Spectral width of SuperDARN echoes : measurement , use and physical interpretation

The Doppler velocity and spectral width are two important parameters derived from coherent scatter radar systems. The Super Dual Auroral Radar Network (SuperDARN) is capable of monitoring most of the high latitude region where different boundaries of the magnetosphere map to the ionosphere. In the past, the spectral width, calculated from SuperDARN data, has been used to identify the ionosphere footprints of various magnetosphere boundaries. In this paper we examine the way the spectral width is presently estimated from the radar data and describe several recommendations for improving the algorithm. Using the improved algorithm, we show that typical spectral width values reported in the literature are most probably overestimated. The physical interpretation of the cause of various magnitudes of the spectral width is explored in terms of the diffusion and dynamics of ionospheric plasma irregularities.


Introduction
The Super Dual Auroral Radar Network (SuperDARN) is an international scientific consortium that operates and maintains high frequency (HF), over-the-horizon radars for conducting research in ionosphere and space physics (Greenwald et al., 1995).SuperDARN is designed to map ionospheric plasma convection over the auroral and polar cap regions.To achieve this, 2-D Doppler velocity (V D ) vectors are estimated from pairs of radars with overlapping fields of view.In normal mode each beam is formed under computer control by a phasing matrix, to scan over 16 sequential beam directions.This provides an overall azimuth coverage of 50 • .A multi-pulse transmit sequence enables Su-perDARN radars to measure Doppler velocities up to 2 km/s over distances of ≤3500 km (Greenwald et al., 1985;Hanuise Correspondence to: P. V. Ponomarenko (phpp@alinga.newcastle.edu.au)et al., 1993;Baker et al., 1995;Barthes et al., 1998).The resulting autocorrelation function (ACF) for every range gate is used to estimate V D , the scattered signal power and the spectral width, W .The Doppler velocity is derived from the temporal gradient of the ACF phase.The spectral width is estimated from the ACF power decorrelation time.These parameters are calculated at each radar site by a computer algorithm known as FITACF, which, among other tasks, fits model functions to various experimental ACF characteristics.
Since the late 1990s, the spectral width has been widely used to infer ionospheric footprints of various magnetospheric boundaries, such as the cusp (Baker et al., 1995) and open-closed magnetic field line boundary (OCB) (Dudeney et al., 1998).Cusp echoes are usually characterised by a relatively large spectral width, W ≥200 m/s, while equatorward of the cusp it appears to be limited to W ≤50 m/s (Baker et al., 1995).Night-time radar returns regularly exhibit a boundary around 70 • MLAT (e.g.Parkinson et al., 2004), which is accompanied by a similar change in W .A recent statistical study by Chisham et al. (2004) confirmed previous case study results, that the spectral width is a good proxy for the OCB over most of the nighttime sector.
Using the spectral width to identify magnetosphere boundaries mapped onto the ionosphere appears to have become standard procedure, as shown, for example, by the extensive referencing on this topic by Chisham et al. (2004).Despite this, large spectral width values (W ≥200 m/s) obtained from SuperDARN radars still lack an adequate physical interpretation.Ponomarenko and Waters (2003) have briefly discussed some of the physical factors that affect the spectral width.In general, these factors are related to either the ionospheric drift velocities or the lifetime of the irregularities.The fastest process determines the ACF correlation time, τ c , and therefore the spectral width, W ∼1/τ c .There have been several attempts to clarify factors that control the measured spectral width.The most general approach, based on a combination of the random velocity distribution of electron density irregularities and ionospheric diffusion was used by Hanuise et al. (1993) and Villain et al. (1996).They found that for large W the value of τ c depends on the average irregularity lifetime, and the ACF power, as a function of lag, is exponential in shape, regardless of the type of the irregularity decay process.However, estimates of the plasma diffusion coefficient, D 100−200 m 2 /s, derived from the measured spectral width