Diagnostic of the temperature and differential emission measure (DEM) based on Hinode/XRT data

We discuss here various methodologies and an optimal strategy of the temperature and emission measure diagnostics based on Hinode X-Ray Telescope data. As an example of our results we present determination of the temperature distribution of the X-rays emitting plasma using filters ratio method and three various methods of the calculation of the differential emission measure (DEM). We have found that all these methods give results similar to the two filters ratio method. Additionally, all methods of the DEM calculation gave similar solutions. We can state that the majority of the pairs of the Hinode filters allows one to derive the temperature and emission measure in the isothermal plasma approximation using standard diagnostic based on two filters ratio method. In cases of strong flares one can also expect well conformity of the results obtained using a Withbroe - Sylwester, genetic algorithm and least-square methods of the DEM evaluation.


Introduction
The solar corona with temperatures greater than a million degrees emits mainly in X-ray domain of the spectrum. Its emission contains both the continuum from thermal plasma and line emission of highly ionized elements. The earliest phases of solar flares are dominated by non-thermal emissions (hard X-rays), the remainder of the events manifests itself primarily by thermal emissions at wavelengths ranging from X-rays through visual. The thermal radiation comes from plasma having a wide range of temperatures.
Temperature and emission measure diagnostic of the active region and flaring loops is very important to model the loop structures, and analysis the heating sources and their (heating) distribution along the loop.
Correspondence to: M. Siarkowski (ms@cbk.pan.wroc.pl) For the optically thin emission, assuming isothermal plasma temperature, the well known method of filter ratio, based on ratio of the observed fluxes in two selected energy (or wavelengths) bands (Vaiana et al. (1973), Thomas et al. (1985)) can be used. In the case of multi-temperature plasma the filter ratio method derives the line-of-sight average isothermal temperature and emission measure. Therefore, in this approach it is more adequate to use many filters or line in terms of differential emission measure distributions DEM (T )/dT (Pottash et al. (1964), Sylwester et al. (1980)). Extended discission of the DEM versus isothermal approximation can by found e.g. in Schmelz et al. (2001), Cirtain et al. (2007) and references therein.
Hinode is a new advanced optical, UV and X-ray solar satellite launched on 22 September 2006. It is equipped with three instruments: Solar Optical Telescope (SOT), EUV Imaging Spectrometer (EIS) and the X-Ray Telescope (XRT). XRT is a high resolution Wolter-I type grazing incidence telescope, which is sensible to the emission of plasma in the temperature range 5.5 < log T < 8. Such wide temperature coverage is realized by nine X-ray filters, each with its own passband and therefore a different response to plasma temperature. These filters are mounted on two filter wheels which allow many combinations of filters and thus can be used for a temperature diagnostic. Figure 1 presents the theoretical temperature response functions of the ten XRT filters combination for unit volume emission measure of 10 44 cm −3 for each filter used in this paper. XRT has 2048x2048 backilluminated CCD array with 13.5 µm pixel size. It allows the Sun soft X-ray images to make in 34 x 34 arcmin FOV with spatial resolution of 1 arcsec. Exposure times are in range from 1 ms to 16 sec. Detailed description of the instrument can be found in the paper by Golub et al. (2007).
In this paper we will check if XRT response characteristics allow diagnostics of temperature (T) and emission measure (EM) of the coronal plasma. We also describe the observations selected for the analysis. Additionally we present and discuss filters ratio method in isothermal approximation and three different methods for differential emission measure (DEM) analysis in multitemperature approximation, respectively.
In all calculations we used the temperature response function as given by SolarSoft and presented in Figure 1.

Data analysis
For our work we selected three typical X-ray emitting active regions observed by Hinode/XRT with 6 or more different filters. To prepare the selected data we used standard SolarSoft routine xrt prep.pro which is similar in nature to sxt prep.pro (Yohkoh software). This routine converts raw data into index and structure arrays. Additionally, this process removes several instrumental effects. We could mention some of the most important, for example: replacement of the near-saturated pixels for value greater than 2500 DN with 2500 DN, remove radiation-belt/cosmic-ray and streaks, calibration for read-out signals, removing of the CCD bias, calibration of the dark current, normalization of each images for exposure time. Using processed data we calculated their differential emission measure distributions and using two filters ratio method, temperature and emission measure maps in isothermal approximation.

