Verification of an Approximation Method for Calculating Multiple Scattering of Sky Radiation Elisabeth de Bary

Inst i tut für Meteorologie, Mainz D 6500, Postfach 3980, Germany. Received 15 June 1972.

Eschelbach's1 computations of the sky radiance, which exactly account for the effect of multiple scattering, enable one to verify the accuracy of an approximation method tha t was developed for taking the scattering of higher order into account.2

This verification was based upon the comparison of the quotients primary scattering to total scattering obtained from the approxi­mation method of de Bary and from Eschelbach's computations:

(PS/TS) deB/(PS/TS) Esch.

In the solar almucántar at the zenith distance of Z0 = 66°, the error decreases with increasing wavelength and decreasing tur­bidity (from T = 4 to T = 2) (limits: + 6 0 % and - 7 % ) . Furthermore, the maximum error is shifted with increasing wave­length from the scattering angle 90-50°. Both these statements' are valid for the albedo values of 0.0 and 0.25 as well. However, the error decreases with increasing albedo values. The approxi­mation method mostly yields an overestimation of the proportion of primary scattering in the total scattering, whereas an under­estimation takes place in case of four extreme conditions only— namely, long wavelength, large scattering angle, great turbidity, and higher values of albedo [Figs. 1(a) and (b)].

The comparison in the sun's vertical (Figs. 2 and 3) at the zenith distance of Z0 = 66° yields results tha t are very similar to those in the solar almucantar. The underestimation of the proportion

November 1972 / Vol. 11, No. 11 / APPLIED OPTICS 2717

Fig. 3. Sun's vertical, Z0 = 66°, A: 0.25.

Fig. 1. Solar almuncantar, Z0 = 66°, A: 0.0, A: 0.25.

Fig. 2. Sun's vertical, Z0 = 66°, A: 0.0.

of primary scattering in the total scattering is greatest in case of great turbidity and short wavelength; the maximum deviation is shifted with increasing wavelength from the scattering angle φ = 90° (Z = 24° in the countervertical) toward smaller scat­tering angles. The approximation method fits better for the albedo value 0.25 than 0.0. The verification curves show the striking fact of a dent at ψ = 100-110°. I ts physical explanation is the use of numerical values of the molecular atmosphere in the approximation method. In a molecular atmosphere the mini­mum of scattering occurs at φ = 90°, whereas in an atmosphere containing aerosol particles the minimum occurs at φ = 100-110°. There is a secondary maximum in the direction toward larger scattering angles. Close to the horizon, the error of the approximation method is generally markedly reduced. Here, the use of the Rayleigh values results in an overestimation of the increase of scattering of higher order (Figs. 2 and 3).

The approximation method yields satisfactory results for wave­lengths greater than 0.55 μ also in case of high turbidity. The error is less than 25%. For short wavelengths, however, it is inadequate. The deviation of the approximation method from the exact computations is greatest at a zenith distance of the sun Z0 = 66° that the present computations were based upon. The accuracy of the approximation method increases with in­creasing solar elevations. A comprehensive survey on the pro­portion of primary scattering in the total scattering will be pub­lished in the near future.

References 1. G. Eschelbach, J. Quant. Spectrosc. Rad. Transfer 11, 757

(1971). 2. E. de Bary, Appl. Opt. 3 , 1293 (1964).

2718 APPLIED OPTICS / Vol. 11, No. 11 / November 1972