values, were much larger than typical values for the ambipolar diffusion coefficient in the F-region, which are D∼1 m 2 /s (e.g.Hysell et al., 1996).André et al. (2000) discussed Bohm diffusion as a possible alternative to the classical mechanism.The semiempirical expression for the Bohm diffusion coefficient is (Chen, 1984) where k is Boltzmann's constant, T temperature, e electron mass and B magnetic field.Equation (1) predicts values of D B ∼200 m 2 /s for typical ionospheric conditions, while many of the experimental estimates exceed 300 m 2 /s.Another interpretation was based on small-scale electric field fluctuations that produce random velocity variations in the cusp region (Baker et al., 1995).However, in order to be a viable mechanism, the electric fields need to be rather large at around a few mV/m.
Explanations for large values of the spectral width have also considered possible violations of the assumptions used to develop the FITACF algorithms.For example, multicomponent ACFs have been reported in the large-W sections of backscatter data from the cusp (Baker et al., 1995;Moen et al., 2000).These authors suggested that since the algorithm for estimating the spectral width assumes a single component ACF, multi-component ones may cause an overestimation and high variability of W .In a study by Moen et al. (2000) the frequency spectra were obtained via the Fourier transform of the experimental ACF values, assuming that the cross-range interference was negligible.However, the effect of this interference was recognised by Baker et al. (1995), who attempted to remove contaminated ACF lags but still observed multi-peak spectra in cusp echoes.André et al. (1999) suggested that large spectral width values might be caused by a nonlinear response of the Super-DARN autocorrelation routine to Pc1-2 ULF waves with periods smaller than the integration time t i 1−7 s.To support this, the authors modelled a ULF-modulated radar time series and processed these with the standard FITACF package.They obtained large spectral width values even for relatively small-amplitude waves.However, the reported nonlinear response in the correlation process prompted Ponomarenko and Waters (2003) to examine these results.The modelled spectral broadening in André et al. (1999) was found to be caused by a mathematical error.
Among other possible causes for large spectral width values, André et al. (2000) considered large-and meso-scale velocity gradients within the scatter volume.However, their model calculations showed that these factors were insufficient for producing either large W values or multi-component ACFs.
The generation of large spectral width values via restructuring of the ionospheric plasma at decameter scale lengths by particle precipitation was proposed by Moen et al. (2000).Furthermore, Chisham et al. (2005) established inverse proportionality between W and precipitating electron energy flux measured by DMSP satellites.To explain this fact, the authors assumed that the large spectral width values are produced by the small-scale convection electric field structures in the vicinity of the OCB, but they become effectively suppressed equatorward of the OCB by the increased ionospheric conductivity produced by the high electron energy fluxes (Parkinson et al., 2004).
The effect of forward scatter of the HF waves from the radar by ionospheric turbulence on W was modelled by Valliéres et al. (2004), but this mechanism seems to be only capable of producing large values of W at operating frequencies close to the critical frequency of the ionospheric F-region.
Despite the number of proposed mechanisms, it is unclear if any can explain routinely observed spectral width values of W ≥200 m/s reported in the literature.With this in mind, we examined the estimation of spectral width in the data processing procedures used by all SuperDARN radars as coded in the latest available version (1.09) of the FITACF software.In Sect. 2 we outline the FITACF routines that affect spectral width estimates and describe several modifications that improve the calculation.Analytical functions for fitting the ACF are examined in Sect. 3 within the context of the physics involved in the HF scatter.A representative data set from the TIGER (Tasmania) radar is used to illustrate the effects of the modifications to FITACF.In Sect. 4 we examine physical explanations for the variation in magnitude of the spectral width computed from SuperDARN data.

