An inverse algorithm is developed to retrieve hyperspectral absorption and backscattering coefficients from measurements of hyperspectral upwelling radiance and downwelling irradiance in vertically homogeneous waters. The forward model is the azimuthally averaged radiative transfer equation, efficiently solved by the EcoLight radiative transfer model, which includes the effects of inelastic scattering. Although this inversion problem is ill posed (the solution is ambiguous for retrieval of total scattering coefficients), unique and stable solutions can be found for absorption and backscattering coefficients. The inversion uses the attenuation coefficient at one wavelength to constrain the inversion, increasing the algorithm's stability and accuracy. Two complementary methods, Monte Carlo simulation and first-order error propagation, are used to develop uncertainty estimates for the retrieved absorption and backscattering coefficients. The algorithm is tested using both simulated light fields from a chlorophyll-based case I bio-optical model and radiometric field data from the 2008 North Atlantic Bloom Experiment. The influence of uncertainty in the radiometric quantities and additional model parameters on the inverse solution for absorption and backscattering is studied using a Monte Carlo approach, and an uncertainty budget is developed for retrievals. All of the required radiometric and inherent optical property measurements can be made from power-limited autonomous platforms. We conclude that hyperspectral measurements of downwelling irradiance and upwelling radiance, with a single-wavelength measurement of attenuation, can be used to estimate hyperspectral absorption to an accuracy of ±0.01 m^-1 and hyperspectral backscattering to an accuracy of ±0.0005 m^-1 from 350 to 575 nm.