5 Summary

Most physical models of solar phenomena require the knowledge of the 3D geometry of the object, which instigates reliable 3D reconstruction methods, such as stereoscopy and tomography. Stereoscopy requires a view from two different vantage points, which is now most suitably provided by the dual STEREO spacecraft, but has also been applied to other multi-spacecraft observations before, or using the solar rotation for quasi-static phenomena. Tomography is a technique to combine imaging informations from multiple slices (with different aspect angles) into a 3D density distribution, which can only be carried out in a minimal way in the solar case, by using two or three spacecraft views, unless the solar rotation is used and the object remains quasi-stationary during the time interval of 3D reconstruction. We see that both the stereoscopy and tomography method share the trade-offs of (i) stationarity for solar rotation based methods, or (ii) minimal aspect angle coverage for simultaneous observations, due to the low number of available spacecraft. Nevertheless, stereoscopic and tomographic 3D reconstruction methods became more and more refined by taking stock of auxiliary geometric information from magnetic field models and opacity models, methods that also called “magnetic stereoscopy” or “frequency tomography”. Also solar rotation based methods became more sophisticated by incorporating the dynamics of solar phenomena to first order, such as the “dynamic stereoscopy” or “Kalman filter tomography” methods. Of course, the biggest boost came from the launch of the STEREO mission in 2006, which produced a tenfold increase of publications dealing with solar stereoscopy and tomography. In this review here we focused on stereoscopic and tomographic reconstruction of phenomena in the solar corona, but omitted applications in the heliosphere (interplanetary CMEs) or in the solar interior (helioseismic tomography).

The stereoscopic and tomographic 3D reconstruction methods are wavelength-dependent and require a proper treatment of the opacity in each wavelength. Observations in EUV and soft X-rays involve optically thin free-free emission and have an additive characteristic of the squared electron density integrated along each line-of-sight. Observations in white-light are governed by Thompson scattering and scale in brightness proportional to the (scattering) electron distribution along a line-of-sight. Observations at radio wavelengths can be dominated by free-free emission in the optically-thin regime, or by gyroresonance emission in the optically-thick regime and, thus, requires a 3D density and temperature model for each line-of-sight. Classical stereoscopy represents a simple geometric triangulation from two view points to obtain the 3D coordinates of a point-like or curvi-linear structure, but solar applications are somewhat more complicated, since the intersection point is defined by the emission measure-weighted brightness distribution for optically-thin emission, or by an opaque surface for optically-thick emission.

Let us summarize the basic physical results that have been or could be obtained for coronal observations by means of stereoscopic or tomographic analysis methods:

