Spectral Characteristics UHF Radar Aurora

Spectral Characteristics

UHF Radar Aurora

Brian J. JackeI

Graduate Program in Physics

Submi tted in partial fulfillment of the requirements for the degree of Doctor of Philosophy

Faculty of Graduate Studies The University of Western Ontario London, Ontario October 1997

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Bistatic radar observations at 440 and 933 Megahertz ha .ve been used to study the spectral characteris t ics of CHF scat ter from the auroral E-region. Scattered power spectra were prirnarily composed of a singie peak? and were distributed asymrnet- rically about the peak with a sharper cut-off at higher speeds. Using an approach not previously applied to auroral echoes, spectral moments were estimated directly from the autocorrelation functions. avoiding problems caused by truncation in the lag domain. ACFSwere also characterized in terms of t heir "correlation time- (related to the spectral width) and -decay exponent" (related to the spectral shape). Spectral shapes were neither Gaussian nor Lorentzian, but had an intermediate form that was well characterized by a simple model. Mean Doppler shifts ranged from O to 600 m/s. with a significant number of low speed observations. Large aspect angle echoes were wider than those from srnall magnetic aspect angles, although the range of Doppler shifts was similar at al1 aspect angles. Results for 16 and 34 centimetre irregularity wavelengths also showed the same range of speeds. as well as correlation times (at smali aspect angle). .A small number of 'narrow" (long-lived) echoes were observed at bot h frequencies: these had correlation t imes of nearly 1000 microseconds. more than twice the "typical" values.

Keywords: auroral, radar. spec t ra. moment es tirnation Acknowledgements

Many thanks to Don Moorcroft for his advice and support.

Thanks also to the rest of the Space and Atmospheric research group for their important contributions. Contents

Certificate of Examination IL..

Abstract 111...

Table of Contents v

List of Figures VI~I...

List of Tables ix

1 Introduction 1 1.1 Feat ures of the radar aurora ...... 1 1.2 VHF and CHF ...... I- 1.3 Bistatic radar notes ...... 4 1.1 Review of previous experimerits ...... 6 - 1.4.1 Millstone Hi11 (1393 LIhz) ...... 1 1.4.2 Homer ...... - ...... 8 1.4.3 EISCATand COSCAT...... 9 1.4.4 Millstone Hill (140 Mhz) ...... 11 1.4.5 Summary ...... 12 1.5 Structure of this thesis ...... 12

2 Spectral Characterization 14 2.1 From LPMSto ACFSto spectra ...... 15 3.2 Selected Observations ...... 20 2.3 Spectral Moments ...... 21 2.3.1 Estimation in the frequency domain ...... 25 2.3.2 EstimationfromA~~s...... 27 3.4 Models of the .4 CF magnitude ...... 2s 2.5 Otherfeatures ...... 32 2.5.1 Peak frequency ...... 32 2-52 Full-width half-maximum (FWHM)...... 33 2.53 Double peaks ...... 31 2.6 Summary ...... 36

3 Results at 34cm: MIDASC 37 3.1 MIDASC ...... 39 3.1.1 Algonquin ...... 12 3.2 Some results ...... 34 3.9.1 Double peaked spectra ...... 11 3-22 Very strong echoes ...... 37 3.2.3 Results frorn London ...... 39 .3.3.4 Xarrow echoes ...... 53 3.2 ..5 Monostatic satellites ...... 51 3.3 Discussion ...... 56

4 Results at 16cm: EISCATand COSCAT 58 4.1 Data from 1989 ...... 59 4.2 Datafrom1992 ...... 63 4.2.1 Large aspect angle ...... 66 4.2.2 Srnailaspectangle ...... 67 4.3 Discussion ...... 69 5 Discussion 73 5 .1 Doppler Speed ...... 74 5.2 Spectral width or correlation time ...... 76 5.2.1 Xarrow spectra ...... 79 a.3 Skewness ...... 81 5 Spectral Shapes ...... 85

rr 3.3 New analysis methods ...... 8'; 5.6 Futureexperiments ...... 91 5.7 Summary ...... 92

A Mat hematical Details 94 4.1 Fourier transform relations ...... 94 .A.? Moments in the frequency and lag domains ...... 95 A.3 Derivat ives of the auto-correlat ion function ...... 97 4.3.1 Determining deripatives from discrete data ...... 97 A.4 Standard line shapes ...... 99 A.5 Full width half maximum ...... 101 A.6 Standard deviation as a width estimator ...... 104 4.7 Logarithmic ACF magnitude ...... 106 A.8 Locating a single peak ...... 108 A.9 Twin peaks ...... 109 A . 10 Spikiness or RMS Spectral Power ...... 110

References 112

Vita 117

vii List of Figures

Lag Product Matrix examples ...... 1; Display of -4~~sand power spectra ...... 21 Exampies of coherent ACFS and their power spectra ...... 23 Example XCF magnitudes on a logarithmic scale ...... 31 Cornparison of .4 CF magnitudes to the exponential mode1 ...... 32 Oscillating ACF with single spectral peak ...... 35

Location of MIDASCradar sites ...... 30 E-region aspect angles ...... 41 Complex spectral shapes from multiple scat tering regions ...... 47 Experiment geometry for an F-region target ...... AS Coherent echoes from Algonquin April 7 1995 ...... 19 Comparison of Millstone Hill and Algonquin results ...... 50 Coherent echoes from London !darch 8 1994 ...... 51 Example of a satellite echo ...... 55

Cornparison of moments for the remote EISCAT1989 data ...... 61 Other spectral parameters for the EISCAT1989 data ...... 63 Decay exponent vs . SNR and correlation time ...... 66 Correlation tirne. decay exponent and SKEWB vs . velocity ...... 68 Spectral parameters at small aspect angle ...... 70

Superposition of two Gaussian spectra ...... 84

Standard line-shapes ...... 102 Cornparison of three spectral width parameters ...... 104 Polynomialfit to the ACF magnitude ...... 107 List of Tables

1.1 Summary of previous experiments ...... 13

2.1 Example Observations ...... ')3 2.2 Moments of example data ...... 29 2.3 Correlation times and decay exponents ...... 30 2.1 Peak velocity ...... 33

3.1 -UIDASCtarget parameters ...... 45 3.2 Spectral characteristics for !vIarch 8 1994 ...... 52 3.3 Narrow echo characteristics ...... 53

4.1 EISCAT1989 experiment details ...... 60 4.2 Summary of EISCAT1989 spectral characteristics ...... 62 1.3 E1sc.4~1992 pointing information ...... 65 4.4 Narrow spectra from June 28 1992 ...... 71

5.1 Correlation times ...... -"r r First detected by Harang and Stoffregen [1938], the radar aurora is a phenornenon in which enhanced radar echoes are received from regions of the earth's ionosphere where visible aurora are often observed. In subsequent research it has been found that these anomalous echoes are due to structure in the distribution of free electrons. which can result from a variety of plasma instabilities. As a consequence, so cailed "coherent- returns can be observed at a range of altitudes and locations not necessarily associated wi th the optical aurora. Hoirwer. the high latitudes where visible aurora generally occur are also regions where high electric fields and intense fluxes of charged particles provide considerable energy for plasma instabilit ies. As well, the eart h's dipole-like magnetic field geometry is mostly vertical at auroral latitudes. which affects not only the plasma physics which lead to the echoes. but also the ways in which they can be observed. For these and other reasons. observations of the high latitude radar aurora are sufficiently unlike lower latitude results that their study has developed into a separate enterprise.

1.1 Features of the radar aurora

Data have been collected with a wide variety of experimental configurations, and although they share a common subject, the goals of individual experiments have varied considerably. Generally speaking, these goals can be divided into four groups, although some experiments fa11 into more than one category.

Location/Time when and where do echoes occur? Height Profiles what is the distribution of echoes aith altitude?

Aspect Sensitivity how does echo power vary with magnetic aspect angle?

Spectral Features what are frequency characteristics of scattered signals?

This classification of echo characteristics is strongly influenced by experimental considerations. as the features which can be studied depend largely on the radar system used to observe t hem. While an ideal experiment would provide information about al1 aspects of the radio aurora. this is difficult to achieve in practice. However. experiments with restricted scope can sometimes be easier to analyze, as a more narrowly focused problem may be more easily solved. This thesis will concentrate on the details of spectra features, which have received less attention than the other issues, probably for two reasons. First. it is difficult to make detailed spectral measurements w hile avoiding spat id and temporal averaging. Secondly. there is an unfortunate lack of theory to predict or describe spectral characteristics. While incoherent scatter t heory is sufficiently advanced t hat detailed spectra can be determined from physical parameters the state of coherent scatter theory is not as immediately applicable. That said. technical improvements in measurement and analysis provide an opportunity for more detailed spectral characterization. while recent advances in theory make some interesting predictions which can be compared wi t h the data.

1.2 VHF and UHF

The carier frequency used to observe radio aurora has a significant effect on the nature of the received echoes. One important set of reasons for this are effects due to transmission of an elec t ro-magnetic wave t hrough a collisional plasma (ie. the ion* sphere). This is especially important at the frequency ranges of 3-30 Mhz (referred to as High Frequency or HF) and 30-300 Mhz (Very High Frequency, VHF) as refraction can cause significant beam deflections (Watermann [1990]). As a result, it can be difficult to determine the precise echo location. Frorn an instrumental perspective. HF and VHF systems usually have bearns formed from a small number of antenna elements, and thus have relatively broad (> 5 degree) horizontal bearns and very poor vertical resolution. Consequent ly, large scat tering volumes are observed and t here can be significant effects due to spatial averaging or multiple echo regions. There are also clear differences between the basic characterist ics of echoes observed at differect frequencies. Most obviously. the scat tering cross-section is a strong function of frequency (Moorcroft [1957]). In addition. it has been noted that there are several "types" of echoes at VHF (Haldoupis [1989]). while echoes observed at UHF can perhaps be better viewed as variations on one basic theme. In this thesis we will take advantage of the (potentially) better spatial resolution available at UHF. and concentrate on examining the spectral characteristics of the echoes observed at these higher frequencies. Some comparison will be made to lower frequency (VHF) results. but the primary focus rvill be the study and classification of C'HF echoes. From a t heoretical perspective. an important transition occurs at scale lengths observed by CHF sjPstems.When the plasma irregularity wavelengt h is large then it is possible to use fluid theory to describe the instability mechanisms. However. the lowest radar CH F radar frequency (3OOMhz ) observes i iistabili ty rvavelengt hs of about

O.5m or wavenumbers k = $ zz 13m-'. For these cases the condition kAD << 1 no longer holds and the effects of Landau darnping musc be considered. Consequently at UHF it is necessary to adopt a kinetic mode1 which accounts for the effects of particle trajectories. a much more complicated approach than used in fluid theory. Since the majority of observations have been obtained at VHF, typically between 50-150 Mhz or k=2-6 m-'. it is only natural that the bulk of theoretical work should have been done using the Auid theory framework. Xdditional kinetic treatments have shown that while there are changes at shorter wavelengths, the general features of the fluid results are still valid. Reviews of theory and comparison to experiment can be found in Fejer and Kelley [19SO]and Keskinen and Ossakow [1983]. Results from linear cdculations have shown t hat the two-st rearn inst abili ty is the best candidate for the source of most Eregion auroral echoes. In the simplest fom. the oscillation frequency - - ;zk-vd (1.1) should vary as the cosine of the angle between the radar line of sight and the relative electron-ion drift velocity. Requirements for a positive growth rate give

ie. the component of the drift velocity in the direction of the wave must be larger than the ion-acoustic speed (typically 300-500 m/~). This can be used to explain observations with Vd > c, in a cone around the flow direction. Some additional mechanism is required for echoes observed nearly perpendicular to the mean flow: the two step mode1 of Sudan et al. [19731 uses long waveiength modes as a source to produce shorter wavelengt h instabili ties at oblique angles. However. this does not address the observations of very small Doppler shifts. Other difficulties with current linear theories include a failure to explain large aspect angle observations. or to provide any information about the shape of the scattered power spectrum. It should be noted that these limitations exist for both auid and kinetic theories. and that some sort of non-linear treatment is required. .A number of recent papers have exarnined different non-linear effects including anomalous diffusion (Robinson [1986]). frequency broadening (Robinson and Honary [1990]; Hamza [1992]) and mode-coupling (Hamza and St-Maurice [1993b]). Direct cornparison of theory to experiment is often quite difficult. especially since fluid models are often used in the theoretical work. and may not necessarily apply to UHF.

1.3 Bistatic radar notes

A single station (monostatic) radar system has the transmitter and receiver located close to one another, often sharing the sarne antenna. Pulses of radiation travel outward in hemi-spherical shells. and echoes observed at some delay T rnust have come from a region at a distance (range) of

c(T - Tp)5 2 x Range 5 cT (1-3) where Tp is the pulse length. and c is the speed of light. Gsually a highly directive antenna is used to increase the likelihood that signals are detected only from a small solid angle. which when combined wit h the range information defines the volume of a scattering region. For the case of a physically separated transmitter and receiver pair. scattering will be observed from targt regions which fulfill the two conditions

1. Scattering can only occur when the volume is illuminated by the transmitter pulse.

2. Signals received at some time T can only come from scatterers that are a distance cTl away, and were ihminated at time T - Tl or equivalent ly

This means that the signal observed at a given time colild come from anywhere on an ellipsoidal shell. or some oval at a fixed altitude. Of course, directive antennae are used to produce a smaller effective volume. but the existence of sidelobes can in troduce the occasional complication. In general, the spectral frequency shift A f due to a target with velocity is given by (Sheffield [1975])

where fo is the radar frequency, c is the speed of light, k terrns are the transrnitter- scat terer and scatterer-receiver wave-vectors, y is the scattering angle between the wave-vectors, and VDoppieris the component of velocity dong the scattering wavevector. For a monostatic radar 7 = O and Equation 1.5 can be rearranged to give 6

where X is the radar wavelength. Bistatic links have a slightly more complicated relation

which can be important for large scattering angles ie. a 50 degree scattering angle will produce a 10% change in the Doppler velocity corresponding to a given frequency shift . A related feat ure of bistatic systerns is t hat the receiver niook" direct ion is act ually the bzsector of the transrnitted and received wave paths. Using the previous exarnple, although two sites may have wave vectors separated by 50 degrees. the "lookn direction of the remote site is only 25 degrees different from that of a monostatic system. While t his is a relatively small change when measuring velocity components. it can produce significant differences in aspect angle, allowing two sites to observe an identical t~getunder very different magnetic geometries. There are a variety of other issues which must be considered for bistatic radar experiments. such as polarizat ion. beam posi t ioning. timing and frequency synchro- nization. and system sensi t ivity (see Folkestad [1979. 19811 for more details). How- ever. these generally affect the total received power or SWR. and do not change the spectral characteristics. The task of signal level calibration is a difficult one and very little use will be made here of absolute power figures. although some attention will be paid to relative values.

1.4 Review of previous experiments

As discussed previously. observaiions of the radio aurora have been carried out under a wide range of conditions using a variety of experimental configurations. This section contains a review of the UHF experiments for which spectral information is available. Relevant system parameters are presented, and the experimentd results are

summarized. Spectra are generally identified as singly peaked. in which case moment estimates are provided, or more complicated (ie. multiple peaks). This approach is typical of the literature: a more detailed discussion of spectral characterizat ion is given in Chapter 2.

1.4.1 Millstone Hill (1295 Mhz)

The first detailed spectral measurements of the radio aurora were made by .4bel and Newell [1969]. Using a narrow beam (0.6 degrees) and relatively short pulses (500 microseccnds) . t hey were able to minimize the effects of spatial inhomogeneities. The operat ing frequency of 1295 Mhz is at the upper end of the range for which spectral data are available. but the basic features are the same as observed in later UHF experiments. In fact, it is worth noting that Abel and XeweIl [1969] contains useful srnail aspect angle measurements of virtually dl important radio auroral characteristics: height profiles. aspect sensitivit- spectral shapes and azimuth Doppler scans. Power spectra were obtained using a bank of 48 (500 hz wide) filters providing good frequency resolution. The shape was typically that of a single peak, with widths (actually values of full-width haif maximum. FWHM'see 5.4.5) of approximately 240 to 960 m/~(2 to 8 khz), aithough broader composite spectra were observed between regions of opposite Doppler shifts. Further, Abel and Newell [1969] note that some of the spectral peaks had a "negative tail whicb makes the overall spectrum quite skeweb. although no quantitative estimates of asymmetry were provided. Finally. the Doppler shift at peak echo intensity was measured at a variety of elevation and azimut h angleso and the typical magnitude was found to be 2-1 khz (230-160m/s). At a fixed eievation the velocity variation with azimuth was not sinusoidal (as might be expected from line-of-sight observation of a fixed vector velocity ), but rather tended towards a velocity plateau over at least 20 degrees of azimuth, followed by a rapid transition through zero velocity, then ânother extended plateau at a speed similar to the first but with opposite sign. Further observations using the same radar during 1970-71 are described by Hagfors

[1972]. In this case, spectra were obtained through a set of double-pulse observations with a range of lags. As before, most observed spectra were singly peaked, with Doppler shifts of 3-4 khz and widths (FWHY)of 230 to 810 % (2-7 khz). While some of the spectra appear to be slightly skewed. the effect is not clear. and it is only rnentioned briefly. The work contains one peculiar result. where the "signal level is quite high. and the spectral width is nearfy 20 kkz if we [Hagfors] may venture to apply a width to this sort of spectra'. The caveat is understandable. as the anomalous spectra do not have any weil defined structure. but contain a considerable amount of power distributed apparently at random over the frequency range.

1.4.2 Horner

The next detailed set of CHF spectral data corne from the Homer radar (see Tsunoda et al. (19741 for details) during the early to mid 1970s. In its standard mode. the Homer system obtained scattered power and Doppler shift at 395 Mhz, using a large steerable phased array to quickly measure azimuth/range &rnaps3. The beam widt h of 2.3 degrees (between the 112 power points) is significantly Iarger than that used in most UHF svstems. but is still adequate for avoiding large scale (100 km) spatial aliasing. Whilt the standard mode provided only mean Doppler shift. a more sophisticated configuration was used on several occasions during 1974 and 1976. This mode provided more spectral detail. although due to experimental limitations sophisticated signal processing (Maximum Entropy. see Moorcroft [EK8]j was required. While the Homer data set suffers from limited spatial and spectral resolution, it has the corn- pensating advantage of size. Several hundred thousand rneasurements were made at small aspect angles ( < l degree). allowing statistical analysis to minimize limitations in any given spectrurn. Examples of individual spectra are given by Moorcroft and Tsunoda [1978], along with estimates of the Doppler velocity associated with the spectral peak, and the width at half-power (FwHM).They note that single peaked spectra have widtha of 100 to 500 mis, with narrow spectra tending to have higher Doppler velocities. For the single peaked spectra, it was found that the SNR and Doppler speed were correlated, but tests by Moorcroft [1978] found that the wider spectra observed at srnall Doppler shifts were not simply due to a spurious increase in apparent width at small signal ievels. In addition, there were a significant number of double peaked spectra. with

the peaks separated by 500 to 1500 m/s (typically 1000 m/s). A study by Hall and Moorcroft [19SS] found similar results. with the majority

(more than 70%) of echoes being single peaked. It aas also found that the peak Doppler velocity ranged over f800 m/s. with no clear preferred value. Spectral width was found to increase slightly at srnall Doppler shifts. but the differences were relatively small (300 versus 500 m/s). Double peaked spectra were found to occur more frequently with velocities of f400 m/s. although other combinations were observed. Most recently. Moorcroft [1996] carried out a large scale analysis of some 97,000 spectra with the goal of studying flow angle effects. As in previous studies. it was found that the average spectral width (FWHM) at a given Doppler shift varied from 400-600 m/s. with the largest widths occurring near zero Doppler shift. .4 histogran of spectral widths shows a cutofT at about 120 m/s. with the most probable value around 223 m/s. Only single peaked spectra were used for this study, so no features of double peaked spectra are considered.

