3.5 Stereoscopic tomography and 3D forward-fitting

We turn now to “true tomographic methods”, which require simultaneous observations from multiple aspect angles, in contrast to the “pseudo-tomographic methods” using the solar rotation to vary the aspect angle over time as we described in Section 3.2. One of the simplest methods is the backprojection method (Figure 4View Image). The principle of the backprojection method in tomographic reconstruction, as used in medical tomography, was first demonstrated for a solar image (from Skylab) using 80 projections (Davila and Thompson, 1992). The application to a multi-spacecraft configuration was then simulated (Figure 13View Image), which demonstrated that a high fidelity of 3D reconstruction can already be achieved for a small number of (>∼ 4) spacecraft (Davila, 1994, 1996, 1998). In reality, however, we have only two spacecraft with a stereoscopic view since the launch of STEREO, or three if we combine with a near-Earth spacecraft such as SOHO or SDO. Three viewpoints are not enough to permit tomographic reconstruction with classical methods, but adding some a priori information on the geometry of a structure may allow to determine the density structure with only three viewpoints (Frazin et al., 2009a).
View Image

Figure 14: Rendering of solid flux tubes computed with a potential magnetic field model and hydrodynamic 1D models for each loop. The rendering displays a 3D view by shading, but can be scaled to electron densities, temperatures, or brightness in a particular temperature filter (from Gary, 1997Jump To The Next Citation Point).

One a priori information for tomography is a magnetic field model, possibly constrained by stereoscopic triangulation, as we described in the previous section on magnetic stereoscopy. The subset of loops (or loop segments) than could be triangulated in an active region, serves then as a skeleton, while the magnetic field model can fill in an arbitrary set of auxiliary field lines to fill the entire coronal volume. In the case where no fitting theoretical magnetic field line can be found (or be trusted), auxiliary field lines can also be generated by 3D interpolation (Aschwanden et al., 2009cJump To The Next Citation Point), although this method works only well for a sufficient number of reliably triangulated skeleton field lines, and may contain unphysical solutions that do not fulfill the Maxwell equation of divergence-freeness (∇ ⋅ B = 0) in underpopulated regions. Once a full 3D magnetic field is established for some coronal volume, each field line [x(s),y(s),z(s)] can be populated with a hydrodynamic loop model of the electron temperature Te(s) and electron density ne(s). The emission measure dEM (T)∕dT can then be integrated along each line-of-sight z, convolved with a filter response function R (T) (Eqs. (4View Equation) and (5View Equation)), and EUV and soft X-ray images can be rendered for arbitrary instrument filters (Figure 14View Image). Such simulations of coronal images with rendering for particular instrument filters were already simulated in the pre-STEREO era (Gary, 1997; Alexander et al., 1998) and forward-fitted to observed coronal images (Schrijver et al., 2004; Mok et al., 2005; Lundquist et al., 2008a,b), as well as to image pairs from STEREO in three wavelength filters (Aschwanden et al., 2009cJump To The Next Citation Point).

Alternative methods of 3D reconstruction methods of coronal structures that use auxiliary information involve the line-of-sight velocity measurements of plasma flows in flare loops (Nitta et al., 1999Jump To The Next Citation Point) or in coronal loops (Alissandrakis et al., 2008), time-evolving tomographic 3D reconstruction of polar plumes (Barbey et al., 2008Jump To The Next Citation Point), and multiscale optical-flow methods applied to erupting filaments (Gissot et al., 2008Jump To The Next Citation Point).

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