Two filters ratio diagnostic
The filter-ratio method is a technique commonly used for plasma diagnostics based on data from broad-band X-ray imaging systems. This method requires analysis of two or more intensity maps (images) of the investigated structures obtained with various broad-band filters. It delivers maps of temperatures and line-of-sight emission integrals or emission measure distributions as a function of temperature.
For a multi-band instruments like XRT, under assumption that the emitting plasma is optically thin, the observed flux F k in filter k in a single pixel is: (1) where f (T, λ) is coronal plasma emissivity as function of the plasma temperature T and the wavelength λ, calculated at distance of 1 AU using CHIANTI spectral code (including both emission lines and continuum). η k (λ) is an effective area of the instrument which includes the geometric area and the reflectance of the telescope mirror, the entrance and the focal plane filter transmissions and the CCD quantum efficiency. EM(T) is a collection of emission measures over a range of temperatures located along the line of sight. Defining temperature response function P k (T) of filter k as a convolution of the instrument effective area η k and emissivity of solar plasma f : we can rewrite Equation 1 as: For isothermal approximation this Equation can be written as: Similar to Thomas et al. (1985) using Equation 4 one can calculate ratio R of the observed fluxes (F ) in two filters labeled "1" and "2" for a co-aligned pixel area. This can be written as: According to Equation 5, the ratio R of observed fluxes depend on plasma temperature T only. In the next step, one can calculate the emission measure EM (T ) = F k /P k (T ). This procedure can be used on all combinations of the XRT filters, but it has a limited usability. Especially, ratio R can be sometimes an ambiguous function in the whole range of temperatures.
Using procedure described above we made temperature and emission measure maps for an area limited by an isoline 0.3 of the brightest pixel on the image.

Analysed events
The first analysed region was a relatively faint active region NOAA 0955. Observations were made on 13 May 2007 at around 18:00 UT. Images taken in ten filters combination are available. Temperature and emission measure maps were calculated using two pairs of filters Al poly / Al thick and  Ti poly / Al thick. As an example the resulted maps for Al poly / Al thick pair are presented on Figure 2. For both pairs of filters the filter ratios are sensitive for temperatures from log(T ) = 6.1 up to log(T ) = 7.7. However, for all structures of this quiet active region we obtained temperatures in ranges 2.0-3.3 MK and 1.8-2.4 MK for the Al poly / Al thick and Ti poly / Al thick pairs respectively. The range of the observed temperatures is typical for solar quiet active regions (see e.g. Golub and Pasachoff (1997)).
An active region NOAA 0923 was seen on the edge of the West solar limb between 19 and 23 November 2006. Observations analyzed here were made on 20 November 2006 at around 21:16 UT. Images taken in six filters are available. Temperature and emission measure maps were calculated for two pairs of filters: Al poly/Al mesh and Al poly/Be thin. The resulted maps for Al poly/Be thin filters pair are presented on Figure 3. The highest temperatures range of 3.5 MK are observed in the brightest loop, seen just on the solar limb.
The same active region NOAA 0923 was observed one hour later with seven filters. Temperature and emission measure maps were calculated for two pairs of filters: Al poly/Al mesh and Al poly/Be thick. The resulted maps for Al poly/Be thick pair are shown in Figure 4.
Due to the construction of the filter wheels, XRT images in various filters are only obtained consecutively. Thus an important assumption underlying the filters ratio diagnostic (and also DEM analysis described below) is that there was no substantial change in the morphology and energetics of the event between two images using various filters. This assumption is usually fulfilled in the case of quiet Sun and active region observations. Checking the light-curves for different filters we proved that this assumption is also true in the observations above described. Moreover, in the case of small B1.5 flare images with different filters were obtained first around the maximum and the second one on decay phase of the event. We use this pair because the second filter was only one "thick" filter available. The exact co-alignment of images is likewise important, especially for the DEM calculations in the individual pixels.

