Tools for 3D Solar Magnetic Field Measurement

Transcription

1 **Volume Title** ASP Conference Series, Vol. **Volume Number** **Author** c **Copyright Year** Astronomical Society of the Pacific Tools for 3D Solar Magnetic Field Measurement R. Casini High Altitude Observatory, National Center for Atmospheric Research, 1 P.O. Box 3000, Boulder, CO , U.S.A. Abstract. In this paper we describe some of the challenges that solar physicists face in the application of polarized radiative transfer to the modeling of the emergent radiation from the outer layers of the solar atmosphere, where the plane-parallel approximation breaks down, and 3D atmospheric modeling becomes essential. We review the various plasma conditions occurring in the photosphere, chromosphere, and corona, which determine the different regimes of atomic excitation of these regions. Depending on the relative importance of anisotropic irradiation of the gas over collisional thermalization of the atomic populations, the description of the atomic excitation states may necessitate a full quantum-statistical treatment, which exacerbates the numerical complexity of an already computationally intensive problem. Special emphasis is placed on forward modeling and inversion techniques that mitigate this difficulty, making feasible the interpretation of polarization signals in terms of the magnetic field and its connectivity throughout the solar atmosphere. 1. Introduction Our understanding of the effects of solar phenomena on the terrestrial environment relies on the knowledge of the energy input of the Sun into the inner solar system, from the surface of the Sun to the Earth s magnetosphere. This energy is contained mostly in the electromagnetic radiation emitted by the solar atmosphere. However, highly energetic particles carried by the solar wind also continuously impact the upper layers of the Earth s atmosphere. It is rather well ascertained that both short-term ( 0.1 yr) and long-term ( 1 yr) variations of the magnetic activity of the Sun can lead to important changes in the solar energy output, which can affect the Earth s environment in multiple ways. The 70-year long Maunder minimum in the second half of the 17th century, and the changes in the terrestrial climate that coincided with it, are perhaps the most suggestive example of the importance of studying long-term solar variability (e.g., Eddy 1976). On the other hand, solar flares and coronal mass ejections, which are associated with rapid reconfigurations of the magnetic topology in the solar atmosphere, are demonstrated causes of important effects on Earth, which go from changes in the atmospheric chemistry to electro-magnetic storms that have the ability to disrupt radio communications and power grids (e.g., Lanzerotti 1979). Therefore, our understanding of the Earth-Sun relations, and the potential for predicting their effects, depend on our knowledge of solar magnetism, and ultimately rely on our ability to measure the solar magnetic field and its modifications at different spatial and temporal scales, from the photosphere into the heliosphere. 1 The National Center for Atmospheric Research is sponsored by the National Science Foundation. 1

2 2 R. Casini Observational efforts aimed at measuring magnetic fields on the Sun have been going on almost routinely since about the 1960s (e.g., Babcock 1963). These measurements have steadily improved, not only because of the technological advances in instrumentation, but also, and perhaps foremost, because of the advancement in the modeling of the formation and transport of polarized radiation in magnetized plasmas, which has enabled solar scientists to identify optimal radiation diagnostics to probe different physical environments of the solar atmosphere. Despite this continuous progress, the solar community has at times displayed a tendency to complacency, with a risk to identify the science objectives of solar magnetism investigation simply with perfecting the same type of observations of the same targets. While it cannot be denied that the ability to access increasingly smaller spatial and temporal scales in the Sun s atmosphere brought forward by the availability of larger telescope and faster detectors can expand our previous understanding of well-studied solar phenomena, it is also of fundamental importance that new diagnostic methods and scientific targets be explored, which also may have become accessible thanks to the advance of technology. In conclusion, while there is no doubt that the current studies of solar magnetism are essential for the understanding of the Sun-Earth relations, it is also important that we keep asking new questions. The following action items can be reasonably proposed, since they are slowly coming within reach of the theoretical, modeling, observational, and instrumental capabilities of the present-day solar community: to measure and model magnetic flux emergence at the various spatial and temporal scales occurring on the Sun; to understand the role of magnetic fields in energy transport, from the convective photosphere, through the chromosphere, into the low-β corona; to measure the magnetic field at the true base of the corona, i.e., the chromosphere, in order to validate field extrapolations into the corona and heliosphere; to measure the magnetic field directly into the corona. This program poses many challenges, especially where the measurement of the magnetic field needs to be pushed outward, into the solar chromosphere and beyond. This paper focuses on the fundamental difficulties associated with this specific effort. 2. The importance of atomic polarization The privileged means to measure the vector magnetic fieldb on the Sun is represented by solar spectro-polarimetry, that is, the modeling and interpretation of polarized radiative transfer in spectral lines formed in the magnetized solar atmosphere. On the other hand, the complexity of the radiation hydrodynamic processes occurring in the convective and turbulent layers of the magnetized solar atmosphere makes the detailed modeling of the transport and emergence of polarized radiation a problem of incredible difficulty already in the photosphere. The transition to a low-β, partially ionized plasma regime in the atmospheric layers above the photosphere does not help this situation. However, at the microscopic level, the processes that describe the interaction of radiation with matter in the Sun s atmosphere are characterized by a very general condition, which is the coexistence of two competing mechanisms of atomic excitation and de-excitation, respectively by photons and by collisions with other gas particles.

