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Measuring ionospheric movements using totally reflected radio waves
Journal cf A/mosphrrrc and Twrarrial Printed m Great Bntain.
Physics, Vol
SO, No. 2. pp. 153-16%
1988
cix-9169/X8 %3.00+ .cil Pergamon Press plc
Measuring ionospheric movements using totally reflected radio waves W. R. FROM%,* ELAINE M. SADLER? and J. D. WHI~~AD Department of Physics, University of Queensland, St. Lucia, Queensland, 4067, Australia (Received in,final form 8 September 1987)
Abstract-It is shown that for radio waves of a particular frequency reflected totally from the ionosphere the effect of refraction as well as reflection can be simulated by an effective reflecting surface. This mirrorlike surface will give the correct angle of arrival and Doppler shift for all radars operating at this frequency. It is theoretic~ly possible for the effective retlecting surface to be folded back on itself, but this is unlikely except for F-region echoes refracted by sporadic E-clouds. If the surface is not folded and exists everywhere, it is always possible to describe its motion and change in terms of wave undulations. Experimental data for F-region echoes show that these wave undulations are very dispersive. However, the matching between the best fitting model and the experimental data is worse than expected for reasons we do not understand.
1. INTRODUCTION
The methods of measurement of the movements of travelling ionospheric disturbances which occur mainly in the F-region of the ionosphere fall into three main classes. The first uses closely-spaced stations in which the velocity of the diffraction pattern over the ground is measured. The second uses widely-spaced stations at which the time difference of the occurrence of disturbances is used to calculate the velocity of the disturbance. A sub-group of this experiment was described recently by MAEDA and HANDA (1980) for which the phase difference of waves of a particular period are used. This method is capable of measuring the dispersion of the waves, though the spacing of the stations sets natural limits of the range of wavelengths of disturbances for which the velocity can be measured. It is interesting to note that the coherency was not statistically significant (at the 75% probability mark) over the whole network for periods shorter than 50min for which the wavelength was about 1800 km with a maximum station spacing of 2000 km, only a little more than one wavelength. One is forced to ask whether the measurements are meaningful: naturally for stations close enough together, the coherency will be high and statistically significant. But the error in the velocity measurement must also be high. The third method is to measure the angle of arrival and the Doppler shift and is used in this paper, If there was no dispersion, the ratio of Doppler shift to _. * Killed on Mt. Everest, October 1984. t Present address : Kilt Peak National Tucson, Arizona, U.S.A.
Observatory,
the sine of the angle would give the velocity (PFISTER, 1971). However, in all our measurements the plot of sine of the angle vs. Doppler shift consists of a series of more or less complicated loops, showing that the wave velocities are very dispersive. If it were possible at a single station to measure the tilt and range of the reflecting surface directly overhead, Fourier analysis of both would provide the complete answer assuming that not more than one wave existed at any one frequency. For suppose one of the complex wave amplitudes was A, its wave number k for a certain frequency CL),the Fourier component of tilt at the frequency cct would give Ak, the Fourier component of the rate of change of range would give Aw, the ratio of components would give the phase velocity w/k of the wave. If the two Fourier components were not in phase, one interpretation could be that k is complex, that is, the wave suffers attenuation or amplification. Another possibility is that we have two waves of the same frequency w, but differing in velocity. In the simple case discussed here, it is not then possible to find the parameters A and k for both waves. Essentially an earlier analysis (BROWNLIE et al., 1973) went no further than finding Fourier components which were in phase and calculating the phase velocity. Those components for which the imaginary part of k was more than a modest fraction of the real part were rejected. There are two major problems to the analysis. The first is that movement of ionization well below the actual reflection height, and which causes refraction, may invalidate the analysis : a solution to this problem is given below by establishing that there exists an effective reflecting surface which, acting like a mirror, would reproduce exactly the angles of arrival and Doppler shift of any real ionosphere in which refrac153
154
W. R. FROM,ELAINEM. SADLERand J. D. WIIITEHEAD
tion as well as reflection occurs. It is recognised that some of the apparent movements of the effective reflecting surface may apply to the refracting layer rather than where the radio waves are reflected. However, the analysis is still valid in the sense of determining the movement of the effective reflecting surface. The second problem is that a ground-based sounder observes an echo, not from overhead, but from a direction determined by the tilt fl of the effective reflecting surface. If R is the phase range, the horizontal displacement of the reflecting point (on the effective reflecting surface) is Rsin8, and the simple temporal Fourier analysis of angle of arrival and Dopler shifts fails if this exceeds a small fraction of a wavelength of any waves which are present. A sinusoidal reflecting surface does not produce purely sinusoidal temporal changes in 9 and R. A method of iteration was given by BROWNLIEet al. (1983) for overcoming this problem, and it worked with artificial data. Unfortunately, it did not always converge with real data. The method presented here provides a least squares fit to the data and always converges. The method has been extensively tested also with artificial data-data from calculations of angles of arrival and Doppler shift for a reflector distorted by a collection of known waves. The fit for artificial data is always much better than for the real data even after experimental errors are allowed for. It indicates something wrong with the model used, but the reasons are not understood. Real data always requires a model in which waves with a wide range of velocities are present. The classical picture of travelling ionospheric disturbances as distortions travelling vast distances without much change of form (e.g. HEISLER,1958) is certainly incorrect for the medium scale disturbances. Rather we see a turbulent spectrum with no recognisable feature moving more than its own scale size.
2. EFFECTIVEEQUIVALENTSURFACEFOR REFLECTION
The purpose is to demonstrate that for a given spatial distribution of refractive indices p (at a chosen radio frequency) in the ionosphere including a contour at which p = 0, it is possible to construct an equivalent reflection surface which acts like a mirror and provides the correct angle of arrival and phase path for all radars on the ground operating at the same frequency. The general proof is as follows. For a particular point Q on the p = 0 contour, construct a ray path for a wave reflected back along its path at Q (a normal reflection) until the path
Fig. 1. Construction to find a point X on the equivalent surface. TX = phase range of echo.
reaches the ground at T (Fig. 1). Below the ionosphere, the ray path is straight. This part is extended to X so that TX = phase path TQ. T is the position of a radar which receives a reflection from Q in the direction of TX. The wavefront of the reflection touches AT drawn perpendicular to TX. AT is small and in the limit becomes a wavefront. Complete the rectangle A TXX’. At reflection, the wavefront of the incident and reAected waves must be parallel to each other and the /2 = 0 contour. P is the reflection point for a radar at A, and if A is sufficiently close to T, P will be close to (2. Therefore as P is on the p = 0 contour and lies close to Q, then PQ is a wavefront. The phase paths TQT and APA are thus nearly equal, and XX’ touches a mirror which could be used instead of the ionosphere to give the correct phase and angle of arrival. Therefore around X we can draw a surface perpendicular to TX at X which will have the correct slope (so that its normal lies along TX) and for which the distance 7X’ gives the correct phase path for the actual reflection, This surface is part of the effective reflecting surface: the complete surface is found by considering normal reflections from all points like Q on the /i = 0 surface. Thus we have established that the effective reflecting surface exists and given the rule for finding points on it. The proof is more convincing if we deal with some special cases. The first example is of a wedge of angle SIof nondispersive refractive index p with a mirror on its upper surface (Fig. 2). The refraction of a ray from T reflected at Q is refracted Q’. Let O.@T = 90” X&Q’ = /I. Then XQ’ = OQ’ sin/2 and and
155
Measuring ionospheric movements
Fig. 2. A point X on the equivalent surface when there is a refracting wedge. QQ’ = OQ’sincr. However Snell’s Law gives psincr = sin/I for the refraction at Q’. Thus XQ’ = QQ’ sin /?/sin c( = QQ’ * p. Hence TX is equal to the phase path TQ’Q, and OX is the equivalent surface. It is also possible to prove the result for a wedge for which the refractive index at a point P varies as a linear function of the angle Q6P to second order in CL(a general proof for wedges using geometry and calculus eludes us) and a wedge similar to the first example but with a mirror above it and parallel to OQ, rather than on OQ itself. The equivalent surface for the ionosphere for a particular radar frequency will be just below the p = 0 surface and will be displaced magnetic north or south due to the bending of the group path which occurs even if all the isoionic contours are horizontal. The proof is still valid in the presence of a background magnetic field giving double refraction. The idea of having an equivalent surface is also consistent with PFISTER’S (197 1) theorem which relates the off-vertical angle to the Doppler shift. If all parts of the ionosphere move with the same velocity, Pfister’s theorem is obeyed. The equivalent surface moves without distortion at this same velocity. If the ionosphere changes as it moves, the equivalent surface will likewise change. Since the distance from any radar to a point of reflection on the equivalent surface is exactly the phase range for the corresponding ionospheric echo, the Doppler shift for the equivalent surface will be the same as that for the ionospheric echo. The equivalent surface is not necessarily a simple surface, but may fold back on itself. For example, strong refraction in E-region of F-region echoes could give a fold in the equivalent surface (Fig. 3). All parts of the fold act like a mirror, even though the equivalent ray passes through other parts of the mirror. We must establish the condition for such folding since when it happens, the method of analysis will fail,
Fig. 3. Folding of the equivalent surface in the presence of refraction.
