The FWHM method requires three or more stations that straddle the magnetic footprint of the detected resonant frequency. For the 25 November 2001 interval Rae et al. (2005), Ozeke et al. (2009), the magnetometer stations from the CARISMA chain listed in Table 1 were used. The process used by Ozeke et al. (2009) required the maximum time series amplitudes. The maximum amplitude in the time series occurred at 0235 UT ( ≈ 72  nT) in the GIM magnetometer data (Figure 2C). The corresponding peak amplitude values in the time series from the other magnetometers around this time were selected. The FWHM was then calculated via a linear piece-wise fit to the amplitude versus the station MLat. However, Warden et al. (2021) showed that a better method was to use the spectral amplitudes at the resonance frequency in order to capture the resonance wave power rather than using a single value from the time series.

In order to calculate the spectral amplitudes, the 5 s sampled data between 0200 and 0300 UT, 25 November 2001 were multiplied by a 3,600 s boxcar window. A fast Fourier transform (FFT) of these windowed data was calculated, and the spectral amplitude at 1.5 mHz was recorded for each magnetometer. These spectral amplitudes plotted against station MLat are shown in Figure 2A. The FWHM was calculated by finding the MLat of the linear piece-wise function where the amplitude was 1 2 of the maximum. The MLat difference was 2.9° between these two points which was converted to km to yield a latitudinal resonance width, 2ɛ = 323 ± 138 km. The uncertainty was obtained from an analysis of the 2ɛ values at the 2σ level and is described in Section 4.

A Gaussian function was fit to the amplitude with MLat data, bounded by the most equatorward and poleward stations, PIN and FCC respectively. This was used to investigate variations in the latitudinal resonance width for different curve fit functions to the amplitude profile with latitude. The FWHM was calculated using the 1 2 levels from the maximum amplitude of the Gaussian fit. The uncertainty in the resonance width from the Gaussian fit was obtained from the 1.96 σ values of the Gaussian curve fit coefficients. This yielded 2ɛ = 373 ± 163 km. This was repeated using the FLR amplitude profile given by Eq. 3. Since Eq. 3 provides the amplitude profile in units of km, not MLat, the relative distance in km from the most equatorward station (PIN) to the other stations was calculated using the great-circle distance. The uncertainties were obtained using the same method as the Gaussian curve fit. The FLR amplitude fit yielded 2ɛ = 294 ± 80 km. The amplitude points and curve fits are shown in Figure 2A.

The APM and ADM estimates require a magnetometer pair that straddles the location of a resonance. For the 25 November 2001 interval the GIM and ISL stations were selected for analysis. The time series data were processed using an FFT and five typical window functions were considered. These were boxcar, Hann, and Hamming windows (Harris, 1978), a frequency domain 3 point exponential taper smoothing algorithm (Samson and Olson, 1981), and a multi-taper average (Slepian, 1978; Thomson, 1982). These are common FFT smoothing algorithms used to reduce variance in the spectral estimates.

The GIM and ISL north-south magnetic field time series data for 0200–0300 UT, 25 November 2001 were processed by an FFT with no applied window (i.e., boxcar). There was no unit crossing near the cross-phase peak so no resonant width estimate for the ADM and APM. However, Green et al. (1993) described a method for determining a new reference level to ensure a unit crossing between G ₊ and G ₋. The reference level is defined as M = ( G + G − ) 1 / 2 . New values of the spectral amplitude ratio, G + ( − ) ′ , are defined by G + ( − ) ′ = G + ( − ) / M . This modification also sets the condition G + ′ G − ′ = 1 , and has been used for all subsequent analyses. For the boxcar FFT window, the reference level was M = 1.848 (Figure 8A), which resulted in G + ′ = 1.218 and G − ′ = 0.821 . The cross-phase peak of 147° (Figure 8F) was observed at a resonant frequency of 2.78 mHz. From these values, 2ɛ was 54 km using the APM estimate and 1,411 km for the ADM estimate. The large discrepancy between these resonance width estimates is discussed in Section 4.

The same time series data were processed using a Hann FFT window. From this analysis, G + ′ = 1.47 , G − ′ = 0.68 (Figure 8B) with a peak cross-phase of 178° at 2.5 mHz (Figure 8G). The Hann FFT window analysis yielded an APM estimate of 2ɛ = 6 km and an ADM estimate of 2ɛ = 711 km. The ADM estimate is approximately half the value obtained using the boxcar FFT window, while the APM estimate is an order of magnitude smaller.