  1. Large-scale corona: Tomography yields the 3D density distribution n (l,b,r) e in the corona in a typical height range of r ≈ 1.0– 2.5R ⊙, which can be rendered as a synoptic map ne(l,b) at a given height level r, or in form of radial density scale heights, ne (r) = n0 exp (− h∕ λ), for every heliographic position (l,b). Synoptic density maps are useful to characterize the source function of the solar wind, while local density scale heights can verify hydrostatic gravitational stratification, or in deviating cases, serve as diagnostic of non-equilibrium hydrodynamic processes.
  2. Coronal streamers: Tomography of the large-scale corona usually reveals (long-lived) streamers as most prominent features. Detailed 3D reconstructions of the streamer belt may reveal the folds of the interplanetary current sheet, double plasma sheets and triple current sheets and, thus, provides essential information on the lower boundary for global magnetic field models in the corona and interplanetary space.
  3. Active regions: The 3D density ne(l,b,h ) and temperature distribution Te(l,b,h ) could be reconstructed with tomography using the two STEREO spacecraft with triple EUV temperature filters, applying stereoscopic triangulation and differential emission measure (DEM) modeling to a skeleton of some 100 loops, and interpolating the 3D magnetic field in a space-filling volume. Such a 3D model can provide the DEM distribution in any subvolume of an active region, which can be used to localize and track the heating energy input as a function of space and time.
  4. Coronal loops: Many coronal loops can be stereoscopically triangulated, yielding the 3D coordinates [x(s),y(s),z(s)] along the loop length coordinate s. DEM modeling of the loops can then be carried out straightforwardly after suitable background subtraction, yielding the density ne (s) and temperature profile Te (s ) along the loop. This information can be used (i) to determine the hydrostatic density scale height, which depends on the loop plane inclination angle; (ii) to model the 1D hydrodynamic evolution of loops, as a function of the spatio-temporal heating function and conductive and radiative cooling processes, yielding diagnostics of (RTV) equilibrium and dynamic heating processes; and (iii) to test theoretical (potential, linear, and nonlinear force-free) magnetic field models.
  5. MHD loop oscillations: MHD fast kink-mode oscillations show a lateral displacement of the loop positions [x(s,t),y(s,t),z (s, t)] that can be stereoscopically triangulated as a function of the loop length coordinate s and time t, yielding geometric information on the orientation of the oscillation loop plane (from horizontal to vertical), the coplanarity, the circularity, and helicity of oscillating loops. This dynamic spatio-temporal information constrains the polarization of the kink mode, torsional modes, and asymmetries and the time dependence of the exciter.
  6. MHD waves in loops: Stereoscopic triangulation of loops or fans that contain propagating waves, such as MHD (acoustic) slow-mode waves, yields the absolute spatial direction of the propagating waves and, thus, allows us to correct the observed (projected) wave speeds. This method permitted to measure the true phase speed of sound waves and the temperature of the wave guide, independent of DEM modeling with multi-temperature filters.
  7. Erupting filaments and prominences: The 3D trajectory of erupting filaments can be stereoscopically triangulated, which yields information on the asymmetry and timing of the magnetic destabilization process, the speed, acceleration, and driving forces of the erupting filament.
  8. Bright points, jets, and plumes: Stereoscopic triangulation of features in small-scale phenomena such as bright points, jets, and plumes, reveals their height in the solar corona, the topology of the magnetic reconnection process (i.e., dipolar, tripolar, quadrupolar, null-point, fan surface, fan separatrix, helical, or twisted configuration). The verification of their spatial location at the bottom of the solar corona puts also solid constraints on the spatial distribution of the coronal heating function. Moreover, the importance of small-scale (such as nanoflares) versus large-scale phenomena (such as large flares and CMEs) can only be decided based on the occurrence frequency distribution of their energies, which requires accurate measurements of their 3D volume.
  9. Solar flares: Stereoscopic triangulation during solar flares is somewhat more difficult due to high brightness contrast, saturation, and diffraction pattern effects in EUV and soft X-ray images, but is feasible for a part of the flare loops. The brightest structures during flares turned out to be low-lying non-eruptive filaments in confined flares, or high-lying, expanding postflare loops in eruptive flares, which are two types of flares that can be distinguished by stereoscopic triangulation. Stereoscopy plays also an important role in partially occulted flares, where the flaring emission can be differentiated as a function of the altitude (e.g., coronal hard X-ray sources are often only visible when the bright footpoint emission is occulted). Stereoscopically-aided 3D reconstruction of flare regions should yield accurate measurements of the thermal energy (which scales with the flare volume). Flare energies are the most relevant quantity to establish physical scaling laws and the powerlaw-like occurrence frequency distributions, which are the hallmark of complex nonlinear dissipative systems in the state of self-organized criticality.
  10. CME source regions and EUV dimming: The most prominent manifestation of CMEs in the lower corona is the EUV dimming, whose location constraints the propagation direction of the CME, whose footprint area yields the volume and mass of CMEs, and whose temporal evolution yields the speed, acceleration, and force that drives the CME. Spatio-temporal 4D modeling of the expanding CME structure (e.g., bubble, ice-cone, flux rope) reveals the magnetic topology of the instability (e.g., tether-cutting, shearing, break-out, kink, or torus instability) and, thus, can reveal the physical mechanism that leads to a CME and/or flare.
  11. Global coronal waves: Stereoscopic observations, especially in quadrature, reveal the spatial relationship between the expanding CME structure and the associated global coronal waves that propagate concentrically over the solar surface. The geometric height of the propagating wave, as well as the magnetic field along the trajectories, can verify the physical nature in terms of the local magneto-acoustic phase speed of the wave.

Stereoscopy and tomography are just two special types of 3D reconstruction methods, but we can envision more general methods based on parameterized 3D or 4D physical models that can be forward-fitted to observations. Forward-fitting of such physical models to observations from multiple spacecraft, and using multiple temperature filters, yields even more powerful constraints, and represents an even cleaner way to deal with the underlying opacity effects in each wavelength than simple inversions of stereoscopically triangulated depth or line-of-sight coordinates. Forward-fitting of parameterized physical models should also yield more accurate 3D density models of the solar corona than direct tomographic inversion methods, which are extremely under-constrained (with only two or three slices) in the case of solar observations. We anticipate that forward-fitting of 3D or 4D physical models to multiple spacecraft data will still be called “stereoscopic” or “tomographic” methods in future, but we should be aware that the meaning of these terms progressively deviates from the original definition of simple stereoscopic triangulation and tomographic inversion. We anticipate powerful results from combined modeling of triple spacecraft data from STEREO and SDO/AIA.

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