Fig. 4. The rays to the effective reflecting surface will intersect below the surfaces if &I@,-h,) > 6x approximately.
3. CONDITIONS FOR THE FOLDING OF THE EFFECTIVE SURFACE
Suppose refraction occurs at a height h, (naturally to be thought of as the height of the E region), and reflection occurs from a horizontal plane at a phase height h,. From Fig. 4, folding will occur if
II
de>dx
1
- h,-h,’
where 0 is the small refraction angle at a horizontal distance x. If the refraction angle varies sinusoidally with a horizontal wavelength 1 and maximum value BO,this condition becomes 0, > 1/2n(h, - hE).
(2)
1.56
W. R. FROM, ELAINEM.
SADLER and
It is interesting to compare this inequality with the condition for a split echo from the refracting region itself at a lower radio frequency. Suppose the isoionic contours at height h, have a slope 61where ff = tl, cos 2x,X/A.
(3)
Then taking the isoionic contours to be parallel to the effective reflecting surface for a radio echo for this region, a split echo will occur if G!fl> ;1/2&.
(4)
For a model of the refracting region in which it is a medium of constant refractive index p sandwiched between an upper horizontal plane surface, and a lower surface with a slope given by equation (3), the refraction angle is 0 = (1 --~)a approximately. (If, however, the upper surface is parallel to the undulations in the lower surface, 0 would be zero : therefore in practice for E-region we can expect 8 to be appreciably less than CL) Our measurements show that normal E-region is very flat. 8, is typically less than 2” and J_greater than 25 km. Split echoes are not seen, although sometimes inequality (4) must be nearly satisfied. Folding of the effective reflecting surface for F-region reflection must he fSrdirlyrare. On the other hand, cloudy sporadic E, giving rise to partial reflection, often shows echo splitting and the F-region echoes show clearly the effect of refraction by the sporadic E-cloud. (Sometimes this refraction is very obvious and at other times apparently absent: this has important consequences to the structure of sporadic E and will be pursued in a later paper.) The refraction angles 0 may be as large as IO” and scale sizes in the sporadic E-clouds 10 km : therefore from inequality (21, folding of the effective surface is likely. The folding presents considerable potential difficulties in the analysis of data, and the crucial assumption is made that folding does not occur. The data used for analysis has been selected from days when sporadic E was weak and certainly gave no echoes on the operating frequency of the radar.
4. METHOD OF ANALYSIS
The height 2 of the effective reflecting surface is a function of x,_r (the horizontal coordinates) and I (the time}. If z is single-valued for all X, 4’ and t of interest, Fourier analysis allows us to assume that
J. D.
WHITEHEAU
where the wavenumber k has real components k,, ky and imaginary components I,, Z,,,in the x and y directions respectively, and A and k are functions of w. The summation is taken over all w. Tn theory, we could put j, and & equal to zero, since a single attenuated wave over a finite range of x and y may be represented by a collection of unattenuated waves of various real wavenumbers. However, if the waves in the upper atmosphere are attenuated, then we might expect to be able to represent the whole disturbance with fewer parameters by allowing finite values of I, and &. In the event it proved to be of no advantage as will be discussed later. In a finite observing period, the integral is replaced by a summation. A dcu is replaced by A the complex amplitude of one wave and the summation is taken over all the waves. By having the spacing between the angular frequencies w less than 271divided by the total observing period, we effectively allow more than one wave (with different k’s) at nearly the same frequency, as is discussed below. The observing station is at the origin, and the angle of arrival is defined by 0, and Q,,,where 8, is the angle between the ray direction and the y,r plane. It may then be shown that the point of reflection is (R sin 8, cos 8,,., R sin 0,cos 0,) and the observed tilts at a time t are given by the real part of
-k,,,,R sin 0,. cos 8,) exp(&JXsin 0, cos 8,
+ L,,R sin I!?,.cos O.,).
(6)
with a corresponding expression for tanfI,., and the Doppler shift written as the rate of change of range, dRjdt, by the real part of dR dr set 8, see O,,?= c iu,A, exp i(w,f
- k,
R
sin 8, cos e,
- k,,,R sin @j.cosO,,.)exp(i,,R sin #,Ycos O! +I,,,.Rsin#,,cosO,).
(7)
The problem may be stated as follows. For a series of frequencies w covering the appropriate band, we need to find A and k to fit equations (6) and (7). In order to reduce computer time, advantage is taken of the fact that 0, and 8,: are typically less than IO;‘, so that sin O,V2: O.,, cos 0, N 1,
(8)
and that we might expect to find that 1, and & are small enough to allow the approximation
2 = z. -+- Aexpi(ot-_.,x-k,,y).exp(I,x+Z,..~?)do, s (5)
expU,,, RB, + Q@$) 2 1 -I-LSO, -i-&,.R@,.. (9)
157
Measuring ionosp lheric movements Hence, we arrive at these simplified expressions the real part of the right hand side
taking
8, = 1 i(k,, + &.)A,,( 1+ l,,,RQ, + l,,YRe,) xexpi(w,t-k,,R8,-k,,RB,.), with a corresponding
expression
(10)
for Q,, and
dR dt = Ciw,A,(l+l,,RB,+f,,.RO,.) xexpi(w,t-kk,,RB,-k,,YRB,.).
(11)
The presence of the i!, in the first term in brackets for 0, (equation 10) causes the 8, variation for any particular frequency to be out of phase with dR/dt.
5. GENERAL REMARKS ON DISPERSION
0, N c i(k,, +$,,)A,
exp(iw,t)
(12)
One important and vital piece of information is obtained from the Doppler shift data. If all the waves move with identical phase velocities (V,, V,.) (i.e. the whole reflecting surface moved without change of shape), then Pfister’s theorem states that
0,. 1: C i(k,, + &)A,,
exp(iw,t)
(13)
dR -= dt
4.1. Solution ifall k,, k, are small enough If
changing from roughly sinusoidal to a sawtooth variation and finally to multi-valued functions with 0 and dR/dt taking on an odd number of values at any one time. The ratio of the Fourier amplitudes previously used to find k gives only an approximate value and new Fourier components not present in the reflecting surface will appear. The nature of the data redundance becomes more difficult to specify, but the period, velocity and amplitude of a single wave may now be estimated from the angular variation alone, using BRAMLEY’S (1953) method and from the Doppler shift alone, the period, speed and amplitude using CORNELIUS and ESSEX’S(1979) method.