1.4.3 EISCATand COSCAT

The third family of UHF spectral observations have been collected using the EISCAT facility (see Baron [1986] for details) at 933 Mhz with a fairly narrow beam (0.6 degrees). Initial results were reported by Moorcroft and Schlegel [1988], who present some sample spectra and estimates of peak widths and velocities. These spectra are al1 single peaked, with Doppler shifts from 260-150 m/s, and widths of 450 m/s or less. it should be noted that these results were obtained using the ELSCATsystem in a monostatic (single station) mode, similar to the previously discussed Homer and hlillstone Hill configurations. However, the EISCATdata were observed at much larger aspect angles (> 5 degrees). Subsequently, Schlegel and Moorcroft [1989]used the EISCATsystem in a tristatic configuration, still at large (3-6 degrees) aspect angles. They present several examples of ACFs and spectra observed at a remote site. using the second station to achieve high spatial and spectral resolution. The .4CF hdf widths were found to range from 200100 microseconds. which correspond to spectral half-widths of 350-750 m/s). As well, it is noted that -the spectra are generally 'skewed' in such a way that a steep fall-of occurs toward higizer Doppler shifts and a decrease that is les steep toward zero Doppler sfzift,extending even to dues with the opposite signn. Further work by Schlegel et al. [1990] used a novel mode to collect spectral data with a time resolution of 100 milliseconds. significantly shorter than the 5 second integration t ime used in previous Ersc.;~ experiments. Spect ra measured on t his time scale were found to be very similar to previous observations. All were single peaked with widths (standard deviations) of 500-700 m/s, mem Doppler shifts of 200-300 m/,,and a skewness of approximately -1. The addition of a low power CW transmitter dlowed the EISCATreceivers to observe small aspect angles: the new COSCATsystem and initial results were presented by McCrea et al. [1991]. Spectra shown in this paper have a narrow (stdev=2OO to 300 m/s) skewed shape (skewb=100 m/s) over a range of mean speeds (300-500 m/s). One interesting feature is t hat the narrowest echoes (stdev = 170 m/s) occur exclusively at speeds of rt4-0 m/.. Other COSCATresults are contained in Eglitis et al. [1995], who show phase speeds of -600 to 600 m/, and widths (FWHM)of 100 to 600 m/s, where the most typical values were 150 to 250 m/s.

Most recent ly Jackel (19921; Jackel et al. (19971 studied sorne EISCAT data collected at very large (4-8 degrees) aspect angles. Spectral shapes were consistent with those of previous experiments: a single peak with mean Doppler speeds ranging from O to 700 m/S. Direct calculation of spectral moments indicated that the width (standard deviation) was essentially constant at 400 m/s, independent of scattering cross section or mean Doppler shift. Skewness values were about -1, but with considerable scatter. 1.4.4 Millstone Hill (440 Mhz)

The final group of observations were collected using a 440 Mhz incoherent scatter radar dso located at Millstone Hill. This system has a narrow beam (1 degree) and can operate with a range of pulse lengths. Most spectral observations were obtained using a 1000 microsecond pulse or longer. so spatial resolution is no better than 50 kilometres. Typical integration times are on the order of 10 seconds to 1 minute. First results of coherent spectra are contained in St.-Maurice et al. [1989] which also deals with height profiles and scattered power. Power spectra were found to be mostly single peaked. with Doppler speeds in the range 0-500 m/s. Spectral shapes were similar to those observed in other CHF experiments, although no explicit mention was made of width or skewness. An example of a double peaked spectrum is given: it appears during an azimuth scan and is a superposition of the single peaked spect ra observed at neighboring locations. Further observations are given by Foster and Tetenbaum [1991], who observed echoes at small !< 1 degrees) aspect angles on six different days. 30 individual spectra are presented, but several sets of mean Doppler shift are shown. During some of the most disturbed times the speeds approach 700 m/s, with more typical values in the range 0-300 m/.. A related study by Foster and Tetenbaum [1993] examines the correlation between cross-section and velocity, during which phase speeds of 0-750 m/, are observed. Interesting results are presented by Foster et al. [1992], who found a tendency for the Doppler velocity to decrease with increasing aspect angle. This pattern is not completely unambiguous, as the data were collected by scanning in elevation, and there may be contributions from spatial and flow angle variations. Despite this? there seems to be a clear difference between the velocities observed at aspect angles above and below 3 degrees. Unfortunately, no other spectral moments or sample spectra are provided. Further work by del Pozo et al. [1993] at small aspect angles contains several exarnples of coherent spectra. most of which are singly peaked. The velocity of the spectral peak ranges from -300 to +300 m/s. and some of the spectra are clearly asymmetric. Further analysis was carried out by assuming that the spectra were Lorentzian in shape. although no detailed tests of this assurnption were provided. Plots of the velocity versus azirnuth show indications of plateaus on either side of a quick transit ion region. alt hough the non-sinusoidal nature is less pronounced t han in the Horner results. Perhaps the most interesting results are contained in cornparisons of velocity with spectral rvidth which suggest that the narrowest (FwHM=YOOm/.) spectra occur at large velocities. while small Doppler shifts are generally associated with wider (FwHM=~~~m/sj spectra.

1.4.5 Summary

The majority of spectra show a single peak. and most of the more complex results can be interpreted as a superposition of two peaks. often in situations where spatial aliasing could have occurred. As a consequence, Chapter 2 shall concentrate on the characterizat ion of single peaked structures. while remaining alert to the occurrence of more complicated features. Doppler velocities cover a wide range (k1000 m/s j. although values of 300 to 100 m/s are most typical. Spectral widths are less variable. generally between 300 to 600 m/s. Many spectra exhibit a skewed shape, but quantitative results are not generally available. Similady, issues such as detailed line shapes (ie. Lorentzian or Gaussian) have not been well explored.

1.5 Structure of this thesis

This chapter has presented a basic introduction to the kind of observations made with UHF radar systems. with a particular emphasis on spectral characteristics. In the second chapter the physical significance of power spectra will be discussed, then examples of observed spectra will be used to present several different methods of characterization. These methods will be applied to data collected by the MIDASC Table 1.1: -4 summary of the spectral characteristics given in previous CHF observations of the radar aurora. Widths reier to the full-width half maximum. except for those marked (ij which indicate the standard deviation. Site Frequency Year Aspect Doppler Width Skewed (-Mhz) Angle Speed Mills tone 1395 196ga small 230-460 210-960 -ves 1972~ small 345-460 240-810 yes Homer 398 1978' small 0-800 100-900 198gd small 0-1000 300-600 1996' small 0-800 100-600 EISCAT 933 1988' large 200-350 300-600 1989g large 100-300 350-790 ves 1990h large 200-400 500-700i ves 1997' large 0-700 300f ves COSCAT 933 1991' smail 900-500 200-300i yes 1995~ small 0-600 100-600 -ves -Millst one 340 1989' smdl 0-500 300-1000 yes 1991" small 0-700 1992" small 0-150 1993" small 0-700 large 0-400 1993P small 0-600 200-300 _ves a) Abel and Newell [1969] b) Hagfors [1972] c) Moorcroft and Tsunoda [1978] d) Hall and Moorcroft [1988] e j Moorcroft [1996] f) Moorcroft and SchIegel [1988] g) Schlegel and Moorcroft [1989] h) Schlegel et al. [1990] i) Jackel et al. [199f] j) McCrea et al. [1991] k) Eglitis et al. [1995] 1) SC.-Mauriceet al. [1989] m) Foster and Tetenbaurn [1991] n) Foster and Tetenbaum [1992] O) Foster et al. [1992] p) del fozo et al. [1993]

system in Chapter 3: Chapter 4 will contain a similar analysis of E~SCATand COSCAT results. Finally, a cornparison of the results wiil be given in Chapter 5, along with suggestions for future work. Some definitions and calculations have been placed in an .i\ppendix, and are referred to throughout. Chapter 2 Spectral Characterization

-4s mentioned in Chapter 1. the received signal (electric field magnitude) of radio wave scattering from the ionosphere can be expressed in terms of a power spectral distribution. Such a representation is useful because it is closely related to the distribution of electrons in the scattering volume (Sheffield (19733; Hagfors [1977])

at a particular wavevector C (spatial scale). In turn. the electron distribution can be calculated from a governing set of equations and assumptions. or simpiy used to infer basic characteristics of the scattering source. For example. if the received signal is Doppler shifted by some amount &, then the scattering electrons may be in motion relative to the radar. The width or spread of the received spectrum can give indications as to the range of apparent electron velocities or alternatively may be related to the lifetime of a plasma instability. Multiple peaks rnay be due to multiple scattering regions, or to some more cornplicated coupling phenornenon. In the case where the scattering is due to thermal fluctuations in a plasma (incoherent scatter), it is possible to calculate the power spectral density as a function of physical parameters such as electron and ion velocities, temperatures, and collision frequencies (Sheffield [1975]; Hagfors (1977); Trulsen and BjmnA [1977]). As a result. it is possible to fit a mode1 spectrum to the data, with the resulting properties providing direct physical information about the region of interest. Coherent scatter is not currently so well understood. There are equations which can be used to predict the instability phase velocity under certain conditions, but the linear theory used to derive them may not always be valid. A considerable amount of effort has been spent on including non-linear effects. but so far these efforts have met with limited success. at least when it cornes to the task of describing observed spectral shapes. Some recent work bu Harnza [1992]: Hamza and St-Maurice j1993b.a] has addressed some issues related to spectral features. specifically the relationship between spectral width and Doppler velocity. This chapter will present some "typical" examples of observed power spectra from a variety of experirnental configurations. These exarnples will be used to motivate a discussion of the kinds of information which can be determined from a power spectrum. As well. the relationship between power spect ra and autocorrelation functions (ACFS)will be used to enhance the study of spectral characteristics. For the sake of clarity much of the mathematical detail required for this chapter has been placed in Appendix A. .4s well. while methods discussed here will be applied to the sarnple spectra. discussion of the physical significance of any results will be left to Iater chapters.

2.1 From LPYS to ACFSto spectra

Although the object of physical interest may be the power spectral distribution of the scattering process, the directly rneasured quantity is a sequence z = {q,12,. . . z,~) of samples related to the scattered signal as a function of time. Generally the pulse length is larger than the sarnple rate. so that a number of samples rnay share contributions from the same region. If the pulse repetition frequency (PRF) is high and/or the scattering region is extended then the required data bandwidth may be becorne unmanagable, making it impossible to simply store every sample for later analysis. One method of reducing the data volume assumes that the scattering process is "wide-sense stationary" (see Papoulis [1984], page 220) so that the expectation value is a finite constant E{z(t))= r and the autocorrelation depends only on T = t2 - tl

If this assumption holds. at least for the time-scales of interest, then for any given pair of sarnples the corresponding element of the lag product matrix

is randomly distributed about the "tme" value of the autocorrelation function at lag (k - j)k. It should be noted that the samples z are usually complex valued, due to the signal processing methods used for most auroral radar systems. During each pulse period the N sarnples will produce N2 LPM values. although the symmetry of lag products (L~hl(j,k) = LPM'(~,j)) rneans that only $N(N - 1) unique numbers must be stored. While this may seem like an unwelcome inflation, LPMSfrom M successive pulse periods may be averaged together, so the cornparison between storing LPMSand storing al1 the sarnples becomes $V(N - 1) versus MN or

'(N2 - 1) versus M. For typical values of !V = 300 and iCI = lOhz x 30 seconds = 300. the LPMwill be 50% smaller. In practice the reduction in data volume is even greater. as the number of useful lags L is usually much smaller than the total number of samples: lags at greater delays do not contain information about the desired signal and need not be calculated, much less stored. -4s well, averaging has the added benefit of reducing the variance of any given lag product, which can be helpful for small signal levels. For stochastic signals the averaging must be applied to the power spectrum or .!CF,the random phase of the signals rneans that it is not possible to sirnply average A4 consecutive sets of sarnples. -4 lag product rnatrix contains al1 of the information required to construct the autocorrelation function. For a signal which contains the same spectral content throughout this is particularly simple, as the average of the main diagonal gives the zero lag, the first upper diagonal the first lag (as shown in Figure 3.1 b), and so on Another simple case is that of a single discrete (small volume) target. which will only

contribute to samples z,~. . . zn2?where (722 - nl)Ar is the pulse length. Clearly. the ACF for this case should only be constructed from the lags which were produced ent irely from the signal samples

such as in Figure 2.1~.

Figure 2.1: Examples of a Lag Product Matrix for five samples and four lags. The full LPW is shown in (a), al1 the products for the first ACF lag in (b), and the products constructed from samples 2 through 4 in (c).

The general case of multiple overlapping signals is more complex. While the largest possible number of lags are required for maximum spectral resolution. it may be necessary to limit the range of samples in order to isolate a single source. Partially overlapping signals may be separated reasonabiy well, but if two signals occupy the same sarnple range then it is impossible to determine their individual characteristics, or even to distinguish between a single "complexn signal or multiple "simplen components. For these reasons bistatic radar systems can often obtain better spectral resolution t han a monostatic system. Assuming the same sampling rate, bot h systems can collect the same number of lags from any given target volume. but the monostatic returns may contain contributions from other sources, while a (narrow beam) bistatic receiver will be preferentially observing a smaller region. Of course, this will ody be significant if t here are multiple closely spaced scat terers. and the monost atic system will always have the advantage of being able to study the structure of spatial variations. which is more difficult for a bistatic link, Once some set of lag products have been selected the CF can be assembled. Using the exarnple in Figure 2.k, there are three available lags {po, pie'm1,pze'"). In addition, the symmetry of the ACF gives two negative lags, as the true "twe

sided" ACF is {pze-'*, pl e-'*~ ? po, pi eiQ1, p2ee*); these additional lags were implicit in the construction and storage of the LPM,but must be explicitly included in the production of a power spectmm. .At t his stage it is wort h commenting on the fact that N measurements have been used to produce a two-sided ACF with 2N - 1 elements. While this apparent doubling of information may seem questionable, it is due to the fact that the N original samples are complex values. and the components of the ACF are even or odd about the origin. Transformation of the ACF to a power spectrum is not completely straightforward. The discrete version of the Fourier transform

is not computationally efficient, which may be an issue for large data sets. The usual response is to use a Fast Fourier Transform (FFT) algorithm. which scaies as O(N ln N) rather than 0(X2).One problem with this is that FFTs typically work best for a number of points M that is factorable into smaller values, ideally M = P. Clearly 2.V - 1 will always be odd, and rnay be a prime number for unfortunate choices of N. Even if the FFT algorithm will accept an odd nurnber of elements, the results may be subtly different than for even numbers of elements. Consequently. it is preferable to arrange matters so that the number of points to be transformed is even. The simplest way to achieve this is the removal of the largest negative ACF log, so that there are 2N - 2 values passed to the FFT. These are typically arranged with the positive Iags first

ACF = {po, pl P" .p2et@. . . . .p~-~e'"~-~, pN- 1 eleN-' .pN-Ze-'Qx-2 . . . . .ple-'mi) (2.8) which produces a power spectrum at the discrete frequencies

where the first element is the zero frequency ( DC) value and the middle element is the

Nyquist frequency fNypuist = &. These values can be arranged into a more familiar order.

which makes explicit the assumption of periodicity, and the fact that frequencies at

Ifivyquiat map to the same bin. In this case the frequency resolution is given by

Bandwidth 1 1 Af = --- 2-Y - 2 AT 2(N - 1)

.-2 subtle complication arises because the input XCF is assumed to be periodic. with the midpoint value (at NI2 + 1) occurring at both -:VAT and NAT. For the arrangement given above this is not actually true, and any sharp jumps introduced in this way will introduce "ringingr into the resulting power spectrum. One response to t his is discussed by Brigham (19881, and involves the introduction of a new midpoint value which is the average of the latest positive and negative lags

to give an CF with 2N points

which transforms to a power spectrum with 2N intervals, for a resolution of

Af= Bandwidth - --1 1 2N AT 2N The difference between Equations 2.11 and 2.14 is smd for large IV, but may be significant if there are only a few lags. One advantage of dealing directly with the ACF is that it avoids many of the issues associated with actually calculating a power spectrum. alt hough of course there are operations which are best applied in the frequency domain, such as peak location (sA.8) and estirnating the FWHM(f A.5). Figure 2.2 illustrates some issues regarding the display of ACFSand power spectra. The first two puiels cont ain a well resolved two-sided autocorrelat ion funct ion. plot ted as real and imaginary components. and the corresponding spectral peak. Expressing the .!CF in terrns of its magnitude and phase (Figure 2.2~)more clearly separates the correlation timescale contained in the magnitude from the mean Doppler shift which dominates the phase term. -4s well. since the magnitude and phase are symmetric and anti-syrnmetric about iag zeroo negative lags need not be shown, allowing increiged detail in the same plotting interval. Since the dominant phase component is often a linear term. t his can be removed to show more subtle variations in the phase such as the cubic curve in Figure 2.Zd. This linear phase shift can be plotted on a narrow common scale, and spectral shapes can be more easily compared. Displays similar to panels d and e will be used in this study with the spectral scale changed to velocity units. as is commonly done for auroral echoes.

2.2 Selected Observations

In order to illustrate the issues which arise when analyzing power spectra and aute correlation functions, some examples of actud measurements are shown. Data are selected to display a range of features, as observed by different radars. The radar systems used (MIDASC,EISCAT and COSCAT)are described more fully in Chapters 3 and 4; only selected details are given here. As well, detailed discussion of results will be left to later chapters; the focus here is on generd concepts of display ond analysis of data. Table 2.1 contains a list of the six different data sets along with the date and time of measurement, the sampling interval and carrier frequency, and (a) Two-sided ACF with real and imaginary (dots) components

(b)Power spectrum of (a) (c)One-sided ACF with magnitude and phase (dashes) (d)Same as (c). linear phase removed (e) Power spectrum of (d), zoorned in to show detail

O 300 600 -15.0 -7.5 0.0 7.5 15.0 Lag time (microseconds) Frequency (khz)

Figure 2.2: -4 well resolved CF in panel (a) is plotted in terms of real and imaginary components, the power spectrum is shown in panel (b). Panels (c) and (d) display the ACF in more useful ways, a zoomed view of the shifted power spectrum is in (e). Table 2.1: Selected exarnples of observed data. ACF lag spacing (Ar) is given in microseconds. radar frequency ( fo ) in Megahertz.