Multi-pulse radars and the ACF
SuperDARN is required to measure V D up to ∼1−2 km/s over distances up to d max 3500 km.This would be impossible using a single-pulse transmit scheme, where the pulse repetition frequency derived from the maximum return range would be f r ≤40 Hz.Therefore, the maximum Doppler shift that can be measured with a single-pulse scheme is equal to the Nyquist frequency f n =f r /2≤20 Hz, while ionospheric convection can produce Doppler shifts up to f max D =2V max D /λ∼100−200 Hz.In order for SuperDARN to measure the required V D values, a seven-pulse coded sequence was implemented (Greenwald et al., 1985;Hanuise et al., 1993;Baker et al., 1995;Barthes et al., 1998).It produces ACFs with an elementary lag of 2.4 ms (f n 200 Hz).The transmit pulse length of 300 µs gives a spatial resolution of 45 km over 70-75 range gates (maximum range 3550 km).
The multi-pulse transmit sequence overlaid with the receiver samples is shown in Fig. 1.The receiver data is sampled at 300 µs intervals for a time that allows the last pulse from the sequence to arrive from the furthest range ( t max 0.1 s).The ACF is computed from the 17 possible time lags available via the combinations of pairs of pulses, giving a maximum lag value of τ max =43.2 ms.For each beam, the process is repeated for the specified integration time, t i 3−7 s.This sets the number of ACFs that are averaged, N avg =t i / t max (usually N avg 70).The timing of data sampled at the receiver sometimes coincides with the time a transmit pulse must be sent.For this case, the receiver is "blanketed" around that time for a duration of 2-3 pulse lengths (see insert in Fig. 1).This effect is easily identified from the radar timing information and the appropriate samples are rejected.While the multi-pulse transmit scheme allows measurement of larger V D , it has some disadvantages.The major problem is cross-range interference (CRI), when returns from different pulses in the sequence arrive at the receiver simultaneously from different ranges.Figure 2   from left to right.The pulses approach two individual targets A and B separated by cτ/2, where c is the speed of light (i).After the first pulse, p1, reaches the first target, A, it produces an echo p1A, which propagates back to the radar (ii).In the same way p1 also generates a second echo, p1B, which reaches target A from the right at the same time as the second transmitter pulse p2 arrives there from the left and generates another echo, p2A(iii).From that point, both echoes, p1B and p2A, travel back to the receiver, interfering with each other (iv).
In ionospheric sounding, a plasma volume with intensive small-scale irregularities presents a continuous target that may stretch over ≤1000 km, providing favourable conditions for CRI.An excellent illustration of CRI is presented in Fig. 1F of Yukimatu and Tsutsumi (2002), which shows how the echoes received from all seven pulses overlap in the time domain.For a known multi-pulse sequence, it is quite straightforward to calculate when and at which ACF lags CRI might occur.However, it is a more complicated matter to detect if CRI is significant for a given range.This aspect of the ACF processing is discussed further in Sect.3.
To estimate W and V D , the data processing algorithms in FITACF deal with power and phase calculated from real and imaginary parts of a complex ACF, R(τ ). Figure 3 shows a modelled single-frequency ACF (top).The real part of the ACF, [R(τ )], is an even function with a maximum at zero lag, and the imaginary part, [R(τ )], is an odd function of the lag.Both the real and imaginary parts of the ACF decay in amplitude with increasing lag as the received data decorrelate.This is illustrated in the bottom panel of Fig. 3, which shows the ACF power |R(τ ] used to estimate lag zero power and spectral width W ∼1/τ c via fitting an exponential or Gaussian model.The fitting procedure is based on the assumption that the Fourier transform of the ACF has a single spectral component, i.e. Further details and illustrations on the fitting procedures may be found in Hanuise et al. (1993), Baker et al. (1995), and Barthes et al. (1998).