3 Solar Magnetic Field Measurement 3 On one side, the temperature stratification of the solar atmosphere and the presence of local plasma inhomogeneities are responsible for the non-isotropic illumination of the plasma at a given depth. This condition is such that atomic polarization (that is, the presence of population imbalance and quantum coherence among the magnetic sublevels of the various atomic species; see Landi Degl Innocenti & Landolfi 2004) is naturally created in the solar plasma. On the other hand, the local plasma density drives the thermalization of the atomic populations through (isotropic) collisions, so that the atomic polarization created by anisotropic illumination is partially destroyed. This ubiquitous competition between atomic excitation by anisotropic radiation and collisional thermalization of the atomic populations has different outcomes depending on the various plasma conditions of the diverse regions of the solar atmosphere. In the photosphere, the gas density is such that isotropic collisions are relatively frequent, and thus very efficient at thermalizing the atomic populations. At the same time, the region of unit optical depth around a given atom is only of the order of 10 2 km (the mean free path of a photon), that is, smaller than the convective spatial scale associated with solar granulation. Therefore, the radiation anisotropy in the photosphere is not very large, and is predominantly determined by the radial thermal gradient i.e., the center-to-limb variation (CLV) of the solar radiation as seen by the atom. As a result, atomic polarization is typically negligible in the photosphere, although there are some notable exceptions for certain atomic and molecular species forming in the top part of the photosphere. 2 Thus the transport of polarized radiation occurs mainly in a regime of local thermodynamical equilibrium (LTE). As the plasma density quickly falls with height above the photosphere, the collisional rates also drop accordingly, while the photon mean free path increases by several orders of magnitude with respect to the photospheric value. Hence, radiation anisotropy becomes very important, as the local distribution of radiation is affected by the largescale inhomogeneities of the solar atmosphere. In addition, an atom in the chromosphere is able to see the entire solar disk, and so its height above the photosphere becomes an important contribution to the radiation anisotropy, because of the decrease of the solid angle subtended by the Sun for increasing heights. In the chromosphere, this geometric effect starts being competitive with the contribution from the CLV, and it finally becomes the dominant source of radiation anisotropy at coronal heights. The transition from high- to low-β plasmas also favors the appearance of inhomogeneities at smaller spatial and temporal scales than in the photosphere, and these also become important for a correct modeling of polarized radiative transfer in the chromosphere, at the spatial resolutions that will be accessible with future large-aperture telescopes. This small-scale structuring of the chromosphere is one of the main drivers for the development of full-3d modeling tools for polarized radiative transfer. As a result of the changing plasma conditions, atomic polarization becomes an essential ingredient for the modeling of polarized radiative transfer in the upper layers of the solar atmosphere. In fact, the signature of atomic polarization is the dominant contribution to the polarized radiation that we receive from the solar corona. The presence of atomic polarization in a gas characterizes evidently a condition of departure from LTE. However, unlike the non-lte problem that is commonly described 2 It suffices to mention the Sr I line at nm, or the C 2 molecular lines of the Swan system (Trujillo Bueno et al. 2006). The well-known photospheric lines of Fe I at nm also show evidence of atomic polarization when observed at the solar limb with high spatial resolution (Lites et al. 2010).