Typical skewness 1 - - 7 E89b Troms0*Troms0 89/06/ 14 03:58 Moaostat ic volume C92a Oulu* Kiruna 92/06/27 21:15 Double peaks C92b Oulu-.cKiruna 91/06/28 2136 Long correlat ion Strong echoes 1 Long correlation

a note describing features of interest. Figure 2.3 shows the six example ACFS.presented as magnitudelphase plots. The linear phase trend has been removed, which is equivalent to shifting the mean power spectrum to zero frequency This more clearly shows subtle details in the phase curves, and allows a common scale for the power spectra. The same lag scale is used for al1 records to illustrate the range of correlation times, and the fact that certain records contain a truncated ACF. Similady, there are clear differences in spectral width. UThilemost of the records appear to be composed of a single peak. example C92a shows an asymmetric double peak and a corresponding oscillation in the ACF magnitude (see §A.9 for a discussion of the relationship between double peaks and oscillating ACF magnitudes.) The more complex spect.ra1 shape of C92b is a consequence of the ACF truncation. this introduces "ringing" which can be reduced by applying a window or taper to the ACF, at the cost of decreasing spectral resolution. Finally, it should be noted that the mean velocity (given in the upper right hand corner of each power spectral plot) covers a wide range, from -500 to 500 m/s. Figure 2.3: Six examples of coherent scatter autocorrelation functions and the corresponding power spectra, see Table 2.1 for data collection details. The left hand panel in each plot gives the magnitude (solid line) and phase (dashed line) of the ACF. a common scale of zero to 1000 microseconds was used on dl plots and the mean (lin- ex)phase shift has been removed to more clearly display higher order behavior. The right hand panel in each plot contains the resulting power spectrurn, dl on cornmon velocity scales of -1400 to 1400 m/s. The mean Doppler shift in metres per second is given in the upper right corner of each panel. 2.3 Spectral Moments

.4 very common way to describe spectra is in terms of their moments. As defined in section A.2. the zeroth moment is simply a measure of the total area under the power spectrum

the first moment is the mean value about which power is evenly divided

and higher order moments relate to the more subtle details of the spectral shape. Calculations are simplified if the zeroth moment is normalized to 1. which is also equivalent to normalizing the CF so that the zero lag is equal to 1. Once the spectrum is normalized. each moment is the result of a polynomial weighted integral of the spectrum

so if al1 the moments were known then the spectrum might be reconstructed from an appropriate combination of orthogonal polynomials. In practice it is not possible to obtain more than a few low order moment estimates from data. However. even these few moments can provide information in a manner that is easily understood and rnay even be p hysically relevant. For example, the first moment (mean) gives an indication of how far the spectrum has been shifted from the origin. which can be due to the bulk plasma motion or instability phase velocity of the observed scattering region. The second moment (variance) depends on how closely the rest of the spectrum is bunched up around the mean value; it is often a convenient estimator of spectral width, which in turn may be related to the collision frequencies or growth rates in the target volume. Asymmetries about the mean are contained in the third moment, however it is customary to use a dimensionless quantity which may be more appropriate for identifying similar shapes among spectra with different widths. .4n alternative choice developed for this study

is the "SKEWB-

where the narne was chosen to indicate that it is a skewness-like parameter based on the cube root of the third moment. It shares the same units as the mean and standard deviation. It should be noted here that the standard spectral line shapes ($4.4) are

ail symmetric. so a significant non-zero duefor skewness or SKEWB means that a profile cannot be characterized as being simply Gaussian or Lorentzian. Finally, the fourth moment is related to how "peaky" the spectrum is. X scaled version, jM4 kurtosis = -(du2)Z- 3 may be more useful, as it is zero for a Gaussian, positive for profiles with broader *shoulders". and negative for more Lorentzian type shapes. Beyond fourth order. moments become increasingly difficult to estimate accurately. and their physical significance is less obvious.

2.3.1 Estimation in the frequency domain

Actually calculating the moments of a power spectrum is not always a straightforward task. When dealing with a discrete set of measurements, the corresponding spectrum

1. Band-limited, with a finite frequency range given by the inverse of the sample int erval

2. Resolution limited, wit h a finite number of frequency bins

The first feature rneans that when calculating moments the range of integration is reduced frorn infinity to some limi ted spectral band. This will obviously be a problem if the spectral feature is wider than this range. Even narrow features may be troublesome if they are highly Doppler shifted towards either edge of the band. Secondly. there will be a finite number of frequency Obins'. and it can be difficult to resolve complicated details (such as multiple peaks) in a narrow spectral feature which only spans a few bins. -4 related problem is that narrow spectra correspond to wide ACFS. In extreme cases the ACF magnitude at the last lag may still be significant. which will cause "ringing" in the power spectrum. The choice is then dealing with negative values in the power spectrum or the loss of resolution from application of a window (taper) to the ACF. Even if the sampling effects are negligible, there is the ever present spectre of noise. Any measured spectrum will contain a contribution from undesired sources such as man-made interference, cosmic background and thermal effects in the receiver itself. Most modern radar systems have some kind of noise subtraction capability, but t hese are never perfectly effective. Finally, the apparent ly stochastic nature of the signal (and noise) spectrum introduces scatter in measured spectra which can only be countered by averaging over a large number of measurements. It is instructive to model the power spectrum as being composed of a signal corn- pooent (with some intrinsic shape). a random component due to the stochastic nature of the signal and noise spectra, and a DC component left over from an? error in the noise subtraction.

This simple model helps to illustrate some of the problems faced when estimating an arbitrary moment

Starting with the last term, it is clear that for an integration region which is symmetric about the mean, the contribution will be zero for al1 odd n, but not for n even (ie. standard deviation and kurtosis). If the limits of integration are not symmetric about the mean. then the odd moments will also be biased in some way. Effects due to this problem can be minimized by selecting syrnmetric limits of integration. or by shifting the spectral mean to zero. The middle term will be zero on average. but not for any particular measurement. For example, a small random variation at a frequency far from the mean will be heavily weighted. and contribute greatly to the moment estimate. Over an infinite integration range this effect will cancel out. but given a limited number of frequency bins it can actually be helpful to reduce the range of integrat ion so as to include only the signal region. This can significantly reduce the variance of moment estimates. but care must be taken to avoid truncating the signal component. which can have a significant impact on higher order moments.

2.3.2 Estimation fkom Acrs

As shown in section A.2. the moments of a power spectrum are related to the derivatives of the autocorrelation function at lag zero

a result which is well knoan in Fourier analysis (ie. Bracewell [1986]; Papoulis [1981]) and statistics (ie. 526.1.3 in hbramowitz and Stegun [1972]). From a practical perspective, this approach has been used with weather and MST radars to find Iow order moments such as power, Doppler velocity, and width (ie. Woodman and Hagfors [1969]; Zrnic [1979];Woodman [1985]: Keeler and Passarelli [1987]) for spectra which have approximately Gaussian profiles. Although moment estimation via derivatives of the ACF has proven effective for these applications, it seems not to have been applied to the study of auroral spectra. This oversight is puzzling, as this method seems ideally suited for overcoming some of the problems which arise when characterizing auroral spect ra, part icularly sampling and truncat ion effects. It may be that the obviously nomGaussian nature (ie. skewness) of many auroral power spectra rnay have been considered an impediment. but this is not generdly tme. Equation '2.23 can be greatly simplified by expressing the ACF in terms of magnitude and phase. king the symmetry and asymmetry of the magnitude and phase. combined with the normalization convention for the magnitude. it can be shown (54.3) that to fourth order the moments are simply

-4s a consequence. the mean and SliEw~cm be determined frorn the phase curve alone. while the standard deviation and kurtosis depend only on the shape of the magnitude. Even bet ter. al1 t hese moments require only a knowledge of the ACF near lag zero. so in theory only a few lags at small delay are required to obtain al1 the moments of interest. Of course, in practice the situation is slightly more complicated. Noise is a significant issue. as broadband noise will be mapped to the first few lags of the ACF. precisely those of interest. Deri vat ive estimation is not trivial: 9.4.3.1 addresses some of the problems which can arise when dealing with discrete noisy data. More subtle questions. such as whether certain moments actually exist, may need to be considered (see SA.4, A.6). Despite these concerns, it is possible to obtain reliable moment estimates from most ACFSup to 4th order. Table 2.2 contains moment estimates for the six example spectra. using the 2nd through 6th lags of the ACF. It should be noted that estimating moments in the spectral domain would require the use of al1 available lags to obtain the power spectrurn. Even then, ringing due to truncation effects in examples C92b and M95b would render such results useless.

2.4 Models of the ACF magnitude

While moments may provide useful information about the power spectral distribution. a more detailed description of the spectral shape would be welcome. In order to accomplish this it is necessary to make some assurnptions about the characteristics Table 2.2: Spectral moments of example data. Al1 values are in m/s, except Kurtosis which is dirnensionless. Note the sign difference between mean velocity and SKEWB. II E89a 1 E89b 1 C92a 1 C92b 1 M95a 1 M95b 1 Mean -404 -531 -62 -236 47 3237 Standard Deviation 363 434 239 121 256 105

SKEWB 37.5 400 111 163 -249 -150 Kurtosis 3.2 2.0 -2.3 13.6 1 0.1 -3.3

of the spectrum and to select a mode1 which has parameters that cm be adjusted to fit observations. These parameters may or rnay not correspond to some physical property of the scatterers. but can be helpful in classifying different features. Two models which are commonly used to describe spectral features are Gaussian and Lorenztian line shapes (5A.4). While they have quite different shapes in the spec- t ral domain. when transformed to ACFSt heir magnitudes take on a similar character. Both can be described by the equation

where the decay exponent R, = 1,2 for the Lorentzian and Gaussian cases respectively. A theoretical basis for the first mode1 is mentioned by del Pozo et al. [1993] who note that in na homogeneous and stationary turbulence regime. such peaks can be interpreted as Lorentzian distributions whose half power widths dehe the relaxation times". For the second model, the signal is assumed to be due to a large number of independent scatterers. and the central limit theorem is used to obtain a Gaussian result. At UHF, the literature is silent on which of these models is more appropriate, although some work has been carried out at HF (ie. Hanuise et al. [1993]). Looking at the examples of p(r) shown in Figure 2.3, there does seem to be a generally exponentid character to the ACF magnitude. However, the details are complicated by the range of decay rates (spectral widths). One way to remove this Table 2.3: Correlation times and decay exponents corresponding to the six sets of example data. Note that the value of 3.5 for C92a is the upper limit for the the fitting routine. the "true" value is actually larger. 7 E89a E89b C92a C92b M95a M95b

Correlation Time 144 122 17 764 395 573 DecayExponent 1.52 1.36 3.5 1.22 1.10 1.98

dependence is to introduce a scaled time

7- r, = - where b(re) = Te While there is probably no physical significance to the choice of r,, it is mathematically convenient (when r = T,. T,"~ = 1 for any n,) and easily applied to the data. Using this scding, the example data becorne the curves shown in Figure 2.4, where the dashed lines denote the Lorentzian and Gaussian models. Interest ingly. the data falls between the two model curves, suggesting that n, ranges from 1 to 2. with a slight preference towards the larger value. By carrying out a non-linear fit to the data. it is easy to determine what value of n, best characterizes any given record. This procedure will be applied to data in later chapters, and the resulting estimates of n, will be compared to other parameters. It should be noted that r, will be estimated directly from the data (using linear interpolation); although it might be possible to improve the fit to data by varying it along with n,? there is no justification for this increased complexity, and the model is already sufficientiy ad-hoc. Table 2.3 gives the estimated values for re and n, for the exarnple ACFS. Cor- relation times (re)are roughly proportional to the inverse of the standard deviation (see 5A.5). Decay exponents (n,) are al1 between 1 and 2, indicating that neither Gaussian nor Lorenztian lineshapes are appropriate. Not surprisingly, while this model may describe the data quite well, it is not perfect. Some limitations can be seen by plotting the magnitude of the measured ACF against the modelled values, which is best done on a logarithmic scale to more 1 Scaled lag time

Figure 2.4: Four of the example ACFmagnitudes displayed with a logarithmic y-axis. The x-axis is scaled lag time so that al1 magnitudes equal -1 at T, = 1. Curves for the Gaussian and Lorentzian spectral models are given as dashed and dotted lines respectively. clearly display features over a wide range of magnitudes. Figure 2.5 shows the six example ACFSdisplayed in this way. where a perfect fit between data and the model would be a straight line along the diagonal. Both the C92a and M95a records deviate considerably from this ideal. not surprisingly as they both have more complex spectral shapes. The other four records show good agreement with the model. with the E89a and E89b data matching the rnodel ive11 down to the 10% level, after which there is some slight deviation. Since magnitudes for the M95b and C92b -4~~sare truncated, al1 that can be said is that the model describes the data very well within the limited range. The other component of the ACF deserves some attention. The most prominent feature of the phase curve is a linear term, which corresponds physically to the Doppler shift of the spectral mean. If the linear trend is removed, the phase curve generaily has a shape which resembles a low order polynomial. Fitting a cubic term to the detrended phase works well for most of the ACF magnitudes greater than about 0.1, 0.1 0 Measurernents

Figure 2.5: Measured ACF magnitudes compared to the model exp(-(r/~~)"~) using values of T. and n, estimated from the data. Results are poor for C92a, which has a clearly double peaked spectrum. and M95a, which also appears to have a more complex spectral shape. Both the E89 records agree well down to the 10% level: the C92b and M95b ACFSalso match well within their limited ranges.

but there is usually a significant residual at later lags. Introduction of increasingly higher order polynomials (x5' xi etc.) provides some irnprovement. but a very large number of terms are required to produce a fit that is significantly bet ter t han a simple linear plus cubic model.

2.5 Other features

2.5.1 Peak hequency

When spectra are both broad and asymrnetric, there rnay be a significant offset between the mean frequency (first moment) and the location of the spectral peak (often called the peak frequency). Alternatively, if there are multiple peaks in the spectnim then the mean value may not have any physical significance. Currently, there is no reason to believe that for simple spectra the peak frequency is any more or less useful Table 2.1: Location of the spectral velocity peak for the six sets of exarnple data. E89a E89b C92a C92b M95a M95b Peak Velocity -524 -590 -34 -300 134 502

than the mean frequency. However. it is a visually appealing feature t hat can be easily estimated for well resolved spectra, and has seen occasional use in the literature. As shown in 5.4.8 the spectral peak is best obtained in the spectral domain. even for poorly resolved spectra: unlike moment estimation there is no advantage to operating in the lag domain. Table 2.4 contains the spectral peaks for the example data: these are similar to the spectral mean except for the more complex spectra in C92a and M95b. Given the similarity between spectral peak and mean. and the lack of any reason to prefer either one. the more easily obtained mean will be used throughout this study.

2.5.2 Full-width half-maximum (FWHM)

A commonly used spectral widt h estimator is the full-width at half-maximum (FWHM? $A..j)?also referred to as the distance between half-power points. FWHMdepends only on the width of the spectral peak, unlike the standard deviation which is sensitive to small variations far from the mean. Consequently FWHMis a much more robust estimator, although it does require that the spectral peak be well resoived. The FWHM can be easily determined for the standard line shapes (5A.4); a Gaussian has FWHM=2.350, more than twice the standard deviation. The result for a Lorentzian, FWHM = 29 illustrates another advantage of this estimator, as it can be determined for spectra whose second moments are infinite. In general, the FWHMis relatively insensitive to the detailed spectral shape, whereas the standard deviation for any given correlation t ime depends strongly on the decay exponent. For t his reason it is difficult to compare values of standard deviation and FWHMunless other information about the spectral shape is available. Overall, the FWHMis probably a more useful measure of spectral width than standard deviation. but it may be difficult to obtain from very narrow spect ra, unlike the correlat ion t ime.

2.5.3 Double peaks

Although the majority of CTHFauroral echoes have single peaked spectra, there are also a significant nurnber which have multiple (usually two) peaks. -4s well. some of the spectra have a more cornplex single peak which could be interpreted as the superposition of two or more components. Double ~eakedshapes are usually easily identified in the frequency domain and estimates of the peak locations and spacing can be obtained manually. Other indicators are provided by the standard deviation, which will be much wider, and kurtois, which should become more positive. Measures such as the FWHMcm also be useful, but may run into trouble if there is a deep valley between the peaks. Looking at the lag domain, it can be shown (3A.9) that identical symmetric spectral features separated by a frequency Aw have the Fourier transform

-ACF(T) = p(r)2 COS (FT) = p(T)etG7 where p(~)is the ACFmagnitude of a single feature. and ;i is the average of the two mean frequencies. This simple mode1 suggests that multiple peaks will produce an oscillating ACF magnitude P(r);the peak spacing can be estimated either by fitting a cosine of angular frequency % or simple visual identification of the nulls at T = with n an integer. Given that double peaks produce oscillations in the ACF magnitude, it is ternpting to view any oscillation as a sign of multiple peaks. Figure 2.6 shows an example of how this simplist ic approach can fail, where obvious oscillations in the ACF magnitude results in a spectrum with a complicated but single peaked shape. This failure is not surprising, as although there may be multiple component peaks, they can have different widths and shapes, or be so close as to be indistinguishable as separate peaks . OdB

-1 OdB

-20dB

O 300 600 -1000 O 1 O00 Lag tirne (us) Velocity (m/s)

Figure 2.6: .4n autocorrelation function with nulls in the magnitude (solid linel displayed on a logarithmic scale) and the corresponding power spectrum. The notch at about 380ps corresponds to a peak separation of about 220 mis. much less than the width of the spectral feature. The CF phase (dashed line, with a linear scale) shows two regions of approximately linear phase. with a sharp transition at the ACF null.

