3 Fourier analysis

We briefly review our evaluations of Fourier quantities from the TRACE data. The analysis is standard following Edmonds & Webb (1972), Paper I, and Lites et al. (1998), but there are various display choices that require discussion. We write the temporal Fourier transform of a time sequence f₁ (x,y,t) for solar location (x,y) as

$$F_1(x,y,f) = a_1(x,y,f)+i \, b_1(x,y,f)$$ (1)

where a₁ and b₁ are real numbers, t represents time and ftemporal frequency. The subscript 1 indicates a particular TRACE sequence. The cross-correlation spectrum between two simultaneous sequences 1 and 2 per location (x,y) is

$$F_1(x,y,f) \,\, F_2^*(x,y,f)$$ F₁₂(x,y,f) =

$$c_{12}(x,y,f)+i \,d_{12}(x,y,f)$$ = (2)

where the asterisk indicates the complex conjugate and c₁₂ and d₁₂ are real numbers.

Power normalisation.

Standard choices for the display of power spectra are to plot the modulation energy

 

P_E(x,y,f) = a(x,y,f)²+b(x,y,f)², (3)

the fractional modulation

$$P_f(x,y,f) = \frac{a(x,y,f)^2+b(x,y,f)^2}{a(x,y,0)^2+b(x,y,0)^2},$$ (4)

or to use Leahy normalisation

$$P_{\rm L}(x,y,f) = \frac{a(x,y,f)^2+b(x,y,f)^2} {\sqrt{a(x,y,0)^2+b(x,y,0)^2}}$$ (5)

(Leahy et al. Leahy et al. 1983, cf. van der Klis 1989, Appendix of Doyle et al. 1999). The latter serves to estimate the significance of periodic signals but is not used here.

This choice in normalisation does not affect the relative shape of an individual power spectrum but becomes important when comparing or averaging different signals. For example, if some wave process dissipates the same amount of energy at the wave frequency both in network and internetwork locations, using (3) without normalisation is appropriate. Fractional power normalisation (4) would underestimate the network dissipation in that case because network is consistently brighter, but it becomes the right measure when both network and internetwork are affected by the same multiplicative modulation process. We illustrate such differences below.

Phase differences.

The phase-difference spectrum is

$$\Delta\phi(x,y,f) = \arctan \left( \frac{d_{12}(x,y,f)}{ c_{12}(x,y,f)} \right)$$ (6)

where positive values of $\Delta \phi$ imply that signal 1 is retarded with respect to signal 2. Different strategies exist to display and average phase differences $\Delta \phi$. The simplest one is to simply display all samples per temporal frequency in an unweighted scatter diagram (e.g., Gouttebroze et al. 1999), neglecting the amplitudes of the contributing Fourier components, or to display the scatter point density as brightness (e.g., Kneer & von Uexküll 1993). Another extreme is to display only spatial averages per frequency through averaging over a spatial wavenumber k_h segment or an annulus in the (k_x,k_y)spatial transform plane (e.g., Deubner et al. 1992). We prefer, as in older work, to visualise also the scatter itself in order to permit appreciation of its distribution. Lites & Chipman (1979) applied weighting per (x,y,f) sample by the cross-power amplitude $\sqrt{P_1 \, P_2}$ (with P₁ = a₁²+b₁² and P₂ = a₂²+b₂²) to produce binned greyscale $\Delta \phi$ displays with normalisation per temporal frequency bin. In Paper I only the samples with the highest mean Fourier amplitude $(\sqrt{P_1}+\sqrt{P_2})/2$ were plotted as scatter diagrams.

In this paper we show binned greyscale scatter plots with cross-power amplitude sample weighting following Lites & Chipman (1979).