4 4 R. Casini Figure 1. The basic structure of the inversion problem for polarized radiative transfer. The red boxes at the left indicate the model parameters that are used as first guesses of the forward modeling calculation, and are successively updated in response to the fitting of the observed Stokes profiles (red box at the right). The self-consistency loop shown at the center must be iterated until the local radiation field is in equilibrium with the atomic system, that is, the feedback of the (polarized) output radiation (gray box at the right) on the atomic excitation (gray box at the left) is completely and self-consistently accounted for. in textbooks on stellar atmospheres (e.g., Mihalas 1978), the treatment of atomic polarization involves a remarkably higher level of complexity. To understand why this is the case, it is useful to draw a parallel with the case of light polarization. From a quantum-mechanical point of view, a photon is described by a complex, two-component wave function (a spinor). The two basis states (1, 0) and (0, 1) correspond to the two possible states of helicity of the photon, and any pure state of polarization can be expressed as a linear combination of those two fundamental states. 3 We note that there is a close formal analogy between the description of photon polarization and that of a quantum system of angular momentum (spin) J= 1/2 (e.g., Fano 1957). In a stellar atmosphere we obviously must deal with a very large number of photons and atoms, which can occupy many different states of polarization that are statistically uncorrelated with each other. This type of statistical ensemble must be treated in quantum mechanics through the formalism of the density matrix,ρ(fano 1957). A gas of electrons or photons is then completely described by giving the elements of the corresponding 2 2 density matrix, but because of the conjugation properties of this matrix, this requires the specification of only four real quantities. In a classical description of 3 Jones classical formalism for the description of light polarization (e.g., Shurcliff 1963) is closely related to this concept, but the two basis states correspond in that case to the two orthogonal states of linear polarization of the electromagnetic field on the plane perpendicular to the propagation vector.

5 Solar Magnetic Field Measurement 5 polarized light, for example, this corresponds to the completeness of representation by the four Stokes parameters. 4 Similarly, a gas of atoms of angular momentum J requires in general (2J+ 1) 2 real numbers for its description. In the more traditional non-lte problem, instead, the density matrix becomes proportional to the unit matrix, and the same gas of atoms is thus described simply by providing the relative populations (occupation numbers) of the various J states. In the case of LTE, the population of a J level is completely determined by its energy, and therefore one has only to specify the density of the atoms in the gas and its temperature. In conclusion, the extension of polarized radiative transfer from the traditional non-lte problem to the inclusion of atomic polarization determines a drastic increase in the dimensionality of the statistical equilibrium problem: each level J must be described by (2J+ 1) 2 numbers instead of just one departure coefficient. This part of the problem can become easily the most computationally intensive, especially for high-j atomic and molecular states, and in the presence of external fields. 3. Forward modeling and inversion tools The complexity of the interaction between polarized light and polarizable atoms that was illustrated in the previous section, and the strong non-locality of the transfer of polarized radiation through an optically thick and diversely magnetized atmosphere, make the inversion of the Stokes profiles observed on the Sun a fundamentally ill-posed problem. For this reason, Stokes profile inversion can only be based on the efficient implementation of the forward problem for the generation and transport of polarized radiation. This is schematically represented in Fig. 1. The input parameters (guesses) describing the thermodynamic and magnetic properties of the model atmosphere are represented by the red boxes at the left. Together with the prescribed boundary conditions for the illuminating radiation, these are used to initiate the forward calculation of the emergent radiation. The self-consistency loop (Landi Degl Innocenti & Landolfi 2004) pictured at the center must be iterated until the output radiation is in equilibrium with the atomic excitation condition of the gas of atoms, that is, the Stokes vector of polarized radiation and the density matrix of the atomic system (gray boxes at the right and left, respectively) have converged to stable values at each point in the atmosphere. Then the output radiation at the surface can be compared with the observations, and the input parameters corrected if necessary. The convergence of the self-consistency loop is an incredibly difficult problem, which may even have no practicable solution in general. A common way to circumvent this difficulty is to ignore the feedback of polarization on the atomic system, so the state of atomic excitation is brought to equilibrium only with the intensity of the output radiation. The condition behind this approximation is that the polarization of solar radiation must be weak enough that atomic excitation (also including atomic polarization) is essentially determined only by the intensity distribution of the local radiation field. State-of-the-art numerical models of polarized radiative transfer (e.g., Manso Sainz & Trujillo Bueno 2003) do account instead for the feedback of light polarization on the atomic system, at least for the simpler atomic structures or atmosphere geometries. 4 It is important to observe that each direction of propagation of radiation in the atmosphere corresponds to a different statistical ensemble of photons, so that each line of sight carries its own set of Stokes parameters.

