The radiative transfer equation has a formal solution in the form:

where is the evolution operator, which needs to be evaluated at every layer in order to perform the integration. During the 1960s and early 1970s, the first numerical solutions to the radiative transfer equation for polarized light became available (Beckers, 1969a,b; Stenflo, 1971; Landi Degl’Innocenti and Landi Degl’Innocenti, 1972; Wittmann, 1974a; Auer et al., 1977). Techniques to solve Equation (1) have continued to be developed even during the past two decades (Rees et al., 1989; Bellot Rubio et al., 1998; López Ariste and Semel, 1999a,b).Figure 1 shows an example of how the Stokes vector varies when the magnetic field vector changes. In that movie we use spherical coordinates to represent the three components of the magnetic field vector: , where is the strength of the magnetic field, is the inclination of the magnetic field with respect to the observer, and is the azimuth of the magnetic field in the plane perpendicular to the observer’s line-of-sight. In Figure 1 we assume that the observer looks down along the z-axis, but this does not need to be always the case.

A major milestone was reached when these methods to solve the RTEs (1) and (4) were implemented into efficient minimization algorithms that allow for the retrieval of magnetic field vector in an automatic way (Ruiz Cobo and del Toro Iniesta, 1992; Ruiz Cobo, 2007; Socas-Navarro, 2002; del Toro Iniesta, 2003a; Bellot Rubio, 2006). This retrieval is usually done by means of non-linear minimization algorithms that iterate the free parameters of the model (Equation (2)) while minimizing the difference between the observed and theoretical Stokes profiles (measured by the merit function ). The that minimizes this difference is assumed to correspond to the physical parameters present in the solar atmosphere:

Here and represent the observed and theoretical (i.e., synthetic) Stokes vector, respectively. The latter is obtained from the solution of the RTE (4) given a particular set of free parameters (Equation (2)). The letter represents the total number of free parameters in and, thus, the term represents the total number of degrees of freedom of the problem (number of data points minus the number of free parameters). In Equation (5), indexes and run for the four components of the Stokes vector (I,Q,U,V ) and for all wavelengths, respectively. Finally, represents the error (e.g., noise) in the observations .

Traditionally, the -minimization has been carried out by minimization algorithms such as the Levenberg–Marquardt method (Press et al., 1986). However, more elaborated methods have also been employed in recent years: genetic algorithms (Charbonneau, 1995; Lagg et al., 2004), Principal-Component Analysis (Rees et al., 2000; Socas-Navarro et al., 2001), and Artifical Neural Networks (Carroll and Staude, 2001; Socas-Navarro, 2003).

Due to our limited knowledge of the line-formation theory, these investigations have usually been limited to the study of the photospheric magnetic field, where Local Thermodynamic Equilibrium and Zeeman effect apply for the most part. However, recent advancements in the line-formation-theory under NLTE conditions (Mihalas, 1978), scattering polarization (Trujillo Bueno et al., 2002; Manso Sainz and Trujillo Bueno, 2003; Landi Degl’Innocenti and Landolfi, 2004), etc., allow us to extend these techniques to the study of the chromosphere (Socas-Navarro et al., 2000a; Asensio Ramos et al., 2008; Trujillo Bueno, 2010; Casini et al., 2009). Indeed, some recent works have appeared where the magnetic structure of sunspots in the chromosphere is being investigated (Socas-Navarro et al., 2000b; Socas-Navarro, 2005a,b; Orozco Suarez et al., 2005).

Techniques to study the coronal magnetic field from polarimetric measurements of spectral lines are also becoming available nowadays (Tomczyk et al., 2007, 2008). These observations, carried out mostly with near-infrared spectral lines, are recorded using coronographs (to block the large photospheric contribution coming from the solar disk) and, therefore, limited to the solar limb. Other possibilities to observe polarization on the solar disk involve EUV (Extreme Ultra Violet) lines, which are only accessible from space, and radio observations of Gyroresonance and Gyrosynchrotron emissions, which can show large polarization signals: White (2001, 2005), Brosius et al. (2002), and Brosius and White (2006). Unfortunately, so far radio measurements have allowed only to infer the magnetic field strength in the solar corona. Interestingly, opacity effects in the gyroresonance emission (see Equations (1) and (2) in White, 2001) might also permit to infer the inclination of the magnetic field vector with respect to the observer’s line-of-sight, i.e., . However, this possibility has not been yet successfully exploited.