In general. the spectra and ACFSused in this study are described by the rules

0 Smoothly decaying .&CF magnitudes correspond to smoot h single peaked spectra

Double spectral peaks always produce oscillations in the ACF magnitude

Oscillations in the ACF magnitude correspond to a complex spectral shape. possibly with double peaks

-4n automated method for detecting oscillations in the CF magnitude would be quite useful for classifyirig spectral shapes as simple or complex. Unfortunately, while this task is easily carried out by eye. it has proved more difficult to develop a general algorithm which works for a wide variety of correlation tirnes and noise levels. 2.6 Summary

Spectral moments axe useful for characterizing the frequency distribution of a scattered signal. Although they are usually determined from the power spectrum. an alternative method of estimation from the autocorrelation function is also possible. Working directly with the ACF has some advantages. particularly for very narrow spectral features. Remaining in the lag domain. models of the .!CF can be fit to data. rather than studying spectral shapes directly in the frequency domain. A simple mode1 of the ACF magnitude provides estimates of the correlation time and decay exponent, which are related to the spectral "width" and 'shape'. Ail these methods are quite general. but t heir results are most easily interpreted when the spectra are single peaked. In the next two chapters many of these techniques will be appljed to a variety of data. and the resulting parameters used to describe spectrd characteristics. Q Chapter 3 Results at 34crn: MIDASC

Located near Boston. Massachusetts. the Millstone Hill radar is well suited for st udies of the radio aurora. Although originally designed and primarily used as an incoherent scatter radar. it can view as extensive region of zero aspect angle E-region targets as well as the entire range of larger aspect angles. While coherent echoes are merely un- desirable interference when studying incoherent scat ter, in recent years the Millstone Hill system has also been deliberately operated as an "auroraln radar. No fundamen- ta1 changes to the system are required for these studies, only the use of attenuation to ded wi th the increased scattering cross-sections and possi bly some changes to the pulse length and sampling rates. The current system can easily accommodate these requirements with a flexible design that allows a wide variety of pulse configurations and sampling schemes. In a series of papers (St.-Maurice et al. [1989]; Foster and Tetenbaum [lW1. 19921; Foster et al. [1992]; del Pozo et al. [1993]) results have been presented for a large number of coherent echo observations of instabili ty wavelengths around 34 cent imet res (the main transmitter frequency is 140 Mhz). Results of these studies (summarized in 5 1.4.4) were consistent wit h previous observations. but also exposed some interesting new features. Perhaps the most significant results are contained in Foster et al. [1992] who observed that aspect sensitivity and phase velocity appear to behave differently in the aspect angle ranges above and below three degrees. While not directly relevant to the topic of spectral characteristics, it is evidence in support of models which have unstable modes at small aspect angles and at larger aspect angles a separate regime of damped waves that are fed by some non-linear process. Other interesting results include studies of layer thickness and aspect sensitivity. as well as a wide range of observed phase veloci ties. wi t h the largest speeds generally occurring at large cross sections. Very little has been ~ublishedon the finer details of the received spectra. for good reasons. The 150 foot steerable antenna used at Millstone Hiil has a two-way beam width of about one degree. which corresponds to 14 kilometres in arc at a 1000 kilometre range. More significantly? the pulse lengths of 1000 to 2000 microseconds required for adequate spectral resolution provide very poor range resolution. A lack of spatial resolution is potentially troublesome. especially with the possibility of returns from a sidelobe at some other (Iikely smaller) aspect angle than the main beam. This is not to suggest that spectral studies of Millstone Hill data are impossible. but there are unavoidable tradeoffs in spectral and spatial resolution for a monostatic site, and some care must be taken in the analysis. Pubiished examples of the coherent spectra observed at Millstone Hill (St.-Maurice et al. [1989]; del Pozo et al. [1993]) have the same basic features:

a single peak or the apparent superposition of two peaks

widths of a few kilohertz

mean Doppler shifts between O and 3 kilohertz

More complicated shapes exist, but these can generally be viewed as the superposition of a narrow peak on the incoherent scatter background. Line of sight Doppler velocities were studied by St.-Maurice et al. [1989]; Foster and Tetenbaum [1992]; Foster et al. [1992]; del Pozo et al. [1993], dl of whom observed velocities over the range f600 m/s (some as high as 800 m/s), with no clear preponderance of speeds near 350 m/~(a value typically assumed for the ion-acoustic speed Cs). Although good quality spectra are presented in two of the Millstone papers, only del Pozo et al. [1993] undertook a study of more detailed spectral characteristics. Using a Lorentzian model, spectra were fit and the behavior of the widths (FWHM) was examined. There appeared to be a pattern where the widest spectra (300 m/s) were found at smd phase velocities (line-of-sight Doppler shifts), and narrow (200 m/s) spectra at the highest Doppler velocities (600 m/s). These results are complicated by the extremely poor range resolution produced by the use of a 2000ps pulse, and the fact that 30s integration periods were used during a continuous azimuth scan. resulting in a 5 degree angular resolution. As noted in section 1.2 of del Pozo et al. [1993]: " Many observed spectral feat ures do not match the Lorentzian that we have assumed can be used to describe the spectrum of the waves. Most significant is that the majority of the spectra observed are asymmetric, a feature which cannot be described by a simple Lorentzian shape.' In an attempt to fit the residuals from the Lorentzian model. they introduced an additional (usually broad) spectral component which they label "Type D", although it is not clear what physical significance should be attached to this. The additional data presented in this Chapter has been subjected to a more thor- ough analysis than was applied by del Pozo et al. [1993]. Furthermore, most of the data were obtâined bistatically using the MIDASCsystem described in the next section. The resulting increase in spatial resolution can reduce complications due to multiple scat terers, and provide enhanced spectral resolution.

The Millstone Hill radar is composed of a powerful transmitter, a large steerable antenna, and a combination of oscillators, filters, A/D converters and cornputers known as MIDAS(mllstone Hill -ta kquisition System). Recent advances in comput- ing and the availability of small stable time references made it possible to build a portable version of MIDAS,called MIDASC.A fully funct ional radar receiver, MI-

DAS~cm take the signal provided by any antenna, apply several stages of filtering and mixing, then produce a digitized version which can be subjected to various ma- nipulations before being stored for later analysis. Control software was designed to operate along wi th -MIDASat Millstone Hill; the systems can use ident ical or different sampling strategies. A reference signal provided by a GPSunit dlows MIDASCto re- main synchronized with the puises transrnitted from Millstone. and has the additional benefit of providing accurate location information. To date. MIDASC has been used at two different sites (see the following sections for details). Their geographic locations are shown in Figure 3.1 aiong with the transmitter position. The "horizons' shown are due to minimum elevat ion limits at each site. For Millstone Hill the limit of about 4" is required to avoid illuminating nearby homes with rnegawatt pulses, while at Algonquin the constraints are due to mechanical steering restrictions. The elevation horizon for London (not shown) is approximately zero degrees. so virtually dl of the region shown in Figure 3.1 is accessible.

Algonquin

Figure 3.1: Location of the MIDASCsites on a geographical map, and the E-region (110 km) horizons for minimum elevation, which are 4' and 12' respectively for Mill- s t one and .4lgonquin.

One of the advantages of bistatic observations is that each radar may be observing the sme volume at a different aspect angle. Figure 3.2 shows some contours of magnetic aspect angle as seen from Millstone Hill and Algonquin. There are regions where zero aspect angle can be attained at both sites, other positions where both 41 radars are at large aspect angles. and some targets that provide opportunities for large/small aspect angle combinations.

Figure 3.2: E-region aspect angles observed frorn Millstone (solid lines) and Algonquin (dashed lines) using the IGRF95 model. Contours are for 1.3:5,7 and 9 degrees. Symbols marking the MIDASCsites are given for reference.

When fully assembled '\?IIDAsCfits in two stacked equipment racks. about the size of a refrigerator. -4 monitor and keyboard for the Sun workstation require some additional space. and room is required for cable access to the back, but MIDASCis still extremely compact. The system can be completely disassembled and packed a a couple of hours, re-assembly is only slightly more time consuming, with most of the additional time due to the number of cables connecting the various components. After assembly it is generally wise to run some tests with a frequency generator to ensure that the basic functionality is intact. This should be followed by several runs with the pre-amplnoise source package, to check the noise pulse triggering and to establish a reference of the noise level. A similar check should be carried out when the front end is connected to the antenna. Depending on the accessibility of the antenna, and on the successful completion of the basic tests, the MIDASCsystem can be ready to collect data in about five hours after arrival. Designed and built at Millstone Hill. the MIDASCsystem was completed in the fall of 1993. Incoherent scatter was successfully observed at &lillstone Hill in an essent ially monostatic configuration. with the zeni t h antenna used for transmission. the MIDASsystem receiving on the same antenna, and the MIDASCsystem using the smaller steerable antenna pointed vertically. This test confirmed t hat the MIDASC system was properly synchronized in frequency and timing, and that noise levels were acceptably low. MIDASCwas then packed and moved to London, Ontario. where it was installed on the third Boor of the Physics and Astronomy building at the University of Western Ontario, using a small (3m) steerable parabolic antenna on the roof. Witb such a srnall gain (z10 dB) incoherent scatter could not be detected. but auroral echoes were expected, especially considering the large power-gain provided by Miilstone Hill. In fact, coherent echoes were detected on the first night of observations. although technical problems and interference were significant. The second attempt on March S 1994 was unambiguously successful: 53.2.3 contains some results from that day.

3.1.1 Algonquin

.4fter further tests at London, the MIDASCsystem was moved to the Algonquin site. Located in the middle of the -4lgonquin provincial park in northern Ontario. the Algonquin Radio Obser~torywas established by the National Research Council in the 1960s. -4 variety of equipment was installed at this site. which was well isolated from anthropogenic noise sources. The single laxgest structure is a 150 foot (43.7 metre) steerable parabolic antenna. Some details of the construction cm be found in Jeffery [1969]; completed in 1966 the dish has a surface accurate to a few millimetres. and can function efficiently at centimetre wavelengt hs. Originally a Cassegrain feed system was used; the current configuration requires that the antenna feeds be placed in a cabin above the prime focus. Access to the focus cabin can be achieved by climbing up one of the support arms while the dish is pointed vertically. This is extremely time consuming and can be quite unpleasant during poor weather conditions. .An alternative method involves the use of a -cherry- picker' while the dish is at a low elevation angle. This is somewhat more convenient but the cherry-picker is not always available. Ideaily the number of trips to the feed cabin should be kept to a minimum. Tracking of ceiestial objects has been carried out using a complex arrangement involving several older cornputers. which is aonetheless capable of the arc-second accu- racy required for radieast ronomy at extremely high frequencies. Current ly manage- ment of the Algonquin site is in a state of flux, but the large dish is still operational, and is actually undergoing an upgrade with the control systems being replaced by a single modern cornputer. Hopefully the funding situation for this excellent and unique facility will stabilize. as the large area and high efficiency of the Algonquin dish are useful for coherent scatter observations. and absolutely essential for measuring incoherent scatter. The location of Algonquin provides several interesting opportunities for combined inlcoherent scatter experiments that sirnply cannot be achieved with any other existing radar system. Several campaigns have ben carried out using MIDASCat the Algonquin site. The first of t hese, during May 16-22 1994. was not successful at the goal of observing incoherent scatter. but several valuable lessons were learned about the system at Al- gonquin. It became very clear that a calibration noise source was absolutely essent ial in the front end package, not only as a reference level, but as a valuable diagnostic tool in order to minimize the number of troubleshooting trips to the feed cabin. With this addition. the installation at Algonquin for the next campaign during April 5-7 went much more smoothly. Observations of the radio star Cassiopea A indicated that the system noise levels were acceptable, and that the beam was well formed. However, no incoherent scatter was observed on either April 6th or 7th. As if to compensate for this setback, ionospheric conditions were extremely disturbed on the 7th, with Kp=i-9 during the afternoon and evening providing an excellent opportunity for bistatic observations of coherent echoes, some results of which are presented later in this Chapter. The next campaign in November 2-3 1995 was another futile attempt at observing incoherent scatter, although study of the radio sources Cygnus -4 and Casseopia .A seemed to show that the system was operating perfectly. During the last campaign from October 1-4 1996 several radio star tests were followed with bistatic moon- bounce experiments, designed to test the timing, frequency stability. sensitivity, and airning capabilit ies of the MIDASC / Algonquin combinat ion. These were al1 successful. and so were the subsequent observations of bistatic incoherent scatter, but the ver): low signal levels (SNR zz 0.01) required unacceptably long (> 10 minute) integration times for the recovery of geophysical parameters. Urith appropriate modifications to the feed and front end systems used at Algonquin, and a careful selection of experiment geometry it should be possible to improve signal-tcmoise ratios by a factor of five, which would dow incoherent scatter observations on a reasonable tirnescde.

3.2 Some results

As mentioned in the previous sections' there have been a number of coherent echo observations made using the MIDASCsystem. Rather t han attempting to provide a comprehensive catalog of al1 rneasured spectra, a smaller selection of results will be presented. These will include -typicaln observations, as well as some nanomalous' results, some of which appear to be of geophysical origin, while others illustrate potential pitfalls for auroral radars. Table 3.1 gives a list of the relevant experiment configurations used. wi t h the t arget Iocat ions, required antenna orientations, and resulting magnetic aspect angles.

3.2.1 Double peaked spectra

Before proceeding with the study of typical spectral data, it is instructive to examine a peculiar set of records which demonstrate some of the pitfalls which can dict observations of auroral echoes. On April 07 1995 the Algonquin site was attempting to observe bistatic incoherent scatter from an F-region tuget (Algonquin4f). After an Table 3.1: Positions. required antenna orientations. and result ing aspect angles for some of the target locations observed with the MIDASCsystem. The first part of each target name is the receiver location and the letter at the end indicates E or F-region alti tudes. The "wavelength" entry refers to the instabilit y wavelengt h observed by the bistatic receiver. at Millstone Hill this value is alwavs 34.1 cm. Londonle 1 Algonquin4f 1 Algonquin7e - Target Height (km) Latitude Longitude

- Transmitter Elevat ion Azimuth Range (km) Aspect Angle Receiver Elevation Azimut h Range (km) Aspect Angle Beam Angle Waveiength (cm) extensive period wit hout detectable signal levels. srnail (SNR < 0.3) enhancements in the range-power profiles were observed at a delays between 4000 and 8000 microseconds. Power spect ra from t his region were double peaked, although frequent ly qui te asymmetric. Short ly after t hese observations began, Millstone started observing signs of coherent scatter at ranges well beyond the target region. probably through a sidelobe. Upon closer examination, the bistat ic echoes displayed several t roubling characteristics

1. Delays (ranges) of 4500~swere too large. as the expected value was 2700~s

2. Power profiles were broad. not sharp boxcar-like shapes

3. SNR was too high. rising to levels exceeding 0.5

4. Spectra were too narrow. widths of 5 khz should have been closer to 11 khz

Figure 3.3 shows one of the peculiar range power profiles, along with three spectra corresponding to different delay times. These clearly show that the spectral content changes significantly with delay tirne? which is not the expected behavior for incoherent scatter from a single F-region target.

-4n alternat ive explanat ion is t hat Algonquin was act ually observing coberent E- region echoes through some combination of sidelobes. Figure 3.4 is a map of the experiment geometry for the intended F-region target, dong with the E-region zero aspect angle regions. Delays bracketing the observed power are also given for E region (110 km) heights. This shows that the range of locations which the echoes could have corne from included a large region of very small aspect angles. Since the strongest echoes are typically observed at zero aspect angle, it is quite likely that Algonquin was in fact observing Eregion echoes through a combination of Millstone sidelobes and feed spillover at Algonquin. This is consistent with the nature of the observed spectra which could be interpreted as a superposition of two narrow peaks with opposite Doppler shifts, a view which is further strengthened by the variation in the spectra as a function of range shown in Figure 3.3. While primarily a novelty, -5 O 5 Frequency (khz)

Figure 3.3: Complex spectral shapes from multiple scattering regions. Top plot shows the range (actually delay in microseconds) power profile, with a feature that is wider than the 1000ps pulse length. ACFSfrom the three numbered regions (each with 20 lags) correspond to the three power spectra shown below (normalized to peak heights). Note the dominant negatively Doppler shifted peak in the first spectrum, the positive peak in the third? and the double peaked result in the middle. these events do show that coherent scatter may be observed from apparently unlikely targets. As well. close examination of the spectral variation with delay can, under some circumstances. identify multiple scattering regions.

3.2.2 Very strong echoes

Once it was clear that coherent "contaminationn would make incoherent scatter observations impossible, it was decided to take advantage of the opportunity for bistatic observations of coherent echoes. The experiment geometry Algonquin7e was selected to provide a srnall aspect angle at Algonquin, and a moderate value at Millstone Hill. As very strong echoes were observed by both receivers, attenuation was introduced in an attempt to avoid saturation effects. Range-power profiles were generally quite sharp at Algonquin, indicating that a small scattering volume was being observed. Figure 3.1: Experiment geometry for an F-region target (*) midway between Millstone and Algonquin. Dotted lines are the bistatic delay ellipses for 4400~sand 6200~sat 1lOkm. and the shaded regions are the E-region zero aspect angle zones.

Millstone Hill had somewhat more extended returns, probably due to the larger volume formed by the low elevation beam passing through a scattering layer. Spectra observed at both sites were single peaked: an example of the Algonquin data is shown in Figure 3.5. Truncation in the lag domain is apparent, only lags up to 800 ps were calculated, but even extending to 1000 ps (the pulse length) would not have been sufficient. Another noteworthy feature is that the mean Doppler shift is quite small (- 50 m/s); this is typical of the echoes observed at both sites during this time. Echoes were observed simultaneously at both sites over a period of more than one hour. One interval (21:11 to 21:37 UT) has been selected for cornparison of spectral characteristics. Figure 3.6 shows plots of various parameters for the Millstone data plot ted against Algonquin results from the same time. Spectral widt h (standard deviation) at Millstone Hill is - 50 m/s larger than Algonquin, similarly, correlation times are - 150ps lorger at Algonquin. Decay exponents from Millstone are clustered near - 1.8, while the Algonquin results are more scattered around a mean near 312. The difierence in Doppler speeds is probably due to the different line-of-sight 300 600 900 m icroseconds

Figure 3.5: An example of the data observed using the MIDASCsystem at Algonquin, during the afternoon of .4pril 7 1995. directions for the two radars; while the LOS vectors differ by about 25 degrees, both sites appear to be looking roughly perpendicular to the flow direction. The other differences are likely due to the two factors which apply only to the Millstone Hill data

0 Larger scattering volumes. which may produce wider spectra that have a more Gaussian shape

Larger aspect angles, (4' at Millstone versus l0 at Algonquin) which may influence the width and shape of the scattered spectra

3.2.3 Results Fom London

During the evening of March 8 1994, the MIDASCsystem was used to observe small aspect angle echoes from London, Ontario. A small (3 metre) dish was used for reception. The resulting low gain (10dB) and wide beam (40+ degrees) rneant that the scattering region was largely defined by the Millstone beam and pulse. The target Mean Velocity Standard Deviation l

Correlation Time Decay Exponent I ...... O....'.'...

Cl...9e..-

200 300 400 500 1 .O 1.5 2.0 Millstone

Figure 3.6: -4 cornparison of the spectral parameters observed at target Algonquin7e from Millstone Hill and Algonquin during the afternoon of April7 1995. Mean velocity and standard deviation both have units of m/s, the correlation time is in rnicroseconds. and the decay exponent is dimensionless. Londonle was selected to give the smallest possible bistatic aspect angle. in order to maximize the received power. kakechoes were observed intermittently throughout the early evening (23 to 03 UT). with a short intervai of much stronger echoes from 350 to 4: 10 UT.

0.0 300 600 900 microseconds

Figure 3.7: An example of the data observed using the MIDASCsystern at London. Ontario. during the evening of March S 1994. Note the relatively sharp range-power profile, and the ACF truncation.