6 6 R. Casini Figure 2. A list of forward modeling and inversion tools for spectro-polarimetry, currently supported through the Community Spectro-Polarimetric Analysis Center (CSAC) initiative of the High Altitude Observatory ( These codes span from traditional non-linear least squares fitting of Milne-Eddington atmospheres in the Zeeman regime of the magnetic field, to pattern-recognition based inversion of scattering polarized radiation from slabs in an arbitrary regime of field strength, applicable to the diagnostics of prominences and filaments. Once the conditions of atomic excitation are determined for the entire atmosphere, the emitted polarized radiation can be calculated at each point along any line of sight (LOS), and integrated by means of standard radiative transfer schemes. For simpler models of the solar atmosphere the convergence problem of the selfconsistency loop can be greatly simplified. For example, in the case of the traditional non-lte problem, at each iteration one has only to recalculate one departure coefficient for each atomic level in the model, rather than a density matrix that depends on both the illumination condition and the applied magnetic fields. Since light polarization cannot modify the overall population of a level, it is clear that the simplified iteration scheme described above, where the feedback of polarization on atomic excitation is neglected, is perfectly applicable in this case, and provides in fact an exact answer to the problem. In the case of an LTE atmosphere, of course, no iteration of the self-consistency loop is necessary, since the state of atomic excitation is provided directly by the model. The optically thin case is also of great interest, since it can often be applied to the modeling of the scattering polarization in prominences, and in the corona. In this case, the assumption is that the photon mean free path is larger than the observed structure, and therefore in average a photon is scattered only once within the gas, before escaping towards the observer. In this single-scattering approximation, there is evidently no feedback of either intensity or polarization on the statistical equilibrium of the atomic system, and therefore the self-consistency loop is traveled through only once. The program outlined in the Introduction involves probing the entire visible solar atmosphere with spectro-polarimetric instruments, in order to determine the connectivity of the solar magnetic fields from the photospheric regions, where magnetic flux emerges out of the convective layer, to the far corona, passing through the region of transition from high- to low-β plasmas. This requires that we consider all the various