k,RB, and k,.RO,. are always much less than unity,
then
v,o,+V,O,, (15)
dR dt
V,sin0,cos8,+V,sin0,cos8,?i
=
Ciw,A,
exp(iw,t),
(14)
and A, and k, may be found by Fourier inversion. This is the simple method discussed above. Naturally one of the requirements is that for the solution found, k must satisfy the condition that 1kR0 1cc 1 for all 0. Even when this is satisfied, it is found that k is complex, that there is attenuation or amplification. There is just sufficient information (the complex Fonrier components of Q,, O,,.and dR/dt at each frequency o) to find the complex A and the real k,, k,, I,, I, for each n. As an alternative to the attenuated waves, we might consider the possibility of more than one wave at the one frequency. With this simple method of analysis there is not sufficient information to find the two complex amplitudes, the two real k, and k.P(eight real numbers in all). The condition that 1kR0 1cc 1 implies that not all data may be analysed using this method. It also suggests that not all data need be self-consistent, since the k calculated from the data by this method may not satisfy the inequity. For example, artificial data derived from a reflecting surface which folded on itself, could be made to fit an effective reflecting surface which did not. 4.2. Data redundancy with IkRBj > I If (kRB[ - 1 or greater for some of the time, the raw data takes on a markedly different appearance,
and the Doppler shift would be a linear function of 0, and Q,.. The experimental results show variability from day to day as to how well Pfister’s theorem is obeyed. There are days when the linear trend is obvious, though not perfect, and other days when it is difficult to pick by eye. Clearly the waves are dispersive to a greater or lesser extent and methods such as Bramley’s must fail under such circumstances, as will, of course, the measurement of the diffraction pattern velocity (BROWNLIE et al., 1973). The dispersion means that the ripples change in form as they move overhead and this renders invalid the methods previously used to determine the wave parameters from the angular or Doppler shift variations alone. However, one could envisage using only the angular variation by some sort of iterative method to deduce the phase velocity and wave amplitude as a function of frequency. Thus, the Doppler data is largely redundant and we do not expect to obtain perfect fitting between data and theory due to experimental errors and incomplete models. Indeed it will become clear that the difference between data and theory is systematic. It is not due to the disregard of higher frequencies, nor experimental errors. 6. DATA REDUNDANCY
AND THE LIMITS
TO WAVE
ANALYSIS
An important question must now be raised. Given arbitrary Q,, 0,. and dR/dt as functions of time is it
158
W. R. FROM,ELAINEM.
SADLER
always possible to find models which will fit equations (10) and (1 l)? The answer to this question is No! Consider for example the following data set : 0, = 0 (so that we restrict ourselves to two dimensions for simplicity)
(16)
6, = @= Akcosot
(171
dR/dt = Adcoswt,
(18)
and Ak’R >>1 (implying that the distortions are large and ought to produce ‘Z’ traces not present in the data). Or as a second example, the data set : a, = 0
(19)
f3, = 0 = Ak’cos(wt-kR6)
(20)
dR/dt = Aw cos(wt - kR0)
(21)
where Ik’[ K Ik I and Ak’kR >>1. The second inequality implies that multiple echoes will exist, but the first does not allow the angles of arrival to be large enough. It may be shown that for either of these two data sets, it is not possible to construct a reflecting surface model. The proofs are somewhat involved and will not be given here. They are available in an internal report (WHITEHEADand FROM, 1987). Thus, it is possible for data (contrived or real) to be inconsistent with our model represented by equation (5) and provide a test of the model itself. Very few measurements of ionosphere movement have this self-checking feature. Are there ionosphere models which would produce such data, inconsistent with the dispersion model? One obvious one is a single cloud of ionization. If vertical motion as well as horizontal motion were allowed, dR/dt and the B,, 0, angles could be varied in any chosen way. The reason this does not have to be consistent with the dispersion model (equation 5) is that z does not have any value at all beyond the edges of the cloud. Several clouds and models in which one cloud splits into two or more may also be inconsistent with the model. Sporadic E often seems to have this structure and other methods of analysis must then be used. However, whenever z has a single value for all x1 y and t around the radar and for the observing time, z must be expressable in terms of equation (S), and thus under these conditions, the data must be consistent with a dispersion model. The only assumption made in the analysis to be presented is that the effective equivalent surface is not folded and that it exists at all x, )’ and l. In particular the assumption that ]kRf? I CC1 will not be required.
and J. D. WHITEHEAD 7. CHOICE
OF FREQUENCIES
In Fourier analysis of a finite length of data, it is usual to choose the fundamental frequency (with a period equal to the length of data) and the harmonics of this. if only a few specific, but non-ha~oni~ frequencies are present, the amplitudes of these frequencies may be determined by interpolation between the harmonic frequencies. We cannot directly use Fourier analysis (except as an initial approximation) because the terms k,RO, and k,RB, inside the exponentials generate all the harmonics and mixed frequencies. Nevertheless the anatysis assumes that certain wave periods dominate the distortion of the effective reflecting surface. As to the choice of wave periods, if the waves are allowed to be attenuated, the simple analysis using the approximation that I kROl CCI seems to justify only the use of the harmonic wave periods, certainly no greater number of periods. The additional redundancy with split echoes could allow the use of some ad~tional periods and indeed, if the Fourier component of a particular period remains sign&ant after the corresponding best choice has been subtracted from the data, during the required iteration, a second wave of the same period could be introduced by the programme. If attenuation is not permitted, the number of frequencies up to a chosen harmonic could be increased substantially. With lkR@l CC1, there is insu~cient info~ation to double the number of frequencies, but it is clear that the data becomes more redundant (and thus allows more frequencies to be chosen) if IkRO 1is larger. If the data redundance could be properly assessed, a more precise choice could be made. We could have used two or more waves with identical frequencies but this would have increased the technical difficulties with the calculations. Instead wave periods equal to a number (NPF) times the total record length divided by the integers 1, 2, 3, etc. are used. If the actual waves in the ionosphere have specific periods, one of our chosen periods will be very close to it. Also since in the atmosphere there are no vertical planes to reflect horizontally travelling waves, we would not expect to find two travelling waves of exactly the same period. The purpose was to match as closely as possible the mode1 with the data, and so NPF was chosen to minimise the mean square error. Extensive tests were carried out on particular data sets : if the time plots of Doppler shifts and angles appeared nearly sinusoidal, then NPF should be about 2. For more sawtooth time variations, NPF should be larger, and if ‘Z’ traces are evident, NPF should be about 6. Sometimes the obvious presence of short periods and computer size
Measuring ionospheric movements forced a choice of NPF smaller than ideal. In testing models with and without attenuation, care was taken to restrict the total number of frequencies so that the same number of (real) unknown parameters was identical. Ifwe were really able to measure wave attenuation it would be an important parameter for checking against theory and for locating sources of the waves. For example, we expect waves to be attenuated in the direction in which they travel. For either model we may hope to find some pattern in the measured dispersion-the speeds being consistent with expected internal gravity waves and the directions indicating perhaps a small number of sources. Thus there are ways of checking that the answers we find from the analysis are reasonable. The details of the iterative procedure will not be given here but are available (FROM et nl., 1987). A least squares method adapted to the form of the parameters is used. The root mean square remainder after subtracting the 0 and dR/dt for the model is minimised. This remainder falls rapidly initially in the calculation and tends to a finite value: further iterations make little difference. The amplitudes of the higher frequencies tend to fall during the iteration, as the programme correctly takes into account the kRQ terms. Standard programmes exist for finding least square fits to any set of equations : these were tried but after ten times as much computer time had not reduced the remainder as much as the programme above. It is often more efficient to use a special procedure adopted to the particular equations to be fitted.