Calculation of spectral width
The FITACF software used routinely by all SuperDARN radars consists of a number of subroutines and those relevant to this study are described in Appendix A. The major tasks performed by FITACF are to: 1. Determine ACF lags contaminated by cross-range and pulse-overlap interference and exclude them from further processing.
2. Find lag zero noise, R n (0), and non-zero lag noise, R n (τ =0) , using the ACFs with the lowest lag zero power.
3. Use the above noise levels to check for any coherent interference, i.e. "noise ACF".If R n (τ =0) exceeds the expected fluctuation level, R(0)/ N avg , by ≥60%, then the "noise ACF" is estimated and subtracted from all ACFs.
5. Subtract fluctuation level, σ R =R(0)/ N avg , and R n (τ =0) from the ACF power.All ACFs with a negative zero lag power are excluded.All non-zero ACF lags with negative power or those considerably deviating from a single-component ACF shape are marked as "bad lags" and excluded from power fitting.
6. W is estimated by fitting linear or quadratic functions to the logarithm of the ACF power, for exponential or Gaussian models, respectively, with contributions from different lags weighted by the linear ACF power.
We found that a number of processes currently implemented in the code adversely affect the estimation of the spectral width.These processes and recommendations for changing the algorithms are summarised below.

Fluctuation level
Firstly, we focus on the use of the statistical fluctuation level, σ R =R(0)/ N avg .In the original FITACF package, this value is treated as a positive offset to all ACF lags and is subtracted from the ACF power, |R(τ )|, before applying the fitting routines.All resulting ACF lags with negative power are effectively excluded from the power fitting process.We would like to point out that by definition σ R is a magnitude of statistical fluctuations of the measured ACF power around its expectation value.These fluctuations can both increase and decrease the power, so they cannot be treated as a positive offset.With subtraction of σ R the ACF power becomes biased towards lower values, and this bias increases with decreasing N avg .This, in turn, leads to a systematic bias of W towards larger values.We illustrate this effect in Fig. 4, where we reproduce the FITACF estimation of W for a model ACF with an exponential |R(τ )|, characterized by a correlation time τ c =20 ms (squares).We used the same N avg =31 and transmitter frequency, f 0 =11.9MHz as for the specialmode experimental data presented later.The above values for τ c and f 0 correspond to W 200 m/s.In Fig. 4 triangles show ACF power with the subtracted fluctuation level, |R(τ )|−σ R (lags with negative ACF power are ignored).In FITACF, W is estimated by fitting linear or quadratic functions to the logarithm of the ACF power for exponential and Gaussian models, respectively.During the fitting procedure the results are weighted by the ACF power.In the bottom panel of Fig. 4 we show results of a linear fit applied  To illustrate the effect of extracting σ R on real radar records, we analysed data obtained between 12:00 and 14:00 UT (∼23:00-01:00 MLT) on 10 December 1999 from the TIGER (Tasman International GeoEnvironmental Radar) SuperDARN radar (Parkinson et al., 2004).Figure 5 shows a time-range map for W obtained for this interval from data processed using the original, unmodified FITACF code.The radar was programmed for high temporal resolution with 3 s/sample (N avg =31) on beam #4, which points towards the south AACGM Pole.Starting from ∼12:30 UT, the data show a pronounced boundary between low and  high W values around 67−68 • MLAT.Sea/ground scatter echoes have not been removed from the data.All ACFs with less than three "good" lags were rejected.
In Fig. 6 the same data were re-processed with FITACF modified to not remove σ R before the curve fitting routines.
An overall decrease in the magnitudes of W is clearly seen while the location of the sharp latitudinal gradient in W remains essentially the same.This effect is quantified in Fig. 7, which shows histograms calculated from the original and modified data.The modified FITACF produces values for W that are essentially confined to below 200 m/s while the unmodified routine gives many values well beyond this limit.Furthermore, the modified FITACF generates considerably smaller fitting errors over the range W =50−300 m/s, as shown in Fig. 8. Finally, following the same arguments as for σ R , we want to point out that there is no justification for FITACF extracting R n (τ =0) .Due to the fact that all coherent noise has already been removed (see point 3 at the beginning of this section and Sect.A2 in the Appendix), this parameter represents a fluctuation level for the incoherent background noise.However, its subtraction has little effect on the spectral width estimate compared with that caused by the removal of σ R , so we left that part of the FITACF code unchanged.

Zero lag noise
The next modification to FITACF involved the zero lag noise, R n (0) (for details see Sect.A2 in the Appendix).Because all coherent noise has been removed from the ACFs, R n (0) represents a δ-correlated addition to the ACF power at τ =0.The presence of a δ function at τ =0 effectively underestimates the correlation time, which causes an overestimation of the spectral width.This effect is illustrated in Fig. 9, where we used the same basic model as in Fig. 4 but modified it by adding R n (0)=0.2R(0)at τ =0.While the initial spectral width value is W 200 m/s, the modified one is larger by 30%, W 260 m/s.Furthermore, our model calculations show that the combined effect of subtracting σ R (N avg =31) and retaining 20% zero lag noise leads to a 100% overestimation of the spectral width, giving W 400 m/s for W 200 m/s.
The lag zero noise effect was mostly found to influence the sea/ground scatter rather than ionosphere scatter.To quantify the change in the experimental sea-scatter spectral width, we analysed data from TIGER beam #4 recorded on 20:00-22:00 UT on 28 September 2000, which consisted of seascatter echoes only (Ponomarenko et al., 2005).Comparing the original estimates of W with the FITACF modified to extract the zero lag noise, we found that the original algorithm incorrectly classified up to 20% of the echoes as ionospheric rather than sea scatter.