We overlay spatially-averaged phase difference curves following Lites et al. (1998), given per sampled frequency by

 

$$[\Delta\phi]_{xy}(f) \arctan \left( \frac{[d_{12}(x,y,f)]_{xy}}{ [c_{12}(x,y,f)]_{xy}} \right)$$ = (7)

where the square brackets express averaging over locations (x,y). This procedure equals vector addition of the individual cross-correlation samples with each vector length set by the product of the two Fourier amplitudes so that the cross-power amplitudes again act as weights in setting the slope of the summed vector. The procedure avoids wraparound errors that occur in straightforward averaging from the $\arctan$ evaluation, for example when a value just above $\pi$ is transformed into one just above $-\pi$ and then averages erroneously with one just below $\pi$ to $\Delta\phi \approx 0$ instead of $\Delta\phi \approx \pi$.

Figure 5: Partial cutouts from the May 12 1700Å sequence illustrating temporal variations at 30s cadence. The numbers specify elapsed time from 15:26:32 UT in seconds. Axes: X and Y in arcsec from disk center. The greyscale is logarithmic. The first panel is a central cutout of the top panel in Fig. 2. The white markers along the sides of this panel specify the horizontal cut location used in Figs. 6-7. The white box marks the yet smaller subfield which is duplicated in the third column using a sign-reversed logarithmic greyscale to display the slowly-varying internetwork background pattern. Taking the inverse emphasizes the internetwork features by darkening the network and makes the brightness minima of the three-minute oscillation appear as brightest features. These reversed extrema appear to travel fast along the strands of a more persistent background mesh which we attribute to gravity-wave interference. Similar behaviour is seen, in counterphase, for the bright internetwork grains on non-inverted but rapidly displayed movies such as the ones on URL http://www.astro.uu.nl/~rutten/trace1 from which this figure is derived. The grain-to-mesh superposition is also visualized in Fig. 11.

Coherence.

There is also a choice for the evaluation of the degree of coherence between two signals. It requires some sort of local temporal or spatial averaging, because without any smoothing the coherence between two sinusoidal Fourier components at given (x,y,f) is unity regardless of the corresponding Fourier amplitudes and phase difference. The Würzburg practice of annular k_h averaging has the advantage that adjacent frequencies are treated independently, but the disadvantage that the modulations are assumed isotropic. Note that in this case the mean coherence goes to zero for pure noise. In contrast, Lites and coworkers treat each spatial pixel as an independent sample of solar behaviour but average over a frequency interval (Paper I, Lites et al. 1998), a tactic necessarily followed also in the one-dimensional phase modelling of Skartlien et al. (1994). For pure noise this procedure yields positive coherence $C = 1/\sqrt{n}$ when averaging a sufficiently large sample, with n the number of frequency resolution elements per averaging interval. We use the latter method, selecting boxcar frequency smoothing that is represented by angle brackets in writing the coherence as

 

$$\equiv \frac{<F_{12}><F^*_{12}>} {<F_1^2><F_2^2>}$$ C²(x,y,f) (8)

$$\frac{<c_{12}>^2+<d_{12}>^2} {<a_1^2+b_1^2>\,<a_2^2+b_2^2>}\cdot$$ = (9)

The spatially averaged coherence per frequency is then:

$$[C^2]_{xy}(f) = \frac{[<c_{12}>^2]_{xy}+[<d_{12}>^2]_{xy}} {[<a_1^2+b_1^2>\,<a_2^2+b_2^2>]_{xy}}\cdot$$ (10)

Fourier reduction.

We determined and Fourier-transformed the temporal brightness variation per pixel using equidistant time sampling with closest-neighbour image selection as discussed above, 10% cosine bell windowing, and replacing the zero-frequency transform values by the original mean brightness. We applied frequency smoothing for the coherence evaluations with n=5 so that pure noise has C = 0.45. The phase differences between different TRACE passbands were corrected for the temporal shifts between their respective sampling scales. The shifts result from the sequential TRACE image taking and produce artificial phase shifts that increase linearly with frequency when measured as phase-difference angle. Other effects from the sequential sampling are discussed in Sect. 6.

The resulting Fourier power spectra, phase-difference spectra and coherence spectra per solar location were spatially averaged over the network and internetwork pixel categories, respectively. We do not show results for the intermediate pixel category when these are indeed intermediate between network and internetwork.