8. ASSESSMENT OF THE METHOD
One significant advantage of the method is that it is possible to assess its effectiveness in a reasonably objective manner. The other methods of measuring movement might have given an answer, but there is no method of checking the self-consistency of the data model. The first test is how well the model fits the data. The redundance implies that a perfect fit need not be obtained if the data contains experimental errors. For a particular model of wave-like distortions of the effective reflecting surface, it is not practical to work out what the angles of arrival and Doppler shift would be, since the calculation of each ray path at each time involves iteration. Instead what is calculated is the slope of the model ionosphere at theplace where the rejection actually took place (not where it would take place with the model ionosphere), and the cor-
159
responding Doppler shift. This slope and Doppler shift will be called the theoretical slope and Doppler shift. In the calculation, for reasons of computer space (a PDPI l/34), 200 data points (spread over a time of l-2 h) are analysed at one time with up to 74 wave frequencies being used. For waves for which 1kRQ 1cc 1, perfect fitting would be achieved with 200 wave frequencies. However, with real ionospheric data it is obvious that the difference between the theoretical and experimental data contains low frequencies so that the reason for ill-fitting is not due to taking insufficiently high frequencies. The observed and theoretical data shown in Figs. 5 and 6 is typical of the data analysed. Although the models appear to match quite well (as would be expected with a large number of adjustable parameters), it is also obvious from these figures that the error is systematic. The largest errors tend to occur where the observed slope is large and then it is larger than is accounted for theoretically. Other than stressing the point that more than one or two waves of one frequency may be present, it is not obvious what causes this discrepancy. By comparisons of this sort between the two methods of analysis, slightly better fitting to the data occurs with the non-attenuated waves. The root mean square remainder for this model is 95% of that for attenuated waves with identical numbers of adjustable parameters. 8.1. Testing the programme with model data For a particular model of reflecting surface, it is possible to calculate what the observed angles of arrival and Doppler shift should be. In calculating this model data, an iteration procedure must be used to find those rays reflected back to the radar. Random errors equal to the experimental errors are added to the calculated data. The model data is then used with the analysis programme to see how satisfactory it is. The two major results which arise from this test are that the models fit the model data much better than experimental data fits, but that the system of waves which is calculated to fit the model data best is not identical to the system of waves used to generate the data. A large variety of models were used. A measure of fit was the ratio of the root mean square remainder to the spread of the original data. This ratio would then be zero if a perfect fit were found, and unity if the model was a flat horizontal sheet. The ratio was found to be approximately R = 0.03 + 0.07 x log, ,, (number of waves in initial model) to within a factor of two.
160
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162
Because the programme which generates the data from the initial model takes longer when a larger number of waves is used, the maximum number used was 80. The models include a few extreme cases of multiple traces, and when these are not counted, the ratio for models with a large number of waves was about 0.08. By comparison, the mean value of R for real data is 0.172. Whereas the fits for real data always include groups of points taken over periods of about 2 min when the data and model do not fit, the same is not true for most of the fitting to the artificial data. Some artificial data was generated with an error in the programme so that the angular positions of horizontal sections of the ionosphere, rather than resection points, was found. This incorrect model data gave errors similar to that observed for real data, that is an increased value of R (to about 0.30) and groups of four or five points in the data which did not fit the derived model. This accident shows not only that arbitrary data cannot necessarily be made to fit any model, but that the real data behaves like incorrectly generated artificial data! Whilst correct artificial data can be fitted quite well to models (thus demonstrating that the computer programmes work), the match between the input model of waves used to generate the data and the output model which is fitted to the artificial data is not so
Fig.
good. This is a result of the equations being ill-conditioned. In situations where the wave frequencies of the input models are separated by at least the reciprocal of the total time interval, the matching is reasonable. Speeds do differ by an average of 20%, but the modelIing programme picks out the dominant waves without too much difficulty. However, when the input model frequencies are closer than this, the matching becomes worse. The reason is that the equations are ill-conditioned: as was noted earlier, for situations where the off-vertical displacement of the reflection point is much less a wavelength of any of the waves, we have insufficient data to determine the parameters of the two waves we know must exist when the corresponding Fourier components in the Doppler shift and angles are out of phase (by other than zero or 180“). Thus in general we might expect to find difficulty determining the wave parameters even though we know there must be many waves since reducing the number in our model makes the fitting worse. We have the same situation with real data. Though we know that a satisfactory model (in terms of fitting the data) requires a large number of waves, the illconditioned equations will not allow us to determine the wave parameters at all accurately. The best we can hope for is to see trends in the models. Because of size limitations on the computer, only
7. A mass plot from 60 h of data (mostly daytime F-region) showing the dispersion
of the waves
Measuring
ionospheric
movements
164
W.
R. FROM, ELAINEM. SADLERand
200 experimental data sets could be analysed at the one time. If 300 data sets were available, models for the first 200 sets and the last 200 (using a common hundred sets) would be used. Thus several models were often generated for the same run of data. Even more overlap could be used: two models found for data sets of which 190 out of the total 200 points were common would yield significantly different models because the equations are ill-conditioned. The dominant waves have similar velocities, but the background of small amplitude waves which we need to obtain the best fit change considerably. Thus at the cost of considerable computer time, it is possible to generate several wave models from one set of data which may then be used for statistical studies.
9. RESULTS OF THE ANALYSIS The mass plot of speed vs. period (Fig. 7) shows the considerable dispersion required. Some of the spread is due to errors in the analysis, but we know from the angles vs. Doppler shift plots that dispersion must be severe. Analysis of this data shows that the speed is least around the dominant 10-15 min period. These waves have speeds of about 150 m s-‘. However, significant waves are found also at other periods. Longer period waves have slightly greater speeds (180m s- ‘) and short period waves (say - 5 min) have speeds of about 240m s-’ on the average. For some data runs, the short periods were not significant, but a recent run (Fig. 8) shows a dominant 5min period with a horizontal trace speed of 130 m s-‘. Whilst the period of this wave suggests it lies on the sonic branch of the dispersion curves (HINES. 1960), the speed suggests otherwise. It is likely that the waves are in moving air and it is proposed to interpret the data generally as waves in moving air. To find out more about the waves, we need to know also the vertical structure. The effective equivalent surface must be found for two or more radio fre-
J. D.
WHITEHEAD
quencies. Work carried out so far indicates that it is almost as difficult to trace the vertical structure for these waves as it is the horizontal structure (JONES, private communication). Whilst a statistically significant regular phase shift of each wave with height does occur, it is not easy to measure the vertical wavelength with any confidence. But major difficulties remain with determining the behaviour of one effective equivalent surface of explaining why the fitting to the models has systematic errors and how to overcome the difficulty of ill-conditioned equations. In spite of these difficulties, the method is superior to other ways of measuring movement in the F-region. For example, movement of the interference pattern over the ground has been shown to be a poor measure (BROWNLIE et al., 1973) ; methods using only angular measurements or only Doppler shift measurements fail to detect disperion ; methods using widely spaced measurements measure the dispersion, but that the data still do not fit the models shows that we have information which cannot yet be utilised. One important parameter which has not been used in the present analysis is the echo amplitude. There are some technical difficulties, such as amplitude changes due to absorption, but additional information is present and ought to be used.
10. CONCLUSION It has been proved that an effective equivalent surface exists which acting like a mirror would give the same angle of arrival phase path and Doppler shift for all radars operating at a chosen frequency as does the real ionosphere. Using the measurements of angle of arrival and Doppler shift of high frequency radar echoes from the ionosphere, it is possible to deduce that the waves which distort this equivalent reflecting surface are highly dispersive. However, the data does not fit the best models as well as might be expected and this remains a mystery.
REFERENCES
BRAMLEY E. N. BROWNLIE G. D., DRYBURGHL. G. and WHITEHEAD J. D. CORNELIUS D. W. and ESSEX E. A HEISLERL. H. HINES C. 0. MAEDA S. and HANUA S PFISTERW.
1953 1973
Proc. R. Sot. A 220, 39. J. atmos. ten-. Phys. 35,2147
1979 1958 1960 1980 1971
J. atmos. terr. Phys. 41, 48 I. Aust. J. Phys. II, 79. Can. J. Phys. 38,441. J. atmos. terr. Phys. 42, 853. J. armos. terr. Phys. 33, 999.
Measuring Reference
ionospheric
movements
165
is also made to the following unpublished reports.
WHITEHEAD J. D. and FROM W. R.
1987
FROM W. R.. SADLER E. M. and WHI~HEAD J. D.