Similarly to the Hann FFT window, the GIM and ISL north-south magnetic field data were multiplied by a 3,600 s Hamming window. The spectral amplitudes and cross-phase were calculated from the FFT. The resonant frequency from the spectral amplitude unity crossing and cross-phase peak was 2.2 mHz with a cross-phase peak value of 49.7° (Figure 8H). From the spectral amplitude ratios, G ^′ = 1.61 and G − ′ = 0.63 (Figure 8C). From these values, 2ɛ = 540 km using the APM, and 2ɛ = 574 km using the ADM estimate. This ADM estimate is approximately half the ADM boxcar FFT estimate (1,411 km), while the Hamming window, APM estimate is an order of magnitude larger than the APM boxcar FFT estimate.

Applying a window function in the time domain smooths in the frequency domain. A smoothing algorithm may be applied directly in the frequency domain. An example is the exponential taper (e.g., Samson and Olson, 1981). The boxcar windowed time series was processed by the FFT. An 3-point wide, exponential taper was used to smooth the complex spectra. This FFT window method was applied to the 25 November 2001 data. The spectral amplitude ratios were G + ′ = 1.09 and G − ′ = 0.919 (Figure 8C), with a unity crossing at ≈ 1.82  mHz for the scaled G + ′ and G − ′ . With a cross phase value of Δϕ = 24.6° (Figure 8I), the APM estimate was 198 km and ADM estimate was 3,306 km, with the ADM estimate approximately twice the boxcar ADM estimate and four times the Hann and Hamming window estimates.

The multi-taper method for improved estimates of the spectrum was described by Thomson (1982) and has been used on ULF time series (e.g., Stephenson and Walker, 2010). An algorithm for generating the window functions was described by Gruenbacher and Hummels (1994). Percival (1993) discussed a choice of parameters from a time bandwidth product, NWΔt, where N is the number of samples spaced in Δt and W is the half bandwidth. The parameters chosen for this interval were based on NWΔt = 4, with N = 720, ΔT = 5. Suitable window tapers for spectral energy within the window have eigenvalues ≈ 1 (Percival, 1993) and the number of windows, K, where this condition holds is given by K < 2 NWΔt. For the multi-tapers applied in this study, the tapers with eigenvalues > 0.985 were selected, resulting in four window tapers. The orthogonal time series windows were generated and the spectra of each of the windowed time series data were averaged to obtain the spectral amplitude ratio and cross-phase. From this analysis, a unity crossing in the spectral amplitude ratio occurred at ≈1.9 mHz, with a corresponding cross-phase Δϕ = 24°, as shown in Figures 8E,I, respectively. The multi-taper processing yielded APM and ADM estimates of 2ɛ = 259 km and 2ɛ = 282 km, respectively. A summary of this interval is shown in Figure 3.

The data for 1 October 2012 are shown in Figure 3. The 1 s sampled data between 1800 and 1830 UT for the north-south magnetometer components were analysed using the same processes as described above. The data for the 19 June 2015 are shown in Figures 4–7 and were analysed in a similar way. The 2ɛ results for all intervals and methods are listed in Table 2 for the FHWM and Table 3 for the APM/ADM. The FWHM width estimates for 25 November 2001 agree within uncertainty bounds. However, this is not the case for 19 June 2015 cases.

Given the variation in the resonance width values, without prior knowledge of the damping coefficient, it is difficult to determine which estimation method or FFT window accurately estimates the resonance width of FLR resonances. In order to progress, a computer simulation of FLRs using a magnetohydrodynamic ULF wave propagation model (Waters and Sciffer, 2008) was considered. The difficulty with this approach is the unknown FLR damping coefficient in the simulation. A simpler approach was to consider the driven damped harmonic oscillator model. Orr and Hanson (1981) and Gough and Orr (1984) described the FLR magnetic field time series response, driven at angular frequency, ω _D , modelled as b p ̈ + 2 γ b p ̇ + ω 0 2 b p = ω 0 2 b D ⁡ cos ω D t , (6) where b _p is the transverse magnetic field time series response, γ is the damping coefficient, ω ₀ is the natural angular frequency of the resonant field line, b _D is the magnetic field driver amplitude, and ω _D is the angular driving frequency. The steady state solution to Eq. 6 is b p = ω 0 2 b D ω 0 2 − ω D 2 2 + 4.0 γ 2 ω D 2 1 / 2 cos ω D t − ϕ , (7) where ϕ is given by ϕ = tan − 1 2 γ ω D ω 0 2 − ω D 2 . (8)

The APM, ADM and FWHM provide resonance widths in distance (km or L) while Eq. 7 has the damping coefficient in s ⁻¹. The relationship between the spatial and temporal damping factors depends on the resonance model and the latitudinal resonance profile as discussed, for example, by Yumoto et al. (1995). In order to determine the resonant frequencies, ω ₀, for Eq. 7 and provide the mapping between the spatial resonance widths and temporal damping coefficient, a radial Alfvén velocity profile was generated using the International Reference Ionosphere (IRI) and the Field Line Interhemisphere Plasma model (FLIP; Richards et al., 2009) for the 19 June 2015 interval. A quadratic polynomial was fit to a low-latitude portion of this profile, between 2.07 and 2.92 R _E . This provided the mapping of resonant frequency to L shell as shown in Figure 9. By generating the amplitude and phase with frequency using Eq. 7 over a range of L values, according to the mapping provided by the IRI and FLIP plasma mass densities, amplitudes (and phases) with both frequency and L were obtained. The damping coefficient can be mapped from s ⁻¹ into km or L by scanning in latitude, at f _R , for the 1 / 2 amplitudes.