According to Equations (1) and (2) the solution to the radiative transfer equation depends on the stratification with optical depth of the physical parameters. The range of optical depths in which the solution will be valid depends on the region of the photosphere in which the analyzed spectral lines are formed. In the future we will refer to this range as . can be determined by means of the so-called contribution functions (Grossmann-Doerth et al., 1988; Solanki and Bruls, 1994) and the response functions (Landi Degl’Innocenti and Landi Degl’Innocenti, 1977; Ruiz Cobo and del Toro Iniesta, 1994). In the literature, it is usually considered that the range of optical depths, that a given spectral lines is sensitive to, is so narrow that the physical parameters do not change significantly over . This can be mathematically expressed as:

where refers to the -component of (Equation (2)). When the conditions in Equation (6) are met for all ’s, a Milne–Eddington-like (ME) inversion can be applied. The advantage of ME-codes is that an analytical solution for the RTE (1) exists in this case. ME-codes assume that the physical parameters are constant in the range . One way to determine the magnetic field at different heights in the solar atmosphere is to perform ME-inversions of spectropolarimetric data in several spectral lines that are formed at different average optical depths ’s, with each spectral line yielding information in a plane at a different height above the solar surface.As an example of the results retrieved by a Milne–Eddington-like inversion code we show, in Figures 2 and 3, the three components of the magnetic field vector, for two different sunspots, in the observer’s reference frame. or magnetic field strength is shown in the upper-right panels, or the inclination of the magnetic field vector with respect to the observer’s line-of-sight in the lower-left panels, and finally, or the azimuthal angle of the magnetic field vector in the plane perpendicular to the observer’s light-of-sight in the lower-right panels. The first sunspot, AR 10923 (Figure 2), was observed very close to disk center () on November 14, 2006. The second sunspot, AR 10933 (Figure 3), was observed on January 9, 2007 very close to the solar limb (). In both cases, the magnetic field vector was obtained from the VFISV Milne–Eddington-type inversion (Borrero et al., 2010) of the Stokes vector recorded with the spectropolarimeter on-board the Japanese spacecraft Hinode (Suematsu et al., 2008; Tsuneta et al., 2008; Ichimoto et al., 2008a). The observed Stokes vector corresponds to the Fe i line pair at 630 nm, which are formed in the photosphere. As explained above, Milne–Eddington inversion codes assume that, among others, the magnetic field vector does not change with optical depth: (see Equation (2)). Therefore, Figures 2 and 3 should be interpreted as the averaged magnetic field vector over the region in which the employed spectral lines are formed: .

When the conditions in Equation (6) are not met, it is not possible to perform a ME-line inversion. If we do, the results should be interpreted accordingly, that is, the inferred values for correspond to an average over the region where the spectral line is formed. A different approach consists in the application of inversion codes for the radiative transfer equation that consider the full dependence of the physical parameters . In this case, the solution of the radiative transfer equation can only be found numerically (cf. López Ariste and Semel, 1999b). Examples of these codes are: SIR (Ruiz Cobo and del Toro Iniesta, 1992), SPINOR (Frutiger et al., 1999), and LILIA (Socas-Navarro, 2002). This allows to obtain the optical depth dependence (-dependence) of the physical parameters with one single spectral line. Ideally, in order to increase the range of validity of the inferred models, one still wants to employ different spectral lines.

The elements of the propagation matrix (Equation (1)) for the linear polarization (see, e.g., del Toro Iniesta, 2003b, Chapter 7.5) can be written as:

where corresponds to the azimuthal angle of the magnetic field vector in the plane perpendicular to the observer’s line-of-sight. Equations (7) and (8) also hold for the dispersion profiles (magneto-optical effects) and present in the propagation matrix . Note that these matrix elements remain unchanged if we take instead of . Because of this the radiative transfer equation cannot distinguish between these two possible solutions for the azimuth: []. This is the so-called 180°-ambiguity problem in the azimuth of the magnetic field. Because of this ambiguity, the azimuthal angle of the magnetic field (as retrieved from the inversion of spectrolarmetric data) in Figures 2 and 3 (lower-right panels) is displayed only between 0°and 180°. A number of techniques have been developed to solve this problem. These techniques can be classified in terms of the auxiliary physical quantity that is employed:- Acute-angle methods: these techniques minimize the angle between the magnetic field vector inferred from the observations (see Section 1.3) and the magnetic field vector obtained from a given model. The question is, therefore, how is the model magnetic field obtained. Traditionally, it is obtained from potential or force-free extrapolations of the observed longitudinal component of the magnetic field: , which is independent. The extrapolation yields the horizontal component of the magnetic field, which is then compared with the two possible ambiguous solutions: and . Whichever is closer to the extrapolated horizontal component is then considered to be the correct, ambiguity-free, solution. Potential field and force-free extrapolations can be obtained employing Fourier transforms (Alissandrakis, 1981; Gary, 1989). Some methods that solve the 180°-ambiguity employing this technique have been presented by Wang (1997) and Wang et al. (2001). In addition, Green’s function can also be used for the extrapolations and to solve the ambiguity (Sakurai, 1982; Abramenko, 1986; Cuperman et al., 1990, 1992).
- Current free and null divergence methods: these methods select the solution, or , that minimizes the current vector and/or the divergence of the magnetic field: . The calculation of these quantities makes use of the derivatives of the three components of the magnetic field vector. Because the vertical (z-axis) derivatives are usually not available through a Milne–Eddington inversion (see Sections 1.3.1, 2.1, and 2.3) only the vertical component of the current is employed. In addition, the term is neglected in the calculation of the divergence of the magnetic field. The minimization of the aforementioned quantities can be done locally or globally. Finally, note that current free and null divergence methods usually rely on initial solutions given by acute-angle methods and potential field extrapolations.

