Surface albedo retrieval based on high spatial resolution data

The paper aims to compare the accuracy of three methods of solving the atmospheric correction problem for a Lambertian surface using high spatial resolution remotely sensed data. Three couples of formulas are presented. Each couple contains a formula for expressing the reflectance in a target pixel at the upper boundary of the atmosphere via albedo in target and adjacent pixels of the surface and a formula for the surface albedo in a target pixel via reflectances. The derivation of each couple of formulas is given. Derivation conditions are presented. Formulas of only one couple are found by solving the radiation transport equation in $1D$ geometry. Formulas of two other couples include values obtained by solving the transport equation in $3D$ geometry. Results of testing the accuracy of all formulas when solving an atmospheric correction problem on data of high ($30$ m) spatial resolution are given. Problems with aerosol optical depths from $0.2$ to $2$ for a wavelength of $\lambda=0.55~\mu$m for all possible albedo values (from $0.1$ to $0.9$) in target and adjacent pixels are considered. It is shown that only one couple of formulas out of the three gives high accuracy under any condition. Formulas of the two other couples give sufficient accuracy (with less than $10\%$ error) only for a small value of the aerosol optical depth and a small difference of the albedo of the target and adjacent pixels.