.4n example of the stronger echoes is given in Figure 3.7, which shows a well defined scattering region. Table 3.2 contains a summary of al1 the strong (SXR < 0.1) echo parameters. Doppler shifts are in the range 120 to 350 m/s. the average decay exponent is about 312, and al1 but one of the SKEwB values are of opposite sign to the mean. The wide range of correlation times (and spectral widths) is interesting; an examination of the last three records shows that the SNR level was low, particularly at 4:06, but the -4~~swere still well formed. Table 3.2: S ummary of echo spectral charact erist ics observed wi t h experi ment geometry Londonle on March 8 1994. Delay is the round-trip time to London (in

microseconds). T, is the ACF correlation time (in microseconds). n, is the ACF decay exponent. the last three columns contain the spectral moments (in m/s).

Time (UT) Delay T, n, Mean Stdev SKE~~ 0353 3500 336 1.84 -335 303 253 0354 4440 326 1.78 -127 288 240 0356 4460 418 1.55 -336 278 233 O358 4560 336 2.02 -122 354 -454 04:OO 4460 285 1.33 -140 1 323 295 04:06 3380 930 1.23 -344 138 333 04:08 3320 585 1.43 -248 206 310 04: 10 4320 778 1.12 -289 213 195

3.2.4 Narrow echoes

An interesting set of relatively narrow spectra were observed from the Algonquin site during an interval on April07 1995. The nominal target for these data (Algonquinge) was selected to give moderate aspect angles at both sites after unsuccessful attempts observing echoes at moderate and large aspect angles (Algonquin8e). Returns were sporadic during this interval (late evening LT ), following a sustained period of st rong widespread echoes in the late afternoon and early evening. At Algonquin, signal levels for the narrow echoes were quite low, on the order of 0.1, but the measured spectra were generally quite well defined. Table 3.3 contains a spectral summary of al1 the echoes observed at Algonquinge with SNR of 0.1 or greater. Alt hough t here were also clear examples of similar spectra with SNR les t han 0.1, the parameter estimates are very scattered, and will not be included. With a sampling rate of 40p, the 25 lags used for this experiment provide a spectral resolution of 500hz= 187mls. This is larger than the width (standard deviation) Table 3.3: Summary of echo spectral characteristics observed wi t h experiment geometry Algonquinge on April 7 1995. Delay is the round-trip time to Algonquin (in microseconds), T. is the ACF correlation time (in microseconds). n, is the ACF decay exponent. the last three columns contain the spectral moments (in m/s).

Time (UT) Delay T. n, Mean Stdev SKEWB

03:08 4980 625 1.43 -403 262 403 03:12 4620 753 1.97 -470 165 193 03: 14 4700 869 1.96 -481 104 150 03: 17 4620 160 3.13 -478 108 -104 03:18 4740 883 2.19 -473 96 103 03:19 5060 916 2.38 -424 119 1-27 03:48 4860 674 2.34 -429 123 183 03:39 5230 760 2.30 -431 126 105 0351 4540 744 2.46 -466 107 5 6 0353 4620 751 2.34 -455 117 91 0353 5060 728 2.26 -438 122 133 03:56 4540 1088 1.94 -414 146 383 0357 4740 931 2.01 -333 97 -74 1 of most of the observed echoes: most of the power is in a single frequency bin. which makes analysis in the spectral domain fairly tricky. Correlation times generaily fa11 in the range 700-SOOps, much larger than for most of the data shown in other sections, and the estimates for spectral width (standard deviation) cluster around a low value of 120mfs. Although the large estimates for n, (2.0 to 3.4) are interesting. the long correlation times correspond to ACFmagnitudes which have not decayed very much by the end of the 1000ps pulse length, so there is no significant ;tailn to fit. which may bias the results. Mean Doppler velocities are fairly constant at about 430 m/s. and most of the estimates for sKEWB have a sign opposite to the mean and a magnitude comparable to the standard deviation. Using only the eleven records where

SKEWB and mem had opposite signs, the average value for skewness was 1.19 ( with a sample standard deviation of 0.40). These results are complicated by the fact that al1 the narrow echoes are observed at delay times of 4600 to 4900ps, considerably later than the expected value of 3650~s associated with the intended target volume. Examination of the 4600~sdelay oval shows that it passes though the bistatic zero aspect angle region over an extent which includes both radar azimuths. While there is no way of proving that the echoes corne from a target at zero aspect angle which is close to one of the beams, it is the optimum configuration for maximizing the power returned away from the direct intersection volume. It should be noted that the sporadic echoes observed by Millstone during t his interval came from delays of about 5400ps, close to the expected value of 5265~s. This suggests that the Millstone echoes were coming from the intended target region, and should not be compared to the Algonquin results.

3.2.5 Monostatic satellites

Very narrow spectra can be found in the Millstone data, although with features that differ from the Algonquin results. These other narrow features are only observed in isolated (not consecutive) records, and axe sharply defined in range, with a width exactly equal to the pulse length. ACF magnitudes for these lags decay very slowly, and the phase term is essentially linear. An example of this kind of feature is shown in Figure 3.8: this particular record was chosen because the SNR level was high, and the ACFcould be well determined. Other similar records with lower SXR levels exist and the mean Doppler shift of the power spectra is highly variable, values as high as 20 khz have been observed. Al1 of these features are consistent with a 'hard target" of some kind. probably a satellite. Such a target would appear as a point scatterer. occurring in only a few range gates and producing essentially monochromatic returns. The Doppler shift would depend on the relative orientation of the satellite path and the radar line of sight. and since high latitude satellites axe moving at high relative speeds they would pass quickly through the beam.

1-00

0-95

0.90 50 150 250 -1 5 O 15 microseconds khz

Figure 3.8: A presumed jhard target" echo observed at Millstone on April 08 1995 3:36:00 UT. Note the sharp definition of the range-power profile and the very slow decay of the ACF magnitude. Estimated spectral standard deviation was 30 metres per second, estimated ACF correlation time was 1775 microseconds.

It is worth noting that these sorts of echoes have not been observed with the MIDASCsystem. This is not unexpected, as a monostatic system is much more likely to have a satellite pass though the main beam at some range. In contrast? a bistatic system has a single primary intersection target, and additional secondary regions determined by the intersection of the transmit ter and receiver sidelobes. These secondary intersections are associated with much Iower gains than a single monostatic bearn. and will not be as cornmon at large ranges. Consequently. bistatic satellite observations should be relatively infrequent. as indeed they appear to be.

3.3 Discussion

Spectra presented in this chapter were generally single peaked. with the limited number of multiple peaks occurring only for extended scattering regions. Most echoes of geophysical origin share some basic parameters:

a Spectra are consistently skewed. Values for SKEWB are comparable to the standard deviation. and have signs opposite to the velocity.

a Decay exponents for most of the records center on 2. but there appear to be two distinct types.

a "Broad" echoes, with r, =3OO-4OOps (standard deviations of 290-330 m/s). These have Doppler speeds ranging from 0-300 m/~,with no obvious change of shape over that range.

a "Narrow" echoes with r,=700-900ps (stdev= 100-130 m/s). The limited set of these observations have speeds of 300-450 m/s, and have decay exponents around

Both kinds of echoes were probably observed at small aspect angles, although there is some uncertainty in the location of the narrow spectra. .4dmittedly, this classification is based on a small nurnber of measurements. The large set of observations previously made at Millstone Hill are unavailable for re- analysist but published results seern to indicate t hat the spectral shapes were generally al1 of a single type which correspond to the "broad" results obtained here. This conclusion was reached by examining the figures in St.-Maurice et al. [1989] and del Pozo et al. (19931. and by assuming the FWHMin del Pozo et al. [1993] correspond to ACFS with decay exponents of 3/2. Further support is provided by the relative frequency of occurrence of the two echo classes. It would be surprising if al1 previous Millstone results were of the "narrow" kind, with &broadœspectra only obtained when cooperating wit h the MIDASCsystem. Assuming that the typical Millstone spectra are of the "broad" type greatly extends the fange of Doppler velocities at which t hey have been observed. This is quite important. as it means that the results shown here cannot simply be classified as slow and wide. The "narrow" class of echoes are noteworthy for several reasons. First. they do not seem to bave been observed with the Millstone Hill system alone (ie. rnonostatically). These results may be due to range smearing eKects from long pulses. or perhaps they have been discarded during processing as hard target echoes. The large phase speeds are suggestive of an ion-acoustic speed for a heated plasma. but the very limited data set does not allow any definit ive conclusions. 4 Chapter 4 Results at 16cm: EISCATand COSCAT

-4nother excellent source of spectral data is the EISCAT(firopean lncoherent Scatter) radar system. .4rranged with a tristatic geometry, the main site near Troms~(Nor- way) has a powerful two megawatt transmitter that operates in the frequency range around 933 Mhz, a sensitive receiver, and an efficient 32 metre steerable parabolic antenna. Two additional receivers are located in Kiruna (Sweden) and Sodankyla (Finland) with antennas identical to the transmitter. but lower system noise temperatures. The three sites are synchronized so that it is possible to observe scattering from a cornmon volume using a variety of pulse coding schemes. Data collection and analysis is similar to the MIDASsystem, and the final product is usudly a complex autocorrelation function. More technical information about the EISCATsystem is available from a number of sources, such as Folkestad [1977], Rishbeth and Williams [1985]. or Baron [1984. 1986). Because the EISCATsystem was originally designed as an incoherent scatter radar. some care was taken to avoid "contamination" from the radio aurora. Most significantly, the transmitter at Troms0 was placed so that the local topography would block any direct viewing of srnall aspect angle regions; thus the smallest aspect angle that can be observed monostatically is about 6 degrees. Bistatic observations using

either remote site cm achieve minimum values of about 4 degrees. If the primary consideration is to avoid detecting coherent echoes in the sidelobes while observing incoherent scatter at some other target, then the EISCATsystem has a comfortable margin of error, as discussed by Egeland [1977]. However, if the main beams are directed at those visible target locations wit h the smallest aspect angles, coherent echoes can be observed under disturbed conditions. as was first shown by Moorcroft and Schlege1 (19881. There are two advantages to using a tristatic system for studies of the radio aurora. First , each receiver observes a different orientation of scat tering wavevector, and the t hree measured Doppler shifts may be used to estimate a vector velocity for the target region. Second, each site will view the common target volume at different aspect and flow angles. which cm be very significant factors for coherent echoes. However. the geomagnetic location of the EISCATsystem is such that coherent scatter can only be observed in a limited region to the north of al1 sites. As a consequence. the scattering wavevectors are not widely separated. This rneans that it is difficult to reconstruct a full vector velocity. and the flow angles observed at each site vary by only about 30 degrees. Despite these limitations, echoes can be observed at a wide range of aspect angles (4-11+ degrees). and useful results cm be obtained even with the limited set of wavevectors.

4.1 Data fkom 1989

During an esperiment on June 1, 1989. the EISCATsystem was used to observe coherent scatter from the auroral Eregion. Several volumes were studied: Table 4.1 listç the target regions (ail at 105 km altitude) along with the aspect angles ( IG RF89) observed at the t hree receivers; the smallest transmit t er elevat ion angle

(10")corresponds to the smallest aspect angles that cmbe observed at any of the three sites. Coherent echoes were observed at al1 five target volumes, although the signal levels were extremely low at the largest elevation (largest aspect angles). Sampling rates were 10~sat al1 sites, with Trornsg collecting 36 lag ACFS(from a 360ps pulse) and both remote sites collecting 80 lags (from a 1000ps pulse). The integration was 5 seconds at al1 sites, with care taken to avoid effects due to antenna motion. Further details can be found in Jackel [1992] and Jackel et al. [1997]. Although some information about the spectra is presented in these two references, the primary Transmit ter Target .Aspect angle Elevat ion Latitude Longitude Tromsa Kiruna Sodankylii 1O" 73.95 27.69 5.8 42 3.2 P- 12' 72.55 36.43 7.0 5.0 5.0 14" 72.21 25.46 8.3 5.9 6.1 16" 11.93 24.69 9.8 6.9 7.1 18" 71.70 24.07 11.4 8.0 8.4

Table 1.1: Some details of the EISCAT1989 experiment emphasis was on height profiles and aspect sensitivity. This section contains an improved analysis of the measured ACFS which clarifies the previous results and provides a body of large aspect angle observations for cornparison to the smder aspect angle results in other sections. Various combinations of the zeroth through third moments for the two remote sites are presented in Figure 4.1. This is essentially the same as Figures 5 in Jackel et al. [1997]. but moment estimates here were obtained directly from the ACFS.and produce results that are rnuch less scattered. The main features of these plots

0 A limiting width (standard deviation) of 350 m/s, independent of cross-section. and with only a slight increase at the smallest Doppler shifts.

a Skewness values of about 1 (SKEWB is equd to the standard deviation. and has a opposite sign to the velocity ), wi t h no apparent dependence on cross-sect ion or Doppler shift.

Sirnilar widths and skewness values at both remote sites, despite a difference of nearly 30' in flow angle. are typicd of the EISCAT1989 measurements at dl aspect angles, although there are some interesting minor variations. Aspect angle =

O Sodankyla

A Kiruna

I i a l 1 0-1 10- -600 -300 O Cross section (m") Doppler Shift (m/s) Figure 4.1: Cornparison of moments for the remote EISCAT1989 data at aspect angles of 6". Filled symbols are used to highlight records which are clearly dominated by coherent scatter (a, > 1 x l~-"rn-'). .4 summary of the spectral parameters is contained in Table 4.2, with the results arranged in groups of increasing aspect angle. Records with cross-sections less than 1E-17 have been removed in order to minimize the efFects of noise. while cross- sections greater than le-16 have also been eliminated in an attempt to correct for the fact that large levels of scattered power are not observed at al1 aspect angles.

The average values for the spectral velocity. width, and SKEWB are given, along with the generalized decay exponent. The sample standard deviation is dso included (in braces), to indicate the amount of variability.

Velocity (m/s) Width (m/s) SKEWB (m/~) n~

- -- Sodankylii (4.2") -242 (135) 353 (39) 189(201) l.jî(0.14) Kiruna (4.2O) Sodankyla (5.0") Kiruna (5.0") Sodankyla (6.1") Kiruna (.5.g0) Tromse (5.8") Sodankyla (7.2") Kiruna (6.9") Tromsa (7.0") Sodankyla (8.4") Kiruna (8.0") Tromsa (8.3")

Table 4.2: Summary of spectral characteristics for the EISCAT1989 experiment. Only values with 1 x 10-'' 5 q,5 1 x 10-l6 were used in order to minimize differences in power due to aspect angle variations. The first value in each colurnn indicates the mean of the selected parameter, while the second (in braces) is the standard deviation.

Looking at the average velocities, it should be noted that there was considerable variation in line-of-sight velocity during the experiment, covering a range of more than 600 m/s. The prima- flow direction (ExB) direction was eastward (see Jackel

[1992]), and the smaller average velocit ies at Sodankyla are due to the fact that it was observing more nearly perpendicular to the flow direction than the other sites. Widt h (standard deviation) estimates at the remote sites increase slightly at the larger aspect angles. from 3.50 to 400 m/s. Results from Trorns0 are at the upper end of the range for al1 aspect angles. which may be due to spatial averaging effects from the larger observed scattering volume. Averages of SKEWB changed considerably at the remote sites, from 200 to 400 m/s with increasing aspect angle, while the Troms0 values were consistently smaller. Finally, the decay exponent shows a clear decrease from about 1.55 at the smallest aspect angles to 1.43 at the largest. This occurred consistently at al1 sites. while the (derivative based) moment estimates tended to be different at Troms0 . Figure 4.2 displays the relationship between the decay exponento correlation time and (mean) Doppler shift. Interestingly, the first two parameters seem to be independent of the spectral mean. which suggests that the "shape" (as determined by the ACF magnitude) is not a strong function of LOS velocity. -4 few low (< 100 m/i) speed values complicate the picture. with slightly reduced values for n,, but low speeds generally also correspond to smaller scat tering cross sections, and t herefore

Iower signal to noise ratios. The tight cluster of points on the T, vs. n, plot indicate that al1 the ACF magnitudes have roughly the same shape, while the lack of structure indicates that these two parameters are basically uncorrelated. To sumrnarize. it appears that the spectra are fairly well described by a single shape, which is the same at al1 Doppler shifts.

4.2 Data fkom 1992

The COSCATsystem uses the EISCATreceivers with an additional transmitter located in Oulu, Finland. Described by McCrea et al. [1991], it is a 500 watt CW unit that Aspect angle = 6" . *- - Sodankyla A Kiruna

il: : -600 -300 O 50 150 250 Doppler Shift (m/s) Correlation Time (us)

Figure 4.2: Some other spectral parameters for the remote EISCAT~~~~data at aspect angles of 6 degrees. Filled symbols are used to highlight records which are clearly dominated by coherent scatter (0,> 1 x l~-~~rn-~). operates at frequencies around 930 Mhz. An array of dipoles produces a beam with a width of 4 degrees in azimuth and 8 degrees in elevation. Although the power and gain are considerably smaller than that of EISCAT.the magnetic geometry is much more favorable, as Eregion aspect angles as small as 0.5 degrees can be observed from Sodankyla and 1.5 degrees from Kiruna.

Aspect Bearing Azimut h Elevation Beam Wavelengt h Angle Angle Kiruna 1.5 23.7 11.60 7.45 21.9 16.4 cm Sodankyla 4.6 -3.4 -2.62 7.03 39.0 17-0 cm Tromsa 6.4 36.7 30.48 11.78 O 16.1 cm

Table 4.3: Sumrnary of antenna orientation. aspect angles, and inter-beam angles for target C2 during the EISCAT1992 campaign. Several nearby targets were also used. their aspect angles are the same to within O.S, and the bearings and inter-beam angles are the sarne to within 3'.

In a campaign during June 25-23 1992 the COSCATsystem was used to illuminate a target where Kiruna could observe small aspect angle E-region echoes. At the same time. data was collected with EISCATfrom the same region using the Tromse- Sodankyla link. These observations were collected as part of a more complicated beam scanning experiment. whose purpose was to study the spatial structure of the echoing region. .4 preliminary analysis of the range-power profiles showed that the scattering layer was quite variable throughout the observation period; no results regarding t hat aspect of the experiment will be presented. As well, only the rernote sites will be used in order to avoid complications arising from the larger monostatic volume and lower spectral resolution amilable at Trorns0. Looking strictly at the spectral data, there were hundreds of records (with 5 second integration times) collected during the campaign period. These data will be presented in two sections, the first of which will contain large aspect angle observations similar to the 1989 experiment, while the small aspect angle COSCATresults will be placed in the second.

4.2.1 Large aspect angle

The Tromsa-Sodankyla link observed coherent echoes during four intervals ( 1:30 to 1 UT June 25, 9 to 10 UT June 27.9 to 10 UT June 28) with the vast majority of useful signals occurring on the first day. Data from al1 periods was grouped together. as no significant differences could be found between the (admittedly few) pre-midnight and post-midnight records. To reduce scatter, only records with SNR> 0.2 were used: ACFSwhose magnitudes near the origin were poorly fit by a low order polynornial were also excluded. These two critera removed about 100 records from the initial set of more than 1000. Doppler speeds were considerably lower than from the 1989 campaign. with a large number of small (C100 m/s) velocity observations.

1 10 1 O0 100 150 200 SNR Correlation Time

Figure 1.3: Decay exponent vs. SNR and correlation time for the Sodankyla 1992 data. On the first graph filled symbols denote correlation times greater than l5Ops, while on the second they indicate SNR greater than 5.

