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Estimation of Propagation Parameters of Non-uniform Ionospheric Disturbances From HF Phase Measurements
Weixing Wan and Jun Li
Wuhan Ionospheric Observatory
Wuhan Institute of Physics
the Chinese Academy of Sciences
Wuhan 430071, P.R. China
This paper develops a new technique used to estimate the phase velocities of non-uniform ionospheric gravity waves from single station observation of both HF Doppler frequency shifts and angles of arrival. Based on the time-frequency analysis, we first derive the observation equations which reveal the relationship between the observables and the propagation parameters of the ionospheric disturbances. Then, the detailed calculation procedure of the new method is proposed to separate the individual gravity waves from the non-uniform disturbances, and to estimate the phase velocities and azimuth angles of horizontal propagation of each separated gravity wave packets. As an example, the new method is used to the data analysis of Digisonde drift measurements. The results show that the suggested technique is useful in the study of the propagation, evolution and dispersion of the causative gravity waves.
The ground-based phase measurements of ionospheric HF echoes, known as the observation of the Doppler frequency shifts and the angles of arrival, have been widely used in the experimental study of large scale ionospheric disturbances such as gravity waves. It is believed that the simultaneous observation of both Doppler shifts and arrival angles, even at a single station, can be used to determine the horizontal propagation parameters of ionospheric gravity waves. Accordingly, in the data analysis, two kinds of methods have been proposed. The first may be called the time domain method[1,2]. In this analysis the measured Doppler shift and two arrival angles at any certain time t are directly used to estimate the horizontal phase velocity, Vph(t), and the azimuth angle, s (t), of the horizontal propagation of the gravity waves. The second method is the frequency domain method. In this analysis the time sequences of the observables are transformed into their frequency spectra, then for certain frequency the spectral components are used to estimate the phase velocity, Vph(W ), and s (W ) as functions of frequency W .
It is easy to confirm that the above two methods are equivalent when there exists only one gravity wave, which is called the uniform ionospheric disturbance. While in the case of non-uniform ionospheric disturbances, the results of the two methods may be quite different. We assume that the non-uniform ionospheric disturbances consist of several individual gravity waves, so the time domain method is suitable for the case that the individual gravity waves are not overlapped each other in time domain. On the other hand, the frequency domain method is useable if the gravity waves are not overlapped each other in frequency domain. While the gravity waves contained in the non-uniform disturbances are overlapped in both time and frequency domain, as is the real case in most time, the above two methods may both lead to mistakes.
The purpose of the present work is to develop a new technique to separate the individual gravity waves from non-uniform ionospheric disturbances, and to estimate the phase velocities and azimuth angles of horizontal propagation of each separated waves. Unlike both the time domain method and the frequency domain method, the present method is really a time-frequency method based on the theory of time-frequency analysis of Signal Processing. In the following part of this paper, we first derive the observation equations which connect the observables and the propagation parameters of non-uniform ionospheric disturbances. Then we give the detailed analysis procedure, as well as the calculated results of the new method when applied in the data analysis of the Digisonde drift measurement. Finally a brief summary for this work is given.
We first introduce the normal velocity V, of the radio reflection surface (an iso-electron density surface in the ionosphere). The amplitude, V, of V, and the horizontal components, nx and ny, of the unit vector along V are defined and connected with the HF Doppler frequency shifts, d w , and the angles of arrival (indicated by the deviation, d kx and d ky, between the wave number vectors of the received and transmitted radio waves) as
where N is the electron density on the reflection surface, w is the radio frequency.
Here it is assumed that the reflection point does not itself move along the reflection surface. This requires this distance moved to be much less than the smaller horizontal structure present. The problem of analysis when this assumption is not valid has been dealt with by From et al (1988). Unfortunately the requirement for this assumption means that the ionosphere has to be rather quiet. It can be recognised by the saw-tooth appearance of V, as a function of time, and is clearly evident in figure 1. Then only the dominant features of the ionospheric disturbances can be deduced by this method.
We assume that, (1) the non-uniform ionospheric disturbances are composed of many individual gravity wave packets; (2) the time-frequency spectra of each individual gravity wave packets are not overlapped each other in the time-frequency plane (t,_), that is, each wave packet occupies a certain sub-region. These assumptions make possible for us to adopt the following model to express the time-dependent spectrum, N(_,Kx,Ky,t), of the electron density fluctuation,
where the time-frequency functions Kx(W ,t) and Ky(W ,t) indicate the time-dependent dispersion relations, and N(W ,t) is the time-dependent frequency spectrum of electron density in the observation point which is assumed to be the original point of our coordinate system. It is obvious that the model of Eq.(2) satisfy the definition of N(W ,t) as,
We assume that the ionospheric disturbances are so weak that the departures of the reflecting point from overhead of the station is negligible. Then, by using equations.(1), (2) and ignoring a factor 1/1/2 Ñ N| which is approximately a constant, N(W ,t) can be used to express the time-dependent frequency spectra of V(t), nx(t) and ny(t),
or equivalently, the time-dependent auto spectrum, F NN(W ,t)=N(W ,t)N*(W ,t), can be used to express the time-dependent auto and cross spectra of V(t), nx(t) and ny(t),
In the above equations (Equations.(4) and (5)), the terms on the left hand side, i.e, the time-dependent frequency spectra, V(W ), nx(W ) and ny(W ), and the auto and cross spectra, F vv(W ,t), F xx(W ,tF yy(W ,t), F vx(W ,t), F vy(W ,t) and F xy(W ,t), may be experimentally obtained by the time frequency analysis (such as wavelet analysis and the short time Fourier analysis). On the right hand side, the time-frequency functions, Kx(W ,t), Ky(W ,t), F NN(W ,t) and N(W ,t) are all the parameters to be found. So these equations are called the observation equations.
