Inversion formulas and range characterizations for the attenuated   geodesic ray transform
  We present two range characterizations for the attenuated geodesic X-ray transform defined on pairs of functions and one-forms on simple surfaces. Such characterizations are based on first isolating the range over sums of functions and one-forms, then separating each sub-range in two ways, first by implicit conditions, second by deriving new inversion formulas for sums of functions and one-forms. 
Yernat Assylbekov, Fran\c{c}ois Monard and Gunther Uhlmann
math.AP math.DG