Bias in a solar constant determination by the Langleymethod due to structured atmospheric aerosol:comment

Bruce W. Forgan

A recent article by Schotland and Lea [Appl. Opt. 25, 2486 (1986)] discusses the errors in radiometercalibration by the Langley method and the detail required for very precise work. Their numerical methods onaerosol air mass can be simplified for use in the calibration of the worldwide sunphotometer network by usinggeometric considerations, assuming aerosol scale heights of -1 Km and using the apparent solar zenith angleat the instrument as the independent variable in these formulas, so that refraction effects are incorporated.By using multiple-pass Langley analyses the errors due to air mass assignments can be reduced significantly.

"It is generally accepted that the calibration is theparamount problem of sunphotometry". 1 In their re-cent article, Schotland and Lea2 (SL) continue theseries of recent articles3 4 that quantify the errors inradiometer calibration by the Langley method. TheSL article tabulates the intrinsic errors in using inap-propriate species air masses in Langley analysis. Thisnote is an addendum to the SL article and will com-ment on (1) the small differences found by SL in thesolar constant bias errors for the radically differentaerosol distribution; (2) the use of apparent solar ze-nith distance for air mass calculations, (3) using ascaled air mass for Langley extrapolations.

In their discussion on aerosol model sensitivity, SLexpressed surprise at the small difference between theerrors generated by two of the aerosol models. Model1 characterized a mixed atmospheric boundary layerwith an exponential decrease with height and corre-sponding scale height of 1 km. Model 2 was used todescribe a stable atmosphere to 3 km and a rapidexponential decrease above 3 km. While the solarconstant errors, when using a molecular air massrepresentation, were high for both aerosol models, the

The author is with Cape Grim Baseline Air Pollution Station, P.O.Box 346, Smithton, Tasmania 7330, Australia.

Received 23 July 1987.0003-6935/88/122546-03$02.00/0.© 1988 Optical Society of America.

percentage difference in the errors from model 2 tomodel 1 was <11%.

The small difference is a direct consequence of theoptical center of extinction of the two models. In fact,it will be shown that an estimate of the maximum solarconstant bias error difference between aerosol model 2and a very thin surface layer of aerosols is only -35%.

SL's optical air mass equation [their Eq. (2)] formodel i can be expressed as

= J a[l - (nr 0 sin0 o/nr)2T 0 5dr (1)

with dependencies on refractive index n, extinctioncoefficients a, T the optical depth, r the earth radius,and H the height above the earth's surface (H = r - r).

For most species i, o- is confined to a finite heightrange (<35 km) with 0.99 < (ro/r)2 < 1. For G > 10°,atmospheric curvature effects dominate, and in thisdomain, if atmospheric refractive-index terms are ne-glected, there should exist some r so that Eq. (1) can beapproximated by

M = 1 - (ro sinO0 /ri)2 ] 0 5dr,

which reduces to

Mi = [1. - (ro sino0o/r,)2 ] 0.5, (2)

where r represents the optical center of extinction.Equation (2) can also be written in terms of H as

Mi = (1 + Hi/ro)[cos2 O0 + 2Hi/r + (w/rO) 2 J 0 5 , (3)

which, apart from the (Hilro)2 term, represents thegeometric air mass equation for a species concentratedat Hi above the earth's surface.5

2546 APPLIED OPTICS / Vol. 27, No. 12 / 15 June 1988

Using the data provided by SL in their Table I forMm (molecular) and Ma (aerosol model 2), least-squares fitting techniques were used to fit the form ofEq. (3) and find Hm and Ha; the results were 6.58 and1.54 km, both with standard errors for all 00 < 800 of<0.0001. Similar calculations could not be done foraerosol model 1 as the data were not provided, soapproximations were used to estimate a representativeoptical center of extinction.

SL stated the expression for solar constants biaserrors derived from paired air mass data, namely,

ln(F*IF) = r(MmMa 2 - Mm2Mai)(Mm2 - Mmi)1 , (4)

where F is the true solar constant and F* the derivedvalue. They also derived from Langley analyses the'bias errors for various models and apparent solar ze-nith distance ranges. Their Langley analyses showedproportionally to Ta but with higher coefficients, thehigher coefficients being the result of the induced cur-vature in modeled Langley data.

Again, using the data in SL's Table I, Eq. (4) wasused to derive the errors for a paired zenith range of 45and 800 for aerosol model 2. For Ta = 0.100 the biaserror in the solar constant was found to be 0.44%.Using Eq. (3) a H was found for the same pair, whichwould give an increase in the bias error of -11% repre-sentative of the aerosol in niodel 1; not surprisingly, itwas found to be '1.0 km and corresponds well with thescale height. With H = 0 km, which gives Mi = sec~o,there was only a 35% increase in the solar constant biaserror from aerosol model 2; that is, the bias error wasfound to be 0.59% for Ta = 0.1.

