### 3.1 Solar-rotation stereoscopy

First we are describing here stereoscopic and tomographic 3D reconstruction techniques that use only one single viewpoint (at Earth), while the aspect angle change is produced by the solar rotation.

#### 3.1.1 Static solar-rotation stereosocopy

For quasi-stationary structures in the solar corona, the solar rotation with a synodic period of  = 26.24 days provides a natural change in aspect angle that can be exploited for stereoscopic measurements. If an image of the Sun is aligned with the solar rotation axis in y-direction and the heliographic latitude of the Sun center is small (), the solar rotation rate introduces a displacement of a source at heliographic longitude and latitude in east-west direction as

where is measured relative to the longitude of the observer and is the differential rotation rate that slightly depends on the latitude ,
with the coefficients A = 14.71 ± 0.05 deg/day, B = –2.4 ± 0.2 deg/day, and C = –1.8 ± 0.3 deg/day, yielding an equatorial (sidereal) rotation period of  = 24.47 days, which has to be multiplied with a factor of  = 26.24/24.47 = 1.0723 for the synodic coordinate system. The height dependence of the displacement in Eq. (1) yields a radial altitude measurement above the solar surface with a radius of , since all other parameters can be measured at two different times and , which gives a unique value for the altitude , supposed that the altitude of the source does not change during the observing interval. The lower time limit of a stereoscopic measurement is given by the desired accuracy of altitude measurement and the spatial resolution of the instrument, because the stereoscopic parallax effect has to exceed the spatial resolution of the instrument, which is given by
Say, for a spatial resolution of and an accuracy of we need , which obviously exceeds most evolutionary time scales of coronal structures (typically from minutes to hours).

The first solar-rotation based stereoscopic height measurements are described in Berton and Sakurai (1985), who measured the 3D coordinates of a set of coronal loops identified in Skylab XUV images from 1973, measuring the stereoscopic parallax effect during 1 – 2 days. They estimated the measuring error to and reported the 3D coordinates of a large inter-active region loop with a height of , for which the loop plane inclination angle of could be determined.

The same method of solar-rotation based stereoscopic parallax measurement has been applied to radio maps of active regions at a wavelength of (Figure 2), observed with the Very Large Array (VLA) over the course of several days (Aschwanden et al., 1992; Aschwanden and Bastian, 1994a,b; Aschwanden, 1995). While the VLA has a spatial resolution of at this frequency , the accuracy of the source positions could be optimized to by cross-correlating partial source maps, which yields sub-resolution accuracy for the locations of the source centroids. Radio emission from active regions at is dominated by free-free emission and, hence, the 3D centroid position corresponds to the altitude where the source becomes optically thick.

Sunspots are also relatively stable over a time interval of a day and, thus, the iso-Gauss surfaces in the lower corona. Radio mapping of sunspots at microwave frequencies of is dominated by gyro-resonance emission, which originates from dome-like iso-Gauss surfaces above sunspots, typically at harmonics of of the fundamental gyrofrequency of the magnetic field. Measuring the stereoscopic parallax of the dominant gyroresonance source above a sunspot over the course of 4 days with the Owens Valley Radio Observatory (OVRO), the altitudes of the gyroresonant layers could be determine for a set of 7 frequencies in the range of , which were found in a height range of , and allowed to constrain the local magnetic field with a potential field model and to identify the correct harmonics at every frequency (Aschwanden et al., 1995). Of course, stationarity of the magnetic field over time scales of 4 days can only be expected for stable large sunspots, while small-scale magnetic fields change during much shorter time scales in active regions.

#### 3.1.2 Dynamic solar-rotation stereosocopy

Coronal loops are generally not stable over a time period of a day, because the heating rate appears to be discontinuous and intermittent and cooling by conductive and radiative loss occurs on time scales of less than an hour. This means that loops at the same location observed a day apart are not identical, but continuously replaced by plasma upflows and downflows. However, what stays more permanent is the magnetic field, say over time scales of a few days, especially the large-scale dipolar fields that represent the main magnetic field structure of an active region, produced by the leading sunspot and the trailing region with opposite magnetic polarity. The property of the quasi-stationarity of the magnetic field can thus be used to constrain the 3D geometry of near-cospatial loops at different times, regardless how often the plasma is flushed through the magnetic conduits. A technique that uses the stable 3D geometry of the magnetic field and is not sensitive to the fast-paced plasma dynamics has been developed for the 3D reconstruction of a set of dipolar loops that make up an active region, called dynamic stereoscopy (Figure 3), which was applied to a few days of SOHO/EUV images (Aschwanden et al., 1999, 2000). This method allows us to determine the approximate 3D geometry of coronal loops under the assumption of planarity, which requires only one free parameter (the inclination angle of the loop plane) to determine the 3D coordinates of loops from the observed 2D projections as a function of the loop length coordinate .