O a0 OP m I

Estimates of the decay exponent, as shown in Figure 4.3, are essentially independent of signal power (for SNR>1), but there appears to be a decrease in n, as r. goes from 100 to 200~s.This behavior is more complex than was observed in the 1989 data. where the correlation times were al1 in the range 100 to 150~s.When viewed as a function of mean Doppler velocity (Figure 4.4) some other patterns in the 1992 data become apparent. Most obvious is a clear decrease in r, with decreasing Doppler speed; although the minimum seems to be slightly offset from zero there are not enough positive velocity values to be certain. It should be noted that there is a corresponding increase in stdev at srnail speeds, but the effect is very small, from 310 to 370 m/s. Special attention should be paid to the ten low speed (- 100 m/,)records with long (- 200ps) correlation times. These corne from a short episode of scattering on June 28. and correspond to some noteworthy COSCATdata discussed in the next section. Decay exponent estimates follow a more complicated pattern, remaining roughly constant (at about 1.4) over the range -400 to -200 m/s, then increasing to about 1.6 at the smallest speeds. The maximum seems to occur at Ivl = O? in contrat to the offset r. minimum. Finally. SKEWB is fairiy constant at about 250 m/s. but there is an additional scatter of points at about -300 m/s.

4.2.2 Smd aspect angle

Data collected at Kiruna were obtained from a small aspect angle region illuminated by the transmitter at Oulu. The COSCATsystern was o~eratedin cycles with a period of CW transmission followed by an interval where the transmitter was turned off. This allowed noise spectra to be estimated and subtracted from the data. The resulting signal -4~~swere of slightly lower quality than provided by the nearly simultaneous noise and signal measurements of most pulsed systems. Coherent echoes were observed at Kiruna during three periods

1. June 25 2.5 to 4.0 UT

2. June 27 21.1 to 21.5 UT

3. June 28 21.5 to 21.7 UT Figure 4.1: Correlation time, decay exponent, and SKEWB as a function of rnean Doppler velocity for the Sodankyla 1992 data. Filled symbols are used to indicate the (relatively few) records from pre-midnight periods, al1 the rest come from 1.5 to 3.5 UT on June 25. and as with the Sodankylii data, a rnajority of the good records are from the first period. ACF magnitude curves were generally smooth during this time. with approx- irnately 10% showing signs of more complex behavior: a few of these have obvious notch-like minima indicative of double peaked spectra. Correlation t imes ranged from 200 to 600ps. and were typically 300~s.The few (about 10) good records on June 'Li have a slowiy decaying ACF magnitude which aiso has a "wiggle" (oscillation that doesn't reach zero). Finally the small number of records from June 28 (see Table 4.1) fall into two groups: those with long correlation times (500 to 600ps) and those with even longer correlation times (700 to 800~s).While the SNR values for these records were dl less t han 1. it is worth noting that the longest correlation times were actually associated wit h the strongest echoes. Figure 4.5 shows some of the parameters for the June 25 data versus mean velocity. Long correlation times (greater than 350~s)are given by filled squares, these do not appear to follo~vany clear pattern in decay exponent or SKEWB. However, there is a suggestion that the longest correlation times occur for speeds between 400 and 500 m/~.which is similar to the other COSCATresult by McCrea et ai. [1991]. The significance of this is not clear. especially since there are several records on June 28 with even longer correlation times. but velocities of 200 m/s. Decay exponents very clearly increase from about 1.1 at 300 m/s to 1.6 at 600 m/s, which seems to be quite different from the large aspect angle results. Having said that, the velocity ranges covered by the 1992 Sodankyla data are generally smaller, and it could be argued that the decay exponent dips slightly around 300 m/s. Furthermore. there was some indication that the 1989 data decay exponent began decreasing for speeds of 300 m/s or Iess.

4.3 Discussion

Results from larger aspect angies (4-6 degrees) during the 1989 and 1992 campaigns display several common features, and some interesting differences. To reconcile the Figure 4.5: Spectral parameters as a function of mean Doppler velocity for the small aspect angle (Oulu-Kiruna) 1992 experiment. For the second two graphs, filled symbols denote correlat ion t imes greater t han 350us, smaller squares are for correlat ion times less than 350us) Table 4.4: Spectral characteristics of narrow spectra observed on June 25 1992 from Kiruna during 21:35:35 and %1:37:05 UT. r, is the CF correlation time (in microseconds), n, is the ACF decay exponent, the last three columns contain the spectral moments (in m/s).

1.11 -234 113 1S4 1595t 527 1-12 -216 223 214 i differences, it should first be noted that most of the 1989 spectra had large Doppler shifts (200-600 m/s), while the speeds associated with most of the 1992 results are smaller (0-400 m/s). With that in mind, the correlation times of GOps from the 1989 campaign are consistent with the 1992 results for large speeds, and the decrease in correlation time as velocity approaches zero displayed by the 1992 spectra would not be apparent in the 1989 results simply because there were no small speeds in the earlier data set. Sirnilarly, the decay exponents of 1.5 agree for large Doppler shifts; more complicated behavior near small speeds cân only be observed in the more recent data set. One intriguing feature of the 1992 results is a tendency for the narrowest spectra (longest correlation times) to occur for v- 300 m/s. while for larger speeds the correlation times set tle down at 150ps, alt hough this feature is not unambiguous. The most striking feature about the smdl aspect angle results from 1992 is the nmow spect rai widt h, wit h correlat ion t irnes ranging from 200 to 5OOps for "typical" echoes and 500-800ps for a few 'narrow" spectra. Decay exponent and skewness values are similar to the large aspect angle results, dthough there are slight trends with velocity that are not apparent over the comparable range in the 1989 results. From the June 25 data it would appear that there is a tendency for the narrowest spectra to occur for v=400-500 m/s, similar CO the results shown by McCrea et al. [1991]. However. the set of extremely narrow echoes from June 28 have phase speeds of only 250 m/s. Not only is this much lower than the other results. it is less than any reasonable estimate of the ion-acoustic speed. Returning to the larger aspect angle results. it turns out that the cluster of long correlation time (r,.- 200ps) slow (Ivl -- 150 m/s) echoes were measured at the same tirne (21.6 UT) as the slow narrow echoes at smdl aspect angle. Since the COSCAT and EISCATlook directions only differ by 25 degees. similar velocities are expected if both receivers are observing the same target. If these echoes are from the same scattering region, than it is interesting to note that the smdl aspect angle correlation times are 3-4 times larger than for the larger aspect angle data (800~svs. 2OOps). This is in contrast to the ratio of about 2 between the range of "typical" correlation times observed at small and large aspect angles (100ps vs. 2OOps in Figures 1.4 and 4.5 ). Q Chapter 5 I Discussion

Aithou& the universe is under no obligation to make sense, students in pursuit of the P6.D. are

-Robert P. Kirshner

An interest ing variety of UHF coherent echoes have been presented in earlier chap ters. As well. some new ways of characterizing power spectra have been introduced, wit h a particular emphasis on feat ures of the autocorrelation function. Est imat ing spectral moments in the lag domain has proven quite useful. especially for the very narrow spectra associated with coherent echoes. An alternative measure of spectral broadening is given by the correlation time scale of the CF magnitude, which is a robust and easily deterrnined quantity. Subtle details of the spectral lineshape can be identified using the decay exponent. These tools for characterizing power spectra work best for single peaked profiles. which comprise the majority of the results shown in Chapters 3 and 4. More complex

spectral shapes are often composed of two narrow peaks, which cm be interpreted as the superposition of signals from multiple scattering regions. This view is supported by the relatively infrequent appearance of multiple peaks in the MIDASCand EIS- CAT bistatic data, which are obtained from relatively small (- IO3 km3)intersection volumes. -4 study of the MIDASCdata has shown that complex spectra are only observed when the range power profile is broader than the pulse length (indicating the existence of an extended scattering region); records with sharp range-power profiles (corresponding to echoes from a single scattering region) always produce spectra with a single peak. Monostatic systems tend to observe larger scattering regions, while the CW nature of COSCATmeans that a very large volume is continuously illuminated: bot h these configurations increase the opportunities for some sort of spatial averaging. This chapter will focus on the features of single peaked spectra. Data presented in earlier chapters will be drawn toget her. paying special attention to the similarit ies and differences of results at different frequencies and aspect angles. An effort will be made to place these results in the context of previous experiments. and to compare wit h some t heoretical predictions. The order of present ation will roughly follow the moments, starting with speeds, followed by widths, then skewness. After the summary and discussion of results. some suggestions for future experiments and analysis methods will be presented.

5.1 Doppler Speed

The mean frequency of a coherent power spectrum is easily determined regardless of any truncation in the lag domain, provided that estimates are obtained directly from the ACF using the rnethods described in Chapter 2 and Appendix A. Spectra that are both wide and skewed will require some attention to higher order terms in the CF phase curve, otherwise a linear fit is usually adequate. For narrow spectra the mean and peak frequencies are very similar; in the absence of any reason to prefer the peak frequency. the mean ha, been used to characterize the spectral shift. .4ssuming that the Doppler shifts correspond to physically meaningful speeds (ie. wave phase velocity ), the relationship V = wlk (5.1 ) can be used to convert from Doppler frequency to Doppler speed. From the 933 Mhz data in Chapter 4 it is clear that the measured speeds range from 200 to 600 m/s at small aspect angle (COSCAT)and O to 600 m/s at large aspect angles (EISCAT).The Iack of smdl phase speeds in the first set of data are likely due to the limited range of flow angles during the amilable observations; other data from COSCATby Eglitis et ai. [1995] extend the range of observed speeds down to zero. .4lthough the relative frequency of small speeds may be somewhat iow. slow speeds are definitely not an unusual occurrence. Furthemore. while there appear to be sorne variations in other spectral features with speed (see later sections). these effects are relatively minor and there is no clear distinction between the characteristics of low and high speed spectra. Similar results are seen at 440 Mhz. Data in Chapter 3 rnay only provide examples of small (0-100 m/s) and moderate (400 m/i) Doppler speeds, but there ore a large number of other observations (ie. Hall and Moorcroft [1988]; Foster and Tetenbaum [1992]; Foster et al. [1992]; Moorcroft [1996]) throughout the range O to 800 m/s. Again, smdl speeds rnay be less common, but they are far from unusual. .4n obvious, but interesting, conclusion is that the range of speeds is roughlÿ the same at both 933 and 440 Mhz. Given that the data at each frequency were obtained under a variety of conditions, it seems reasonable to assume that the overall range of plasma veloci ties and flow directions were roughly similar. If true. this would mean t hat the relationship between drift velocity (or electric field) and coherent phase velocity is similar at 16 and 31 cm scale lengths. This is not surprising, as both are well beyond the transition from fluid to kinetic regimes, but it does suggest that no essential changes occur over a factor of two in irregularity wavelength. A more subtle detail is that the range of observed speeds at 933 Mhz is also the same at small and large aspect angles. This is noteworthy, especially in light of the results of Foster et al. [1992] who found indications of decreasing phase velocity with increasing aspect angle. Ot her studies at large aspect angles (Schlegel and Moorcroft (19891; Jackel et al. [1997]) have found that the ratio of coherent phase velocity to drift velocity is about 112, so the wave speed is always less than the mean flow. In contrast, at smail aspect angle the ratio of phase to drift speed changes significantly, with extensive "plateaus" where the phase speed is essentially constant separated by a sharp transition region. Since the relationship between phase and drift velocity at small speeds is so different for small and large aspect angles, it is intriguing that the limiting value for Doppler shift (- 800 m/s) should be the same. Another important result is that a significant number of echoes have Doppler shifts much less than the ion-acoustic speed Cs,a feature that is not predicted by linear theories of the two-strearn instability. One typical response to this issue is given by Robinson and Honary [1990]:

Radar aurord backscat ter caa dso arise wben the ion-acoustic tbreshold is

not exceeded. however. in t his case the gradient drift instability is probabky responsible for the waves which cause the backscatter.

This may be possible at longer wavelengths (VHF). but the gradients required at CHF are extreme. Even if slow moving echoes were due to gradients. then it would be reasonable to expect that spectra with speeds < 300 m/, should be somehow different than higher speed echoes. Whiie this may be the case at lower observing frequencies (VHF), such a distinction is less obvious at CTHF. Admittedly, slow echoes are observed less frequently (ie. EgIitis et al. [1995]), but this could be a geornetric effect which indicates how rarely some auroral radars look nearly perpendicular to the flow direction. This interpretation is supported by the results of Moorcroft [1996] which show a roughly bell-curved distri but ion of speeds from a much Iarger data set collected from a larger geographic area.

5.2 Spectral width or correlation time

Although the second moment of power spectra can be estimated reasonabiy well from the ACF, some care must be taken to remove the influence of higher order derivatives. As well, even the existence of a standard deviation may be questionable for certain spectral shapes. While the Full-Width Half-Maximum may be a more robust quantity, it can be diEcult to obtoin when spectra are very narrow. Correlation time has the advantage that it can be easily deterrnined directly from the CF magnitude, unless truncation in the lag domain is very severe. It can be obtained without any assumptions about the detailed spectral shape, but may also be related to the width Small aspect angle (< la) Large aspect angle (> 4') Xarrow Typical

440 Mhz 700-900 300-400 n/a 933 blhz 600-800 200-500 100-100

Table 5.1: Correlation times in microseconds for UHF coherent echoes. grouped by frequency and aspect angle. A further distinction has been made between "oarrow' and "typical" echoes at small aspect angles.

pararneters for Gaussian and Lorentzian spectra. Quantitative comparisons of spectral width require more information about the spectral shape, but in general ACFS wi th long correiation times correspond to 'narrowœ spectra, while shorter times produce "widern spectra. This basic principle will be used hereafter in referring to echoes as narrow or broad depending on their correlation times. However. when comparing results from different frequencies it is important to note that two ACFSwith identical correlation times will produce spectra whose widths (expressed in terms of velocity) differ by a factor of 2.1 at frequencies of 410 and 933 Mhz. Spectral widths obtained from EISCAT,COSCAT, and MIDASCtend to fdl into groups according to the radar frequency and target aspect angle. A further distinction can be made between "typical" echoes at smail aspect angles, and a few noticeably narrower spectra. Table 5.1 gives values for the correlation times in each group. summarized from the results of Chapters 3 and 4. One striking feature is the difference between large and (typical) small aspect angles at 933 Mhz, aith the larger aspect angle results being wider by a factor of two. This pattern is suggestive of theoretical work by Hamza [1992]who found that larger aspect angle echoes could be stabilized by frequency broadening, and the required amount of broadening increased with aspect angle. -4 more quantitative cornparison is not possible, as Hamza [1992]uses a fluid theory to obtain his results, and also notes t hat the introduction of anomalous collision frequencies can profoundly reduce the amount of broadening required at a given aspect angle. The lack of width estimates for large aspect angles at 110 Mhz is quite unfortunate: aithough reliable values may be difficult to obtain (due to contamination from smaller aspect angles) this should be a priority of future research. As noted previously Foster et al. (19921 found subtle variations in aspect sensitivity and phase velocity around 3' aspect angle. it would be interesting to see if a similar transition in spectral widt h could be observed. Corn paring the small aspect angle results obt ained at different frequencies, i t appears that the range of correlation times are roughly the same. Viewed in terms of power spectra, this means that echoes from 340 and 933 Mhz have similar widths when measured in hertz. but converted to a velocitp scale the lower frequency results are taice as wide. Alternatively, the conelation time may be interpreted as a measure of the instability lifetime. Considering only single *eventsY, a region of plasma t hat is on the threshold of instability will grow or decay very slowly. producing a long-lived oscillation and a correspondingly narrow spectrum. Quickly growing or decaying waves will have shorter correlation times and produce wider spectra. It is important to realize that a strongly damped mode could potentially be very long lived if an additional source of energy (such as mode coupling) were available. For t his reason. some caution must be exercised when interpreting lifetirnes. as the linear growth rates may not be the most important factor. Having said that, it is still tempt ing to consider the following simple mode1

a Lifetimes for 430 and 933 Mhz echoes at small aspect angles are the same, so the balance of linear growth rates and non-linear saturation effects is the sarne at 16 and 34 cm scales.

a Large aspect angle echoes have shorter lifetirnes, as energy is provided by im- pulsive mode-coupling events, t hen quickly damped.

As noted previously, the theoretical literature is largely silent on the topic of spectral width or lifetimes. Those few papers which do exist are based on fluid theory, so their applicability to our results is questionable. Keeping that in mind. it is pleasantly surprising that Hamza [1992] predicts frequency broadening that is independent of wavelength, although as noted earlier. the precise amount of broadening required at a given aspect angle is strongly influenced by the anomalous collision frequencies. Un- fortunately. the relationship between Doppler velocity and spectral width suggested by Harnza and St-Maurice [1993b.a]

is less consistent with observations. Given that speeds at the two frequencies cover the same range. and assuming that the ion-acoustic speed is the same for observations at both wavelengths. then Equation 5.2 predicts that the spectra at both frequencies should be of comparable width (in units of m/s). which does not appear to be true. At a fixed radar frequency Equation 5.2 is somewhat more successful at predicting variation of spectral width as a function of Doppler speed. Observations by del Pozo et al. [1993] showed a trend for spectral width to increase as Doppler shift decreased. with the narrowest spectra found at large velocities. Large aspect angle results from 1992 at 933 Mhz displayed a sirnilar pattern (Figure 4.4), with the smallest correlation times near zero Doppler shift. However, the 1989 EISCATlarge aspect angle spectra have widths that are nearly identical over a range of 200 to 600 m/s, and al1 the results from 1992 seem to indicate that for speeds above about 100 m/s the width is fairly constant. This is not entirely surprising, as Equation 5.2 cannot possibly be expected to describe veloci t ies greater than the ion-acoustic speed, nor was it ever intended for use at large aspect angles.

