region reported in the literature. Figure a shows the density profiles. All the profiles are based on the combination of an ionospheric exponential law for O+ and a magnetospheric power-law component for H+, with different parametrizations. The parametrizations of , and result from statistical fits to spacecraft observations, whereas the parameters used by and were chosen to study specific plasma environments. In particular, the two profiles from represent conditions inside and outside a density cavity. Figure b shows the magnetic field strength along the field line passing through the general rocket measurement region (taken at 76° N, 3° E and 100 km altitude), computed using the IGRF-14 model . Figure c and d show the corresponding electron skin-depth λe=c/ωp and Alfvén velocity profiles (solid lines), together with inertial Alfvén wave velocity profiles (dashed lines). The inertial Alfvén velocities were computed assuming a perpendicular wavelength λ⟂0=5 km at 200 km altitude following, scaled with 1/B under the assumption of constant magnetic flux. This perpendicular scale should correspond to auroral structures with optical widths of ∼ 1–2 km . Figure e compares these theoretical profiles with the source altitudes inferred from our observations, with the inertial Alfvén wave velocities expressed as equivalent electron kinetic energies E=12mevAI2, which corresponds to the assumption that electrons are accelerated to approximately the local inertial Alfvén speed. As discussed in the introduction, theoretical and modelling studies have shown that electrons can be accelerated through resonant interaction to parallel velocities greater than 1VA, up to 2VA-v∥0 e.g., while we also expect electrons within the resonance window defined by the wave parallel electric field E∥ to be possibly released before reaching the wave speed, i.e. at energies slightly below 1VA. The curves of Fig. e should therefore be regarded as characteristic equivalent energies rather than strict bounds. The scatter points represent all source altitude estimates from Fig.  as a function of electron energy, derived from the pitch-angle method (purple), the quadratic fit (red), and logarithmic fit (green).

Using the parameters described above, the inferred source altitudes from our events are consistent with the IAW velocities predicted using the density profiles of , , , and the cavity profile of . A smaller number of events are also consistent with the profile of , which is the results of a statistical fitting of dayside measurements. In contrast, almost none of our source altitudes follow the outside-cavity conditions profile of . Taken together, the source altitudes inferred from our events are most consistent with profiles with the lowest O+ scale heights, leading to a transition to H+ dominated plasma at lower altitudes. If the plasma on surrounding field lines has a higher O+ scale height, e.g. as in the statistically fitted dayside profile from , measurements from a satellite horizontal fly-by would resemble a locally depleted density region. This picture would be consistent with previous studies reporting observations from satellites between 1000 and 4000 km of altitude of Alfvén waves propagating inside density cavities and carrying intense parallel electric fields . The coupling between density cavities, Alfvén waves, and intense parallel electric fields has indeed been investigated and seemingly established in numerical studies .

The comparison between inferred release altitudes and the Alfvén-wave phase velocity profiles of Fig.  suggests it is possible to fit the inferred energy, time, and height obtained from the pitch-angle dispersion fits described in Sect. 3.2 to the acceleration region parameters that influence the inertial Alfvén wave velocity: O+ and H+ density profiles and perpendicular wavelength. We can build such a fit assuming a plasma density profile in the topside ionosphere of the following form 2nO+=n0e-z/H3nH+=n1z+RERE-14ne=nO++nH+ with n0 the O+ density constant, n1 the H+ density constant, H the scale height of O+, RE the radius of the Earth, and z the geodetic height (vertical distance above ground). Quasi-neutrality is assumed, so that the electron density ne is determined by the total ion density. The inertial Alfvén speed is defined as 5vAI=VA1+k⟂2λe2 where VA=B/μ0(nO+mO++nH+mH+) is the pure MHD Alfvén velocity determined by O+ and H+ mass density and the IGRF magnetic field strength B, and λe=c/nee2/meϵ0 is the plasma skin-depth. The perpendicular wavenumber 6k⟂=2πλ⟂0BB0 is fixed by a perpendicular wavelength λ⟂0 and magnetic field strength B0 defined at an altitude of 200 km and scaled by 1/B assuming constant magnetic flux.

