Multi-filter images from the solar corona are used to obtain temperature maps that are analyzed using techniques based on proper orthogonal decomposition (POD) in order to extract dynamical and structural information at various scales. Exploring active regions before and after a solar flare and comparing them with quiet regions, we show that the multi-scale behavior presents distinct statistical properties for each case that can be used to characterize the level of activity in a region. Information about the nature of heat transport is also to be extracted from the analysis.

The increasing number of space telescopes and space probes that provide
information about phenomena occurring in space is yielding an enormous amount
of data that need to be analyzed to get information about the physical
processes taking place. In particular, images of the Sun obtained from the
Solar Dynamics Observatory (SDO) instruments and the new Interface Region
Imaging Spectrograph (IRIS) mission have a remarkable high resolution that
allows studies of the Sun with great detail not available before. The
analysis of the images usually involves some processing if one has to extract
information not readily obtained from the raw images. Some techniques used
for image analysis and feature recognition in the Sun have been described by

The images can be processed for pattern recognition using techniques such as
wavelet analysis that sort the structures in terms of the scaling properties
of their size distribution

The rest of the paper is organized as follows. In Sect.

In this section we review the basic ideas of the POD method and its application in separating spatial and temporal information as used in the present work.

Singular value decomposition (SVD) is, generally speaking, a mathematical method based on matrix algebra that allows one to construct a basis on which the data are optimally represented. It is a powerful tool because it helps extract dominant features and coherent structures that might be hidden in the data by identifying and sorting the dimensions along which the data exhibit greater variation. In our case, SVD is used to decompose spatio-temporal data into a finite series of separable modes of time and space, which are orthonormal. The modes give the best representation of the relevant timescales and space scales of the data.

The SVD of the matrix

Another useful mathematical expression of SVD is through the tensor product.
The SVD of a matrix can be seen as an ordered and weighted sum of rank-1
separable matrices. By this we mean that the matrix

In our analysis the matrix

Topos and Chronos is a name given to the separation of time and space
variations in the data in a way that the POD method can be applied

The singular value decomposition of

From Eq. (

The data we analyze with the methods of the previous section are obtained
from the observations of the Atmospheric Imaging Assembly (AIA) instrument

The first step is to select appropriate events that contain the phenomenology
of interest. In our case, we focus on the propagation of a heat front
associated with an impulsive release of energy, such as a solar flare. What
we call a heat front is simply an emitting thermal structure that moves
across the solar disk, but we are not interested in the actual identification
of it with known waves in the solar atmosphere. There could be various
possibilities for the propagating front, such as EIT waves

AIA composite of EUV emissions (94, 193, and 335 Å) (ref.:

The chosen event occurred on 31 August 2012 at around 20:00 UT. There were seven
active regions on the visible solar disk. An image at a fixed time is shown
in Fig.

The AIA instrument consists of four detectors with a resolution of
4096

Each wavelength is produced by a temperature distribution that peaks at a
characteristic temperature (

The DEM distribution is typically given by
DEM(

Emission Measure map (left panel) and Temperature map (right panel).

The maps generated in the previous section combining the emission at six
wavelengths can be further processed to extract structures and dynamical
features related to the thermal energy content. The time evolution of the
data space array,

Snapshots at different times of the region where the heat front is moving.

The same POD analysis performed on the temperature maps in Fig.

As mentioned in Sect.

Topos–Chronos decomposition of solar flare activity. Odd columns:
spatial modes

Topos–Chronos decomposition of pre-flare solar activity. Odd
columns: spatial modes

Topos–Chronos decomposition of the quiet-Sun region. Odd columns:
spatial modes

This correlation can be studied using Fourier analysis to extract the
characteristic spatio-temporal scales of each POD mode. For the Topos modes,
we first transform the 1-D vector

The Topos–Chronos plots for the pre-flare and quiet-Sun cases in
Figs.

Length scale vs. timescale for all POD ranks; solar flare.