1987
Data redundancy and limits to the wave analysis. Report No. 2, Department of Physics, University of Queensland, Australia. Iterative procedure in the wave analysis. Report No. 3, Department of Physics, University of Queensland, Australia.
Physics, Vol
SO, No. 2. pp. 153-16%
1988
cix-9169/X8 %3.00+ .cil Pergamon Press plc
Measuring ionospheric movements using totally reflected radio waves W. R. FROM%,* ELAINE M. SADLER? and J. D. WHI~~AD Department of Physics, University of Queensland, St. Lucia, Queensland, 4067, Australia (Received in,final form 8 September 1987)
Abstract-It is shown that for radio waves of a particular frequency reflected totally from the ionosphere the effect of refraction as well as reflection can be simulated by an effective reflecting surface. This mirrorlike surface will give the correct angle of arrival and Doppler shift for all radars operating at this frequency. It is theoretic~ly possible for the effective retlecting surface to be folded back on itself, but this is unlikely except for F-region echoes refracted by sporadic E-clouds. If the surface is not folded and exists everywhere, it is always possible to describe its motion and change in terms of wave undulations. Experimental data for F-region echoes show that these wave undulations are very dispersive. However, the matching between the best fitting model and the experimental data is worse than expected for reasons we do not understand.
1. INTRODUCTION
The methods of measurement of the movements of travelling ionospheric disturbances which occur mainly in the F-region of the ionosphere fall into three main classes. The first uses closely-spaced stations in which the velocity of the diffraction pattern over the ground is measured. The second uses widely-spaced stations at which the time difference of the occurrence of disturbances is used to calculate the velocity of the disturbance. A sub-group of this experiment was described recently by MAEDA and HANDA (1980) for which the phase difference of waves of a particular period are used. This method is capable of measuring the dispersion of the waves, though the spacing of the stations sets natural limits of the range of wavelengths of disturbances for which the velocity can be measured. It is interesting to note that the coherency was not statistically significant (at the 75% probability mark) over the whole network for periods shorter than 50min for which the wavelength was about 1800 km with a maximum station spacing of 2000 km, only a little more than one wavelength. One is forced to ask whether the measurements are meaningful: naturally for stations close enough together, the coherency will be high and statistically significant. But the error in the velocity measurement must also be high. The third method is to measure the angle of arrival and the Doppler shift and is used in this paper, If there was no dispersion, the ratio of Doppler shift to _. * Killed on Mt. Everest, October 1984. t Present address : Kilt Peak National Tucson, Arizona, U.S.A.
Observatory,
the sine of the angle would give the velocity (PFISTER, 1971). However, in all our measurements the plot of sine of the angle vs. Doppler shift consists of a series of more or less complicated loops, showing that the wave velocities are very dispersive. If it were possible at a single station to measure the tilt and range of the reflecting surface directly overhead, Fourier analysis of both would provide the complete answer assuming that not more than one wave existed at any one frequency. For suppose one of the complex wave amplitudes was A, its wave number k for a certain frequency CL),the Fourier component of tilt at the frequency cct would give Ak, the Fourier component of the rate of change of range would give Aw, the ratio of components would give the phase velocity w/k of the wave. If the two Fourier components were not in phase, one interpretation could be that k is complex, that is, the wave suffers attenuation or amplification. Another possibility is that we have two waves of the same frequency w, but differing in velocity. In the simple case discussed here, it is not then possible to find the parameters A and k for both waves. Essentially an earlier analysis (BROWNLIE et al., 1973) went no further than finding Fourier components which were in phase and calculating the phase velocity. Those components for which the imaginary part of k was more than a modest fraction of the real part were rejected. There are two major problems to the analysis. The first is that movement of ionization well below the actual reflection height, and which causes refraction, may invalidate the analysis : a solution to this problem is given below by establishing that there exists an effective reflecting surface which, acting like a mirror, would reproduce exactly the angles of arrival and Doppler shift of any real ionosphere in which refrac153
154
W. R. FROM,ELAINEM. SADLERand J. D. WIIITEHEAD
tion as well as reflection occurs. It is recognised that some of the apparent movements of the effective reflecting surface may apply to the refracting layer rather than where the radio waves are reflected. However, the analysis is still valid in the sense of determining the movement of the effective reflecting surface. The second problem is that a ground-based sounder observes an echo, not from overhead, but from a direction determined by the tilt fl of the effective reflecting surface. If R is the phase range, the horizontal displacement of the reflecting point (on the effective reflecting surface) is Rsin8, and the simple temporal Fourier analysis of angle of arrival and Dopler shifts fails if this exceeds a small fraction of a wavelength of any waves which are present. A sinusoidal reflecting surface does not produce purely sinusoidal temporal changes in 9 and R. A method of iteration was given by BROWNLIEet al. (1983) for overcoming this problem, and it worked with artificial data. Unfortunately, it did not always converge with real data. The method presented here provides a least squares fit to the data and always converges. The method has been extensively tested also with artificial data-data from calculations of angles of arrival and Doppler shift for a reflector distorted by a collection of known waves. The fit for artificial data is always much better than for the real data even after experimental errors are allowed for. It indicates something wrong with the model used, but the reasons are not understood. Real data always requires a model in which waves with a wide range of velocities are present. The classical picture of travelling ionospheric disturbances as distortions travelling vast distances without much change of form (e.g. HEISLER,1958) is certainly incorrect for the medium scale disturbances. Rather we see a turbulent spectrum with no recognisable feature moving more than its own scale size.
2. EFFECTIVEEQUIVALENTSURFACEFOR REFLECTION
The purpose is to demonstrate that for a given spatial distribution of refractive indices p (at a chosen radio frequency) in the ionosphere including a contour at which p = 0, it is possible to construct an equivalent reflection surface which acts like a mirror and provides the correct angle of arrival and phase path for all radars on the ground operating at the same frequency. The general proof is as follows. For a particular point Q on the p = 0 contour, construct a ray path for a wave reflected back along its path at Q (a normal reflection) until the path
Fig. 1. Construction to find a point X on the equivalent surface. TX = phase range of echo.
reaches the ground at T (Fig. 1). Below the ionosphere, the ray path is straight. This part is extended to X so that TX = phase path TQ. T is the position of a radar which receives a reflection from Q in the direction of TX. The wavefront of the reflection touches AT drawn perpendicular to TX. AT is small and in the limit becomes a wavefront. Complete the rectangle A TXX’. At reflection, the wavefront of the incident and reAected waves must be parallel to each other and the /2 = 0 contour. P is the reflection point for a radar at A, and if A is sufficiently close to T, P will be close to (2. Therefore as P is on the p = 0 contour and lies close to Q, then PQ is a wavefront. The phase paths TQT and APA are thus nearly equal, and XX’ touches a mirror which could be used instead of the ionosphere to give the correct phase and angle of arrival. Therefore around X we can draw a surface perpendicular to TX at X which will have the correct slope (so that its normal lies along TX) and for which the distance 7X’ gives the correct phase path for the actual reflection, This surface is part of the effective reflecting surface: the complete surface is found by considering normal reflections from all points like Q on the /i = 0 surface. Thus we have established that the effective reflecting surface exists and given the rule for finding points on it. The proof is more convincing if we deal with some special cases. The first example is of a wedge of angle SIof nondispersive refractive index p with a mirror on its upper surface (Fig. 2). The refraction of a ray from T reflected at Q is refracted Q’. Let O.@T = 90” X&Q’ = /I. Then XQ’ = OQ’ sin/2 and and
155
Measuring ionospheric movements
Fig. 2. A point X on the equivalent surface when there is a refracting wedge. QQ’ = OQ’sincr. However Snell’s Law gives psincr = sin/I for the refraction at Q’. Thus XQ’ = QQ’ sin /?/sin c( = QQ’ * p. Hence TX is equal to the phase path TQ’Q, and OX is the equivalent surface. It is also possible to prove the result for a wedge for which the refractive index at a point P varies as a linear function of the angle Q6P to second order in CL(a general proof for wedges using geometry and calculus eludes us) and a wedge similar to the first example but with a mirror above it and parallel to OQ, rather than on OQ itself. The equivalent surface for the ionosphere for a particular radar frequency will be just below the p = 0 surface and will be displaced magnetic north or south due to the bending of the group path which occurs even if all the isoionic contours are horizontal. The proof is still valid in the presence of a background magnetic field giving double refraction. The idea of having an equivalent surface is also consistent with PFISTER’S (197 1) theorem which relates the off-vertical angle to the Doppler shift. If all parts of the ionosphere move with the same velocity, Pfister’s theorem is obeyed. The equivalent surface moves without distortion at this same velocity. If the ionosphere changes as it moves, the equivalent surface will likewise change. Since the distance from any radar to a point of reflection on the equivalent surface is exactly the phase range for the corresponding ionospheric echo, the Doppler shift for the equivalent surface will be the same as that for the ionospheric echo. The equivalent surface is not necessarily a simple surface, but may fold back on itself. For example, strong refraction in E-region of F-region echoes could give a fold in the equivalent surface (Fig. 3). All parts of the fold act like a mirror, even though the equivalent ray passes through other parts of the mirror. We must establish the condition for such folding since when it happens, the method of analysis will fail,
Fig. 3. Folding of the equivalent surface in the presence of refraction.
