[6] —

Finally, the natural logarithm is taken from the equation and the scattering efficiency also moved into the polynomial, yielding the final equation used in the DOAS retrieval:

[7] —

This is a linear equation between the natural logarithm of the measured quantities (I and I₀), the slant column densities of all relevant absorbers (SC_j) and a low order polynomial with coefficients b_p*.

Because measurements are taken at many wavelengths (50..200) in parallel, the DOAS retrieval consists in solving an overdetermined set of linear equations, which can be done by standard methods of linear least squares solutions.