"Bad lag" routines
The "bad lag" routines in FITACF (Sect.A3 in the Appendix) deal with the distortion of the ACF shape caused by both known and unknown factors.The "known" category contains a low signal-to-noise ratio, transmitter pulse overlap with receiver samples (Fig. 1), and CRI (Fig. 2).Bad lags caused by "unknown" factors are identified as a deviation of |R(τ )| from the single-component ACF shape required for fitting.
An analysis of "known" factors revealed that many ACF lags with high levels of CRI were not rejected by the original FITACF routines.To illustrate this, consider the situation shown in Fig. 2. Assume that the scattered signals from targets, A and B, are given by u A (t) = u A0 e i{ωt+ψ A (t)} and u B (t) = u B0 e i{ωt+ψ B (t)} and are not correlated with each other.In this case the ACF calculated for the range A at lag τ consists of two terms: where ... represents statistical averaging.Due to the fact that u A (t) and u B (t) are not correlated, the last (interference) term will reach zero for N avg →∞.However, for a finite N avg this term produces an additional statistical fluctuation level (see Sect.A1 in the Appendix), σ * R u A0 u B0 / N avg = R AA (0)R BB (0)/N avg , while the fluctuation level from R AA (τ ) is σ R u2 A0 / N avg =R AA (0)/ N avg .Therefore, any CRI left in the ACFs cause an increase in the overall statistical fluctuation level.Importantly, this interference can contribute to the ACF power, both constructively and destructively, producing either positive or negative "spikes".The expected fluctuation levels are equal (σ * R =σ R ) when R AA (0)=R BB (0).It is easy to show that CRI affects the same lag in R * BB in the same way, which makes the last condition a natural threshold for rejecting CRI contaminated ACF lags: if µ=R BB (0)/R AA (0)>1, then the lag R * AA (τ ) is rejected; otherwise, we have to reject the same lag in R * BB .In contrast, the CRI algorithm implemented in FITACF (Sect.A3) applies a much larger threshold, µ=0.3N avg 20.This means that power from the "contaminating" range is allowed to be 20 times larger than power from the examined range.In normal operation mode N avg 70, in which case the expected fluctuation level can be as large as σ R +σ * R =σ R (1+ 0.3N avg ) 5.5σ R .Consequently, many lags with unacceptable levels of CRI remain in the ACFs.
The ACF "shape" routines in FITACF are based on empirical criteria (Sect.A3 in the appendix) and are designed to account for significant ACF power variations from an unknown origin.Keeping this in mind, one might suspect that a considerable part of these deviations originates from an inadequate treatment of CRI, which is capable of causing both a negative and positive depletion of non-zero lag power.In fact, when we changed the threshold value for CRI rejection using the data set in Fig. 5 from 0.3N avg to 1, the number of "bad shape" lags decreased by ∼65%.This means that we could effectively bypass the "bad shape" FITACF routines (Sect.A3).Importantly, after using the lower CRI threshold (µ=1), the effective fluctuation level in the ACF lags can be as high as 2σ R compared with 5.5σ R from the original FITACF.

Summary of FITACF Modifications
We revised a number of FITACF routines that deal with noise and interference, and that affect spectral width estimates, both directly and indirectly.We established that the largest effect is caused by the removal of the statistical fluctuation level, σ R , from the ACF power.Figures 5-7 show that this causes considerable overestimation of W , spawning searches for physical mechanisms that may cause these artificially large spectral widths.We have implemented the following modifications to FITACF: 1.To not remove the statistical fluctuation level from the ACF power.4. To disable the "bad shape" routines dealing with ACF power depletions from an unknown origin.
After applying these modifications, we have obtained timerange maps for spectral width which do not differ much from those shown in Fig. 6 (see, e.g.Fig. 11).Therefore, the major changes compared with the original FITACF can be attributed to retaining σ R (red curves in Figs. 7, 8).However, the other modifications further improved separation between low and high width distributions, as well as spectral width error estimates (green curves in the same figures).

