This work presents an iterative method for the estimation of the absolute dose distribution in patients undergoing carbon ion therapy, via analysis of the distribution of positron annihilations resulting from the decay of positron-emitting fragments created in the target volume. The proposed method relies on the decomposition of the total positron-annihilation distributions into profiles of the three principal positron-emitting fragment species - 11C, 10C and 15O. A library of basis functions is constructed by simulating a range of monoenergetic 12C ion irradiations of a homogeneous polymethyl methacrylate phantom and measuring the resulting one-dimensional positron-emitting fragment profiles and dose distributions. To estimate the dose delivered during an arbitrary polyenergetic irradiation, a linear combination of factors from the fragment profile library is iteratively fitted to the decomposed positron annihilation profile acquired during the irradiation, and the resulting weights combined with the corresponding monoenergetic dose profiles to estimate the total dose distribution. A total variation regularisation term is incorporated into the fitting process to suppress high-frequency noise. The method was evaluated with 14 different polyenergetic 12C dose profiles in a polymethyl methacrylate target: One which produces a flat biological dose, 10 with randomised energy weighting factors, and three with distinct dose maxima or minima within the spread-out Bragg peak region. The proposed method is able to calculate the dose profile with mean relative errors of 0.8%, 1.0% and 1.6% from the 11C, 10C, 15O fragment profiles, respectively, and estimate the position of the distal edge of the SOBP to within an average of 0.7 mm, 1.9 mm and 1.2 mm of its true location.