4 Analysis of a quiet Sun SUMER spectrum

  To test Ga-GA on real data we chose to analyse a spectral region in the SUMER wavelength range that is known to suffer from blending problems, both between spectra of different optical orders as well as just wavelength coincidences. Those problems resulting from blends between lines that happen to overlap in the first and second grating orders can be decomposed experimentally, and thus serve as a limited check on the the GA approach.

The dataset analyzed here was obtained on October 26th 1996, with the $1 \times 300$ arcsecond slit crossing the north polar limb, using SUMER's B detector. Data were acquired in the 1400 Å spectral region, containing strong lines of Siiv, Oiv, and Oiii (in second order), as well as other weaker lines.

The observing sequence was designed to obtain data between 1399 and 1408 Å (and in the second order spectrum with wavelengths at half of this range) on both the bare and KBr coated part of the detector, sequentially. The exposure time on the KBr part was 180 seconds, and 360 seconds on the bare part. The bare and KBr regions of the detector have very different sensitivities to first and second order spectra. Assuming that the spectra did not change significantly between the bare and KBr exposures, the different count rates acquired on the two regions allow one to decompose the spectrum analytically into first and second order components, I₁ and I₂ through the following equations  

$$Cts({\rm KBr}) = k_1 I_1 + k_2 I_2$$ (5)

 

$$Cts({\rm bare}) = b_1 I_1 + b_2 I_2$$ (6)

where $Cts({\rm KBr})$ and $Cts({\rm bare})$ refer to the count rates per pixel per second on the KBr and bare parts of the detector, I₁, I₂ are intensities of the first and second order spectra, and k₁, k₂, b₁, b₂, are (known) instrument sensitivities defined through these equations. Figure 8, top panel, shows $Cts({\rm KBr})$ and its components, k₁ I₁ and k₂ I₂. Values for I₁ and I₂ were obtained using measurements of $Cts({\rm KBr})$,$Cts({\rm bare})$ and instrumental sensitivities discussed by Judge et al. (1997). Figure 8 also shows $Cts({\rm bare})$ and its components, in the bottom panel.

  

Figure 8: The 1400 Å region of the solar spectrum as measured using the SUMER instrument (see text for details). The top panel shows the average spectrum, in counts/pixel/second, recorded on the KBr region of the detector. Positions of known strong lines are marked- the positions of lines of O III are marked assuming that they are formed in the second order. The bottom panel shows the same thing, but recorded on the bare part of the detector. The lines plotted with symbols show the spectral decomposition into first and second order lines using the known sensitivities from SUMER

In each case the count rates are averaged over 300 spatial pixels, including the solar limb, and time during the exposures.

Shown in the top panel of Fig. 9 is a decomposition performed using Ga-GA based only upon the $Cts({\rm KBr})$spectrum shown in the upper panel of Fig. 8. This is simply a "blind" fit, using no prior information about the spectrum, except that we expect between 16 and 20 Gaussians to be present with on constant background. Such "blind" fits show that we can obtain a reliable decomposition of the entire spectrum. An example where a "blind" run is significantly better than one where a priori knowledge is used to aid in the decomposition is given below (see Table 4).

  

Figure 9: Comparison between Ga-GA decomposition and the analytic decomposition of the SUMER spectrum in Fig. 8. The top panel shows the decomposition from the Ga-GA algorithm using only the KBr data from the top panel of Fig. 8. The bottom panel shows the decomposition from a single run of Ga-GA using constrained wavelengths in the fitness calculation. See Table 4 for the details of the runs with constrained wavelengths

4.1 Using additional knowledge

  Usually, extra information about the spectrum is known, and it may be needed for some cases. This information can be "hard-wired" into Ga-GA easily. For example, we could demand that the spectral decomposition must not contain spectral detail narrower than the instrumental width ($\sigma_{\rm inst}$). Or, we could specify that relative positions (or intensities) of lines from the same ion, known to great accuracy from laboratory measurement, be fixed to certain values. Such constraints can be incorporated into the GA through a simple modification of the fitness evaluation, Eq. (3). For such an example we might use:

$$\begin{eqnarray} E(\underline{x}) =& \chi^2 + C_{i} H^2(W_i,\sigma_{\rm inst}) +... ...i} - X_{j}) - (X_{i}^{\rm lab} - X_{j}^{\rm lab})\right)^2+ \ldots\end{eqnarray}$$ (7)

where we introduce the additional constants C_i and D_ij to control the "trade-off" between $\chi^2$ and the newly incorporated information, and where $H(W_i,\sigma_{\rm inst})$ will weight the optimization against features narrower than $\sigma_{\rm inst}$.A future version of Ga-GA may take advantage of this additional information to act as desktop on-line plasma analysis package. Recall however, that the number of parameters in the calculation effects the rate of convergence (Sect. 3.1 and Sect. 3.2).

The lower panel of Fig. 9 shows the results of a Ga-GA decomposition where we have included a line list of all the lines marked in upper panel of Fig. 8, the implementation of this is discussed below. The "fixed" wavelength decomposition (see results in Table 4) tells us additional information about the spectrum; there is an average redshift of 0.070 Å of the lines in the list from their reference position. This corresponds to a velocity of around 10 km s^-1. The comparison of the contributions between first and second order lines in the 1404- 1408 Å region shows that Ga-GA can successfully decompose a real, convoluted spectrum, into meaningful components.

 

Table 4: This table contains the results of Ga-GA analysing the SUMER spectrum of Fig. 8 where the wavelengths, $\langle \lambda_{G} \rangle$ (Å), intensities, $\langle I_{G} \rangle$, and widths $\langle W_{G} \rangle$ (Å) are the mean values of a ten run ensemble. $\dagger$ indicates that, in this wavelength range, a line of Arviii at $\lambda=700.245$ Å (in second order) dominates the emission, as is clear from inspection of images shown by Judge et al. (1997) but this was not given in the line list. This line was detected in the "blind" decomposition of Sect. 4 ($\lambda_{G}=1400.558$ Å, I_G=0.030 and W_G= 0.151 Å) with correspondingly different measurements for the two lines of Siii. This result illustrates that a priori information (in this case, the line list), must be correct or erroneous results will occur. Mean standard deviations in $\langle I_{G} \rangle$ and $\langle W_{G} \rangle$ are 0.002 and 0.001 respectively