Fig. 4. The rays to the effective reflecting surface will intersect below the surfaces if &I@,-h,) > 6x approximately.
3. CONDITIONS FOR THE FOLDING OF THE EFFECTIVE SURFACE
Suppose refraction occurs at a height h, (naturally to be thought of as the height of the E region), and reflection occurs from a horizontal plane at a phase height h,. From Fig. 4, folding will occur if
II
de>dx
1
- h,-h,’
where 0 is the small refraction angle at a horizontal distance x. If the refraction angle varies sinusoidally with a horizontal wavelength 1 and maximum value BO,this condition becomes 0, > 1/2n(h, - hE).
(2)
1.56
W. R. FROM, ELAINEM.
SADLER and
It is interesting to compare this inequality with the condition for a split echo from the refracting region itself at a lower radio frequency. Suppose the isoionic contours at height h, have a slope 61where ff = tl, cos 2x,X/A.
(3)
Then taking the isoionic contours to be parallel to the effective reflecting surface for a radio echo for this region, a split echo will occur if G!fl> ;1/2&.
(4)
For a model of the refracting region in which it is a medium of constant refractive index p sandwiched between an upper horizontal plane surface, and a lower surface with a slope given by equation (3), the refraction angle is 0 = (1 --~)a approximately. (If, however, the upper surface is parallel to the undulations in the lower surface, 0 would be zero : therefore in practice for E-region we can expect 8 to be appreciably less than CL) Our measurements show that normal E-region is very flat. 8, is typically less than 2” and J_greater than 25 km. Split echoes are not seen, although sometimes inequality (4) must be nearly satisfied. Folding of the effective reflecting surface for F-region reflection must he fSrdirlyrare. On the other hand, cloudy sporadic E, giving rise to partial reflection, often shows echo splitting and the F-region echoes show clearly the effect of refraction by the sporadic E-cloud. (Sometimes this refraction is very obvious and at other times apparently absent: this has important consequences to the structure of sporadic E and will be pursued in a later paper.) The refraction angles 0 may be as large as IO” and scale sizes in the sporadic E-clouds 10 km : therefore from inequality (21, folding of the effective surface is likely. The folding presents considerable potential difficulties in the analysis of data, and the crucial assumption is made that folding does not occur. The data used for analysis has been selected from days when sporadic E was weak and certainly gave no echoes on the operating frequency of the radar.
4. METHOD OF ANALYSIS
The height 2 of the effective reflecting surface is a function of x,_r (the horizontal coordinates) and I (the time}. If z is single-valued for all X, 4’ and t of interest, Fourier analysis allows us to assume that
J. D.
WHITEHEAU
where the wavenumber k has real components k,, ky and imaginary components I,, Z,,,in the x and y directions respectively, and A and k are functions of w. The summation is taken over all w. Tn theory, we could put j, and & equal to zero, since a single attenuated wave over a finite range of x and y may be represented by a collection of unattenuated waves of various real wavenumbers. However, if the waves in the upper atmosphere are attenuated, then we might expect to be able to represent the whole disturbance with fewer parameters by allowing finite values of I, and &. In the event it proved to be of no advantage as will be discussed later. In a finite observing period, the integral is replaced by a summation. A dcu is replaced by A the complex amplitude of one wave and the summation is taken over all the waves. By having the spacing between the angular frequencies w less than 271divided by the total observing period, we effectively allow more than one wave (with different k’s) at nearly the same frequency, as is discussed below. The observing station is at the origin, and the angle of arrival is defined by 0, and Q,,,where 8, is the angle between the ray direction and the y,r plane. It may then be shown that the point of reflection is (R sin 8, cos 8,,., R sin 0,cos 0,) and the observed tilts at a time t are given by the real part of
-k,,,,R sin 0,. cos 8,) exp(&JXsin 0, cos 8,
+ L,,R sin I!?,.cos O.,).
(6)
with a corresponding expression for tanfI,., and the Doppler shift written as the rate of change of range, dRjdt, by the real part of dR dr set 8, see O,,?= c iu,A, exp i(w,f
- k,
R
sin 8, cos e,
- k,,,R sin @j.cosO,,.)exp(i,,R sin #,Ycos O! +I,,,.Rsin#,,cosO,).
(7)
The problem may be stated as follows. For a series of frequencies w covering the appropriate band, we need to find A and k to fit equations (6) and (7). In order to reduce computer time, advantage is taken of the fact that 0, and 8,: are typically less than IO;‘, so that sin O,V2: O.,, cos 0, N 1,
(8)
and that we might expect to find that 1, and & are small enough to allow the approximation
2 = z. -+- Aexpi(ot-_.,x-k,,y).exp(I,x+Z,..~?)do, s (5)
expU,,, RB, + Q@$) 2 1 -I-LSO, -i-&,.R@,.. (9)
157
Measuring ionosp lheric movements Hence, we arrive at these simplified expressions the real part of the right hand side
taking
8, = 1 i(k,, + &.)A,,( 1+ l,,,RQ, + l,,YRe,) xexpi(w,t-k,,R8,-k,,RB,.), with a corresponding
expression
(10)
for Q,, and
dR dt = Ciw,A,(l+l,,RB,+f,,.RO,.) xexpi(w,t-kk,,RB,-k,,YRB,.).
(11)
The presence of the i!, in the first term in brackets for 0, (equation 10) causes the 8, variation for any particular frequency to be out of phase with dR/dt.
5. GENERAL REMARKS ON DISPERSION
0, N c i(k,, +$,,)A,
exp(iw,t)
(12)
One important and vital piece of information is obtained from the Doppler shift data. If all the waves move with identical phase velocities (V,, V,.) (i.e. the whole reflecting surface moved without change of shape), then Pfister’s theorem states that
0,. 1: C i(k,, + &)A,,
exp(iw,t)
(13)
dR -= dt
4.1. Solution ifall k,, k, are small enough If
changing from roughly sinusoidal to a sawtooth variation and finally to multi-valued functions with 0 and dR/dt taking on an odd number of values at any one time. The ratio of the Fourier amplitudes previously used to find k gives only an approximate value and new Fourier components not present in the reflecting surface will appear. The nature of the data redundance becomes more difficult to specify, but the period, velocity and amplitude of a single wave may now be estimated from the angular variation alone, using BRAMLEY’S (1953) method and from the Doppler shift alone, the period, speed and amplitude using CORNELIUS and ESSEX’S(1979) method.