7 Solar Magnetic Field Measurement 7 plasma regimes that we previously discussed with regards to the relative importance of atomic polarization for the transport of radiation through the magnetized solar atmosphere. It is essential that the solar community be provided with the right modeling and interpretational tools to tackle efficiently the different tasks of the outlined program. The Community Spectro-polarimetric Analysis Center (CSAC) is a base-funded initiative of the High Altitude Observatory, which aims at filling this need by serving the solar physics community with a set of state-of-the-art forward modeling and inversion tools for solar spectro-polarimetry. A list of the codes currently supported, with a short description of the realm of applicability and of the numerical method employed, is given in the table of Fig. 2. In summary, the simplest forward problems of polarized radiative transfer are well understood and solidly implemented in current numerical efforts geared towards a full 3D modeling of the solar atmosphere. These models span from the Milne-Eddington approximation of a mean atmosphere, through the LTE atmospheric model with both thermodynamic and magnetic gradients, to the simplest non-lte problem accounting for the departure of the total population of an atomic level from LTE. One important limitation of these codes is their restriction to the Zeeman effect, which sets the low boundary of magnetic field strengths that can be probed at around a few 10 2 G. In contrast, the full non-lte problem necessary to tackle the most general atmospheric conditions for an arbitrary regime of magnetic fields which involves the modeling of atomic polarization from the limit of zero field strength, through the regime of the Hanle effect, to the strong-field regime of the Paschen-Back effect and of levelcrossing interferences lags significantly behind in present-day numerical simulations, due to the added complexity and increased computational demands discussed above. The challenge represented by the manifold increase of the dimensionality of the statistical-equilibrium problem in the treatment of atomic polarization can sometimes be mitigated in the case of lines that are formed under a condition of weak anisotropy of the radiation field. In such case, adopting a multipolar expansion of the density matrix, 5 it becomes possible to retain only the first few orders, providing an effective reduction of the dimensionality of the problem. For example, the description of atomic polarization up to order K= 2 (atomic alignment) requires only 1+3+5=9 real numbers, regardless of the value of J (see Note 5). The computational effort associated with the iteration and convergence of the selfconsistency loop, and with the actual integration of the radiative transfer equation for polarized radiation through the atmosphere, does not seem instead susceptible of any significant optimization in the near future. This is because the methods that have been developed over the past several decades in order to attack efficiently the forward modeling of the non-lte problem for the unpolarized case already provide a highly optimized approach also to the more general non-lte problem involving polarization. Thus, a break-through improvement in the computational efficiency of this aspect of the non-lte problem can only be expected from a radical revision of the inversion methodology. 5 Using a multi-level atom in the state J as an example, the (2J+ 1) 2 elementsρ(jm, JM ) of the density matrix for that state can be rearranged into an irreducible representation of the rotation group with elements ρ K Q (J), where K= 0,...,2J and Q= K,..., K. The order K= 0 describes then the total population of the level J, whereas the orders K= 1 and K= 2 are said to represent, respectively, the orientation and alignment parts of the level polarization (e.g., Landi Degl Innocenti & Landolfi 2004).

8 8 R. Casini There is a growing consensus (e.g., Asensio Ramos 2011, these proceedings) that pattern-recognition techniques represent a viable candidate as an alternative approach to non-lte atmosphere modeling, since through them one can be completely dispensed from performing the real-time forward calculations that constitute the bottleneck of the problem. In the following section, we consider a specific implementation of patternrecognition techniques, principal component analysis (PCA). We recall the general ideas behind this strategy, and present some results of its application to the magnetic diagnostics of prominences and filaments. 4. Principal Component Analysis Inversion strategies for Stokes polarimetry based on pattern recognition techniques attempt at removing the computational hurdles posed by the foreword modeling core of the polarized radiative transfer problem in a complex atmosphere. This is achieved by creating a characterization of the physical problem and its observational output in terms of patterns of spectro-polarimetric profiles. The underlying assumption is that these patterns are univoquely determined by the physical conditions of the atmosphere. When that is not the case, an ambiguity arises in the estimation of the set of atmospheric and magnetic parameters determining the observations. However, it is important to remark that such a degeneracy of the physical conditions corresponding to a given set of observables is intrinsic to the problem at hand, and cannot be avoided regardless of the inversion strategy adopted. Such ambiguities, of both local and non-local nature, 6 are common, and are at the very origin of the ill-posedeness of the radiative transfer problem for polarized radiation, which has been hinted to earlier in this paper. The nature and type of the patterns that can be adopted for recognition-based inversion strategies depend on the particular technique. Wavelet analysis and PCA are two well studied techniques of pattern recognition, with potential applicability to the problem of polarized radiative transfer. We should also mention inversion methods that are based on machine learning, such as artificial neural networks and support vector machines (Asensio Ramos 2011, these proceedings), since the training sets for these methods are essentially of the same nature and scope as those used for the extraction of patterns in recognition-based inversion methods. In this section we focus on the application of PCA to the inversion of spectro-polarimetric profiles, since its applicability has already been convincingly demonstrated (e.g., Socas-Navarro et al. 2001; Skumanich & López Ariste 2002; López Ariste & Casini 2002; Casini et al. 2005, 2009). In the remaining part of this section, we summarize the procedure to implement a PCA-based inversion. First one must determine a universal basis of Stokes profiles for the problem at hand. This is by far the most critical aspect of the entire method, because all the physical information about the production and transport of polarized radiation through the model atmosphere is captured at this stage. Basically, one must engage in the computation of the forward model of polarized radiative transfer over many different realizations 6 A well known example of local degeneracy is the so-called 180-degree ambiguity, that is, the nondeterminability of the orientation of the vector magnetic field along the direction of its projection on the plane of the sky, which occurs in the regime of the Zeeman effect.

