A Global Survey of EUV Coronal Power Spectra

Abstract

We present results of an investigation of single-pixel intensity power spectra from a 12-hour time period on 26 June 2013 in a $1600 \times 1600$-pixel region from four wavelength channels of NASA’s Solar Dynamics Observatory/Atmospheric Imaging Assembly. We extract single-pixel time series from derotated image sequences, fit two models as a function of frequency $[\nu]$ to their computed power spectra, and study the spatial dependence of the model parameters: i) a three-parameter power law + tail, $A\nu^{-n}+C$, and ii) a power law + tail + three-parameter localized Lorentzian, $A\nu^{-n} + C + \alpha/ (1 + (\ln\nu-\beta)^{2} /\delta^{2} )$, the latter to model periodicity. Spectra are well described by at least one of these models for all pixel locations, with the spatial distribution of best-fit model parameters shown to provide new and unique insights into turbulent, quiescent, and periodic features in the EUV corona and upper photosphere. Findings include the following: individual model parameters correspond clearly and directly to visible solar features; detection of numerous quasi-periodic three- and five-minute oscillations; observational identification of concentrated magnetic flux as regions of largest power-law indices $[n]$; identification of sporadically located five-minute oscillations throughout the corona; detection of the known global ${\approx}\,\mbox{four}$-minute chromospheric oscillation; 2D spatial mapping of “coronal bullseyes” appearing as radially decaying periodicities over sunspots and sporadic foot-point regions, and of “penumbral periodic voids” appearing as broad rings around sunspots in 1600 and 1700 Å in which spectra contain no statistically significant periodic component.

Appendices

Appendix A: Spectral Fitting

In this appendix, we give details on the considerations and issues involved with fitting a model to spectra computed from time series generated by extracting intensity values from a single pixel over a 12-hour time interval. The nominal image cadence is either 12 or 24 seconds with few missing images (at most ${\approx}\,3\%$). The time series were placed on a uniform 12- or 24-second time grid and the gaps removed using linear interpolation prior to computing the spectra.

To estimate the spectral model parameters, many methods were considered in order to address the following issues:

• i) Non-stationarity – Over a 12-hour time period, the spectra at a given location may change from, for example, power law + tail to power-law dominated. To address this, we can use shorter time segments to compute the spectra with the drawback of a possibly less accurate power-law index because the spectra will have fewer points at low frequencies.

• ii) Noise – The spectra for a given 12-hour time series typically have large noise amplitude, and as a result, “failed” fits often resulted. A failed fit is one in which the curve-fitting routine produces a spectrum that visually does not match the spectra in a sensible manner or does not fit at all. These fits are due to inherent limitations in nonlinear fitting algorithms. We considered two approaches to reducing the noise: i) computing the average of spectra derived from segments of the full time-series and ii) computing the average spectra in a $3\times 3$ pixel box, which has the drawback that neighboring pixels may not have the same spectral type.

• iii) Computation time – Spectra with a large amount of noise take much longer to fit. As an example, when fits for $1600\times 1600$ spectra are computed using no averaging, the time was projected to be greater than ${\approx}\,100~\mbox{hours}$ compared to ${\approx}\,4~\mbox{hours}$ for the averaging method that was used.

Based on these issues and considerations, the method used for computing model parameters in this work is given as follows. We note that this methodology was based on extensive numerical testing and experimentation along with visual inspection of the fits at individual spatial locations (to verify that the fits were consistent with what was expected visually).

• i) For each pixel, average the spectra from six 2-hour non-overlapping time segments from the full 12-hour interval.

• ii) Average nine spectra in a $3 \times 3$ box to compute the final spectra for a pixel at the center of the box.

• iii) Compute parameter estimates using the dog-box method from the SciPy optimize.curve_fit version 0.18.1 package for Python 3.6.4 with parameter bounds given in Table 4 and uncertainties corresponding to the standard deviation of the nine spectrum values used in the averaging described in (ii). For the 1700 Å channel, we used an uncertainty at frequency $f_{i}$ proportional to $\log_{10}(f _{i+1}/f_{i})$ with the uncertainty for the highest frequency equal to that of the next highest. This ad-hoc approach was taken for 1700 Å because it led to fewer failed fits and better fits from a visual perspective.

  Table 4 Parameter bounds used in model fitting. The parameter $\beta$ is provided in both frequency [mHz] and temporal [minutes] units.

• iv) Use the best-fit parameters from the dog-box optimization as initial guesses for the TRF optimization method from SciPy’s $\sf{optimize.curve \_fit}$ optimization package with the same parameter and uncertainties used for the dog-box optimization step.