In recent reviews by Metcalf et al. (2006) and Leka et al. (2009) several of these
techniques are compared against each other, employing previously known magnetic field
configurations and measuring their degree of success employing different metrics when
recovering the original one. It is important to mention that in these reviews, some other very
successful methods (which do not necessarily fall into the aforementioned categories) are also
employed^{2}:
the non-potential magnetic field calculation method by Georgoulis (2005) and the manual utility
AZAM by Lites et al. (private communication), which is part of the ASP routines (Elmore
et al., 1992). In those reviews it is found that acute-angle methods perform well only if the
configuration of the magnetic field is simple, whereas interactive methods (AZAM) tend to fail in the
presence of unresolved structures below the resolution element of the observations. Current free
and null divergence methods tend to work better when both conditions (Canfield et al., 1993;
Metcalf, 1994) are applied instead of only one (Gary and Demoulin, 1995; Crouch and Barnes,
2008), with local minimization being more prone to propagate errors than global minimization
techniques.

Several of these techniques are very suitable to study complex regions, in particular outside
sunspots. However, in regular sunspots (excluding those with prominent light bridges or
-sunspots^{3})
the magnetic field is highly organized, with filaments that are radially aligned in the penumbra. We can use
this fact to resolve the 180°-ambiguity in the determination of the azimuthal angle . This is
done by finding the coordinates of the magnetic field vector in the local reference frame:
^{4}
and taking whichever solution, or , minimizes the following quantity:

Because we aim at minimizing the above value (Equation (9)) this method can be considered as an acute-angle method where the reference magnetic field is not obtained from a potential extrapolation but rather assumed to be radial. Note that if the sunspot has positive polarity, the magnetic field vector and the radial vector tend to be parallel: and, therefore, the (minus) sign should be used in Equation (9). If the sunspot has negative polarity, then the magnetic field vector and the radial vector are anti-parallel and, therefore, the sign (plus) should be employed. However, this is only a convention: we can choose to represent the magnetic field vector as if a sunspot had a different polarity as the one indicated by Stokes V .

As an example of the method depicted here we show, in Figures 4, 5, and 6, the vertical and horizontal and components of the magnetic field vector (Equation (46)), once the 180°-ambiguity has been resolved for two sunspots: AR 10923 and AR 10933 (same as in Figures 2 and 3). , , and are the components of the magnetic field vector in the local reference frame. Note that strictly speaking, the unit vectors and shown in these figures correspond to the unit vectors at the umbral center. Although differences are small, at other points in the image the unit vectors have different directions since those points have different (,) and () coordinates (Equations (29) – (33)). Once the 180°-ambiguity has been solved we can obtain, in the local reference frame, the inclination and the azimuth of the magnetic field, (Figure 7) and (Figure 8) as:

It is important to notice that because the ambiguity has now been solved, the angle varies between 0° and 360° (see Figure 8), whereas before, lower-right panels in Figures 2 and 3, ranged only between 0° and 180°.

As already mentioned, the method we have described here works very well for regular (e.g., round)
sunspots. There is, however, one important caveat: when the retrieved inclination (in the observer’s
reference frame) is close to 0, the azimuth is not well defined. In this case, applying Equation (9) does
not make much sense. Here we must resort to other techniques (Metcalf et al., 2006) to solve the ambiguity.
The region where = 0° occurs usually at the center of the umbra for sunspots close to disk center,
and it shifts towards the center-side penumbra as the sunspot is closer to the limb. A similar
coordinate transformation as the one depicted here have been described in Hagyard (1987) and
Venkatakrishnan et al. (1988), with the difference that no attempt to solve the 180°-ambiguity was made.
Bellot Rubio et al. (2004) and Sánchez Almeida (2005a) employ a smoothness condition
to solve the 180°-ambiguity, however their coordinate transform is done in two dimensions,
whereas here we consider the Sun’s spherical shape. In addition, only one heliocentric angle
was considered in their transformation, whereas here changes for each point on the
solar surface (Equation (39)). One might think that the variation of the angle across the
field-of-view (FOV) are negligible. However, for a FOV with 100 × 100 arcsec^{2} this variation
can be as large as 4 – 5°. These differences can be important, for instance, when searching for
regions in the sunspot penumbra where the magnetic field points down into the solar surface:
.