9 Solar Magnetic Field Measurement 9 of the physical system representative of the particular problem at hand, with all the pertinent constraints and ranges of physical parameters. For example, one might wish to investigate the formation of polarized line profiles in a flux-tube model of photospheric active regions with nearly vertical fields; or the Hanle effect in scattering polarization from an optically thin slab located at coronal heights, and embedded in a weak magnetic field of arbitrary geometry; or the stochastic non-lte formation of chromospheric lines in a 3D atmosphere, with a prescribed average functional dependence with height of the atmospheric parameters and the magnetic field, and a prescribed statistical 2D distribution of atmospheric horizontal inhomogeneities; and so on. Of course, this is exactly the same type of forward model calculation that one would have to perform in a more traditional approach to spectro-polarimetric inversion. The need for an accurate representation of the physical environment under study through an adequate mathematical model is evidently common to any inversion approach. The need for the completeness of the set of representative Stokes profiles (which is emphasized by the use of the term basis proposed at the beginning of this paragraph) is also very important. One must make sure that the parameter space for the problem is exhaustively sampled and represented by the basis set of Stokes profiles. The optimal and efficient sampling of the parameter space is on its own an interesting problem of applied statistics. Brute force Monte Carlo methods are typically very inefficient when applied to the sampling of multi-dimensional parameter spaces. Optimized sampling strategies, such as latin hypercube sampling (LHS; McKay et al. 1979), can help greatly in achieving a sufficiently complete coverage of the parameter space. For example, when applied to the case of scattering polarization in a magnetized, optically thin slab (such as in the modeling of Stokes profiles formed in solar prominences), the sampling of the parameter space (typically, with eight dimensions) requires an order of magnitude less calculations with LHS than with unoptimized Monte Carlo (Casini et al. 2009). The practical way for determining the basis of Stokes profiles from the initial set of synthetic observations has been described elsewhere (Casini & López Ariste 2003), and it will not be repeated here. The next step is to build the actual inversion database. This time the focus is to sample completely and uniformly the space of the observed profiles, rather than that of the model parameters. This can be done efficiently by applying a sieve method on the construction of the database through Filtered Monte Carlo sampling. The natural way of doing so is to use the basis of Stokes profiles previously determined as a projection set for the calculations of the new database. In fact, any four-vector of Stokes profiles corresponding to a realization of the physical model whether this is determined from synthetic or true observations can be regarded as a multi-dimensional vector in the space spanned by the PCA basis. Hence, such four-vector is completely identified by its projection coefficients on this basis. 7 Each synthetic observation can then be represented by a point in a coordinate space where the projection coefficients are the coordinates of the point, and each point is labeled by the configuration of the corresponding point in the space of model parameters. The filtering parameter for the sieve method becomes then the (weighted Euclidean) distance (PCA distance) between two distinct points in the coordinate space of the PCA coefficients. 7 The spectral information of the Stokes profiles (i.e., number of wavelength positions and spectral sampling) is fully encoded in the PCA basis, and so it does not need to appear again in the inversion database. This is another advantage of PCA, since it permits a significant compression of the database information.

