Let us consider first the simplest method of optically thin EUV or soft X-ray emission. The brightness of the solar corona observed in EUV or soft X-rays is produced by optically thin free-free emission, which can be characterized by an emission measure that is proportional to the squared electron density and column depth along the line-of-sight at a given temperature (assuming that the plasma is in local thermal dynamic equilibrium, which is not always the case for loops undergoing rapid heating),

while the intensity or flux measured at a wavelength represents the temperature integral of the differential emission measure multiplied with the filter response function , which defines the general inversion problem for the 3D density distribution , for a given temperature filter , with being the conversion factor of the squared density to the observed brightness (as defined by Eqs. (4) and (5)), and is the data noise. The absolute coordinate system with origin in the Sun center is given in coordinates with the x-axis co-aligned to the solar rotation direction, while is the image coordinate system rotated around the solar axis (y-axis) by the line-of-sight angle according to, Thus, if we have an image with a size of pixels and want to reconstruct the 3D density distribution with voxels, the problem is under-constrained. Using the variation of the aspect angle, for instance by taking multiple images () over a time interval with significant solar rotation (of the aspect angle ), will increase the number of constraints to , as long as the reconstructed volume is static, but a unique solution would theoretically require images, which is not feasible in practice. The inversion of the 3D density distribution is therefore always under-constrained and requires special inversion techniques with additional constraints. One way to reduce the degrees of freedom is to reduce the solution volume by using a spherical coordinate system aligned with the solar surface and by restricting the number or radial voxels to a small number that covers only about the lowest density scale height, see for instance Frazin et al. (2009b) (Figure 15, bottom left).The simplest tomographic reconstruction method is the backprojection method, which yields a probability distribution based on the linear addition of projections from different directions (Figure 4). Another (under-constrained) inversion method that has been used for solar tomography is the robust, regularized, positive estimation scheme (Frazin, 2000; Frazin and Janzen, 2002). The combination of differential emission measure analysis (Eq. (4)) and solar rotation tomography allows in principle to reconstruct the average density and temperature in each voxel (Frazin et al., 2005b; Frazin and Kamalabadi, 2005b). Besides the problem of under-constrained inversion, the time variability is an additional challenge, which could be overcome with Kalman filtering (Frazin et al., 2005a). The first attempt to reconstruct the 3D density distribution in the solar corona by means of solar-rotation tomography was done using a 2-week’s dataset of soft X-ray images from Yohkoh (Hurlburt et al., 1994).

Let us consider the second wavelength regime, i.e., white light, which undergoes Thompson scattering in the solar corona, which is sensitive to the geometry of the distribution of scattering particles and the direction to the observer. The scattering cross-section depends on the angle between the line-of-sight and the radial direction through the scattering electron as (Jackson, 1962),

where is the differential cross-section in units of [] and is the classical electron radius. By integrating over all solid angles we obtain the total cross-section for perpendicular scattering, the so-called Thompson cross-section for electrons, The total scattered radiation can then be calculated by integrating over the source locations of the photons (the photosphere) and the scattering electrons (with a 3D distribution ) along the line-of-sight , as a function of the scattering angle with respect to the observers line-of-sight, which was first calculated by Minnaert (1930), van de Hulst (1950), and Billings (1966). For a recent review see Howard and Tappin (2009a). The degree of polarization , which is observed in the polarized brightness (pB) component of white-light images, is defined as, where and represent the tangential and radial terms of the total scattered radiation, as given in Minnaert (1930), van de Hulst (1950), Billings (1966), or Howard and Tappin (2009a). Many coronagraphs (such as those on STEREO) have the capability to measure linear polarization in three orientations, from which the total and polarized brightness can be derived. The total scattered radiation is proportional to the electron density of the scattering corona, in contrast to the square-dependence of the observed brightness on the density (Eq. (4)) for free-free emission in soft X-rays and EUV.Since the differential cross section (Eq. (8)) varies only by a factor of two with angle , the plane-of-the-sky or plane-of-max-scattering approximations are very poor. It is essential for any tomography approach which aims to reconstruct an extended density distribution to treat the observations as the result of an extended integral along the line-of-sight. The plane of maximum scattering has often been approximated with the plane-of-sky in the past, which is appropriate for locations near the solar limb, but needs to be corrected with the actual plane of maximum scattering, for sources at large distances from the Sun, indicated with the impact radius in Figure 5 (Vourlidas and Howard, 2006). This correction becomes relevant for tomographic 3D reconstruction of CMEs that propagate far away from the Sun. Using a time series of images in polarized brightness taken over a time interval with significant solar rotation (which varies the line-of-sight angles) allows then to deduce the 3D density distribution in a coronal volume.