Equation 7 was used to investigate the effects of data processing and resonance width method (FWHM, ADM, APM) on estimates of Δθ, given the known damping. For the ADM and APM, two L shell locations were selected as ground station locations, a poleward point at 2.55 R _E and an equatorward point at 2.44 R _E with resonant frequencies of 18.3 and 20.0 mHz, respectively. These locations provided estimates of the input damping at a resonant frequency, f _R = 19.1 mHz, mid-way between the locations. The time series responses to a broad-band driver spectrum between 10 and 30 mHz were generated using Eq. 7, with b _D = 1. The phase spectra from the poleward and equatorward locations were unwrapped then subtracted to provide the phase difference with frequency. The phase difference at f _R was used as Δϕ in Eq. 4. The FWHM method was simulated by generating the time series response for every point in the resonant frequency latitudinal profile, and calculating the FWHM for the spectral amplitudes recorded at the 19.1 mHz frequency. Two cases were considered for the damping coefficient; γ ω 0 = 0.1 as typical reported day-time values (Orr and Hanson, 1981; Gough and Orr, 1984) and γ ω 0 = 0.055 (Walker et al., 1979). Table 4 lists the latitudinal resonance widths (km) obtained from Eq. 7 time series resonance response using the same processes described in Section 3.

From the results in Table 4, the boxcar and exponential taper FFT window functions gave the most accurate estimates for ADM methods, close to the actual value for the smaller spatial damping coefficient of 2ɛ = 145 km. The Hamming FFT window function gave the next best estimates. The Hann window and multi-taper algorithms gave the least accurate results, underestimating the resonance width. While the APM estimates followed a similar trend in which processing methods yielded the most accurate results, these estimates tended to underestimate 2ɛ to a greater extent compared with the ADM. The Hann FFT window and APM underestimated the resonance width for the 2ɛ = 267 km case, and gave a non-physical, negative resonance width for the 2ɛ = 145 km case. The multi-taper and APM width estimate was negative for the 2ɛ = 267 km damping case and over-estimated for the 2ɛ = 145 km damping case. These Hann window and multi-taper based estimates are discussed further in Section 4.2. Comparison of the window function methods between the APM and ADM show that the boxcar, exponential taper, and Hamming window smoothing methods give similar results between the two estimation methods.

For the FWHM, Eq. 3, and linear piece-wise curve fits yielded the most accurate estimates of 2ɛ. The Gaussian curve fit gave up to twice the expected values for both 2ɛ cases. The linear piece-wise curve fit depends on the spatial resolution of the magnetometer station chain. Therefore, the accuracy of the estimates obtained from the linear piece-wise fit in this driven damped harmonic oscillator model may not be realised in practice given the relatively sparse spatial measurements.

Comparison of the data analysis techniques for all three resonance width estimate methods shows that the ADM and FWHM tend to over-estimate 2ɛ calculated from the real γ, while the APM underestimates this value. However, the uncertainties for the APM and ADM using the boxcar FFT window function and exponential smoothing taper, and the FWHM linear piece-wise and Eq. 3 curve fits estimates covered the real γ value.

For the APM and ADM estimates, details of the spectra reveal why the Hann, Hamming and multi-taper window functions yielded quite different results. These methods use information from closely spaced magnetometers which translates to a narrow band in the spectrum. Figure 10 shows the amplitude spectra at the poleward and equatorward points, and the corresponding cross-phase for the 2ɛ = 145 km and 2ɛ = 267 km damping cases from the driven damped harmonic oscillator model. For the multi-taper, Hann and Hamming windows, the amplitude spectra have a sharper decrease in amplitude away from the resonant frequency at 19.1 mHz, relative to the exponential taper and boxcar window functions. The convolution in the frequency domain of the FFT window and the data mixes adjacent frequency components into the spectra, which are then divided to obtain G ₊ and G ₋. This combined process for the Hann and multi-taper windows gives larger differences between the estimated and known values for the resonance width for the driven damped harmonic oscillator model.