The first two steps were needed to reduce the noise in the spectra and decrease the amount of time needed for each optimization. Because we wished to keep our spatial features as sharp as possible, we used only a $3 \times 3$ spatial averaging window and then obtained additional smoothing from segmentation of the 12-hour interval. This combination seemed to involve the least amount of averaging required to obtain few failed fits and for the computation to complete in a reasonable amount of time. Time segments of 2 hours in length were used because we found that their power-law indices were similar to those obtained from using the full 12-hour segment, and when segments of 1 hour were used, the power-law indices began to show substantial differences.

The parameter ranges in Table 4 used for optimization were based on those used by Ireland, McAteer, and Inglis (2015) and were refined to those presented through a process of trial and error. The lower bound for $C$ was chosen so that its middle value was near the center of the histogram peak in resulting distributions. Our value of $n=0.3$ is lower than that of Ireland, McAteer, and Inglis (2015) because our region of interest included a coronal hole, which we observe to frequently have such low power-law indices.

The fourth step was introduced because although the dog-box method produced in general the best fits of the optimization methods in SciPy’s $\sf{optimize.curve\_fit}$ optimization package, in some cases, we found unphysical spikes in the histogram of the $1598\times 1598$ fits for one or more of the six model parameters for a given wavelength at values that were at the centers of the parameter bounds given in Table 4. This was found to be due to the fact that the dog-box optimization method uses the centers of the parameter bounds as the initial guesses, resulting in early termination of the minimization algorithm if a local minimum of the function happened to be found near there. This issue was corrected by the use of a two-stage/step curve fit routine that used the dog-box method to compute initial parameter estimates that were then used as the initial estimates for a TRF optimization.

Appendix B: Significance Calculation

Model M1 has a total of $p_{1}=3$ adjustable parameters and Model M2 has $p_{2}=6$. As a result, M2 is expected to provide a better fit to the spectra on average. The $F$ test is used to determine when this is meaningful. The $F$-statistic associated with this test, which applies when M1 is nested in M2, is

$$ F = {\frac{ (\frac{\mathit{RSS}_{1}-\mathit{RSS}_{2}}{p_{2}-p_{1}} )}{(\frac{\mathit{RSS}_{2}}{n-p_{2}} )}} , $$

(6)

where $n$ is the number of data points used to estimate the model parameters, and $\mathit{RSS}$ is the weighted sum of squared residuals. The null hypothesis is that M1 fits the data as well as M2 (stated formally, that the $\alpha$-parameter in M2 is zero). This hypothesis is rejected when the value of the $F$-statistic is above the threshold in the $F(p_{2}-p_{1},n-p_{2})$ distribution associated with a false-rejection probability of $p<0.005$. Stated informally, we claim a spectrum has a periodic component when $p<0.005$, with the expectation that this claim is false at most $0.5\%$ of the time. The threshold of $p=0.005$ was chosen after testing values over two orders of magnitude. We found that $p<0.005$ gave null-hypothesis rejections that were most consistent with visual inspection of the two model fits.

For consistency, the $p<0.005$ threshold was applied to all wavelengths. As noted in the main text, relaxation of the $p<0.005$ constraint to $p<0.05$ revealed additional structure in the coronal bullseye in 193 Å that is likely to be real because of the spatial coherence of the structure. However, the use of $p<0.05$ for all wavelengths would have added significantly more noise to the observations in the wavelengths that had more locations with spectra with a significant periodic component.

Appendix C: Segment Averaging

To illustrate the impact of the segment-averaging procedure on power spectra, Figure 10 has four power spectra obtained from location D in Figure 2a (171 Å with the different degrees of segment averaging discussed in Section 4.2). Specifically, we present averaging of 2 6-hour sequences (Figure 10a), 3 4-hour sequences (Figure 10b), 6 2-hour sequences (Figure 10c), and 12 1-hour sequences (Figure 10d). These spectra are additionally averaged using the $3 \times 3$-pixel average procedure described in Section 2.3. Figure 10e presents these averaged spectra overlaid on a raw (unaveraged) spectrum obtained from the original 12-hour time series.

Figure 10

Demonstration of the effects of the segmenting procedure for the average of spectra to reduce noise. Each of the four upper panels represents a different level of segmenting a full 12-hour sequence using 2 6-hour sequences (upper-left panel), 3 4-hour sequences (upper-right panel), 6 2-hour sequences (lower-left panel, as used in this study), and 12 1-hour sequences (lower-right panel). These averaging choices correspond to those discussed in Section 4.2. The original data time series for these plots was obtained from Sample Point D (Figure 2a) in 171 Å, and was averaged in a $3\times 3$-pixel box according to the methodology described in Section 2.3. The large panel in the lower half of the image overlays all four averaged spectra on top of the raw (unaveraged) spectrum of the original time series (black).