10 10 R. Casini Figure 3. Magnetic map of an active-region filament observed in He I 1083 nm on July 5, 2005, with the Vacuum Tower Telescope and the Tenerife Infrared Polarimeter. This inversion was performed using PCA, with an inversion database of about 250,000 models specifically built for on-disk observations of prominence structures. An example of profile inversion is given in Fig. 4, for the point of coordinate (19,31) marked by a black pixel in the B-map.

11 Solar Magnetic Field Measurement 11 Figure 4. Observed Stokes profiles and PCA-inversion fits for the point of coordinates (19,31) in the map of Fig. 3. For the inversion, the observed Stokes profiles are projected on the same PCA basis, and the corresponding set of vector components is then compared with the coordinate entries in the inversion database. The best fit is provided by the database point that is closest (in the sense of the PCA distance previously defined) to the set of PCA coordinates of the observed Stokes vector. The inverted physical model is obviously the parameter configuration label of the selected point in the inversion database. We would like to conclude this discussion on PCA inversion by summarizing the advantages and disadvantages of the method. The benefit of confining the forward modeling for a given problem to a one-time effort, which is the very rationale of pattern-recognition inversion techniques, has been discussed already. The actual inversion of the observed signals is then just a search of the point in the inversion database that minimizes the PCA distance to the observations. This search involves only a small amount of simple operations, and is thus performed very rapidly. The inversion is also stable, since it could be repeated starting from different points in the same database, and it would still find the same absolute minimum of the PCA distance. (Incidentally, this opens to the possibility of optimizing the search algorithm without affecting the quality of the inversion.) Finally, the search algorithm for the inversion is clearly model independent, and so the coding effort needed to create new inversion codes for different problems is all concentrated in the development of the corresponding forward models. There are nonetheless some important disadvantages associated with this inversion strategy. First of all, the inversion database contains a definite number of models, and therefore there is no easy path to solution refinement. The inversion errors are ultimately determined by the density of the database. Of course, the database can be

12 12 R. Casini augmented by adding new points, but its size can become unmanageably large for an efficient search, depending on the dimensionality of the parameter space, the ranges of the parameters, and the target for the inversion error. The problem of solution refinement for PCA inversion is still being investigated. Possible strategies are the use of Levenberg-Marquardt optimization (this is successfully implemented as an inversion strategy in the HAZEL code; see Asensio Ramos et al. 2008) using the PCA solution as initial guess, or interpolation methods applied to the existing inversion database. Both approaches would require that the PCA solution be sufficiently close to the fully converged solution. This suggests the existence of a critically minimal density of the inversion database for a given problem, which is however difficult to quantify in general mathematical terms. In our opinion, this is where the effort on the investigation of pattern-recognition techniques for spectro-polarimetric inversion should be concentrated in the near future. We finally present an example of PCA inversion applied to the magnetic diagnostics of solar prominences. Figure 3 shows a magnetic map of an active-region filament that was observed spectro-polarimetrically in He I 1083 nm on July 5, 2005, using the Tenerife Infrared Polarimeter at the German Vacuum Tower Telescope in the Canarias (Kuckein et al. 2009). The filament was embedded in the region NOAA around the point of coordinates N13-W29, with an angular distance from disk center θ 23. For the inversion we used a PCA database of 250,000 models with magnetic field strengths between 0 to 2000 G, and LOS inclinations between 20 and 30. The various panels in the figure show (from top to bottom, and left to right): the intensity map at line center, the magnetic strength, the geometry of the magnetic field, expressed by the inclinationθ B from the local vertical and the azimuthφ B measured counterclockwise from the solar radius through the observed point, the estimated optical depth at line center of the slab representing the prominence plasma, the plasma temperature inclusive of micro-turbulence, the LOS velocity with respect to the laboratory frame, and the (not normalized) PCA distance between the observations and the best fit. All nearly 1350 pixels in this map are inverted. On a single CPU (processor Intel Core2 T GHz) the inversion takes about 2 min, not including the overhead time for reading in the PCA database. Because each pixel in the map is inverted independently from all the other pixels, the problem is naturally parallelizable, and the inversion time scales inversely proportionally with the number of CPUs. Figure 4 shows the Stokes profile fit from the inversion of the point of coordinate (19,31) in the map of Fig. 3 (see caption). The legend in the figure reports the inverted values for the magnetic and plasma parameters.δ gives the PCA distance for the inversion of the point, and it measures the goodness of the fit. The estimated height h of the emitting region at the inverted point, and the corresponding inclination θ of the LOS are also derived from the inversion. With these we can compute the geometry of the field in the reference frame of the observer (Θ B,Φ B ), which is also given in the legend. The errors on the inverted values represent the standard deviations of the distributions of the corresponding parameters for the 20 models in the inversion database that are the closest (in the sense of the PCA distance) to the observations. Because the PCA inversion approach is completely transparent to the specificity of the forward model, inversions of the type shown in this section can readily be extended to different plasma conditions and diagnostic lines. At HAO we have developed codes providing the solution of the forward problem for radiative transfer with atomic polarization for various types of energy structure of the atomic system in the presence

