A long-standing issue in solar astrophysics concerns the strength and structure variations with height of the magnetic field in the chromosphere of the quiet Sun. Our empirical knowledge on this issue has remained vague notwithstanding the qualitative information provided by high resolution monochromatic images of the solar atmosphere taken at various wavelengths across strong spectral lines like H-alpha showing a mass of cell-spanning fibrils as a flattened carpet, with upright ones jutting out from network patches. Unfortunately, such images do not provide quantitative information on the
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