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Analysis of artificial fading records
Journal of Atmospheric and Terrestrial Physics, 1968,Vol.30, pp. 975-977. PergamonPress. Printed in Northern Ireland
Analysis
of a r t i f i c i a l f a d i n g
records
D. Jo~Es* and A. D. 1W_AUDE University College of Wales, Aberystwyth (Received 11 November 1967) A b s t r a c t - - B y the analysis of artificially constructed fading curves it is shown that the velocities obtained in Mitra type drift experiments will correspond to the velocities of the higher frequencies present if optical correlation is used in the analysis, and to a velocity averaged over all frequencies if numerical correlation is used. The full result for all frequencies is obtained if a spectral analysis is used.
THERE are several examples in geophysics where two records of signal strength which were recorded at two different places have to be compared with each other and an estimate to be made of the delay of one over the other which will cause them to be most similar. Two examples where this technique is used are in Mitra type drift velocity measurements of the ionosphere, and radio star scintillation measurements on the solar wind. I n these cases each record is a random variation, and while the records of a pair are similar they are not identical. Three methods are commonly used to determine the 'best' value for the delay. Sometimes similar features on each record are compared and the delay for each is estimated, an average being taken over a large number of such comparisons. This will be called the method of similar fades. Alternatively the two records are superimposed, and are then displaced relative to one another until it is judged t h a t the best fit has been obtained. This will be called the method of 'optical correlation'. I f the records are available in a digital form the correlation between records m a y be calculated for various values of delay, and the delay for which this is a m a x i m u m m a y be chosen. This is the numerical correlation method. A fourth method was used by JONES and MAUDE (1965), in which the records were analysed into their various frequency components, and the phase shift, and hence time shift, was calculated for each component. This method will be called the method of 'spectral analysis'. I n the more general case three records of signal strength are made, so t h a t a n y of the four methods m a y be employed to give both the magnitude and the direction of the movement. On some occasions it m a y be suspected from other considerations t h a t the pattern observed is not a randomly changing pattern moving as a whole across the observing sites, but is the result of a train of waves with a continuous distribution of wavelengths. I f waves of different wavelength move with different velocities dispersion is said to take place, and it is the purpose of the present note to indicate what the effect of such dispersion would be on the two correlation methods of estimating the delay. * Now at the Cavendish Laboratory, Cambridge. 975
976
D. JONES and A. D. M_AWD~.
I t was hypothesised that the eye would unconsciously pick out the sharp well defined features for comparison so t h a t the optical correlation would give a delay due to the highest frequency present, while the numerical correlation would give a more equal weight to all the frequencies present. The authors met this problem in connection with Mitra type ionospheric drift experiments, so that the artificial results described below were designed to simulate Table 1 Frequency (c/see)
Velocity (m/see)
Direction (° E a s t of North)
0"0534 0"1335 0.2136 0.3204
10-0 23"0 36"5 54.0
225 225 225 225
this t y p e of measurement, b u t the conclusion is valid for other experiments as well. In order to simulate the two dimensional situation met in ionospheric drift, three fading curves were synthesised which together constituted an artificial fading record such as might be obtained if the strength of the radio signal reflected from the ionosphere were recorded at three positions on the ground. The hypothetical receivers were at the corners of an isosceles right angled triangle of hypotenuse 200 m, with sides pointing North-South, E a s t - W e s t and North W e s t - S o u t h East. Each curve consisted of four sine functions and a random, 'noise' variable. 4
= 0 +
An sin (2 /nt + n=l
0 was a random number uniformly distributed in the range 0-1, the values of A represented the amplitudes of the hypothetical waves of frequency f. The phases ¢ were different for each curve of a record and were chosen so that they represented delays between the curves, which in turn represented velocities for the hypothetical waves. An IBM 1620 electronic computer was used to produce 300 values of _~ at intervals of t of 0.625 sec (an interval chosen to be the same as the sampling interval of drift recordings taken at this laboratory). These points were plotted and joined b y a smooth curve, the resulting curves having much the same appearance as genuine fading curves. These records were then analysed b y optical and numerical correlation, as well as the spectral analysis methods. In the first example the four amplitudes were chosen to be equal, the directions were chosen to be all 225 ° East of North, and the velocities and frequencies were chosen as in Table 1. The results are shown in Table 2. I t will be seen that, as hypothesised, the velocity obtained b y optical correlation approaches that of the highest frequency present, b u t t h a t obtained b y the numerical correlation method is less. The directions obtained b y both methods are correct. The spectral analysis method gave the full picture, b u t the exact correspondence of frequencies is due to the choice of the original frequencies and not to an extreme accuracy of the method.
Analysis of arLificial fading records
977
Table 2
Optical correlation Numerical correlation Spectral analysis
Frequency (c/s)
Velocity (m/s)
Direction (° East of North)
--0.0534 0.1335 0.2136 0.3204
49 40 10.0 23.0 36.7 54.7
223 224 224-73 224.53 224.62 224.60
Table 3
Optical correlation Numerical correlation
Velocity 0n/s)
Direction (° East of North)
47 31.5
224 224
I n a second e x a m p l e the a m p l i t u d e of the highest f r e q u e n c y was reduced b y half. This p r o d u c e d T a b l e 3 where it will be seen t h a t a l t h o u g h the velocity b y optical correlation was still n e a r t h a t of the highest frequency, the numerical correlation m e t h o d showed a r e d u c e d velocity. T h e result of the spectral analysis was unchanged. I t is concluded t h a t w h e n dispersion is present, one of the signs of it will be a difference in v e l o c i t y as m e a s u r e d b y optical a n d numerical correlation methods, a n d t h a t t h e complete picture can t h e n be o b t a i n e d b y m~ans of spectral analysis.
JO~-ES D. and MA~TD~. A . D .