These results show that the small difference in solarconstant bias errors between the very different tropo-spheric aerosol models is a result of small differences inthe optical center of extinction of the distributions,model 1 being 1.0 km and model 2 represented by 1.5km. Furthermore, the approximate analysis usedabove shows that between a uniformly mixed aerosollayer of 3-km depth and an extremely thin layer closeto the surface, the bias error bnly increases by a factorof 0.35. Such small differences in bias errors wereused by Robinson 5 to support secOo as the first approxi-mation for air mass for surface layer aerosol and watervapor.

It should also be noted that the negative bias errorsfor the solar constant shown by SL for the third modelof aerosol (volcanic ash from El Chichon) can be relat-

ed to the effect of stratospheric ozone on solar constantdeterminations in the UV and Chappius absorptionbands. The volcanic ash distribution in their modelwas distributed -20 km similar to the vertical distri-bution of ozone. Using the data from their Fig. 6, onecan estimate that an ozone optical depth in the Chap-pius bands can produce percentage bias errors forozone -1OOo, where To is the ozone optical depth.

All discussion above and in the SL paper concerneddata relative to apparent solar zenith distance 00. Themolecular air mass formulas in common use6 are also afunction of 00. Given that time can be measured withmore accuracy than the apparent zenith distance of thesun's center for the majority of sunphotometer mea-surements, calculations are usually performed to de-rive 0, the true solar zenith distance. While 00 is notequivalent to 0 except at 00 = 0, the differences onlyaccelerate as 0 approaches 90°, with 00 < 0. As a result,reference texts 7 8 on sunphotometry equate 0 with 00for use in molecular air mass formulations expressed interms of 00. Given the extent of the errors applicablethrough the use of Mm instead of an appropriate airmass, it is interesting to note the errors involved when 0is used instead of Oo.

Table I shows the molecular and aerosol model 1 airmasses based on Eq. (3) using Hm and Ha and assumed00. In the right half of Table I the Mm(O) values werecalculated from assumed 0, then Mm(9 0) was calculatedusing the apparent zenith distance derived from 9' = 0- 0(0), where 0 is the refraction correction. For sim-plicity, the tabulations8 for 0(00) were used to repre-sent 0(0).

The differences in air mass from assuming that 0o = °are about half of the magnitude of the difference be-tween Mm and Ma and of opposite sign and hencewould reduce the derived solar constant and total opti-cal depth. However, these errors would apply even if aweighted M were used or Ta is zero. Hence it is appar-ent that for accurate solar constant determinations theuse of air mass functions dependent on apparent solarzenith distance but evaluated using true solar zenithwithout correction for refraction should not continue.

Finally, some comments follow on the use of aweighted air mass function of the type

M=M = E' Miri / Tri,

i=l i=l

(5)

where i represents the various species, typically molec-

Table 1. Air Mass Differences for Different Atmospheric Models and Appropriate Solar Zenith Distancesa

Do Ma(00 ) Mm(00 ) M.(Oo) - Mm(00 ) 0 0 Mm(00) Mm(0) - Mm(00)

40 1.3052 1.3044 0.0008 40 39.987 1.3041 0.000350 1.5552 1.5534 0.0018 50 48.980 1.5525 0.000960 1.9985 1.9937 0.0048 60 59.972 1.9914 0.002365 2.3636 2.3548 0.0088 65 64.958 2.3501 0.004770 2.9185 2.9009 0.0176 70 69.957 2.8928 0.008175 3.8507 3.8088 0.0419 75 74.940 3.7886 0.020280 5.7144 5.5782 0.1362 80 79.912 5.5712 0.0670

a Ma and Mm were derived assuming optical centers of extinction of 1.5 and 6.5 km, respectively (seetext for definitions).

15 June 1988 / Vol. 27, No. 12 / APPLIED OPTICS 2547

ular, aerosol, ozone, and water vapor. SL have dis-cussed the need to be aware of the difference betweenMm and Ma. The above data have shown that for lowlevel tropospheric aerosol, the aerosol air mass can bedescribed by a layer centered around 1.25 km for thetwo disparate distributions used by SL and typical ofthe extremes of the lower atmosphere. The SL biaserror analysis of Ta shows that while the biases inpercentage terms are larger than the solar constant, inabsolute terms for Ta of the order of 0.1 it is a minorcontribution. For example, for a zenith distance pairof 45 and 800, and using aerosol model 2, the percent-age difference between the derived and actual opticaldepth is -3% or 0.003. Then Tr, the biased opticaldepth, can be used as an estimate of Ta for deriving aweighted air mass correction.