Physical mechanisms for large spectral widths
The results obtained from the modified FITACF, as described in the previous section, have important implications for possible explanations for large spectral width values.Physical mechanisms outlined in the Introduction that could not give spectral width values larger than 200 m/s can now be reconsidered.Spectral width values obtained from the original FITACF package are considerably overestimated, while those obtained from the modified FITACF routines are significantly smaller and agree quite well with theoretical estimates for Bohm diffusion.The diffusion coefficient from Eq. ( 1) lies within 150-200 m 2 /s, leading to spectral widths of W ∼150−200 m/s.The more strict treatment of CRI discussed in Sect. 3 also requires revision of previous results regarding multi-component spectra obtained via a Fourier transform of the experimental ACFs by Baker et al. (1995) and Moen et al. (2000).
According to collective scatter theory (Rytov et al., 1988), the major contribution to the scattered HF signal at the reception point is provided by random spatial arrays of irregularities with a spatial period of l =λ/2, according to the Bragg scatter condition.The symbol indicates the direction 123 parallel to the HF wave vector.The longer these structures exist as a whole and maintain their spatial structure, the larger the correlation time of the scattered field.There are two limiting cases to consider.If the irregularity Lagrangian lifetime (lifetime in the irregularity reference frame, Villain et al., 1996), T L , is infinite, the only process causing decorrelation is random velocity fluctuations ("wandering" irregularities).Due to these fluctuations, the resonant arrays will randomly appear and disappear on a characteristic time scale, T V ∼l /δV , where δV is the (root mean square) fluctuation of the velocity of the plasma turbulent motion.Physically, T V is the time required for an average irregularity to randomly "wander" across its own scale size l .In this case the irregularity region ceases to satisfy the Bragg condition, and its scattered field becomes decorrelated.
At the other extreme, if the irregularities are not randomly moving, then lifetime processes are dominant.T L represents a sum of generation time, dwell time (if any) and decay time of an average irregularity with l =λ/2.Usually, the major contribution to T L is given by the decay time due to plasma diffusion processes, T D =1/k 2 D, where D is the diffusion coefficient and k=2π/ l .In reality, the correlation time scale, τ c is determined by the fastest process, so τ c ∼min(T V , T L ).To determine the dominant process, it is convenient to introduce a Lagrangian length, l L =δV •T L , which is the distance travelled by an irregularity during its lifetime due to random velocity fluctuations.If l L l , then τ c T D , otherwise τ c T V .
The most advanced analytical treatment of radar ACFs was given by Hanuise et al. (1993) and Villain et al. (1996).They derived a theoretical expression accounting for both velocity fluctuation and turbulent diffusion processes where D t =δV 2 T L is the turbulent diffusion coefficient.Figure 10 shows the dependence of this model on the Lagrangian lifetime.
At short lags, τ T L , the model curve has a Gaussian shape while at longer lags, τ T L , it becomes exponential.The transition between these two shapes occurs at τ =2T L .The actual shape of the curve is determined by the fastest of the two processes.If l l L , then after applying a second order Taylor expansion, one obtains a Gaussian ACF, To determine if there was any relationship between the ACF shape and the spectral width boundary we used ionospheric scatter echoes from TIGER for 12:00-14:00 UT on 10 December 1999, as shown in Fig. 11.where W err and V err are the fitting errors for the spectral width and velocity, respectively.The resulting histogram for W is shown in Fig. 12.
Following Villain et al. (1996), we applied Eq. 2 to the same data set.First, we calculated the median normalised ACF power across the three spectral width ranges of 20-30 m/s, 60-70 m/s and 150-160 m/s.These were chosen based on the histogram in Fig. 12 and correspond to low, intermediate and high spectral widths.The results in Fig. 13 clearly show that small spectral widths are associated with a Gaussian ACF power shape, i.e. random drift velocity variations δV dominate all other mechanisms, while the large-W ACF power decays exponentially with τ , indicating that diffusion processes are the major contributor to R(τ ).A similar relationship between W magnitude and ACF shape was reported by Hanuise et al. (1993), but their observations were not related to a spectral width boundary.
Figure 14 shows the ratio T L /T V , which may be interpreted as an effective shape factor for Eq. ( 2).Yellow regions (T L T V ) represent Gaussian type ACFs while black regions (T L T V ) depict exponential ones, with the rest of the palette corresponding to intermediate cases.The larger values for W and predominantly exponential ACFs concentrate poleward of the boundary, while the smaller values for W and the Gaussian-shaped ACFs are generally observed equatorward of the boundary.
The spatio-temporal map of the diffusion coefficient, D, is shown in Fig. 15.The values for D are shown for ACFs whose power at the transition lag, τ =2T L , was above the fluctuation level, R(2T L )>σ R (Fig. 10).As expected, the spatio-temporal distribution of D in Fig. 15 looks similar to that of W in Fig. 11, due to the simple fact that W ∼1/τ c ∼k 2 D. The corresponding histogram for D is shown in Fig. 16.
Remarkably, the diffusion coefficient is confined to values, D≤200−250 m 2 /s, i.e. close to the Bohm diffusion estimate.
Random drift velocity spread (Fig. 17) clearly shows that the velocity fluctuations poleward of the spectral width boundary are substantially larger than those equatorward of the boundary.Here we only use values corresponding to the well-defined Gaussian part of the fitting curve, for example, R(2T L )>1−σ R (Fig. 10).The corresponding histogram is shown in Fig. 18, and it has the main maximum around 10 m/s.However, the curve extends well beyond 50 m/s.
Finally, we investigated possible effects of multicomponent ACF (multi-peak spectra) on the estimates for spectral width (Baker et al., 1995;Moen et al., 2000).First, we designed criteria to distinguish between singlecomponent and multi-component ACFs using the fluctuation level, σ R .The criteria were based on the fact that two or more spectral components produce interference minima and maxima in ACF power, and only the components comparable in magnitude might cause significant fading of the power and affect W .As illustrated in Fig. 19, we identify multi component ACFs as those with power that after descending to R(τ )≤σ R at smaller lags, grows back to at least 2σ R at larger lags.After applying this criterion to the TIGER data for 10 December 1999, we found that only a small fraction of ACFs were identified as multi-component (red line in Fig. 12), contributing to 6% of the overall ACFs and not exceeding 20% at individual histogram bins.The above criterion is rather liberal because the fluctuation level σ R may, in fact, be greater than 1/ N avg , due to the presence of CRI, as discussed in Sect.3.3.