21
I:~EFERENCE 1965 Nature, Lond. 206, 177.
Analysis
of a r t i f i c i a l f a d i n g
records
D. Jo~Es* and A. D. 1W_AUDE University College of Wales, Aberystwyth (Received 11 November 1967) A b s t r a c t - - B y the analysis of artificially constructed fading curves it is shown that the velocities obtained in Mitra type drift experiments will correspond to the velocities of the higher frequencies present if optical correlation is used in the analysis, and to a velocity averaged over all frequencies if numerical correlation is used. The full result for all frequencies is obtained if a spectral analysis is used.
THERE are several examples in geophysics where two records of signal strength which were recorded at two different places have to be compared with each other and an estimate to be made of the delay of one over the other which will cause them to be most similar. Two examples where this technique is used are in Mitra type drift velocity measurements of the ionosphere, and radio star scintillation measurements on the solar wind. I n these cases each record is a random variation, and while the records of a pair are similar they are not identical. Three methods are commonly used to determine the 'best' value for the delay. Sometimes similar features on each record are compared and the delay for each is estimated, an average being taken over a large number of such comparisons. This will be called the method of similar fades. Alternatively the two records are superimposed, and are then displaced relative to one another until it is judged t h a t the best fit has been obtained. This will be called the method of 'optical correlation'. I f the records are available in a digital form the correlation between records m a y be calculated for various values of delay, and the delay for which this is a m a x i m u m m a y be chosen. This is the numerical correlation method. A fourth method was used by JONES and MAUDE (1965), in which the records were analysed into their various frequency components, and the phase shift, and hence time shift, was calculated for each component. This method will be called the method of 'spectral analysis'. I n the more general case three records of signal strength are made, so t h a t a n y of the four methods m a y be employed to give both the magnitude and the direction of the movement. On some occasions it m a y be suspected from other considerations t h a t the pattern observed is not a randomly changing pattern moving as a whole across the observing sites, but is the result of a train of waves with a continuous distribution of wavelengths. I f waves of different wavelength move with different velocities dispersion is said to take place, and it is the purpose of the present note to indicate what the effect of such dispersion would be on the two correlation methods of estimating the delay. * Now at the Cavendish Laboratory, Cambridge. 975
976
D. JONES and A. D. M_AWD~.
I t was hypothesised that the eye would unconsciously pick out the sharp well defined features for comparison so t h a t the optical correlation would give a delay due to the highest frequency present, while the numerical correlation would give a more equal weight to all the frequencies present. The authors met this problem in connection with Mitra type ionospheric drift experiments, so that the artificial results described below were designed to simulate Table 1 Frequency (c/see)
Velocity (m/see)
Direction (° E a s t of North)
0"0534 0"1335 0.2136 0.3204
10-0 23"0 36"5 54.0
225 225 225 225
this t y p e of measurement, b u t the conclusion is valid for other experiments as well. In order to simulate the two dimensional situation met in ionospheric drift, three fading curves were synthesised which together constituted an artificial fading record such as might be obtained if the strength of the radio signal reflected from the ionosphere were recorded at three positions on the ground. The hypothetical receivers were at the corners of an isosceles right angled triangle of hypotenuse 200 m, with sides pointing North-South, E a s t - W e s t and North W e s t - S o u t h East. Each curve consisted of four sine functions and a random, 'noise' variable. 4
= 0 +
An sin (2 /nt + n=l
0 was a random number uniformly distributed in the range 0-1, the values of A represented the amplitudes of the hypothetical waves of frequency f. The phases ¢ were different for each curve of a record and were chosen so that they represented delays between the curves, which in turn represented velocities for the hypothetical waves. An IBM 1620 electronic computer was used to produce 300 values of _~ at intervals of t of 0.625 sec (an interval chosen to be the same as the sampling interval of drift recordings taken at this laboratory). These points were plotted and joined b y a smooth curve, the resulting curves having much the same appearance as genuine fading curves. These records were then analysed b y optical and numerical correlation, as well as the spectral analysis methods. In the first example the four amplitudes were chosen to be equal, the directions were chosen to be all 225 ° East of North, and the velocities and frequencies were chosen as in Table 1. The results are shown in Table 2. I t will be seen that, as hypothesised, the velocity obtained b y optical correlation approaches that of the highest frequency present, b u t t h a t obtained b y the numerical correlation method is less. The directions obtained b y both methods are correct. The spectral analysis method gave the full picture, b u t the exact correspondence of frequencies is due to the choice of the original frequencies and not to an extreme accuracy of the method.
Analysis of arLificial fading records
977
Table 2
Optical correlation Numerical correlation Spectral analysis
Frequency (c/s)
Velocity (m/s)
Direction (° East of North)
--0.0534 0.1335 0.2136 0.3204
49 40 10.0 23.0 36.7 54.7
223 224 224-73 224.53 224.62 224.60
Table 3
Optical correlation Numerical correlation
Velocity 0n/s)
Direction (° East of North)
47 31.5
224 224
I n a second e x a m p l e the a m p l i t u d e of the highest f r e q u e n c y was reduced b y half. This p r o d u c e d T a b l e 3 where it will be seen t h a t a l t h o u g h the velocity b y optical correlation was still n e a r t h a t of the highest frequency, the numerical correlation m e t h o d showed a r e d u c e d velocity. T h e result of the spectral analysis was unchanged. I t is concluded t h a t w h e n dispersion is present, one of the signs of it will be a difference in v e l o c i t y as m e a s u r e d b y optical a n d numerical correlation methods, a n d t h a t t h e complete picture can t h e n be o b t a i n e d b y m~ans of spectral analysis.
JO~-ES D. and MA~TD~. A . D .
21
I:~EFERENCE 1965 Nature, Lond. 206, 177.