13 Solar Magnetic Field Measurement 13 Atomic Structure Configuration Level Interferences Example of Application multi-level+b αjm None Fe I nm scattering polarization at the solar limb multi-term+b β(ls )JM J-J He I lines in prominences and filaments multi-level+hfs+b α(ji)fm F-F Na I D 1 and D 2 lines multi-term+hfs+b β(ls )(JI)FM J-J +F-F Na I D-doublet including super-interferences H I+(B,E) β(ls )JM L-L +J-J H I/He II prominence diagnostics H I+HFS+(B,E) β(ls )(JI)FM L-L +J-J +F-F H I/He II prominence diagnostics Table 1. Table of forward modeling codes for magnetic diagnostics with atomic polarization developed and currently maintained at HAO. The multi-term atom in a magnetic field (second entry) corresponds to the forward model of HANLE.FS, which is listed in Fig. 2 among the codes supported through CSAC. of external magnetic and electric fields. These are listed in Table 1. These codes can be used for the creation of PCA databases that cover the diverse conditions of polarized line formation found in the solar atmosphere. 5. Conclusions To this day, the spectro-polarimetric analysis of the radiation that we receive from the different regions of the solar atmosphere remains the privileged means to diagnose the thermodynamic properties of the emitting plasmas, and the topology and energy distribution of the magnetic fields that permeate these regions. Progress in our ability to observe and interpret the spectro-polarimetric signature of solar magnetism in multiple spectral lines, probing the solar atmosphere from the photosphere to the corona, is essential for the ultimate goal of gaining predictive capabilities in space weather, and in particular on the effects of solar phenomena on the terrestrial environment. The diagnostic tools that are necessary to attack this problem under all the diverse thermodynamic and magnetic conditions of the solar atmosphere require sophisticated models of polarized line formation that must go beyond the traditional account of the Zeeman effect. These models must deal with the problem of the formation of atomic polarization, and how this is modified by the magnetic field by a variety of atomic processes, which go from the Hanle effect to the complex phenomena associated with level-crossing interferences (see, e.g., Trujillo Bueno 2001; Landi Degl Innocenti & Landolfi 2004; Casini & Landi Degl Innocenti 2008). The high level of complexity of the forward problem determined by this generalization, and the large and seamless data volume that is expected from both ground-based and space-borne instruments that are now being constructed or designed such as the Advanced Technology Solar Telescope, the European Solar Telescope, and Solar-C require the development of ultra-fast inversion techniques, so to enable the solar community to take full advantage of the discovery potential offered by these multi-national projects. This necessity is reinforced by the aforementioned need to continually advance our capability in the modeling and interpretation of polarized radiation in magnetic diagnostic lines. As we tried to demonstrate in this paper, the true advance lies in the future implementation of the general forward problem including atomic polarization into existing

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