A multipass Langley analysis can be used for thederivation of the solar constant. The steps for a pure-ly aerosol + molecular wavelength could include:Step 1: Use'the molecular air mass as the indepen-

dent variable in the first pass of the Langleyanalysis.

Step 2: From the derived T* of the first pass calculateT = T* Tm and derive a weighted air mass M= (TmMm + TMa)/T*.

Step 3: Using M repeat the Langley analysis to findthe resulting solar constant and T.

To test the multiple-pass method, artificial Langleydata were generated for a number of wavelengths influ-enced by molecular, aerosol (0.025 < Ta < 0.080), andozone ( < TO < 0.040) extinction, over a range ofmolecular air masses from 1 to 6.5. As expected, Lang-ley analyses using a molecular air mass to derive Fproduced bias errors as given by Eq. (4). The biaserrors were dependent on the optical depth of thenonmolecular species and the air mass range. Thestandard errors in these molecular Langley analyseswere typically 0.001. The errors in the derived T* alsofollowed the expected range. 2

Multipass analyses were then conducted on thesame data, and the errors in F* and T* were reducedsignificantly. If the correct air mass forms were used,the bias errors in F were <0.02%, and the errors in T*

were <0.00001. Both errors were not statistically sig-nificant at the 0.1% level. If the aerosol center ofextinction height was in error by 0.6 km the bias errorsin F were still <0.1% and the errors in T* <0.0003. Inall cases, the standard errors of the multipass analyseswere always between 1 and 2 orders of magnitude lessthan the corresponding molecular Langley analyses.The scaling of the bias error by the air mass range wasstill evident. However, it was an order of magnitudeless than when using the molecular air mass.

For step 1, the reason for using the molecular airmass for IR wavelengths is not readily apparent. For aclean maritime location like Cape Grim (41'S, 1450 E),

in northwest Tasmania, the aerosol optical depth isnearly independent of wavelength, and during winterT is near 0.040. For wavelengths >700 nm, this meansthat Ta is always >Tp. Hence a better first approxima-tion would be to use M = sec 00 or an air mass based onan optical center of extinction of -1 km and thencorrect for the molecular atmosphere. In practice, atCape Grim a climatologically derived M is used as thefirst guess in step 1.

For the case when additional species are present, forexample, ozone in the Chappius bands, step 2 needs toinclude an ozone optical depth component using eitherdirect measurements or climatology. 9 However, asshown above and graphically in the SL paper, caremust be taken when large stratospheric concentrationsexist.

In summary, SL have quantitatively described an-other potential source of error in Langley analyses andhave shown the detail required for very precise work,such as the long term monitoring of the solar constantfrom a high altitude ground station. However, theirnumerical methods on aerosol air mass can be simpli-fied for use in the calibration of the worldwide sunpho-tometer network by using geometric considerationsand assuming aerosol scale heights of -1 km. Caremust be taken to ensure the use of Do as the indepen-dent variable in these formulas, so that refraction ef-fects are incorporated. By using a multiple pass Lang-ley analyses the errors due to air mass assignments'canbe reduced significantly.

References1. World Meteorological Organization, "Recent Progress in Sun-

photometry," Environmental Pollution Monitoring and Re-search Programme (1986), Vol. 43, p. 7.

2. R. M. Schotland and T. K. Lea, "Bias in a Solar Constant Deter-mination by the Langley Method due to Structured AtmosphericAerosol," Appl. Opt. 25, 2486 (1986).

3. L. W. Thomason, B. M. Herman, and J. A. Reagan, "The Effect ofAtmospheric Attenuators with Structured Vertical Distributionson Air Mass Determinations and Langley Plot Analysis," J. At-mos. Sci. 40, 1851 (1983).

4. B. A. Herman, M. A. Box, J. A. Reagan, and C. M. Evans, "Alter-nate Approach to the Analysis of Solar Photometer Data," Appl.Opt. 20, 2925 (1981).

5. N. Robinson, Solar Radiation (Elsevier, Amsterdam, 1966).6. F. Kasten, "A New Table and Approximation Formula for the

Relative Optical Air Mass," Arch. Meteorol. Geophys. Bioklima-tol. Ser B 14, 206 (1966).

7. E. Meszaros and D. M. Whelpdale, "Manual for BAPMoN Sta-tion Operators," World Meteorological Organization, Environ-mental Pollution Monitoring and Research Programme Report32, WMO/TD 66 (1985).

8. World Meteorological Organization, International OperationsHandbook for Measurements of Background Atmospheric Pol-lution, WMO 491 (1978).

9. A. T. Young, "Observational Technique and Data Reduction,"Methods Exp. Phys. 12, 123 (1974).

0

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