5 Error analysis

Comparison with October 14 data.

For reasons of space we have only displayed results from the May 12 TRACE sequences so far. The October 14 sequences produce comparable timeslices, Fourier amplitude maps and Fourier spectra, with as major difference that the extreme quietness of the solar field makes the C IV construct yet more unreliable and dominated by noise. Nevertheless, the October 14 sequences provide a different sample that we use here to judge the reliability of the May 12 results, in particular the high-frequency parts of the Fourier spectra which are most sensitive to the nonsimultaneities and the cadence and timing irregularities of the TRACE image acquisition, which differ between the two dates.

Figure 26: Comparison of the spatially averaged Fourier spectra from May 12 1998 and from October 14 1998. The solid curves are the same as in Fig. 18 except that the power spectra are normalised to the zero-frequency value (square of the mean intensity). The October 14 spectra are dashed and have a lower Nykvist frequency (23 mHz).

Figure 26 compares Fourier spectra between the two dates in the format of Fig. 18 (but without scatter clouds and rms curves to avoid overcrowding). The power spectra are again plotted logarithmically but are now made comparable by zero-frequency normalisation as in Eq. (4). The October 14 power spectra have a higher noise level at high frequency. The acoustic peaks are about the same for the two dates.

The phase-difference averages differ much between the two dates from $f \approx 15$ mHz onwards, even in the well-sampled internetwork (upper panels). The averaged coherences disagree appreciably at even lower frequency.

The TRACE fields sample a sizable fraction of the solar disk. The October 14 field was even quieter than the May 12 field but there is no reason to assume that the internetwork was intrinsically different between the two dates in these large spatial averages. Thus, we may expect the solar behaviour to have been similar at the two dates (except for the C IV construct), especially in the internetwork. The fact that the mean phase difference and coherence spectra differ so much above $f \approx 15$ mHz implies that measurement errors and measurement differences are important and upset our anticipation of diagnosing high-frequency oscillation properties exploiting the TRACE virtues of seeing-less and statistically-rich sampling. Various tests were therefore made to elucidate these errors. The principal question is to what frequency the phase-difference spectra are reliable.

Figure 27: Phase-difference closure. The curves show $\vert\Delta \phi (1700{-}1600) + \Delta \phi (1600{-}1550) - \Delta \phi (1700{-}1550)\vert$ where each $\Delta \phi$ is the vector-averaged mean phase difference (solid curves in Figs. 18-19). Solid: internetwork. Dashed: network. Upper panel: May 12 data. Lower panel: October 14 data.

Figure 28: Phase-difference tests using 2500 pairs of artificial solar-like signals with random phase per frequency in the first signal. The top panel results when the second signal is identical to the first one apart from a phase delay of $-90^\circ$ per frequency and both are sampled simultaneously with the actual TRACE 1600 Å timing sequence. The solid curve in the middle panel results when the two signals are identical (without phase delay) but are sampled TRACE-like with the 1600 and 1700 Å timing sequences, which have an average offset of 5s. The dotted curve shows the phase difference for a constant offset of 5s between signals. The dotted curve in the bottom panel shows the combination (two signals $-90^\circ$ out of phase sampled TRACE-like). The solid curve includes correction for constant 5s offset and emulates our phase difference evaluations from the actual data. The curve in the top panel is overlaid for comparison.

Phase-difference closure.

A direct reliability test of the phase differences in Figs. 18, 19 is available in the form of phase-difference closure. If each passband samples a well-defined oscillation phase per frequency, the $\Delta \phi (1700{-}1550)$ differences should equal the summed $\Delta \phi (1700 - 1600)$ and $\Delta \phi (1600{-}1550)$ values. Figure 27 shows that the direct and summed evaluations depart between f=10 and f=15 mHz, and for the network at somehat lower frequencies. This is the same frequency region in which the phase-difference spectra in Fig. 26 depart from each other, indicating that the averaged phase-difference spectra in Figs. 18, 19 indeed become unreliable in this frequency band.

Window taper.

In standard practice, the 10% start and end segments of the brightness sequence per pixel were multiplied by a 10% cosine bell taper to the mean value in order to suppress window edge effects. The original mean was restored as zero-frequency amplitude after the transform. Tests changing or even deleting the edge taper produced more high-frequency noise as expected from edge discontinuities (Brault & White 1971), but no significant change in phase-difference behaviour.

Discrete sampling.

TRACE has no high-frequency filter to suppress noise above the Nykvist frequency. The power curves in Fig. 18 still decrease slightly close to the Nykvist frequency, so that some aliasing seems indeed to occur. However, the Nykvist frequency is quite high in the May 12 data. The power spectra are appreciably higher around f=15 mHz so that the phase-difference spectra should be dominated by non-aliased signal around this frequency. However, at higher frequencies the sampling itself becomes inadequate for phase measurements below the Nykvist frequency. These imply a three-parameter sinusoidal fit (amplitude, offset, and phase) to each signal per frequency and pixel, so that the uncertainties increase rapidly when the signal is sampled less then 3 times per period, i.e. above $f \approx 17$ mHz for the May 12 data.

Non-simultaneous sampling.

The worst phase-difference error source is the non-simultaneity of the imaging in the different passbands. In principle, the temporal offset between the samples in two sequences introduces phase difference at each frequency that equals the offset divided by the period, a linear increase with frequency that was corrected using the equidistant time scale defined for each dataset. (The correction also makes $\Delta \phi$ depart from $0^\circ$ or $\pm180^\circ$ at the Nykvist frequency.) However, the actual timing irregularities and exposure durations cause uncertainties that also increase with frequency and become quite large at the high-frequency end - as demonstrated below.

Sampling tests.

Figure 28 illustrates the effects of the major sampling errors on phase-difference measurements by using an artificially generated signal that possesses a solar-like power spectrum. Statistical variation was introduced by generating 2500 realizations in each of which the phase was chosen randomly at each frequency. The top panel illustrates errors due to discrete sampling and sampling irregularities alone. It uses the actual mid-exposure timing sequence of the May 12 1600Å data to measure phase differences between the 2500 artificial signals and 2500 copies that are phase-shifted over $-90^\circ$ at each frequency. The curve shows the result from vector-averaged phase-difference determination as done above. The deviations from $\Delta \phi = -90^\circ$ illustrate the errors caused by the non-equidistant exposure timing. They are reasonably small out to high frequencies.

The middle panel of Fig. 28 shows a similar test using 2500 pairs of identical (not phase-shifted) signals sampled with the actual 1600 and 1700 Å timing sequences. The noise increases considerably compared with the top panel.

The bottom panel results from a 2500-pair test combining the two error sources. It indicates that the irregular sampling of different signals at unequal times introduces large errors that cause appreciable $\Delta \phi$ distortion for frequencies above f=10 mHz and large $\Delta \phi$ scatter above f=20 mHz. The phase closure test in Fig. 27 indicates a somewhat larger extent of reliability, out to f=15 mHz for errors below $10^\circ$, in the internetwork which provided more samples than the 2500 used in these tests. The internetwork $\Delta \phi$ curves in Fig. 26 also show equality up to f=15 mHz.

The conclusion is that, even though the TRACE data do not suffer from seeing, the sequential nature of TRACE's filter-wheel switching between passbands combined with the sampling irregularity is the major contributor of phase-difference noise and limits the reliability of the $\Delta \phi$ spectra in Figs. 18, 19 to frequencies below $f \approx 15$ mHz.