Figure 10 illustrates our finding that the spectral averaging has the most impact on power-law indices, with n varying from 2.33 to 1.76 for this particular location, but with the Lorentzian contribution essentially unchanged regardless of segmenting. In both cases, we note that noise is far lower in spectra in which the segmented averaging is applied, resulting in almost no failed model fits for large regions of interest. As discussed in Section 4.2, the decrease of 0.57 in power-law index seen in this example is atypically high for the 171 Å channel, where across the entire region of interest we observe that the median variation due to segmenting is 0.04 with a standard deviation of 0.30 (according to Table 2).

Appendix D: Example Spectra: Poor and Good Fits in 171 and 1700 Å

Figures 11 and 12 provide a limited set of examples of good (Panels a and b) and poor (Panels c and d) model fits in the 171 Å and 1700 Å observations, respectively. The pixel locations use the same coordinate axes as those shown in Figure 2d. The presented fits do not necessarily represent the very worst or very best cases of each; our methodology provided us with approximately 6.4 million such fits, of which these are only representative sample in two of the four channels studied.

Figure 11

Examples of good (top row) and poor (bottom row) model fits in the 171 Å observations. Panel a corresponds to a bright foot-point just above the coronal hole, Panel b corresponds to a point inside the sunspot umbra, Panel c corresponds to a point within the magnetic network structure as identified in the magnetogram observations (white region of Figure 2b, just to the northwest of the active region). Panel d corresponds to the boundary of the coronal hole. All point coordinates use the same axes as those shown in Figure 2d.

Figure 12

Examples of good (top row) and poor (bottom row) model fits in the 1700 Å observations. Panel a corresponds to the sunspot penumbra, Panel b corresponds to a point on the boundary of the magnetic network, Panel c is also a point on the boundary of the magnetic network, and Panel d corresponds to a location distant from both the magnetic network and the sunspot. All point coordinates use the same axes as those shown in Figure 2f.

In Figure 11a and b we show two examples of spectra that fit our M2 model well, i.e. include significant Lorentzian components. The spectrum in Figure 11a corresponds to a bright foot-point just above the coronal hole, and Figure 11b to a point inside the sunspot umbra.

Figure 11c and d shows two examples of spectra in which we observe a very broad spectral hump that our model is unable to capture. Figure 11c corresponds to a point within the magnetic network structure as identified in the magnetogram observations (white region of Figure 2b, just to the northwest of the active region) and Figure 11d corresponds to the base of a plage. Both models capture the high-frequency part of the spectrum well, but they fail on the low-frequency observations. Such fits are not typical of our results in this channel, but they highlight the kinds of power spectra that do not fit our model well (both of which would likely be better represented by the Kappa model). It is worth noting that both of these particular spectra are considered statistically significant Lorentzians in our methodology, despite a seemingly poor fit. In limited tests we see that spectra like these are well fit by the Kappa model employed by Auchère et al. (2016a), with one interesting result that the $\rho$-term in the Kappa function, stated as equal to $T/120$, where $T$ is the even spacing of the pulses (in hours), returns values very close to those as located by our Lorentzian component. However, we reiterate that this result is preliminary, and it may be misinterpreted given our limited investigations of the Kappa model.

In the upper row of Figure 12 we show two examples of reasonably well-fit spectra in 1700 Å. The pixel locations use the same coordinate axes as those shown in Figure 2f. Figure 12a corresponds to the sunspot penumbra and is typical of sunspot umbra and penumbra spectra, the majority of which are extremely well fit, while Figure 12b corresponds to a point just inside the magnetic network, very close to its boundary, in which the power spectrum is almost a simple power law with a small but clear bump at ≈ five minutes. The reduced $\chi^{2}$ here is lowered by the falloff of the model at very high frequencies (a common issue with using our model to fit chromospheric power spectra).

The lower row of Figure 12 shows two examples of poor model fits. Figure 12c is a point very close to the boundary of the magnetic network, but just outside the network, presenting a power spectrum that would likely be better fit by a broken power law, and Figure 12d is located distant from both the magnetic network and the sunspot, and highlights a spectrum in which the model fit is visually quite good yet misses most of the very high-frequency points (${\lesssim}\,2.5~\mbox{minutes}$, which generally we do not care about) and several points above the ≈ five-minute line, resulting in a very poor $\chi^{2}$-value of 244.86. Despite this, we can see that the Lorentzian component clearly fits the apparent oscillation at ≈ 4.1 minutes well, and this supports our confidence in identifying global oscillations in both 1700 and 1600 Å concurrent with those identified in other studies. As a general rule, the $\chi^{2}$-values for 1700 Å are impacted most by the model missing the very highest frequencies. The impact of this, however, may be arguably low given that we do not focus on any features with periodicities shorter than three minutes, and the 24-second cadence of the 1700 Å observations means that aliasing artifacts could reasonably be expected in part of this spectrum. Nonetheless, this model is clearly not optimal for describing the majority of chromospheric power spectra.