5.2.1 Narrow spectra

The separation of small aspect angle results into "typical" and "narrow" deserves more discussion than it has received thus far. The primary motivation for this classification was that although the majority of echoes at each frequency fdl into a typical range of correlation times, a small number of much narrower spectra have been observed. Since these narrow echoes are relatively infrequent it is difficult to make definitive statements about their nature, but the few observations have dl occurred during periods of sporadic echo activity. and have correlation times roughly twice the typical values. Results from MIDASChad Doppler speeds of 300-450 m/s. in the neighborhood of the ion-acoustic speed, but the C0sc.4~results were at lower values near 250 m/.. It is tempting to view the narrow echoes as novel results which have never been observed before now. This attractive hypothesis is not directly contradicted by the literature. simply because there are so few pubiished results on spectral width, and the wiety of techniques used to estimate spectral width complicates the task of comparison. If the narrow echoes are indeed a new phenornenon then it is difficult to know how they should be interpreted. At both frequencies these echoes have similar correlation times, but their Doppler speeds vary by more than a factor of two. Interestingly, their Doppler shifted frequencies are quite similar at about 1200 hertz. Of course. it is not difficult to find some convenient relationship between any two numbers, but if the agreement is not simply coincidental then it might be worth considering an instability mechanism with a frequency that is independent of wavelength. However. the hypothesis of a constant frequency is inconsistent with the fact that for the narrow COSCATecboes. simultaneous observations at large aspect angles with Ersc.4~gave similar but not identical Doppler speeds. The srnail difference (- 100 m/s) could be explained as a fixed velocity observed from two sites with slightly different line-of-sight orientations, but is more difficult to reconcile with a source at a single fixed frequency. Alternatively, the narrow echoes may be viewed as part of a continuum of spectral widths where narrow spectra simply occur less frequently than wider ones. This interpretation is consistent with the results of Eglitis et al. [1995]; Woorcroft [1996] who present histograms of spectral width that show a small chance of observing spectra much narrower than the norm. -4 continuum of widths is also shown in McCrea et al. [1991] for COSCATdata, although the narrowest spectra are found at 450 m/,rather than 230 m/s as in this study. Seôrching for an explanation of why some instabilities should appear to be longer lived. one must first consider the features cornmon to the (relatively few) "narrow" echoes observed in this st udy

a small aspect angles

0 sporadic

Or alternatively. there are no "nmow" echoes observed during extended periods of strong scatter or at large aspect angles. One interpretation is that scatter was observed from regions that were only rnarginally unstable, with very slow growth or decay rates. This would also provide an extended period when the waves were at low or moderate amplitudes. and be less likely to lose their energy through mode-coupling. Another possibility is that narrow echoes are produced by fairly small scattering volumes; if a very large volume were unstable it may contain a range of densities and electric fields, and produce a superposition of spectra at a vaxiety of frequencies. These possibilit ies are not mutually exclusive, in fact the combined eRects of small volumes and marginal growth rates ma? be required to produce narrow echoes. Slowly growing isolated instabilities might reasonably be expected to produce relat ively low levels of scat tered power. Unfortunately, none of the available observations of narrow echoes were obtained from configurations where the target location and system gain could be accurately estirnated, so the absolute scattering cross section cannot be reliably estimated.

5.3 Skewness

Asymmetric spectral shapes are common at UHF, typically with a sharp cut-off at higher speeds and a longer "tail" extending towards zero speed. Interestingly, this general shape is observed over the entire range of mean Doppler speeds, and the low speed "tailn can extend across zero Doppler shift into velocities with a sign opposite to that of the spectral peak. In the absence of any theoretical basis for understanding this spectral asymmetry, it has become customary to characterize spectra in terms of their skewness. This is a dimensionless quantity (defined in section -4.2) based on the third moment of the power spectrum. normalized by the second moment so that spectra of the same -shapeV but different widths will have identical skewness. Estimating the second and third moments of measured power spectrum is a difficult task. as higher order moments are especially sensitive to noise. Consequently the ratio of two moments would seem to be a highly uncertain quantity. When moments are estimated directly from the ACF. the second and third moments are determined from the derivatives of the magnitude and phase curves respectively. Given that the ?"' and Ydmoments are obtained from different components of the .~cF.and in the

hopes of reducing uncertainty in the selected parameter. the quantity SKEWB was introduced as a measure of spectral asymmetry. SKEWB depends on the third moment. but cm be converted into the more commonly used skewness through dividing by the standard deviation and cubing the result.

Xmusingly enough, the most prominent feature of SKEwB is that it scales with spectral width, so in fact skewness is approximately constant. That is. wider spectra (shorter correlation times) have larger values of SKEwB (ie. Figure 4.5), while narrower spectral have smaller values of SKEWB. Although estimates of the skewness are more variable than values for either moment sepaxately. it would appear t hat the vast rnajority of observed spectra have a skewness of roughly -1 (for positive Doppler shifts). Variations in SKEWB with mean Doppler shift (such as in Figure 4.1) are generally accompanied by variations in width, the resulting skewness does not show any consistent trend. In fact , within the admittedly considerable limits of uncertainty for any individual estimate, skewness does not seem to depend on frequency, aspect angle, or any other parameter. While it is rewarding to find a spectral parameter that is apparently cornmon to al1 UHF coherent echoes, the significance of a constant value for skewness is unclear. Some work by Hamza [Private communication 19971 indicates that small aspect angle echoes should have skewness values from -1 to -3 (for positive Doppler shifts), but this result is based on fluid theory and does not necessarily apply to iarger aspect angles. Kustov and Uspensky [1995]presented a mode1 for skewness based on altitude integrat ion effects. but i t assumes an inst ability phase velocity t hat varies significant ly with altitude. and requires that scatter be observed over a large (12-21 km) height range. Considering the wide range of scattering volumes provided by the systems used in this study it seems likely that some difference in skewness should have ben observed between the EISCAT.MIDASC, and COSCATresults. which does not appear to be true. Another mode1 for skewness is presented in a paper by Whitehead [1990],who makes some very interest ing points about ACF interpretation. Starting wit h the fact that power spectra often consist of a narrow peak at high velocity and a long tail at the low velocity end, he assumes that the spectrum is composed of a broad (short- lived) and a narrow (long-lived) cornponent. From this it follows that, in the words of Whitehead. " by rneasuring the phase at the longest practical delay of the ACF. we can calculate the velocity of these [long-lived] irreguiarities". Using the linearity of the Fourier Transform a power spectrurn wit h multiple peak-like components. each of which is symmetric about its mean. is equivalent to

Note. however. that this is not the same as saying that there are multiple super- imposed signals. as the power spectrum due to N signals is not necessarily the same as the sum of N individual power spectra

This may not be important if the amplitude spectra have random phases, but there could be interesting effects for coherent signals.

For long delays TL, the longest lived components will have decayed less than the short-lived (broad) terms. This will be true in general, but can also be illustrated with the mode1

If the power Pl in the slowest-decaying (narrowest ) mode is not much smaller than the next mode of comparable width. then it is clear that

and the phase slope at long lags can be used to find the average velocity associated with the narrowest spectral component. For a two cornponent mode1 this may be a useful result. as the areas under the narrow (usually tall) and broad (shorter) peaks are comparable.

Figure 5.1: A superposition of two Gaussian spectra. The result has a skewness of nearly -1, and the ACF has a decay exponent of 312. Note the ACF phase curve at large lags, which follows a Iinear trend characteristic of the longest-lived component.

A superposition of two Gaussian spectra is shown in Figure 5.1. The spectral shape is quite similar to much of the UHF data, and the ACF magnitude has a decay exponent of 312. However. the phase term doesn't match the cubic-like behavior that is generally observed in the UHF data. With the ''mean-trendn (average of fi and fi) removed the phase curve is fiat for early lags, then curves up asymptotically towards the nanow component value. It could be argued that in practice the magnitude or decay times are such that this asymptotic behaviour doesn't occur until the ACF is vanishingly small. but attempts to reproduce this behavior in synthetic data produce spectra which are either too broad or too peaky. Of course. the addition of multiple components might fix things up. but then it is unlikely that there would be a single "long-lived" term. at least for reasonable ACF magnitudes. Before leaving the topic of skewness. some attention should be paid to the characteristics of very slow spectra. As mentioned previousl- these have skewness values similar to their faster moving relatives. This similarity is actually quite puzzling, as spectra with large Doppler shifts have a "tail" extending towards zero, but for a spectrum whose center is near zero, a skewed shape requires a sharp cutoff at some moderate speed, and a "tail" extending towards larger speeds of opposite sign. It was hoped that the 1992 EISCATdata would shed some light on the behavior of skewness for slow echoes. as a range of speeds from -400 to 200 m/s was observed. However, the results (in Figure 1.4) show consistently skewed spectra for al1 negative velocities, zero Doppler shift, and even up to 100 m/s. .As well, there are a few very puzzling results where the spectra are skewed in the opposite direction. These occur over the range -300 to 200 m/s. and have a skewness with the same magnitude as the majority of spectra, but with opposite sign. If there is anything to be learned from these data it seems to be that spectra at small speeds are generally skewed in one direction. except when they occasionally become reversed. It is clear that more study of skewness is required. Hopefully larger data sets collected at a range of speeds will help provide a clearer picture.

5.4 Spectral Shapes

Proceeding to the next spectral moment. kurtosis provides some indication as to whether a spectrum is more or less sharply peaked than a Gaussian. Consequently, it would seem to be an ideal way for characterizing spectral shape. In practice kurtosis estimates have not been widely used in this study, primarily for two reasons. First. it depends on both the 4th and znd moments. or equivalently the fourt h and second derivatives of the ACF magnitude. These are difficult to evaluate precisely and any combination of them is subject to considerable uncertainty. Having said this. it should be noted that with some care it was usuaily possible to obtain estimates for the kurtosis that were not sensitive to minor variations in the ACF and were consistent from one record to the next. Once values for kurtosis were produced, the second problem was that they followed no obvious pattern. Typical values ranged from O to 5. indicating spectra that were more sharply peaked than a simple Gaussian. but there was no clear trend with mean Doppler shift or spectral width. It appears that kurtosis is not a very informative measure of spectral shape. In contrast, the decay exponent proved quite useful for characterizing the shape of the ACF magnitude. Introduced in an attempt to determine whether spectra more closely resembled Gaussian or Lorentzian line shapes. the mode1

was chosen for simplicity and convenience, rather than on any theoretical basis. How- ever, it proved quite successful at describing the basic shape of the .!CF magnitude curve using only two parameters, with T. determined directly from the data and n, ob tained t hrough a least-squares fit. Ignoring the complicating effect s of skewness. the spectral shape is determined directly by the CF magnitude: for the mode1 Equa- tion 5.6 a Gaussian line shape would have n, = 2 while a Lorentzian corresponds to n, = 1. In practice the vast majority of coherent spectra had values of n, between 1 and 2, with an average near 312. Some variation with mean Doppler shift can be seen in some of the data (ie. 1992 EISCATand COSCAT)but other sets of observation (1989 EISCAT)are less variable. Overall it is difficult to see any pattern, which is itself noteworthy, as there is no clear distinction between regimes with specific decay exponents. It could be argued that this lack of significant trends means that the decay exponent is no more useful than kurtosis. However, since the kurtosis is really only the fourth derivative of the magnitude at lag zero, it allows a rough extrapolation. but does not describe the behavior at long lags. In contrat. once it has ben determined. the decay exponent combined with the correlation time can be used to re-construct the ACF magnitude curve quite accurately.

5.5 New andysis methods

Estimating spectral moments in the lag domain can be accomplished with fewer lags than would be required to produce a well-resolved spectrum: reliable values of standard deviation and skewness can be obtained from only six lags. At a 20ps sampling rate this would require a 120ps pulse. which corresponds to a range resolution of 18 kilometres. For monostatic systems, this is a factor of 10 better than provided by typicd 1000 to 2000~spulses, and should help separate the effects of multiple scattering regions. Mult i- pulse schemes could provide even bet ter resolution. as deri vat ive estimation does not require evenly spaced delays. More complicated experiments could measure a few lags at small delays for moment estimation, then further widely spaced lags at longer delays (ie. 300ps, 600ps, 900ps, 1200ps) to provide values of the correlation time and decay exponent. Of course, if double peaked spectra were actually produced by small scattering volumes they would not be well resolved. and the only indication might be increased estimates of the spectral width.

All of the analysis techniques used thus far have been based on the .!CF and power spectrurn pair. Such an approach is typical of auroral radar studies, for reasons which deserve some closer scrutiny. Probably the most important consideration is that power spectra (or -4~~s)are easily collected with radars; frequency andysis occurs naturally from the signal processing systerns used in most radars and is unquestionably correct for describing the signal received from a single point scatterer. Signal variance can be significantly reduced by averaging ACFSor spectra, with the added benefit of a large reduction in data volume. The alternative of simply recording every sample for later analysis requires large fast data storage which has only recently becorne available. For incoherent scatter the spectral perspective has a firm foundation. as the scattering process does in fact have a random phase. and no information is Lost by con- verting the signai to a lag-product matrix. Despite the absence of any sirnilar result for coherent echoes. it is still generally assumed that a spectral analysis is applicable. Certainly any measured signal can be Fourier transformed. but the results are only straightforward if the scat tering process is st ationary. Linear t heory predict s exponential growth or decay which cannot be analyzed as a stationary system. Son-linear equations may produce a system that is stable over very long intervals. but may sti1I undergo structured non-periodic variations. Fourier analysis of these complex variations will always produce some kind of power spectrum. but this may not be the most appropriate method of analysis. Similar problems occur in many fields of research, and in recent years a considerable amount of effort has been expended in the development of new analysis techniques. Although often labeled as "chaotic". a more appropriate classification is that of a "dynamical system" ivhich can include both simple and complex behavior. The number of publications on this topic is huge and growing quickly: no attempt will be made to summarize those results here. only to mention a few topics which may be useful in auroral radar research. Several of the references here were found in Gershenfeld and CYeigend [1993], and the rest were obtained by a random walk t hrough the li terat ure. Most of these new techniques deal with the time evolution of a system as a trajectory through configuration space. This can be illustrated by considering a simple harmonic oscillator with two degrees of freedom (displacerrient and velocity ), so t hat at any given time the state of the system can be described by a single point in a two dimensional space. However, for any part icular set of initial conditions the oscillator will trace out an elliptical trajectory, remaining on a one-dimensional manifold. By an appropriate sequence of measurements it is reasonable to expect that the one- dimensionai dynamics of the original system can be determined. More surprisingly, it cm be show that a system which occupies a multi-dimensional manifold can often be reconstructed from a single time series. This result is exact for well sampled noise-free systems, such as a numerical simulation of the Lorentz attractor. Application to real data sets is not always trivial: a large arnount of high SNR data may be required. In fact. many of the new diagnostic tools (ie. non-linearity in Theiler et al. [1992]. reversibility in Diks et al. [1995])require a fairly long sequence (> 100 points), and cannot be applied to the usual LPM data collected by most radars. However. CW systems are capable of obtaining long sequences at high sarnpling rates, and some pulsed systems cm store data from each pulse separately (ie. Schlegel et al. [1990]). Combining this high-resolution data with new analysis techniques would seem to be a promising avenue for future research. Accepting the constraints of current data collection methods. it may still be possible to obtain some new information. Singular spectrum analysis (SSA, see Broomhead and King [1986]; Vautard and Ghil [1989]: Vautard et al. (19921) attempts to recover a "statistical" dimension from the rneasured time series using a dynamical systems approach. Rather than working directly with the original M measurements xi, a set of M &ged vectors are assembled uj = {xj, x,+~, . . . ,z,+~-~). These are used to construct a covariance matrix - 1 'M T = = -Cu-..f 1 ,=I which is then subjected to singular value decomposition (SVD). The resulting "spectrum" of singular values is used to estimate the number of degrees of freedom, and the eigenvalues can be used as empirical orthogonal functions to produce an estimated

Interestingly? the covariance matrix t is simply the LPM for a purely real signal. This suggests that SSA might be applied to standard auroral radar data, but some issues first need to be addressed.

0 The LPMis cornplex. A superficial examination of the SSA procedure does not reveal any specific restriction to purely reai signals, but a more carefui treatment is required. This potential problem may not be insurnountable. as the signal could be -mixedV up to a higher frequency and the imaginary part discarded.

0 Care must be taken to avoid effects due to range smearing or pulse tapering. LPMSfrom monostatic radars will include some averaging from different ranges which may bias the results. Isolation of a small volume cm be accomplished with a bistatic system. but care must be taken to use only that part of the LPM corresponding to full illumination.

Assuming that these implementation details can be overcome. it should be possible to use currently existing radar systems to obtain estimates of the "statistical" dimension associated wit h coherent scat ter. Two possible outcornes await the use of these new techniques. One is that the data will be found to be consistent with a low dimensional process. This will motivate further study and suggest new ways of collecting data. Estimates of dimension could be usefully compared with results from theory or simulations. Alternatively it ma) turn out that auroral radar data is very high dimensional in nature. This would indicate that a dynamical systems perspective would not be appropriate. Furthermore, it would emphasize the limitations of any theory based on a small set of differential equations (ie. Sahr and Farley (19951). Eitber way, something significant wouid have been learned. .4 comprehensive study of this subject would require an excellent knowledge of the state of the art in dynamical systerns analysis methods. As noted by Gershenfeld and Weigend [1993] "the fiterature in these areas has become fragmented and somewhat anecdo td. The breadth (and range in relia bility) of relevant materiai makes i t diEcult

for new research to build on the accum uIat ed insigh t of pas t experjence (researchers standing on each other's toes rather than shoulders)". Careful selection of radar data sets will also be required in order to isolate features of interest without introducing instrumental effects. However, there is potentidly a great deal to be learned from new analysis techniques, and this should be a topic of future research.

5.6 Future experiments

Srnall aspect angle results at 440 and 933 Mhz suggest that the range of Doppler speeds is sirnilar, as are the lifetimes. skewness and decay exponents. Obviously. data from ot her operat ing frequencies would provide confimation of t hese similarit ies or allow differences to be identified. Enperiments at lower frequencies a.re difficult. as there are no auroral radars currently operating in the frequency range 200-300 Mhz. while systems at 150 Mhz are well into the Buid regime and are more susceptible to refraction effects. However, by a careful choice of experiment, it may be possible to use the MIDASCsystem to observe targets at fairly large (- 80') scattering angles. This would allow the study of wavelengths on the order of 45cm. equivalent to a 340 Mhz operating frequency. Higher frequencies are also accessible. as a 1295 Mhz system is currently operating at Millstone Hill. Although not normally used for auroral observations. it might be possible to apply it to the study of Eregion echoes. This would be more than a repetition of the work by Abel and 'u'ewell [1969] and Hagfors [1973], as current technology would allow the collection of spectra wit h higher resolution in space. tirne: and frequency. Ideally such experiments could be carried out simultaneously with observations of a common volume at 440 and 1295 Mhz. Such an experiment would remove many of the arnbiguities of previous comparisons, and provide data at scales over a wider range than currently available. The other promising area of study would involve simultaneous observations of small and large aspect angles using Millstone Hi11 and MIDASCat a remote location. Although interpretation would be complicated slightly by the difference in line-of- sight directions and the resulting effects on Doppler velocity, it should still help to address the question of how the spectral width (or instability lifetime) changes as a function of aspect angle. Even without the use of MIDASCmonostatic observations could provide valuable information about the variation of spectral characteristics with aspect angle.