Conclusions
There have been many unsuccessful attempts to physically interpret large spectral width values W ≥250−300 m/s measured by SuperDARN radars at high latitudes.We have approached this problem by examining the data processing procedures.A critical revision of the FITACF package that is used for calculating spectral width and other signal parameters at all radar sites revealed inadequate pre-processing of experimental ACFs before using them to estimate W .A revised FITACF package was tested against a representative 2-h data set, which contains a pronounced night-time spectral width boundary.The most important result of the software revision was a decrease in the measured spectral width values.While the spectral width boundary location remained unchanged, corrected W were smaller by 20-40% and confined mostly to 200 m/s.
The smaller W values provided estimates of the effective diffusion coefficient D 200−250 m 2 /s poleward of the spectral width boundary.We consider these values to be in satisfactory agreement with the Bohm diffusion mechanism, which was previously dismissed based on overestimated values for W (André et al., 2000).Further analysis showed that the shape of the ACF power poleward of the spectral width boundary (large W ) is mostly exponential, while equatorward of the boundary it is predominantly Gaussian.Contributions from the multicomponent ACFs appeared to be insignificant, at least for our night-time data set.This implies that in the large-W region the scatter ACF is affected mainly by diffusion processes, for bad lags that are affected by transmitter pulse overlap or cross-range interference.Next, the background noise level is estimated (NOISE STAT), and any coherent interference signal is detected and removed by NOISE ACF, FIT NOISE and REMOVE NOISE.The pre-processed ACFs are passed to the MORE BADLAG routine which identifies bad lags based on the background noise level, statistical fluctuation level, the presence of spurious spikes and assumptions involving decreasing ACF power with lag.After this "clean-up", the ACF power and spectral width, W , is estimated by fitting Gaussian and exponential models to the ACF envelope.The Doppler velocity, V D , is determined from a linear fit to the ACF phase.Routines ELEVATION and GROUND SCATTER determine elevation angle of arrival for the HF signal and ground/sea scatter ACFs (for sea/ground scatter selection criteria see Sect.4), respectively.

A2 Noise level
These FITACF routines are designed to identify which lags of an ACF adversely suffer from different types of noise or interference.These are zero lag noise and non-zero lag noise.
Zero lag noise, R n (0), is the larger of the "clearsky search" noise (passive reception with the transmitter switched off) compared with the average zero lag power obtained from the 10 weakest ACFs along a beam direction.This parameter is calculated in NOISE STAT and used to reject ACFs with small zero lag signal to noise ratio and to calculate the non-zero lag noise.
Non-zero lag noise (NOISE STAT) is obtained by identifying all ACFs for a given beam that have zero lag power, R(0), less than 1.6 R n (0)+σ R and then calculating the average non-zero lag noise power, R n (τ =0) , from all these identified ACFs.This is a single number obtained via summing power from all non-zero lags together and dividing by their total number.In FITACF R n (τ =0) is subtracted from |R(τ )| in all range gates before fitting.