In the present work the two-dimensional distribution of the auto spectrum F NN(W ,t), or the frequency spectrum N(W ,t), is used mainly to separate the individual gravity waves from the non-uniform ionospheric disturbances. To do so, we divide the whole time-frequency plane into several sub-regions so that each sub-region contains only a single peak of F NN(W ,t). We suppose that any peak of F NN(W ,t) is corresponding to only one individual gravity wave packet. Therefore any of the sub-regions is mainly occupied by one individual wave packet, this implies that the individual gravity waves are separated in (W ,t) plane from the non-uniform ionospheric disturbances.
Furthermore, limited to any certain sub-region mentioned above, Kx(W ,t) and Ky(W ,t) indicate the time-dependent dispersion of a individual gravity wave packet, and can be used to determine the phase velocity Vph(W ,t) and azimuth s (W ,t) of the wave packet as functions of both frequency and time.
For a certain individual gravity wave packet, the determined Kx(W ,t) and Ky(W ,t), or Vph(W ,t) and s (W ,t), can be directly used to study both the frequency dispersion and temporal evolution of the wave packet. And, if necessary, we can obtain uniform phase
velocity and azimuth angle of the wave packet from the mean values of W , Kx(W ,t) and Ky(W ,t) averaged in the sub-region with F NN (W ,t) as the power. In this paper, as an example, we take the average of W , Kx(W ,t) and Ky(W ,t) in only the frequency domain to yield the temporal variation, W (t), Kx(t) and Ky(t),
The integrations are carried in the frequency region contains only one spectral peak. Then similar to Eq.(6), the quantities _(t), Kx(t) and Ky(t), are used to determine the phase velocity Vph(t) and azimuth _(t) as functions of time,
In order to exhibit the usage of the new method, we treated the Digisonde drift data observed at Millstone Hill Station. The procedure used and the results obtained are as follows.
First, we draw the Doppler frequency shifts and angles of arrival from the row data of drift measurement by the regression method, and then calculate the time sequences V(t), nx(t) and ny(t) by using equation.(1). Fig.1 is a segment of the curves of V(t), nx(t) and ny(t) obtained from the Digisonde drift measurements. It can be seen from this figure that the fluctuations of the curves are not very sinusoidal as in the case of a single wave, so the disturbances may be non-uniform and consist of several individual gravity waves.
Second, the time sequences V(t), nx(t) and ny(t) are used to estimate the time-dependent auto and cross spectra, F vv(W ,t), F xx(W ,t), F yy(W ,t), F vx(W ,t), F vy(W ,t) and F xy(W ,t), by the dynamic spectrum analysis (a kind of time-frequency analysis developed from the short time Fourier transformation). In the practice calculation, these spectra are estimated by using the multi-channel maximum entropy method in a 2-hour time window advanced in 6-minute time steps.
Then by using the observation equations (Equation.(5)) we calculate the spectrum F NN(W ,t) and the dispersion relations Kx(W ,t) and Ky(W ,t) from the auto and cross spectra estimated in the previous procedure. The contours of F NN(W ,t) corresponding to the data of Fig.1 is shown in Fig.2. Then sub-regions corresponding to individual gravity wave packets are obtained on the (W ,t) plane according to the peak distribution of F NN(W ,t). In these sub-regions Vph(W ,t) and s (W ,t) are calculated by using Equation.(6). The boundary of each sub-regions and the vector phase velocity (formed by both Vph(W ,t) and s (W ,t)) are also overlaid in Fig.2. From this illustration it is clear that the estimated phase velocities and azimuth angles are quite similar in any certain sub-region, while those in different sub-regions may be much different. Again this demonstrates that the sub-regions are mainly occupied by only one individual wave packet.
Finally, by using Equations.(6) and (7), we take the frequency average, W (t), Kx(t) and Ky(t), and then find the phase velocity Vph(t) and azimuth s (t) of each individual gravity waves. As shown in Fig.3, both the phase velocity Vph(t) and azimuth W (t), as well as the average frequency s (t), are all continuously varied in the time domain. This makes possible to study in detail the temporal evolution of the individual gravity wave packets.
This paper proposes a new technique to separate individual gravity wave packets from non-uniform ionospheric disturbances, and estimate the phase velocities and the azimuth angles of the horizontal propagation of the separated wave, from single station observation of both HF Doppler frequency shifts and angles of arrival. As the key technique and the theoretical base of the new method, the time-frequency analysis makes it possible to separate each individual gravity waves in the time-frequency plane, and to estimate precisely the propagation parameters of the separated gravity waves. As an example, the new method is used to the data analysis of Digisonde drift measurements. The experimental results shows that the suggested technique is useful in the experiment study of both the propagation and the evolution of ionospheric gravity wave disturbances.
The data used in this work are provided by Center for Atmospheric Research, University of Massachusetts, Lowell. The authors wish to thank Dr. B. W. Reinisch for his useful suggestion..
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 W. Wan, J. Li, Z. M. Zhang and B. W. Renisch, "Study of ionospheric gravity wave disturbances from drift measurement of a digisonde", Chinese J. Geophys., 36, (in press), 1993.