5.7 Summary

UHF radar echoes from the auroral E-region share a comrnon spectral shape: a single asymrnetric peak. This peak may be Doppler shifted by speeds ranging from O to 800 m/s, and although moderate (200-400 m/s) speeds are most frequently observed. srnall speeds are not uncommon. Small aspect angle results from 933 and 440 Mhz display the sarne range of velocities. suggesting that the governing processes are similar at 16 and 34 centimetre irregularity wavelengths. More surprisingly, large aspect angle observations at 933 Mhz span roughly the same range of speeds. despite the significantly different processes usually assumed to be responsible for echoes from large aspect angles. Comparing spectral widths of large and small aspect angle observations at 933 Mhz. there is a clear difference: spectra at large aspect angles are much broader. alternativel~,srnall aspect angie spect ra have longer correlat ion t imes. For small aspect angles the range of correlation times at 933 and 430 Mhz are comparable. again suggesting that the processes which determine instability lifetimes are the same at 16 and 34 centimetres. The degree of spectral asymmetry is roughly similar for both frequencies and aspect angle regimes. Skewness values near -1 are typical. although there is considerable scat ter no clear dependence on any ot her parameter could be found. As well. the basic spectral shape (as characterized by the decay exponent) is approximately the same for al1 spectra. ACF magnitude curves correspond to neither Gaussian nor Lorentzian spectra, but rother an intermediate form with typical decay exponents of about $. More observations are required to test whether these general patterns extend over a wider range of frequencies and aspect angles. Future studies may use the results given here in order to design experiments that provide spectral parameters with enhanced spatial resolution, and data from previous experiments could be re-analyzed using the methods of Appendix -4. Further study of the .!CF properties may be rewarding, and there are a number of promising new tirne series analysis methods that should be examined. Ideally, the fairly uniform picture presented in this thesis will also assist in the search for t heoretical models of CHF instabilities. - - -

Q Appendix A Mat hematical Det ails

A. 1 Fourier transform relations

When defining the forward and inverse Fourier transforms. there is sorne flexibility in the Iocat ion of normalization constants. A convenient choice is

For any given function f (t)it is possible to calculate the power spectrum

which is purely real and positive. and whose Fourier transform equivalent is the autocorrelation function (AcF)

In general the ACF is a complex quantity, with a symmetric (even) real part. and an anti-symmetric imaginary component. Alternatively, an ACF can be represented as a combination of two real functions

where the rnagnit ude p is positive and symmetric, and the phase cb is anti-symmetric. If the power spectrum is symmetric about some frequency W then the phase term is linear. ie. o(r)= Zr; if the symmetric spectrum is centered at the origin then the XCF is purely real. Using the definition of a Dirac delta function

1 +Oo 6(z - x') = -2T [-dy ëcy(r-z') it cmbe shown that the total area under the power spectrum is exactly equai to the magnitude of the CF at lag zero

as the phase is zero at the origin. Consequentl. normalizing by the zero lag of the ACF will produce a power spectrum with unit area.

Another useful result is the Fourier transform of xn, which can be found by examining the result of repeated differentiation

since when f (x) = 1 the Fourier transform F(y) is the delta function

When dealing with derivatives of the delta function, the following relation is of use

d" d" d" dx f (r)+(x) = (-1)" /+- dx 6(x)- f (2)= (-l)n [d;nf (i) dz -00 dzn

A.2 Moments in the frequency and lag domains

Given some distribution of dues, one can define the zeroth order moment as the total area under the cuve For the rest of this section it will be assumed t hat the distribution has been normalized so that Mo = 1: this is easily done and will simpiify the algebra which follows. The first moment is then the rnean value of the distribution

while higher order moments

are the results of higher order polynomiai weights about the mean. When calculating the first moment of a normalized power spectrum

chmging the order of integration and using equation (-4.8) gives a sirnplified result in terms of the auto-correlation function

which can be further reduced using (A.9) to

Thus the mean value of the power spectrum can be determined from the first derivative of the auto-correlation function at lag zero. Treatment of higher order moments is greatly simplified by the introduction of a shifted frequency scale which in the lag domain is equivalent to a multiplication by a linear phase term

After which the calculation of higher order moments is straightforward

to at least M + 1 lags. and obtaining the derivatives at the origin from

When dealing with entirely even or odd functions. P(x) will contain only even or odd powers: as a consequence the amount of computation can be reduced by fitting only to the non-negative lag times. For example. the magnitude of the ACFcontains only even powers, and four lags would suffice to determine the sixth derivative from

Similady, phase values from the first five lags provide odd derivatives up to the seventh

There are at least two problems with this simple approach. First there may be noise in the measurements. which will introduce errors into the coefficient estimates. Second. there may be significant higher order derivatives, and a failure to fit them may bias the lower order estimates. Effects due to noise can be rninimized by including additional lags. producing an over-determined fit which is equivalent to smoothing the input data. Higher order derivatives are more troublesome, as if they exist then some provision must be made to accouot for their effects. One approach is to include an extra "guard term', for example, if only the first through fifth derivatives were required then Equation A.27 would be used. This would require five lags to produce the three desired estimates along with a guard term which should absorb some contributions from even higher order terms. The magnitude of this guard term can also serve as a useful warning indicator: if it is large then the low order polynomial may be a poor fit to the data, and the resulting derivative estimates rnay be unreliable. In summary, some useful guidelines for derivative estimation are

Estirnate one more derivative than required, so as to produce a guard term

a Include at least one extra lag value, to damp the effects of noise While an additional practical concern may be to

Avoid using certain less reliable lags

For example, the zero lag of the CF may be fairly uncertain, as the majority of the

noise contribution wiIl occur t here. and stat istical uncertainty in the noise est imate will meaa that perfect noise subtraction is not possible. Another less obvious problern arises in some of the EISCATdata. where the first lag seems to have peculiar phase characteristics. Of course. leaving out one Iag will require the addition of a later one, if the effects of noise and higher derivatives are to be accounted for. A more general approach to the problem of unreliable lags would be to introduce some explicit weighting. This would be fairly easy to do, but would have to be tailored to specific noise characteristics and signal to noise ratios.

A.4 Standard line shapes

Two shapes are most commonly used to characterize the nature of power spectral peaks. The first is a Gaussian profile

which is already normalized to unit area. has a mean of 2 and a second moment

(variance) of u2. Since it is symmetric about the mean, the third moment is zero. A Fourier t ransform produces the auto-correlat ion function

(-4.29)

with a magnitude that also has a Gaussian shape. The second standard profile is a Lorentzian (also known as a Cauchy distribution)

and also has unit area, a mean of 3, and a zero third moment. However, the second moment is infinite, so other methods must be used to describe the "width". In the lag domain. the Lorentzian becornes

wit h an autecorrelation magnitude whose logarit hm is linear. compared to a quadratic in the Gaussian case. Both of these curves are symmetric about the mean. so if 2 is set to zero. then

@(T) = O and only the (even) derivat ives of p(r) need to be considered. Using the exponentid expansion

the -kFof a centered Gaussian is

so the derivatives are

V2= -02 D4= 336 (A.34) which produce the expected values for standard deviation and kurtosis. .4s noted previously. the Lorentzian has an infinite second moment, which implies that the second derivative of the ACF at the origin is also infinite. This can be seen by studying the general case of

and expressing the second derivative in the usual way P'(-6) W(6) p"(0) = lim ~'(4- = lim - 6-0 26 s-m 26 C 2 = 6-0iirn -2c {(2~f)"-~- (26)2n-2+ 1(26)3n-2 + . . . 3. ) What is particularly interesting about this result is that the second deri~tivehas t hree possible values

-CG n2 of which ody the Gaussian (n = 2) case is at al1 useful. ACFSwhich decay more quickly are infinitely %ide" in the frequency domain. while a slower than Gaussian CF decay produces an oscillatory power spectrum with negative excursions that give a "width" of zero. This behavior suggests that while the model given by Equation A.35 may be effective for fitting data, it cannot be exactly correct for any real physical system. An alternative model for ACF magnitude corresponds to a Voigt distribution. and has the form p(r) = exp(-air1 - h2) (.4.37) which by appropriate selection of the constants a and 6 can reproduce Gaussian. Lorentzian. or some kind of hybrid behavior. It is more difficult to fit to data, as multiple exponential models are notoriously ill-conditioned, and some care must be taken to ensure that the results are physically reasonable (ie. alb > 0). As if to compensate for the inconvenience, there is an analytic expression for the power spect rum

where W(r)is related to the error function for complex arguments. Figure .4.1 shows the four ACF models introduced in this section. dong with the corresponding power spectra. Parameters for bot h the "generalized exponential" (Equotion A.35) and "composite exponential" (Equation A.37) have been chosen to produce XCFSand power spectra which lie between the Gaussian and Lorentzian models.

~~A.5 Full width half maximum~~

One commonly used method of quantifing spectral width is the Full Width at Half Maximum (FwHM),also referred to as the distance between the hdf-power points. This is simply the width between two points that are selected to be 50% of the peak value. It cm be calculated for any spectrm, regardless of whether or not the Figure A.1: Standard Gaussian and Lorentzian line-shapes, dong with the results of the Voight and generdized exponential models. second moment is finite. There may, however. be practical difficulties in estimating the peak value (the *truc' peak may not fa11 into one of the discrete frequency bins). determining the exact location of the half-power points ( t hey may occur between two bins), and accounting for the effects of noise. These problems are especially severe for poorly resolved (narrow) spectra. For the models given previously. the FWHMvalues are

~~Gaussian FWHM= a2mzz 2.350 (.4.39)~~

~~Lorenztian FWHM = 2B (A.40)~~

An alternative method of characterizing width operates in the lag domain. identifying the correlation time (r.) where the ACF magnitude falls to e-l (from a peak of 1). Like the FWHM,this can be calculated for any ACF.with problems only arising when the magnitude hasn't actudy decayed sufficiently by the last lag and some extrapolation is required. Correlat ion t imes for the model spect ra are fi Gaussian T, = - zz 1.4110 a 1 Lorenztian re = - 4 -4 comparison of the correlation time to FWHM and standard deviation is given in Figure A.3, which shows the results for the model p(r) = exp( -(~/r,)"~ ) for a range of decay exponents. Synthetic ACFSwere generated with high sampling rates and a large number of lags to provide high spectral resolution. Standard deviation and FWHM were estimated in the spectral domain, and are presented in units of khz so that they may be easily scaled to any particular scattering wavelength. As expected, the FWHM varies approximately as l/r.; so does the standard deviation, but it is much more sensitive to the choice of n,. The ratio of FWHMto standard deviation is essent ially independent of the correlat ion time, but does change considerably wit h the decay exponent. For a Gaussian the ratio is quite close to the expected value of 2.35, but it drops to about 1.0 for n, = 1.7 and below 0.8 for n, = i. This will clearly be important when comparing FWHM and standard deviation estimates, as the conversion value depends crucially on the details of the line shape. Decay Exponent - 1.2 1-5 - 1-7 - 2.0 - 2-3

~~1 200 400 600 1 2 3 4 Correlation time (microsecands) WHM (khz)~~

Figure A.2: Cornparison of three spectral width parameters: Full-widt h half- maximum (FWHM), standard deviat ion and correlation time. The ACF magnitude was modeled with p(r) = exp(-(~/r,)"~),for five different values of the decay exp- nent n,. Results were obtained in the spectral domain, from a 512 point CF with a .5ps sampling rate to avoid truncation in the lag domain while producing well resolved spectra (Af = 195 hz).

~~A.6 Standard deviation as a width estimator~~

~~As mentioned previously. a power spectrum with a Lorentzian line shape has an infinite second moment. Furt hermore. the generalized exponential mode1~~

only has a finite non-zero variance for n, = 2, while the vast majority of the data shown in Chapters 3 and 4 have n,~:.This naturally leads to the question: do the standard deviation estimates used so liberally throughout this study have any real significance? One response to this challenge starts with the fact that any real signal must have been bandpass filtered, so the range of frequencies is finite. When combined with the fact that a power spectrum should be non-negative, this means that the second moment must be positive and finite for rneasured data. A more rigorous version of this argument st arts wit h the defini t ion of an autocorrelat ion funct ion for a bandlimited signal

~~J -UN and if POW(W)is finite within the frequency band then it is safe to take a derivative. and move it inside the integral~~

~~so at r = O each moment of the power spectrum corresponds to a derivative of the autocorrelat ion function~~

If the power spectrum is a finite non-negative quantity. then each moment must exist, and so must each derivative of the ACF. Specifically, the second moment will always be a positive number. -4lthough this result is inarguably correct, there may potentially still be a problem. While it may be true that the integral over any finite band will produce a finite result, the concern is that the true power spectrum may have -tailsu which stretch off to infinity. For any given sampling interval the resulting bandwidth will give some estirnate of the standard deviation. but a faster sampling rate will give a larger frequency range. and a different width estirnate. Fortunately, it seems that changing the sampling rates by a factor of two or three (by decimat ing the ACF)does not significantly change the moment estimates. It is difficult to be more precise than that, as the initial sampling rates are not high enough to study changes over a wider range. As well, the effects of noise rnay be masking some more subtle effect . Future experiments may try to take advantage of rapid sampling rates to recover moment estimates using short pulse lengths. In this case the behavior of the derivative estimates would be crucial. One possibility for future work lies in the study of this behavior, which could best be done bistatically (to minimize the effects of spatial variation), but useful results could probably be obtained by a monostatic systern. Another topic which may reward closer examination is the approach of excluding the zeroth and first lags from any derivative estirnates. While it seems entirely plau- sible that these are the lags rendered least reliable by the presence of noise. it is dso tme that they are also the lags that would display any peculiar features which might be an indication of problems with the assumption that the derivatives at zero lag are well defined. It is not clear how such behavior might be identified. but perhaps a careful statistical analysis of the early signal and noise lags might prove helpful.

~~A.7 Logarit hmic CF magnitude~~

The magnitude of the autocorrelation function is a non-negative function with a peak value of p(0) = 1. For the vast majority of practical situations the magnitude is aiways greater than zero. so the substitution

can be used. Although this may not seem like any improvement, the logarithm of p may be polynomial in nature, or at least more so than the magnitude itself. Bot h p and l? are even functions and can be expanded in power series containing only even terms

r(r)= + g4r4 + g6~6.. . where go = O from the requirement that the magnitude be one at lag zero. Although the two functions are very different in form, their derivatives are related in convenient ways. Both have only even derivatives, the first of these is

so the second derivat ive of the magnitude can be determined from the logarithm of the magnitude instead. This can be significant if the estimate of the second derivative is "contaminatedm by higher order terms, as their effects may be reduced by the process of taking a logarithm. The relationship between fourth derivatives is more complicated while the equation for the sixth derivative is still more complex

The potential usefulness of these results can be seen by examining the generalized exponential model introduced in the previous section. Figure A.3 shows both the magnitude p(~)and its logarithm, along with curves representing low order polyne mial expansions about zero lag. This example was chosen to demonstrate how the logarithm may be better fit with a polynomial model t han the magnitude itself. The logarithmic fit is perfect when n, = 2 as the logarithm of a Gaussian can be described perfectly with a parabola, but for n, = i the logarithmic representation is only marginally better.

Figure A.3: Magnitude of the autocorrelation function for exp(-7fs8) in the left panel, the logarithm in the right panel (solid lines). Dotted lines indicate the 2nd order polynomial fit, dashed Iines are for the combination of 2nd and 4th order.

After sorne tests on models and data, it was found that for most cases the logarithmic representation r did not provide significant ly better derivat ive estimates than those obtained directly from p. Furthermore, if the errors in p were normdly distributed. the errors in r would not be. and some care would be required to account for this. Consequently, derivatives in this study are obtained from p alone. It should be noted that the l' representation is useful for display purposes. as it can expose subtle features in the ACFmagnitude which may not be apparent on a linear plot.

~~A.8 Locating a single peak~~

Determining the peak of a power spectral distribution is very simple. Once the maximum sarnple value has been identified, then a simple parabolic fit is usually adequate for estimating the frequency at which the peak occurs. If more sophisticated models of the power spectral shape are available, they may be fit to the data. then the peak value determined from the model. However, given the success of other methods of spectral analysis applied in the lag domain, it is natural to consider the problem of estimating the peak location direct ly from the ACF. It will be shown that the solution is fairly simple for the case of a clearly defined peak. ünfortunately. the procedure is not particularly elegant, and has no clear advantage over working in the spectral domain, except perhaps for extremely narrow peaks. Starting wit h the relationship between the power spectrum and the auto-correlation function, it is easily shown that

~~where symmetry arguments guarantee that the power spectrum will be real, so the sin term can be neglected. Furthermore, since +(T) and wr are both odd. the limits of the integral can be reduced~~

The location of a maximum (or minimum) can be found by setting the fist deri~tive to zero. Given some starting point wo,root finding methods can be used to determine the frequency corresponding to the peak. Since the first derivative is amilable

~~sophisticated minima searching algorithms can be used. When applying this method to actual data sets. it is necessary to replace the integrals with surns~~

and in the majority of cases, the initiai guess can be set to the mean 2. In general. this algorithm works quite well. requiring only about three steps for convergence. However, it is clearly lirnited to those situations where there is only a single peak near the mean. As well. the iterative nature means that the number of operations is not significantly lower than simply performing an FFT. While this method might seem to have advantages when the spectra are very narrow. in those cases the mean and peak frequencies are nearly identical.

~~A.9 Twin peaks~~

~~Consider a power spectral peak that has the following properties~~

~~1. Total power of A.~~

~~2. Mean frequency of wo~~

~~3. .Symmetric shape about the mean~~

~~By itself the third requirement means t hat the corresponding ACF would be purely~~

~~real, while the addition of the second constraint means that the ACF cm be represented as ACF(T)= p(r)exp(zuo~) (-4.57)~~

Because the Fourier transform is linear, a spectnim composed of multiple peaks is equivalent to a superposition of several ACFS. For simplicity, only the case of two peaks with different means dl,LJ~ will be examined here

~~pi(r)ewl' + p2(r)ei*'~~

~~Introducing the quantit ies~~

~~allows the isolation of the midpoint frequency 2~~

~~For the simple case of pl = p, (identical line-shapes of equal power) this reduces to~~

so the original magnitude is modulated by a cosine of period In/Au. For extremely narrow features. p(r) wiil be essentially constant. so even a small part of the cosine curve would aIlow estimation of the peak separation. In practice. the line-shape magnitude will have some structure. and only a clear minimum due to the cosine term will be detectable. If this minimum occurred at the Iast rneasured lag. then

Unfortunately. t his is exactly the same resolution as can be obtained by working in the frequency domain. Consequent ly. unless the spectral peaks are ettremely narrow there is no significant advantage to operating in the lag domain.

~~A.10 Spikiness or RMS Spectral Power~~

By application of Parseval's theorem it can be shown that the total area under the power spectrum squared is reiated to the area under the autocorrelation function magnitude In the discrete case. t his becomes

~~so the RMS value of the power spectrum can be expressed in terms of the ACF magnitude. The possible utility of this result arises when the power spectrum is normalized to unit area~~

~~after which it is instructive to examine two limiting cases. First. for a spectrum with al1 the signal concentrated in a single bin. the peak (and only) power would be~~

~~NAT. Alternatively, if the signal is evenly distributed in frequency, then each bin will -n contain a power of h.-411 other spectra will fdl somewhere between these two cases. The quantity~~

~~( 1.00 single peak. one bin wide 1 .v 0.71 two peaks. each one bin wide (-4.67) flat spectrurn of 2% bins = i can be used to (arbitrarily) define~~

which will vary from O (flat spectrum) to 1 (single peak). An interesting feature of this statistic is that, unlike the standard deviation or half-width, it does not require that large values be nearby. In other words. it is more a measure of how spiky the spectrum is. than how concentrated the power is around some central value. Consequently, it may be useful for identifying double peaked spectra which should have a relatively high spikiness compared to their "width" as estimated by other methods. References

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