k,RB, and k,.RO,. are always much less than unity,
then
v,o,+V,O,, (15)
dR dt
V,sin0,cos8,+V,sin0,cos8,?i
=
Ciw,A,
exp(iw,t),
(14)
and A, and k, may be found by Fourier inversion. This is the simple method discussed above. Naturally one of the requirements is that for the solution found, k must satisfy the condition that 1kR0 1cc 1 for all 0. Even when this is satisfied, it is found that k is complex, that there is attenuation or amplification. There is just sufficient information (the complex Fonrier components of Q,, O,,.and dR/dt at each frequency o) to find the complex A and the real k,, k,, I,, I, for each n. As an alternative to the attenuated waves, we might consider the possibility of more than one wave at the one frequency. With this simple method of analysis there is not sufficient information to find the two complex amplitudes, the two real k, and k.P(eight real numbers in all). The condition that 1kR0 1cc 1 implies that not all data may be analysed using this method. It also suggests that not all data need be self-consistent, since the k calculated from the data by this method may not satisfy the inequity. For example, artificial data derived from a reflecting surface which folded on itself, could be made to fit an effective reflecting surface which did not. 4.2. Data redundancy with IkRBj > I If (kRB[ - 1 or greater for some of the time, the raw data takes on a markedly different appearance,
and the Doppler shift would be a linear function of 0, and Q,.. The experimental results show variability from day to day as to how well Pfister’s theorem is obeyed. There are days when the linear trend is obvious, though not perfect, and other days when it is difficult to pick by eye. Clearly the waves are dispersive to a greater or lesser extent and methods such as Bramley’s must fail under such circumstances, as will, of course, the measurement of the diffraction pattern velocity (BROWNLIE et al., 1973). The dispersion means that the ripples change in form as they move overhead and this renders invalid the methods previously used to determine the wave parameters from the angular or Doppler shift variations alone. However, one could envisage using only the angular variation by some sort of iterative method to deduce the phase velocity and wave amplitude as a function of frequency. Thus, the Doppler data is largely redundant and we do not expect to obtain perfect fitting between data and theory due to experimental errors and incomplete models. Indeed it will become clear that the difference between data and theory is systematic. It is not due to the disregard of higher frequencies, nor experimental errors. 6. DATA REDUNDANCY
AND THE LIMITS
TO WAVE
ANALYSIS
An important question must now be raised. Given arbitrary Q,, 0,. and dR/dt as functions of time is it
158
W. R. FROM,ELAINEM.
SADLER
always possible to find models which will fit equations (10) and (1 l)? The answer to this question is No! Consider for example the following data set : 0, = 0 (so that we restrict ourselves to two dimensions for simplicity)
(16)
6, = @= Akcosot
(171
dR/dt = Adcoswt,
(18)
and Ak’R >>1 (implying that the distortions are large and ought to produce ‘Z’ traces not present in the data). Or as a second example, the data set : a, = 0
(19)
f3, = 0 = Ak’cos(wt-kR6)
(20)
dR/dt = Aw cos(wt - kR0)
(21)
where Ik’[ K Ik I and Ak’kR >>1. The second inequality implies that multiple echoes will exist, but the first does not allow the angles of arrival to be large enough. It may be shown that for either of these two data sets, it is not possible to construct a reflecting surface model. The proofs are somewhat involved and will not be given here. They are available in an internal report (WHITEHEADand FROM, 1987). Thus, it is possible for data (contrived or real) to be inconsistent with our model represented by equation (5) and provide a test of the model itself. Very few measurements of ionosphere movement have this self-checking feature. Are there ionosphere models which would produce such data, inconsistent with the dispersion model? One obvious one is a single cloud of ionization. If vertical motion as well as horizontal motion were allowed, dR/dt and the B,, 0, angles could be varied in any chosen way. The reason this does not have to be consistent with the dispersion model (equation 5) is that z does not have any value at all beyond the edges of the cloud. Several clouds and models in which one cloud splits into two or more may also be inconsistent with the model. Sporadic E often seems to have this structure and other methods of analysis must then be used. However, whenever z has a single value for all x1 y and t around the radar and for the observing time, z must be expressable in terms of equation (S), and thus under these conditions, the data must be consistent with a dispersion model. The only assumption made in the analysis to be presented is that the effective equivalent surface is not folded and that it exists at all x, )’ and l. In particular the assumption that ]kRf? I CC1 will not be required.
and J. D. WHITEHEAD 7. CHOICE
OF FREQUENCIES
In Fourier analysis of a finite length of data, it is usual to choose the fundamental frequency (with a period equal to the length of data) and the harmonics of this. if only a few specific, but non-ha~oni~ frequencies are present, the amplitudes of these frequencies may be determined by interpolation between the harmonic frequencies. We cannot directly use Fourier analysis (except as an initial approximation) because the terms k,RO, and k,RB, inside the exponentials generate all the harmonics and mixed frequencies. Nevertheless the anatysis assumes that certain wave periods dominate the distortion of the effective reflecting surface. As to the choice of wave periods, if the waves are allowed to be attenuated, the simple analysis using the approximation that I kROl CCI seems to justify only the use of the harmonic wave periods, certainly no greater number of periods. The additional redundancy with split echoes could allow the use of some ad~tional periods and indeed, if the Fourier component of a particular period remains sign&ant after the corresponding best choice has been subtracted from the data, during the required iteration, a second wave of the same period could be introduced by the programme. If attenuation is not permitted, the number of frequencies up to a chosen harmonic could be increased substantially. With lkR@l CC1, there is insu~cient info~ation to double the number of frequencies, but it is clear that the data becomes more redundant (and thus allows more frequencies to be chosen) if IkRO 1is larger. If the data redundance could be properly assessed, a more precise choice could be made. We could have used two or more waves with identical frequencies but this would have increased the technical difficulties with the calculations. Instead wave periods equal to a number (NPF) times the total record length divided by the integers 1, 2, 3, etc. are used. If the actual waves in the ionosphere have specific periods, one of our chosen periods will be very close to it. Also since in the atmosphere there are no vertical planes to reflect horizontally travelling waves, we would not expect to find two travelling waves of exactly the same period. The purpose was to match as closely as possible the mode1 with the data, and so NPF was chosen to minimise the mean square error. Extensive tests were carried out on particular data sets : if the time plots of Doppler shifts and angles appeared nearly sinusoidal, then NPF should be about 2. For more sawtooth time variations, NPF should be larger, and if ‘Z’ traces are evident, NPF should be about 6. Sometimes the obvious presence of short periods and computer size
Measuring ionospheric movements forced a choice of NPF smaller than ideal. In testing models with and without attenuation, care was taken to restrict the total number of frequencies so that the same number of (real) unknown parameters was identical. Ifwe were really able to measure wave attenuation it would be an important parameter for checking against theory and for locating sources of the waves. For example, we expect waves to be attenuated in the direction in which they travel. For either model we may hope to find some pattern in the measured dispersion-the speeds being consistent with expected internal gravity waves and the directions indicating perhaps a small number of sources. Thus there are ways of checking that the answers we find from the analysis are reasonable. The details of the iterative procedure will not be given here but are available (FROM et nl., 1987). A least squares method adapted to the form of the parameters is used. The root mean square remainder after subtracting the 0 and dR/dt for the model is minimised. This remainder falls rapidly initially in the calculation and tends to a finite value: further iterations make little difference. The amplitudes of the higher frequencies tend to fall during the iteration, as the programme correctly takes into account the kRQ terms. Standard programmes exist for finding least square fits to any set of equations : these were tried but after ten times as much computer time had not reduced the remainder as much as the programme above. It is often more efficient to use a special procedure adopted to the particular equations to be fitted.
