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Astron. Astrophys. 323, 250-258 (1997)

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3. Data analysis

3.1. Choosing the method

To extract the delay between modes turned out to be a challenging task in statistics. Noise and digitization steps produce considerable effects. Several methods have been developed and tested with an artificial time profile, [FORMULA], sampled from a gaussian form, to which noise, [FORMULA], observed at a time of no bursts has been added:

[EQUATION]

[FORMULA] and [FORMULA] are amplitudes and width, respectively, as approximately observed on 90/02/16 (Fig. 2, bottom). R and L are indices for the two modes. The delay [FORMULA] is a free input parameter to be reproduced by the method extracting it from noisy profiles. Eq. (1) assumes that the noise does not depend on the strength of the signal. This is not the case in reality. However, the bursts were generally weaker than the background and the error is small. Artificial profiles as defined in Eq. (1) were used to test the statistical methods and their effects on the result as a function of the delay and the noise level.

All methods to eliminate the noise have failed. In a first attempt, the data was Fourier transformed, suppressed at the frequencies where only noise contributes, and transformed back. The discrete structure of the data makes this method ineffective since the power spectrum of the noise also contains considerable contributions at lower frequencies. The Fourier method did not improve the accuracy of the delay measurement in the test data. In a second attempt the data was fitted with a smooth curve. However, all fitting routines used just spread the noise over a longer interval. Thus the measured delay and its error remained practically unchanged.

The delay of single spikes has been measured by three methods: (i) the cross-correlation between the two modes, (ii) the 'center-of-mass' time of each mode defined by

[EQUATION]

where the flux density [FORMULA] is observed during the interval with the mean time [FORMULA], and (iii) the median method, i.e. the time at which the area under the time profile splits into equal parts.

All three methods are severely limited by the noise. The center-of-mass method and the median method are able to reliably detect delays of one tenth of the original time resolution. The center-of-mass method weights the wing of an event more than the center. The cross-correlation method is slightly less sensitive, but depends less on the shape of the burst than the other two methods. When applied to real data, the scatter of the delays in single spikes was smallest for the cross-correlation.

The time delay between left and right polarization of 45 artificial, single bursts with different noise has been measured by cross-correlation and is displayed in Fig. 3. The time delay, [FORMULA], in the model (cf. Eq. 1) has been put to zero. The scatter of the measured delays is the result of the noise. The distribution has a standard deviation of 28.2 µs. The mean value of the measured delay is -8.4 µs with a mean error of [FORMULA] s. This test proves that the cross-correlation method can determine the delay with an accuracy of a fraction of the original resolution (in this case 500 µs).

[FIGURE] Fig. 3. The delay of 45 model spikes was measured in artificial, noisy model spikes (cf. Eq. 1) having an input delay of [FORMULA]. The full histogram shows the number of cases per 25 µs bin. The dashed histogram is the distribution neglecting the sign of the delay.

Even better results were achieved when a whole sequence of spikes was cross-correlated together. The sensitivity of this method to detect small delays in artificial data was clearly superior to the other methods. In particular, the deviation from the input delay was significantly lower than averaging the delays of single spikes extracted by any of the three methods.

3.2. The cross-correlation function

The cross-correlation coefficient is defined by

[EQUATION]

The summation in the numerator is over all pairs possible for the lag l ; n is the total number of measurements in the time profile. [FORMULA] and [FORMULA] are the observed values, assumed to be superpositions of the intrinsic spike emissions, L and R, and noise, [FORMULA] and [FORMULA], respectively:

[EQUATION]

Let the standard deviations of the noise be [FORMULA] and [FORMULA]. They are assumed to be unrelated in the following derivations. Tests have shown that the covariance of the noise in the left and right modes is zero except at zero lag. This is a known instrumental effect and will later be accounted for. Neglecting non-zero covariance and using the gaussian law of error propagation, the standard deviation of the cross-correlation is

[EQUATION]

Since the time delay is determined from the central points of the cross-correlation function and the background has been subtracted, the following approximation for small lags can be used:

[EQUATION]

Assuming [FORMULA], consistent with the observed background data and the weakness of the spikes, Eq. (6) can be put into the form

[EQUATION]

The accuracy of the cross-correlation increases linearly with the flux density of the spikes and with the square root of the number of data points, n. Eq. (9) demonstrates how the signal-to-noise ratio is improved by cross-correlating a long sequence of data.