Distribution of the Differential Emission Measure
Differential emission measure, defined as DEM (T ) = d(EM )/dT = n 2 e dV /dT , is a useful diagnostic tool of non-isothermal plasma. It is usually calculated from a number of line fluxes formed at different temperature ranges. Generally the determination of the DEM distribution is an ill-conditioned problem with no unique solution Craig and Brown (1976) and this problem needs some kind of regularization. Calculations of DEM from broand-band filters observations is also possible if a corresponding filters have different temperature response functions (see Figure 1). In the case of Hinode broad-band filter observation such method was worked out and tested by Weber et al. (2004).
In this paper we compare three different methods of the DEM reconstruction using Hinode/XRT observations with ten filters. The methods used are as follows: Genetic algorithm (GA) -This algorithm is based on ideas of biological evolution and natural selection mechanism. Starting from randomly chosen initial populations of different DEMs a new generation of DEMs is produced by crossover and mutations. Process of breeding (and multiplication) of the whole population is controlled by assumed fitness criterion, e.g. DEMs with higher fitness have a higher probability to participate in the process of multiplication using crossover. Examples of DEM distribution calculated with genetic algorithm cam be found for example in papers by Kaastra et al. (1996), Guedel et al. (1997) and McIntosh et al. (2000).
We used an IDL implementation of Charbonnaues 'PIKAIA' Fortran code Charbonnaue (1995). Our population had 1000 individual DEM distributions or 'genes'. After a process of multiplication a new populations was cre- ated consisting of 90% new DEMs and 10% of the old ones with the highest fitness. We used χ 2 as the fitness criterion of the DEM. The process of evolution was stopped after 10 000 -20 000 generations, when the convergence became very slow. The minimum values of χ 2 were around 4.3; typically 15 to 30 'best' newly generated DEMs had χ 2 values below 10. The evolution process was repeated 6 times, each time starting from a new random population. As a result we obtained 136 solutions with χ 2 < 10. In Figure 7 the green line presents the mean of these 136 solutions (χ 2 of the mean is 4.7). The errors were calculated as a standard deviation from the mean. The solution obtained with GA has χ 2 less than those obtained with WS and LS methods.
The Withbroe -Sylwester (WS) method is iterative, multiplicative algorithm relying on the maximum likelihood approach. In our calculations we started from constant value DEM approximation. For detailed descriptions of this method see Withbroe (1975) and Sylwester et al. (1980). Using WS method we performed 200 simulations consisting of 3000 iterations each. The obtained χ 2 were in the range from 5 to 25. The mean solution has χ 2 = 10 (see Figure 7).
Least-square (LS) method -this procedure calculates DEM basing on a least-squares fit of the calculated fluxes to the observed ones. The method is described in more detail by Golub et al. (2007) and Weber et al. (2004) and implemented as a xrt dem iterative routine in SolarSoft. The DEM function is evaluated from some spline points, which are directly manipulated by the χ 2 are fitting the routine. Using LS method we performed 200 LS-simulations. The obtained χ 2 were in the range from 20 to 30. The mean solution has χ 2 = 25.
As an example we have calculated DEM for the loop ob-served on 13th May 2007 (see Figures 5 and 6). The flux used in DEM calculation was averaged over an area delimited by 0.8 contour level on the Al mesh filter image (see Figure 6, left panel).
We have tested all three methods using synthetic isothermal model. The reconstructed width of the solutions allow us to estimate the temperature resolutions of the method. This resolution is low at the edges of considered temperatures range (5.5 < log T < 8) and increase in the temperatures where response functions have their maxima (see Figure 1) with best values of dlogT = 0.5-0.6, 0.2-0.3 and 0.1-0.2 for LS, WS and GA method respectively. DEM errors are similar for all three methods, and we showed in Figure 7 only error bars for LS for readness and transparency.
The uncertainties are in fact very large at lower temperatures (see Figure 7), and for this reason all three methods give solutions lying within error limits of both other methods. Thus one can not evaluate precisely EM in this temperatures range using Hinode/XRT data. However, it is well known that DEM has a minimum around log T ≈ 5.5 for quiet Sun and active regions and for this reason each obtained solution could be right (Brosius et al. (1996), Schmelz et al. (2001)).
There are also differences in DEM solutions obtained for high temperatures. The LS solution indicates the presence of a bigger amount of the plasma having the temperatures in the range (7.0 < log T < 7.3) than the WS and GA methods. The differences are larger than estimated errors. A possible explanation of these differences is that LS models depend on initial approximation of DEM (Weber et al. (2004)). LS method has also the lowest temperature resolution comparing to WS and GA methods. The difference between all three methods should be extensively tested using XRT obser- Obtained χ 2 are rather high. Considering whole loop instead of one or two pixels, the relative errors of flux decrease and thus χ 2 increase (being much above 1.0). On the other side there are still large uncertainties in the fluxes measured with Al thick filter which strongly influence DEM in high temperatures. Large χ 2 value can be also attributed to uncertainness in atomic data which contribute to response functions and to systematic error in background rejection.

Discussion and conclusions
XRT telescope can detect coronal plasma of various temperatures ranging from less than 1 MK up to more than 10 MK. Applying observations made with two or more filters one can analyze coronal temperature and emission measure along a line-of-sight.
The filter-ratio method has been used frequently in the past, but the high quality XRT data give temperature and emission measure maps with unprecedented accuracy and resolution. However, filter-ratio method is not appropriate for multi-temperature structures, giving an average value of T and EM only. For such a case the DEM methods of analysis of the distribution of emission measure should be used.
The DEM distributions obtained with the described methods are presented on Figure 7. All three methods give the main peak of emission for log T = 6.3. Observed differences between results of the three methods lie more or less within error limits (please note the logarithmic scale of DEM). This figure presents also a set of isothermal solutions obtained for different pairs of filters. Their values lie around the main peak of DEM obtained with numerical methods described before. Thus, we can conclude that all three methods give results similar to the two filters ratio method.
We can also conclude that described methods of the DEM calculation gave similar and comparable solutions. Additionally, based on shapes of the emission functions (see Figure 1) we can expect much better conformity of the results obtained with WS, LS and GA methods in a case of strong flares. We can state that the majority of the pairs of the Hinode filters allows one to derive the temperature and emission measure in the isothermal plasma approximation using standard diagnostic based on two filters ratio method. The temperatures evaluated with this method for two non-flaring active region plasma were in the range from around 1.5 MK to about 5 MK.