Paraphrase this: [For corrections to single-scattering calculations, it is useful to calculate average effective optical depths out of the aggregate as seen by individual monomers. These have no physical meaning at size scales comparable to the wavelength of light but give a quasi-physical guide to empirical corrections. Extinction, scattering, and absorption values of tau_out are computed as the average value of the monomer angular cross-section as follows:
tau_coef ¼ X ðF0i=DIST2 i Þ=4=pnðN? 1Þ, (A.7a)
taue_out ¼ tau_coefnCext_mon, (A.7b)
taus_out ¼ tau_coefnCsca_mon, (A.7c)
taua_out ¼ tau_coefnCabs_mon. (A.7d)]

Empirical corrections for multiple scattering

Talk about the T-matrix data we have here. (for the empirical corrections we extended T-matrix runs that were used for the first parameterization model. For every parameter set, we made 25 T-matrix runs with randomly generated aggregate shapefiles. 
The first version of the parameterization did not include the empirical corrections for \(P_{22}\), represented as \(C\left(\theta\right)\) in Eq. (\ref{eq:P22}). It was assumed to be valid as it fits well in the range tested by \citet{Tomasko_2008}.  However, as shown in Fig. (\ref{434001}) without any empirical corrections, \(P_{22}\) fails to match the exact model as the parameters are chosen outside the original test limits. On the other hand, there are systematic deviations from the exact model with the change of each parameter. This makes it possible to correct the model statistically for the deviations from the exact calculations. [ It would be great if I can show some examples for the systematic deviation]