3.3. Measuring the time delay

If the noise is small enough, the resolution of the delay measurement from cross-correlations can be considerably better than the original time step of the data. For this improvement the cross-correlation function is interpolated by a polynomial of third order. A spline interpolation was used taking into account the known standard deviation [FORMULA]. The interpolated function therefore does not exactly follow the values of the cross-correlation function, but allows for a mean deviation of [FORMULA].

The time delay [FORMULA] between the modes is the lag of the maximum value of the interpolated correlation function [FORMULA]. The time delay is defined by the zero point of the derivative, [FORMULA]. The error in the measurement of [FORMULA] is calculated from the Taylor expansion of the function [FORMULA] (e. g. Chatfield 1989),

[EQUATION]

With [FORMULA] sufficiently close to [FORMULA], the derivative can be approximated to the first degree in [FORMULA]. At [FORMULA] the first derivative vanishes, and from from Eq. (10) follows

[EQUATION]

Gaussian error propagation in Eq. (11) yields for the standard deviation in [FORMULA]

[EQUATION]

Since [FORMULA] is a chosen point of reference, [FORMULA]. [FORMULA] near [FORMULA] is of the order of [FORMULA] and can therefore be neglected. Thus only the second term in Eq. (12) is relevant, and

[EQUATION]

The derivative can be approximated by

[EQUATION]

where [FORMULA] is the time resolution of the interpolated cross-correlation function [FORMULA]. Thus

[EQUATION]

Inserting Eq. (15) into Eq. (13) and putting [FORMULA], the standard deviation of the delay measurement finally becomes

[EQUATION]

The value of [FORMULA] has been calculated from test data and found to be constant over a large interval around [FORMULA].

3.4. Other sources of error

The error given in Eq. (16) is only the statistical influence of noise. Two more sources of error must be considered.

Most serious are instrumental effects introduced by the spectrometer. Fortunately, data of two instruments could be used with completely different high-frequency equipment (cf. Sect. 2). As a test of the instruments, the background noise in the two modes was cross-correlated. Fig.  4 (left) shows an example of the results. There is a prominent peak at zero lag, indicating that the noise in right and left circular polarization is not entirely independent. The reason is that the feed is linearly polarized. The two linear polarizations are converted into right and left circular polarizations by [FORMULA] phase changes in a hybrid. All the noise added to the linearly polarized signal before the hybrid is therefore correlated in circular polarization. The effect has only been detected in the 1990 data and is possibly due to one of the additional switches in the signal path of the Phoenix spectrometer. Fig. 4 (right) shows the effect on the data. It enhances the correlation at zero lag, but is completely absent at lag + [FORMULA] and [FORMULA], where [FORMULA] is the original time resolution. The effect was found to be variable in time. The only way to avoid it, was to delete the correlation of zero lag in the interpolation.

[FIGURE] Fig. 4a and b. The cross-correlation coefficient vs. time lag in units of 500µs, Left: from background data on 90/02/15, Right: from a single spike on 90/02/15.

The second systematic effect we have encountered originates if the background level changes with time. A trend in the background moves the peak time into the direction of the higher background. If the trends in the right and left mode are the same, the effect on the time delay between modes cancels. If not, a fictitious delay is introduced. To avoid the effect, the background in the interval was carefully examined before cross-correlation and, if necessary, a minimum envelope was subtracted. Times of high spike rate have been omitted, so that the background could be determined reliably. It is reasonable to assume that the remaining effect cancels in the average over a large enough sample.

3.5. Investigation and results

Table 2 contains the relevant results for the four observations. First, the details of the investigation are summarized for each event.


[TABLE]

Table 2. Measurements of the time delay between modes. A positive sign indicates a delay of the left circular mode of polarization relative to the right mode. The coordinates of the associated H [FORMULA] flares are taken from Solar-Geophysical Data Comprehensive Reports. The second line of 1990/02/16 (b) includes only the 9 strongest spikes.