Length scale vs. timescale for all POD ranks:

Further information can be obtained from the reconstruction error in the POD
representation, defined as the difference

PDF averaged over time frames for energy content in the small scales
of

To quantify the degree of the temperature intermittency, we construct the
probability distribution function (PDF) by dividing the range of values of

In order to determine whether the PDFs have non-standard features like long
tails, it is useful to fit some known function to the data. Normal statistics
produce a Gaussian PDF, so we can fit a stretched exponential

Stretched-exponential fits of the temperature fluctuation PDFs when
energy content in small scales is 10 %;

Stretched-exponential fits of the temperature fluctuation PDFs when
energy content in small scales is 25 %;

In this section we focus on the region affected by the flare and calculate the thermal flux entering there, driven by the released flare energy. This is made using the time evolution of the original temperature maps and comparing the results with those obtained with the Topos–Chronos method. Next, the heat flux is used to explore the properties of heat transport in the corona.

Since the thermal front produced by the flare is a coherent structure, it is
expected that the main mechanism bringing energy into the region of interest
is advective. Using this premise, we use the computed heat flux to estimate
the velocity of the incoming plasma flow as a function of the coordinate

For the analysis of the heat flux we make the following assumptions:
(1) there are no sources of energy inside the region of the temperature maps
since it was chosen to be slightly away from the flare site; (2) the main
direction of motion is along the

The starting point is the heat transport equation in the absence of sources:

Averaged (in

Velocity profiles assuming different levels of constant diffusivity.

Averaged (in

In Fig.

Heat flux profiles as a function of

In addition to the decomposition of the temperature fluctuations in optimal modes, the Topos–Chronos method can be used to do a multi-scale analysis of transport. The separation of space and time allows the extraction of prominent spatial structures persistent over the time span, ordered by rank. Similarly, temporal representation gives information about the time evolution of the structures at the corresponding rank. This separation is amenable to computing the dominant spatial temperature gradients that drive the heat fluxes, and the time variation of the thermal energy, independently.

Substituting the representation for the temperature maps in Eq. (

A similar analysis can be made for the

Averaged (in

Using EUV images from six filters of the SDO/AIA and simple physical assumptions (mainly isothermality within a pixel), we have constructed maps representing the energy content 2-D distribution that we use as approximate temperature maps for the solar corona. The temperature maps for a solar event near an AR in the solar corona were analyzed using POD methods and were compared with similar analyses of other maps: one of the same region but before the flare (pre-flare) and another of a quiet region during the same time period (quiet Sun). The POD method separates time and space information (Topos–Chronos), thus allowing one to determine the dominant space–time scales. The high-rank modes in the decomposition correspond to smaller spatial scales and in some degree to small timescales. An interesting finding is that there is a rough correlation between Fourier timescales and spatial scales when the flare has occurred but not for pre-flare or QS states. This may indicate that flare-driven large-scale heat flows tend to transfer their energy to smaller scales.

Velocity profiles assuming different levels of constant diffusivity.

The Topos–Chronos method was also used to study the statistical properties
of the temperature fluctuations. In particular, we reconstructed the
temperature maps up to a certain rank associated with a given energy content.
The reconstruction error was thus associated with the small-scale temperature
fluctuations. The probability distribution functions (PDFs) of these
small-scale fluctuations were obtained for all times and then averaged in
time for each of the three regions analyzed. One of the main results of the
present paper is the observation that the PDF for the pre-flare state is
significantly broader and has longer tails than the PDFs of the flare and
quiet-Sun cases. The pre-flare activity seems to produce more high-amplitude
temperature fluctuations, characteristic of intermittency, which might herald
the occurrence of the flare. It is interesting to note that this result may
be related to the findings of

A multi-scale analysis of the heat flux was also performed for the region
associated with the flare. The thermal flux profiles along the main
(

We thank Alejandro Lara for his assistance with the use of SolarSoft and IDL. We acknowledge the use of a SolarSoft package developed by M. Aschwanden to obtain temperature maps. This work was partially supported by the DGAPA-UNAM IN109115 and CONACyT 152905 projects. Diego del-Castillo-Negrete acknowledges support from the Oak Ridge National Laboratory, managed by UT-Battelle, LLC, for the US Department of Energy under contract DE-AC05-00OR22725. Edited by: G. Lapenta Reviewed by: two anonymous referees