Traditionally, inversion codes for the RTE (1) such as: SIR (Ruiz Cobo and del Toro Iniesta, 1992) and SPINOR (Frutiger et al., 1999), provide the physical parameters as a function of the optical depth, (Equation (2)). The optical depth is evaluated at some wavelength where there are no spectral lines (continuum), hence the sub-index c. When this is done for each pixel in an observed two-dimensional map, the inversion code yields . However, it is oftentimes convenient to express them as a function of the geometrical height . To that end, the following relationship is employed:

where is the opacity evaluated at a continuum wavelength and depends on the temperature, gas pressure, and electron pressure. Now, these thermodynamic parameters barely affect the emergent Stokes profiles and, therefore, are usually not obtained from the inversion of the polarization profiles themselves. Instead, other kind of constraints are usually employed to determine them, being the most common one, the application of the vertical hydrostatic equilibrium equation: which after applying Equation (12) becomes:Note that, since Equations (13) and (14) do not depend on , they can be applied independently for each pixel in the map. Hence, the geometrical height scale (at each pixel) can be obtained by following the next steps:

- Given a boundary condition for the gas pressure in the uppermost layer of the atmosphere, , we can employ the fixed-point iteration described in Wittmann (1974a) and Mihalas (1978) to obtain the electron pressure in this layer: .
- From the inversion, the full temperature stratification ) and, thus, are known. Since the continuum opacity depends on the electron pressure, gas pressure, and temperature, it is therefore possible to obtain .
- A predictor-corrector method is employed to integrate downwards Equation (14) and obtain . This is done by first assuming that is constant between and : and with , we apply step #1 to calculate .
- Since we also know , we repeat step #2 to recalculate , which is then employed to re-integrate Equation (14) as: Step #4 is repeated -times until convergence: .
- We now have . In addition, ) and, thus, are known. Consequently, we can repeat steps #1 to #3 in order to infer .
- Thus, repeating steps #1 through #5 yields: , , and .
- The equation of ideal gases can be now employed to determine . And, finally, the integration of Equation (12) yields the geometrical depth scale as: . To integrate this equation, a boundary condition is needed. This is usually taken as , which sets an offset to the geometrical height such that the continuum level coincides with .

Applying the condition of hydrostatic equilibrium to obtain the density, gas pressure, and the
geometrical height scale z is strictly valid only when the Lorentz force are small and the velocities are much
smaller than the speed of sound. In the chromosphere and corona this is certainly not the case. In
the solar photosphere the assumption of hydrostatic equilibrium is, in general, well justified.
One exception are sunspots, where the large velocities and magnetic fields might break down
this assumption. In these case, a more general momentum (force balance) equation must be
employed^{5}:

Trying to solve this equation to obtain the gas pressure, density, and geometrical height scale is not an easy task. In the hydrostatic case, the horizontal derivatives did not play any role, thus simplifying Equation (17) into:

However, if the Lorentz force and the advection term cannot be neglected, the horizontal components of the momentum equation must be considered. In addition, the horizontal derivatives of the gas pressure mix the results of the magnetic field and velocity from nearby pixels. Thus, the determination of the gas pressure, density, and geometrical height scale cannot be achieved individually for each pixel of the map. Instead, a global technique must be employed. This can be done by shifting the z-scale at each pixel in the map (effectively changing the boundary condition mentioned in step #7 above) in order to globally minimize the imbalances in the three components of the momentum equation and the term . The shift at each pixel, , represents the Wilson depression. This kind of approach has been followed by Maltby (1977), Solanki et al. (1993), Martínez Pillet and Vazquez (1993), and Mathew et al. (2004). However, changing the boundary condition in step #7 does not change the fact that the vertical stratification of the gas pressure still complies with hydrostatic equilibrium (Equation (13)). A way out of this problem has not been figured out until very recently with the work of Puschmann et al. (2010a,b), who have devised a technique that takes into account the general momentum equation (17) when determining the gas pressure and establishing a common z-scale. Figure 9 shows a map for the Wilson depression in a small region of the inner penumbra of a sunspot (adapted from Puschmann et al., 2010b). Another interesting technique has been proposed recently by Carroll and Kopf (2008), where the vertical height scale can be obtained, instead of a posteriori as in Puschmann et al. (2010b), directly during the inversion of the Stokes profiles. This is achieved by performing the inversion employing Artificial Neural Networks (ANNs; Carroll and Staude, 2001, see Section 1.3) that have been previously trained with snapshots of MHD simulations, which are given in the z-scale.

Living Rev. Solar Phys. 8, (2011), 4
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