A3 "Bad lag" routines
The two routines that identify and deal with "bad lags" are BADLAGS and MORE BADLAGS.The BADLAGS routine marks ACF lags contaminated by transmit pulse overlap and cross-range interference.For cross-range interference, the zero lag power of the current ACF, R cur (0), is compared with the zero lag power of the ACF from the potentially interfering range, R int (0).If R int (0) is sufficiently large, the corresponding interfering lags are marked as "bad".The empirical threshold used for this purpose is µ=R int (0)/R cur (0)=0.3Navg .
The MORE BADLAGS routine is designed to adjust the ACF according to a single-component shape assumed by the fitting routines.Importantly, before this the fluctuation level, σ R , and non-zero lag noise, R n (τ =0) , are extracted from |R(τ )|, which may lead to negative values for ACF power.ACF lags are marked as "bad" according to the following rules: -ACF power is less than zero, |R(τ )|≤0.
-If two consecutive points in the ACF power are less than zero, then the longer lags ("tail") are dismissed:

Fig. 1 .
Fig. 1.Transmit pulse sequence and receiver sampling times used in SuperDARN radars.

Fig. 2 .
Fig. 2.Illustration of conditions for cross-range interference.Transmitter pulses are drawn in blue, reflected ones in red.Time increases from top to bottom.
scheme is equal to the Nyquist frequency f n =f r /2≤20 Hz, while ionospheric convection can produce Doppler shifts up to f max D =2V max D /λ∼100−200 Hz.In order for SuperDARN to measure the required V D values, a seven-pulse coded sequence was implemented(Greenwald et al., 1985;Hanuise et al., 1993;Baker et al., 1995;Barthes et al., 1998).It produces ACFs with an elementary lag of 2.4 ms (f n 200 Hz).The transmit pulse length of 300 µs gives a spatial resolution of 45 km over 70-75 range gates (maximum range 3550 km).The multi-pulse transmit sequence overlaid with the receiver samples is shown in Fig.1.The receiver data is sampled at 300 µs intervals for a time that allows the last pulse from the sequence to arrive from the furthest range ( t max 0.1 s).The ACF is computed from the 17 possible time lags available via the combinations of pairs of pulses, giving a maximum lag value of τ max =43.2 ms.For each beam, the process is repeated for the specified integration time, t i 3−7 s.This sets the number of ACFs that are averaged, N avg =t i / t max (usually N avg 70).The timing of data sampled at the receiver sometimes coincides with the time a transmit pulse must be sent.For this case, the receiver is "blanketed" around that time for a duration of 2-3 pulse lengths (see insert in Fig.1).This effect is easily identified from the radar timing information and the appropriate samples are rejected.While the multi-pulse transmit scheme allows measurement of larger V D , it has some disadvantages.The major problem is cross-range interference (CRI), when returns from different pulses in the sequence arrive at the receiver simultaneously from different ranges.Figure2illustrates CRI for four consecutive time snapshots (i)-(iv) of two pulses, p1 and p2, transmitted time τ apart and propagating

Fig. 3 .Fig. 3 .
Fig. 3. Ideal ACF characteristics from SuperDARN radars.(Top) Th imaginary part (dashed) of a normalised, ideal ACF.The ACF magnit the correlation time scale, τ c which is used to calculate spectral width

Fig. 4 .
Fig. 4. Illustration of the overestimation of W due to the extraction of the statistical fluctuation level, σ r =R(0)/ N avg , from ACF power in FITACF.

Fig. 6 .
Fig. 6.The same data as Fig. 5 but without extracting the fluctuation level σ R from the ACF power |R(τ )| before the curve fitting process.

Fig. 8 .
Fig. 8.The same as in Fig. 7 but for a spectral width fitting error.

Fig. 9 .
Fig. 9.The effect of zero-lag noise on estimating spectral width.

Fig. 10 .
Fig. 10.Physical meaning of the Lagrangian correlation time, T L , for the model described by Eq. (2).

Fig. 12 .
Fig. 12.The same data as in Fig. 7 (green line) but without sea scatter data.

Fig. 13 .
Fig. 13.Median normalised ACF power (red triangles) and the fit to Eq. (2) (blue line).Grey-scale contours show the spread of normalised distribution functions for each lag.Panels (a), (b) and (c) correspond to low (W=20-30 m/s), medium (W=60-70 m/s) and high (W=150-160 m/s) spectral width, respectively.. The vertical dashed line in Fig. 13(b) identifies the transition from a Gaussian to exponential shape at τ =2 T L (see text for details).The horizontal dashed line corresponds to the fluctuation level, σ R .

Fig. 14 .
Fig.14.Ratio between the Lagrangian correlation times, T L , and the Gaussian time constant, T G =T V =1/kδV , the correlation time related to the random drift of irregularities.Yellow colour corresponds mostly to Gaussian shaped ACF power while black shading depicts mostly the exponential shape.

Fig. 16 .
Fig. 16.Histogram for diffusion coefficient data presented in this figure.