2 Method of merging

2.1 Timing

One of the major difficulties in merging various data sets is the need of a common time reference. But even before this step, it appears that each individual long term experiment is experiencing its own timing difficulties. Of course, all these timing difficulties must first be solved before any kind of merging can be attempted, and that is a complicated task. Indeed, the various instrumental clocks are subject to various random and generally not understood jumps, and also the daily starting time is sometimes random itself. The Mark-1 time series is the "oldest'' one, and the relatively long experience acquired by the local management of this instrument makes its timing generally reliable. The Mark-1 timing has then been used as a reference to cross-check the IRIS data sets, until 1996. After this date, GOLF (Global Oscillations of Low Frequencies) (Gabriel et al. 1995) on board of the satellite SoHO (Solar and Heliospheric Observatory) was used as a reference for all ground based data sets. To calculate the timing errors and synchronizing various data sets, the overlapping parts of daily time series were cross-correlated with the reference time series. A fit of the central part of the main peak in the cross correlation gives the time lag. The residual uncertainty is always smaller than 7 s. Whenever no reference data is overlapping with the IRIS data to be checked, an already checked IRIS day is used as a weaker reference to continue the process. After 1996, the use of the GOLF time series as a reference makes it possible to generalize this procedure to all the ground based data sets with an improved accuracy of $\pm$2 s. In any case, all ground based instruments can now easily be equiped with a GPS receiver, so that the timing problem is no longer a problem.
Just to mention an anecdote, the LOWL data set has shown a systematic shift of 12 hours with respect to others, because being not a member of a network, the data set was simply provided in local time. After understanding this point, its synchronization has been made, before 1996, with the IRIS sites of Stanford (California) and La Silla (Chile), that are the only two sites of the network with a significant overlap with Hawaii.

Figure 2: Average daily power spectra of the 3 instruments (IRIS, Mark-1, LOWL) before cross calibration sodium/potassium.

2.2 Cross calibration of sodium and potassium

The different spectral lines observed by the sodium (IRIS, Na I 5896) and the potassium (Mark-1 and LOWL, K 7699) instruments imply that they probe different altitudes in the solar atmosphere. For a given p-mode, the sodium and potassium amplitudes will then be different mostly because of the strong gradient of density with altitude in the solar atmosphere. Moreover, this difference is frequency dependent, as the higher frequencies are less efficiently trapped inside the acoustic cavity. Before merging sodium and potassium data, it is then necessary to cross calibrate the relative p-mode amplitude sensitivities, as a function of frequency.

Figure 3: Calibration function sodium/potassium computed with IRIS and LOWL power spectra ratio.

Figure 2 shows average daily power spectra of the 3 instruments, computed over the same period of 3 years (1994 to 1996). Several peculiarities of this figure require some comments: the high frequency parts of IRIS and Mark-1 display the flat level of the photon statistics noise, while this is not true on LOWL, because of a different raw data sampling procedure, more consistent with the Shannon frequency. The photon noise level appears to be higher on the IRIS power spectra than it is on others. That is due to a significant darkening of the sodium cells used during these years. The higher continuous level of the IRIS power spectrum in the lower frequency range (1 to 2 mHz) is also due to this excess of photon noise. All data sets have been low frequency filtered to avoid the presence of unwanted steps after the merging. This is described in a later section. 3 sharps peaks near 4.6, 7.4 and 9.2 mHz are visible in the LOWL power spectrum. They are due to guiding periodicities that have not been successfully eliminated. The highest two are without consequence, while the 4.6 mHz one implies that any study of the highest part of the p-mode frequency range will need to avoid the use of the LOWL data.

The two ratios IRIS/Mark-1 and IRIS/LOWL have been computed from these power spectra. These ratios have been then fitted by a third order polynomial (see Fig. 3) in the range of frequencies extending from 1.1 to 6 mHz.

These polynomials are taken as the sodium/potassium calibration functions:

$$\frac{{\it DSE}({\rm IRIS})}{{\it DSE}({\rm Mark\!-\!1})}=-0.175x^{3}+2.252x^{2}-8.874x+12.587$$ (1)

$$\frac{{\it DSE}({\rm IRIS})}{{\it DSE}({\rm LOWL})}=-0.095x^{3}+1.353x^{2}-5.260x+7.962$$ (2)

where ${\it DSE}$ is the spectral density.