For each event, the pitch-angle fit provides independent release altitude estimates zi for energy bins Ei. We assume that the release height of a discrete burst of Alfvénic precipitation with energy E is initiated at an altitude where the electron energy is matched to the equivalent inertial Alfvén velocity, i.e., Z(E,n0,n1,H,λ⟂0)=arg⁡min⁡z|E-12mevAI2(z,n0,n1,H,λ⟂0)|. This definition adopts the simplifying assumption that electrons leave the resonant interaction when their parallel speed is equal to the local inertial Alfvén velocity (vAI). In reality, the resonance window allows electrons to leave the interaction with speeds higher (2VA-v∥0) or lower than vAI. An inferred source altitude should therefore probably more accurately be interpreted as an effective release altitude representing a finite local range of altitudes rather than a unique point. Under this approximation of release at 1 vAI, it is now possible to search for plasma and wave parameters that best match the calculated source altitudes by minimizing the variance weighted sum of squares: 7(n^0,n^1,H^,λ^⟂0)=argminn0,n1,H,λ⟂0∑iσi-2zi-Z(Ei,n0,n1,H,λ⟂0)2 where σi is the error standard deviation for the source height of energy bin i (error bars in Fig. ), and (n^0,n^1,H^,λ^⟂0) are the values of the best fit parameters.

The parameters n0, n1, and H describe the background plasma density profile, while λ⟂0 is the only fitted wave parameter. Other wave parameters (e.g. Poynting flux, frequency) as well as characteristics of the electron source population being accelerated (e.g. density, temperature) are expected to affect the morphology of the observed electron dispersions, notably their width, total intensity and maximum energy. However, of these quantities, only the background density and perpendicular wavelength enter the inertial Alfvén speed profile as given by Eq. (). Consequently, we expect these latter parameters to dominantly influence the curvature/slope of the electron dispersions and as such their fits and the inferred release altitudes. Results from test-particle modelling performed by seem to support this hypothesis as only variations in the wave perpendicular scale out of several wave parameters were found to significantly modify time–energy dispersion fits.

Figure  shows this fitting technique applied to the pitch-angle estimated source altitudes of the 338 s event (same event as in Figs. , , and a). Figure a shows the inferred source altitudes and their associated uncertainties (similar to Fig. a). The black curve corresponds to the best fitting IAW speed curve to the points following Eq. (). The values of the best curve parameters are indicated in the top right corner of the panel, with the (25th, 75th) percentile interval of the accepted parameter set given in brackets. The accepted parameter set is defined by sampling the four fit parameters ((n0,n1,H,λ⟂0)) directly from the error landscape explored by the grid search. Retained are all the parameter combinations whose weighted sum of squares (Eq. ) satisfies S<Smin+ΔS with ΔS=4, and each combination is weighted by exp⁡(-(S-Smin)/2). The faint grey curves correspond to 250 parameter sets drawn in this way. For this event, the fitted parameters are remarkably close to the values of the profiles from previous studies shown in Fig. , as well as expected perpendicular IAW scales . The density profiles of O+ and H+ and ntot=nO++nH+ along the field line can be reconstructed from the best fitting parameters following Eqs. () and (), as is shown in Fig. b. The thick solid green line and the dashed blue and yellow lines correspond to the best fit (thick solid black line in Fig. a), while the faint green lines correspond to the 250 sampled parameter sets (faint black profiles in Fig. a). The fitting technique was applied to all the events shown in Fig. , and approximately two-thirds of the cases were found to have fairly good, constrained fits as in Fig. , while the others remained less well constrained.

The parameter estimation presented here is based on the simplified assumption that electrons are accelerated to 1 VA and that the density profiles follow the forms of Eqs. () and (). Future work should determine the most appropriate acceleration assumptions and profile shapes, and assess whether this fitting approach can reliably infer topside plasma structuring.

We also want to highlight that the method relies on source-altitude estimates extracted from pitch-angle–time dispersions seen in multiple energy channels and that it therefore requires sufficient resolution in energy, time and, importantly, pitch-angle, as discussed in Sect. . As such, future missions carrying instruments with higher resolution could improve the precision of the inferred profiles. It could also be of interest to apply the technique on measurements from both rockets and satellites, where short-duration sounding rockets with state-of-the-art high-resolution instrumentation could provide tightly constrained profiles for individual auroral events, whereas satellite measurements may trade some accuracy for repeated observations over longer intervals and across a wider range of geomagnetic conditions and locations.

Both strategies would provide valuable information about the acceleration region, and the method presented here could become one of the few observation techniques capable of constraining plasma density profiles near the transition between O+ and H+ dominated plasma. Repeated measurements could then help characterize the spatial and temporal variability of this poorly sampled region of the auroral topside ionosphere.