The first tomographic reconstruction of the 3D density distribution of the solar corona, based on a time series of coronagraph images from Skylab in white light has been accomplished by Altschuler (1979). Tomographic inversion of coronagraph images from Mark III K-coronameter on Hawaii and from LASCO C-1 were conducted by Zidowitz et al. (1996), Zidowitz (1997), and Zidowitz (1999). Tomography of the solar corona in an altitude range of a few solar radii has been systematically investigated by regularization inversion methods (e.g., Frazin, 2000; Frazin and Kamalabadi, 2005b), and applied to LASCO C-2 datasets (Frazin and Janzen, 2002; Morgan et al., 2009; Morgan and Habbal, 2010), to Mauna Loa Solar Observatory Mark-IV coronameter data (Butala et al., 2005), to STEREO COR-1 datasets (Kramar et al., 2009; Barbey et al., 2011), and by separating contributions of the K- and F-corona (Frazin and Kamalabadi, 2005a).

Solar radio emission has many different emission mechanisms, which moreover have complicated properties depending on the frequency (or wavelength), polarization, opacity, and magnetic field (for an overview see, e.g., Chapter 15 in Aschwanden, 2005, and references therein). For 3D tomography it matters a lot whether radio emission is observed in an optically thin or thick regime. If an optically thin feature is observed (e.g., free-free emission at decimetric frequencies), the radio brightness has to be calculated from a line-of-sight integral with varying opacity, similar as for EUV or soft X-ray wavelengths. In contrast, if optically thick emission is observed (e.g., gyroresonance emission in microwaves), the radio brightness originates from a localized source surface that can be treated like an opaque body in the 3D reconstruction.

Let us consider the case of free-free (bremsstrahlung) emission, which has a free-free absorption coefficient for thermal electrons that depends on the ambient ion density , electron density , and temperature , at position along a given observer’s line-of-sight, and radio frequency as,

where is the Coulomb integral, which yields the free-free opacity as a function of position by integrating over the column depth range , From the free-free opacity we obtain the radio brightness temperature at the observer’s frequency with a further integration of the opacity along the line-of-sight,An example of a quiet-Sun radio brightness spectrum is given in Figure 6 for different coronal temperatures in hydrostatic equilibrium. The observed quantity is the flux density , which can be calculated from the brightness temperature with the Rayleigh–Jeans approximation at radio wavelengths, integrated over the solid angle of the radio source,

These expressions describe how the radio brightness observed at a particular frequency depends on the 3D density and temperature distribution of an observed source (e.g., an active region), and this way describes the inversion problem that has to be solved to obtain the 3D density distribution . One way to obtain a large number of observational constraints, is to synthesize radio images at many frequencies , a method that is called frequency tomography (Aschwanden et al., 1992; Aschwanden, 1995), which was pioneered with the multi-frequency imaging radio interferometer RATAN-600 with up to 36 frequencies in the range (Bogod and Grebinskij, 1997; Gelfreikh, 1998; Grebinskij et al., 2000), and has been simulated for a proposed Frequency-Agile Solar Radiotelescope (FASR) (Aschwanden et al., 2004). However, the frequency tomography method allows only to invert the electron density and temperature as a function of the radio frequency , while an additional opacity model is required to map the radio frequency into a geometric coordinate for each line-of-sight. Such additional information on the absolute height (at an angle from Sun center) can be obtained from solar-rotation stereoscopy (see Section 3.1).
Living Rev. Solar Phys. 8, (2011), 5
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