There is no need of cross calibration outside this frequency range: below 1.1 mHz, all signals have been filtered, so that no information is available. At the high frequency end, our threshold is over the acoustic cutoff frequency of about 5.5 mHz. Beyond 6 mHz is the domain of the so-called pseudo-modes, that can possibly be accessible to sodium data (with a careful selection of the less noisy days), but not to the potassium ones, so that no use of the merged data set can be foreseen at these frequencies.

Figure 4: Average daily power spectra of the 3 instruments (IRIS, Mark-1, LOWL) after cross calibration sodium/potassium.

Potassium velocities $v({\rm K})$ (Mark-1 and LOWL data) are converted into sodium velocities $v({\rm Na})$ (IRIS data) using:

$$\frac{{\it DSE}[v({\rm Na})]}{{\it DSE}[v({\rm K})]}=\frac{\v... ... F}[v({\rm Na})]\vert^{2}}{\vert{\cal F}[v({\rm K})]\vert^{2}}$$ (3)

$$v({\rm Na})={\cal F}^{-1}\Biggl[{\cal F}[v({\rm K})] \times \sqrt{\frac{{\it DSE}[v({\rm Na})]}{{\it DSE}[v({\rm K})]}}\Biggl]$$ (4)

where ${\cal F}[u]$ is the Fourier Transform of the function u and ${\cal F}^{-1}[u]$ is the Inverse Fourier Transform of the function u.

The spectra shown in Fig. 2 before sodium/potassium cross calibration are plotted in Fig. 4 after this cross calibration using Eq. (4). The question must be raised of the contribution of the background noise and of the p-mode amplitudes themselves in the definition of these cross calibration functions. This question is especially relevant in the low frequency domain, well below 2 mHz, where the background differences, that are cancelled by this cross calibration, certainly imply a residual modulation of the amplitude of the p-modes in the merged time series. It is better to accept this residual modulation, or to work harder to adjust the cross calibration to the p-mode amplitudes and thus to accept a modulation of the background noise. The final damage on the performance is presumably comparable. Figure 5 shows an example of a very low frequency p-mode (l = 1, n = 8 at 1.329 mHz) detected on the merged time series, without gap filling, from the average of a few annual power spectra. Its very good SNR (at a 3 mm/s amplitude for each component) indicates that the resulting modulation implied by our calibration is not too severe.

Figure 5: Doublet l = 1, n = 8 at 1.329 mHz (Arbitrary units on the y axis).

2.3 Sampling

Each data set has been recorded with its own sampling time, 45 s for IRIS, 40 s for Mark-1 (actually 42 s before 1984) and 60 s for LOWL. The merging process requires the use of a unique time frame. The two potassium data sets have been resampled at 45 s to fit the sodium by means of a spline interpolation routine, so that the IRIS⁺⁺ data bank contains velocity time series sampled at 45 s (with a corresponding cutoff frequency of 11.1 mHz).

2.4 Low frequency filtering

Before merging, it is desirable to high pass filter the data to remove the unwanted low frequency noise. The solar noise itself would be more or less the same in the various data sets, but the instrumental and atmospheric noises can be quite different, so that the merged data could suffer the presence of significant discontinuities that could damage the performance of the power spectra not only at low frequency since the Fourier transform of a step extends to high frequencies. The same filter has been used for the 3 data sets. It is a Butterworth filter of order 10 with a cutoff frequency of 1.1 mHz, which is an IRR (Infinite Impulse Response) filter. Butterworth filters are characterized by a magnitude response that is maximally flat in the pass-band and monotonic overall. The Butterworth's transfer function is:

$$\vert H(\omega)\vert^{2}=\frac{1}{1+\biggl(\displaystyle{\frac{\omega}{\omega_0}}\biggl)^{2N}}$$ (5)

with N, the filter's order and $\omega{_{0}}$, the cutoff frequency. The cutoff frequency is the frequency where the magnitude response of the filter is $\sqrt{1/2}$.

Figure 6: Power spectra after low frequency filtering (cutoff frequency at 1.1 mHz).

We compute the filter coefficients in vectors b and a of length (N+1) with coefficients in descending powers of z:

$$H(z)=\frac{B(z)}{A(z)}=\frac{b(1)+b(2)z^{-1}+...+b(n+1)z^{-n}}{1+a(2)z^{-1}+...+a(n+1)z^{-n}}\cdot$$ (6)

We then use a zero-phase filtering, which eliminates the non-linear phase distortion of an IIR filter (see Fig. 6) (Porat 1996).