8. ASSESSMENT OF THE METHOD
One significant advantage of the method is that it is possible to assess its effectiveness in a reasonably objective manner. The other methods of measuring movement might have given an answer, but there is no method of checking the self-consistency of the data model. The first test is how well the model fits the data. The redundance implies that a perfect fit need not be obtained if the data contains experimental errors. For a particular model of wave-like distortions of the effective reflecting surface, it is not practical to work out what the angles of arrival and Doppler shift would be, since the calculation of each ray path at each time involves iteration. Instead what is calculated is the slope of the model ionosphere at theplace where the rejection actually took place (not where it would take place with the model ionosphere), and the cor-
159
responding Doppler shift. This slope and Doppler shift will be called the theoretical slope and Doppler shift. In the calculation, for reasons of computer space (a PDPI l/34), 200 data points (spread over a time of l-2 h) are analysed at one time with up to 74 wave frequencies being used. For waves for which 1kRQ 1cc 1, perfect fitting would be achieved with 200 wave frequencies. However, with real ionospheric data it is obvious that the difference between the theoretical and experimental data contains low frequencies so that the reason for ill-fitting is not due to taking insufficiently high frequencies. The observed and theoretical data shown in Figs. 5 and 6 is typical of the data analysed. Although the models appear to match quite well (as would be expected with a large number of adjustable parameters), it is also obvious from these figures that the error is systematic. The largest errors tend to occur where the observed slope is large and then it is larger than is accounted for theoretically. Other than stressing the point that more than one or two waves of one frequency may be present, it is not obvious what causes this discrepancy. By comparisons of this sort between the two methods of analysis, slightly better fitting to the data occurs with the non-attenuated waves. The root mean square remainder for this model is 95% of that for attenuated waves with identical numbers of adjustable parameters. 8.1. Testing the programme with model data For a particular model of reflecting surface, it is possible to calculate what the observed angles of arrival and Doppler shift should be. In calculating this model data, an iteration procedure must be used to find those rays reflected back to the radar. Random errors equal to the experimental errors are added to the calculated data. The model data is then used with the analysis programme to see how satisfactory it is. The two major results which arise from this test are that the models fit the model data much better than experimental data fits, but that the system of waves which is calculated to fit the model data best is not identical to the system of waves used to generate the data. A large variety of models were used. A measure of fit was the ratio of the root mean square remainder to the spread of the original data. This ratio would then be zero if a perfect fit were found, and unity if the model was a flat horizontal sheet. The ratio was found to be approximately R = 0.03 + 0.07 x log, ,, (number of waves in initial model) to within a factor of two.
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Because the programme which generates the data from the initial model takes longer when a larger number of waves is used, the maximum number used was 80. The models include a few extreme cases of multiple traces, and when these are not counted, the ratio for models with a large number of waves was about 0.08. By comparison, the mean value of R for real data is 0.172. Whereas the fits for real data always include groups of points taken over periods of about 2 min when the data and model do not fit, the same is not true for most of the fitting to the artificial data. Some artificial data was generated with an error in the programme so that the angular positions of horizontal sections of the ionosphere, rather than resection points, was found. This incorrect model data gave errors similar to that observed for real data, that is an increased value of R (to about 0.30) and groups of four or five points in the data which did not fit the derived model. This accident shows not only that arbitrary data cannot necessarily be made to fit any model, but that the real data behaves like incorrectly generated artificial data! Whilst correct artificial data can be fitted quite well to models (thus demonstrating that the computer programmes work), the match between the input model of waves used to generate the data and the output model which is fitted to the artificial data is not so
Fig.
good. This is a result of the equations being ill-conditioned. In situations where the wave frequencies of the input models are separated by at least the reciprocal of the total time interval, the matching is reasonable. Speeds do differ by an average of 20%, but the modelIing programme picks out the dominant waves without too much difficulty. However, when the input model frequencies are closer than this, the matching becomes worse. The reason is that the equations are ill-conditioned: as was noted earlier, for situations where the off-vertical displacement of the reflection point is much less a wavelength of any of the waves, we have insufficient data to determine the parameters of the two waves we know must exist when the corresponding Fourier components in the Doppler shift and angles are out of phase (by other than zero or 180“). Thus in general we might expect to find difficulty determining the wave parameters even though we know there must be many waves since reducing the number in our model makes the fitting worse. We have the same situation with real data. Though we know that a satisfactory model (in terms of fitting the data) requires a large number of waves, the illconditioned equations will not allow us to determine the wave parameters at all accurately. The best we can hope for is to see trends in the models. Because of size limitations on the computer, only
7. A mass plot from 60 h of data (mostly daytime F-region) showing the dispersion
of the waves
Measuring
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164
W.
R. FROM, ELAINEM. SADLERand
200 experimental data sets could be analysed at the one time. If 300 data sets were available, models for the first 200 sets and the last 200 (using a common hundred sets) would be used. Thus several models were often generated for the same run of data. Even more overlap could be used: two models found for data sets of which 190 out of the total 200 points were common would yield significantly different models because the equations are ill-conditioned. The dominant waves have similar velocities, but the background of small amplitude waves which we need to obtain the best fit change considerably. Thus at the cost of considerable computer time, it is possible to generate several wave models from one set of data which may then be used for statistical studies.
9. RESULTS OF THE ANALYSIS The mass plot of speed vs. period (Fig. 7) shows the considerable dispersion required. Some of the spread is due to errors in the analysis, but we know from the angles vs. Doppler shift plots that dispersion must be severe. Analysis of this data shows that the speed is least around the dominant 10-15 min period. These waves have speeds of about 150 m s-‘. However, significant waves are found also at other periods. Longer period waves have slightly greater speeds (180m s- ‘) and short period waves (say - 5 min) have speeds of about 240m s-’ on the average. For some data runs, the short periods were not significant, but a recent run (Fig. 8) shows a dominant 5min period with a horizontal trace speed of 130 m s-‘. Whilst the period of this wave suggests it lies on the sonic branch of the dispersion curves (HINES. 1960), the speed suggests otherwise. It is likely that the waves are in moving air and it is proposed to interpret the data generally as waves in moving air. To find out more about the waves, we need to know also the vertical structure. The effective equivalent surface must be found for two or more radio fre-
J. D.
WHITEHEAD
quencies. Work carried out so far indicates that it is almost as difficult to trace the vertical structure for these waves as it is the horizontal structure (JONES, private communication). Whilst a statistically significant regular phase shift of each wave with height does occur, it is not easy to measure the vertical wavelength with any confidence. But major difficulties remain with determining the behaviour of one effective equivalent surface of explaining why the fitting to the models has systematic errors and how to overcome the difficulty of ill-conditioned equations. In spite of these difficulties, the method is superior to other ways of measuring movement in the F-region. For example, movement of the interference pattern over the ground has been shown to be a poor measure (BROWNLIE et al., 1973) ; methods using only angular measurements or only Doppler shift measurements fail to detect disperion ; methods using widely spaced measurements measure the dispersion, but that the data still do not fit the models shows that we have information which cannot yet be utilised. One important parameter which has not been used in the present analysis is the echo amplitude. There are some technical difficulties, such as amplitude changes due to absorption, but additional information is present and ought to be used.
10. CONCLUSION It has been proved that an effective equivalent surface exists which acting like a mirror would give the same angle of arrival phase path and Doppler shift for all radars operating at a chosen frequency as does the real ionosphere. Using the measurements of angle of arrival and Doppler shift of high frequency radar echoes from the ionosphere, it is possible to deduce that the waves which distort this equivalent reflecting surface are highly dispersive. However, the data does not fit the best models as well as might be expected and this remains a mystery.
REFERENCES
BRAMLEY E. N. BROWNLIE G. D., DRYBURGHL. G. and WHITEHEAD J. D. CORNELIUS D. W. and ESSEX E. A HEISLERL. H. HINES C. 0. MAEDA S. and HANUA S PFISTERW.
1953 1973
Proc. R. Sot. A 220, 39. J. atmos. ten-. Phys. 35,2147
1979 1958 1960 1980 1971
J. atmos. terr. Phys. 41, 48 I. Aust. J. Phys. II, 79. Can. J. Phys. 38,441. J. atmos. terr. Phys. 42, 853. J. armos. terr. Phys. 33, 999.
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is also made to the following unpublished reports.
WHITEHEAD J. D. and FROM W. R.
1987
FROM W. R.. SADLER E. M. and WHI~HEAD J. D.
1987
Data redundancy and limits to the wave analysis. Report No. 2, Department of Physics, University of Queensland, Australia. Iterative procedure in the wave analysis. Report No. 3, Department of Physics, University of Queensland, Australia.