S-DIMM+ height characterization of day-time seeing using solar granulation

Context. To evaluate site quality and to develop multi-conjugative adaptive optics systems for future large solar telescopes, characterization of contributions to seeing from heights up to at least 12 km above the telescope is needed. Aims. We describe a method for evaluating contributions to seeing from different layers along the line-of-sight to the Sun. The method is based on Shack Hartmann wavefront sensor data recorded over a large field-of-view with solar granulation and uses only measurements of differential image displacements from individual exposures, such that the measurements are not degraded by residual tip-tilt errors. Methods. The covariance of differential image displacements at variable field angles provides a natural extension of the work of Sarazin and Roddier to include measurements that are also sensitive to the height distribution of seeing. By extending the numerical calculations of Fried to include differential image displacements at distances much smaller and much larger than the subaperture diameter, the wavefront sensor data can be fitted to a well-defined model of seeing. The resulting least-squares fit problem can be solved with conventional methods. The method is tested with simple simulations and applied to wavefront data from the Swedish 1-m Solar Telescope on La Palma, Spain. Results. We show that good inversions are possible with 9–10 layers, three of which are within the first 1.5 km, and a maximum distance of 16–30 km, but with poor height resolution in the range 10–30 km. Conclusions. We conclude that the proposed method allows good measurements when Fried's parameter r0 is larger than about 7.5 cm for the ground layer and that these measurements should provide valuable information for site selection and multi-conjugate development for the future European Solar Telescope. A major limitation is the large field of view presently used for wavefront sensing, leading to uncomfortably large uncertainties in r0 at 30 km distance.