82/06/04: The four recorded frequencies are close enough to be integrated into one channel. This effectively reduces the time resolution to 2 ms. A 27s time interval has been selected and a minimum envelope subtracted to avoid the effect of background fluctuations.

90/01/20: The event has a high rate of spikes. A total of 35 single frequency recordings of 250 ms each have been selected, which contained one full spike each. The background was subtracted individually in each interval. Out of the 35 spikes, 16 were at 1.1 GHz, 9 at 1.4 GHz, and 5 for both 1.7 and 2.0 GHz with an average total duration of 24, 20, 14, and 17 ms, respectively. The 35 intervals were merged into one time series and cross-correlated together.

90/02/15: Same as in the 90/01/20 event. A total of 21 spikes, all at 1.1 GHz, were cross-correlated. Their average duration was 56 ms.

90/02/16: Same as in the 90/01/20 event. A total 49 intervals were merged. The average spike duration at 1.1 GHz was 64 ms. For a second cross-correlation, only the 9 strongest spikes were selected and analyzed separately (average duration 69 ms). The selection increased the accuracy, but the resulting delay between modes was statistically not different from the larger sample.

The method described in Sect. 3.4 yields an average of the delays weighted by the spike flux. The measured delays (Table 2) are statistically significant except for 1990/02/15. Note that the delays cannot be readily compared to the test data in Fig. 3, where a particular model for the spike emission (equal amplitude) has been used.

The error given in Table 2 has been determined from Eq. (16). It is the statistical error of the average value caused by the noise. We expect that the accuracy is reduced considerably by systematic errors in the first two events and that the effective error is more than the value given in Table 2. In the last two events the polarization is high and the noise of the weaker mode is more influential.

The scatter in the intrinsic delay of individual spikes cannot be measured accurately. Nevertheless, there is no indication from the analysis of single spikes that the delays of an event have different signs. In all four cases the weaker mode of circular polarization is delayed. Assuming that the longitudinal component of the magnetic field at the origin of the polarization and in the decisive part of propagation has the same sign, the spikes have been emitted in ordinary mode.

3.6. Polarization

Furthermore, the average degree of polarization can be readily determined from the cross-correlation function at zero lag,

[EQUATION]

The polarization is expressed as

[EQUATION]

where a is the average ratio of left to right circular polarization, [FORMULA]. If the noise is sufficiently small, Eq. (17) becomes

[EQUATION]

Thus,

[EQUATION]

Inserting Eq. (20) into Eq. (18), the degree of circular polarization then is given.

Since the noise is correlated at zero lag, the cross-correlation function, [FORMULA], is interpolated from the adjacent values at [FORMULA] and [FORMULA]. The statistical noise on the result is extremely small and is negligible compared to the systematic calibration errors. Thus the degree of polarization given in Table 2 is rounded off to five percent accuracy.

It is interesting to compare the sense of the observed circular polarization with the hemisphere where the associated H [FORMULA] flare was seen. In 1990 (sunspot cycle 22) the two left circularly polarized events occurred in the Northern hemisphere and the marginally right circularly polarized event originated from a flare in the Southern hemisphere. The right polarized event of 1982 - in the previous cycle, when the global polarity of the Sun was opposite - occurred in the Southern hemisphere.

The magnetic polarity of the leading spot of the active region was determined from magnetograms published in Solar and Geophysical Data. It was found to be consistent with the general trend for positive polarity in the Northern hemisphere during cycle 21, and negative polarity during cycle 22. Assuming that the magnetic polarity of the emitting region is given by the leading spot and no polarization reversals occur during propagation, the observed polarization would indicate that the decimetric spikes of the 1982/06/04 event originated as predominantly ordinary mode, the other three events (one of them marginal) as extraordinary mode.

The trend for extraordinary mode as determined by the leading spot hypothesis is consistent with the findings of Güdel & Zlobec (1991). However, it contradicts the observed delays, when interpreted as a dispersion effect.

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© European Southern Observatory (ESO) 1997

Online publication: June 5, 1998

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