The convolution in the frequency domain also affects the cross-phase. From the damped harmonic oscillator model, the cross-phase values were 70 and 50° for 2ɛ equal to 145 and 267 km, respectively. Figures 10C,F shows that the cross-phase values at f _R for the Hann and multi-taper windows were larger than the known values. The cross-phase value at f _R from the Hamming window for the 2ɛ = 145 km was larger compared to the known value. These larger values of the cross-phase give smaller estimates for the resonance width from the APM. For phase differences between 180 and 360° obtained from the unwrapped phase, the resonance width is negative from the APM. The phase of the harmonic oscillator depends on the driving frequency, natural frequency of the resonant system and the damping coefficient. From Eq. 8, when ω _D = ω ₀, d ϕ d ω D = 1 γ . This inverse dependence on the damping coefficient shows that the phase change over the resonant frequency occurs over a narrower bandwidth for decreased damping. The phase mixing from the convolution of the FFT of the signal and window has a greater impact on the phase difference for smaller damping coefficients.

Independent of the FFT window processing, the uncertainty of the ADM estimate for ɛ depends on the spectral amplitudes and the ratio, G ₊. Figure 11A shows an example of how the ADM estimate for the resonance width changes with G ₊, using a typical station spacing, β, of 200 km. For G ₊ values near unity, there are large changes in ɛ for small changes in G ₊. For G ₊ larger than unity, the rate of change of ɛ is smaller for similar uncertainties in G ₊. This means that the uncertainties in ɛ depend on how close G ₊ is to unity. This section is a description of an uncertainty analysis to determine this dependence.

For the FWHM method, the boxcar FFT window spectral amplitudes at 1.39 mHz were used from the 25 November 2001 interval. The spectral amplitudes were sampled 10⁵ times from a normal distribution. The means were set to the experimental values for each station and the standard deviation was set to different percentages of the amplitude recorded at PIN. The case for the standard deviation set to 15% is shown in Figure 11B. The FWHM was calculated using the linear piece-wise curve fit, and the standard deviation of the resonance width estimates was calculated.

This process was repeated for the APM and ADM resonance width estimates, using the experimental spectral amplitudes from the GIL-ISL station pair at the frequencies, f ₊ and f ₋. The standard deviation of the normal distribution was set to 15% of the recorded spectral amplitude at the f ₊ frequency at the poleward station, GIL. The APM also required the cross-phase distribution. The cross-phase recorded at 1.5 mHz was set as the mean cross-phase with 15% of this cross-phase used for the standard deviation. The uncertainty analysis was repeated for the remaining intervals following the same process. Figure 11B shows the results of this uncertainty analysis for all three estimate methods for all intervals.

As seen in Figure 11B, the FWHM and APM estimates of the resonance width result in the smallest standard deviations. The ADM has the largest standard deviation range in the resonance width estimate, due to the large proportion of G + ′ near unity. In the damped driven harmonic oscillator model, the ADM, using a boxcar FFT window or exponential taper, gave the closest estimates to the known resonance width due to G + ′ being sufficiently far from unity. For the boxcar FFT window, these were 2.6 and 1.6 for the 2ɛ equal to 145 and 267 km, respectively. The results of the uncertainty analysis reinforce the condition that G ₊ should be sufficiently larger than unity for the ADM. How much larger than unity depends on the resonance width.

A major limitation on the FWHM method is the availability of a closely spaced, latitudinally separated chain of magnetometers. This method requires a chain of magnetometers, preferably within 500 km of an FLR footprint, that adequately sample the latitudinal profile of the resonance. The more stations that can be used the better the resolution of the FLR profile with latitude. This requirement is less problematic for FLRs detected in the northern hemisphere over North America, Europe and Asia, but limits the application of this method for studies using southern hemisphere chains where the magnetometer networks are sparse.

Hodograms were also generated for each of the intervals, following the process outlined by Pilipenko et al. (2013). A requirement of the hodograph method is that the complex ratio, H _P /H _E should vary smoothly around a circle in the complex plane over the range of frequencies in the vicinity of the resonance. However, this was not the case for the selected data intervals. Furthermore, the number of complex points is small for resonance signatures in the Pc5 range which limits the number of points to fit. Deviations from the circular resonance profile for the hodograph method were observed for a resonance model that included enhanced driving frequency amplitudes over a narrow band of the excitation profile, a topic for future study.

The driven damped harmonic oscillator model and uncertainty analysis results can be used to guide the choice of estimate method and analysis technique. Where a station pair straddles a resonance and the G + ′ ratio is sufficiently far from unity, the ADM or APM, with both boxcar and exponential taper smoothing could be used to obtain estimates of the latitudinal resonance width. If a sufficiently spatially resolved chain of magnetometers is available, the FHWM estimate using an Eq. 3 curve-fit could also be used.