6. THE BANDWIDTH EFFECT6.1 Introduction
In Section 1.2 the theory of the Dobson instrument measurement was developed for monochromatic spectral bands and no account was made of the spectral variation of parameters, particularly of ozone absorption, across the instrument's finite bandwidths. The principal deficiency of this assumption of monochromaticity is its failure to account for the small monotonic decrease in the effective absorption coefficient of a finite band with increasing airmass and absorber amount. This effect is known as the bandwidth effect, or the Forbes effect. It arises from the relative attenuation, with increasing airmass and absorber amount, of those wavelength intervals within the band which are more strongly absorbed, and hence the relative increase in the energy weighting of the wavelength intervals within the band which are less absorbed, i.e., which have smaller absorption coefficients.
The calculation of the bandwidth effect relies on a knowledge of the slit transmittance functions but these are not in fact well known. The exact widths and shapes of the transmittance functions will, of course, vary among a group of instruments according to the particular adjustment of the slits and of the other optical components. Furthermore, the functions usually assumed (triangular for S2 and trapezoidal for S3) will be convoluted (i.e., smeared) by the effects of lens aberrations and other optical imperfections (see Section 2), and will have low-transmittance "wings" owing to these effects and to stray light. Indeed, the bandwidth effect and stray light effects have the same origin, namely, lack of perfect monochromaticity in the band, and are described by the same mathematical expressions. Generally, the smearing or broadening of the slit transmittance functions will result in greater bandwidth effect errors.
The choice of bandwidths for the Dobson instrument is largely a compromise between the bandwidth effect on the one hand, and the need for high energy transmission and low wavelength sensitivity (see Section 5) on the other. If the slits S1 and S2 remain equal, then the energy transmitted by the pair is proportional to the square of their slitwidth, which means that any increase in resolution is costly in energy transmission. As we shall see, the bandwidth effect for the present bandwidths of about 1.0 nm for S2 and 3.0 nm for S3 is usually less than 1%, and so these bandwidths are quite satisfactory.
6.2 Theory and calculations
The bandwidth effect for various ozone measuring instruments has been discussed by Vanier and Wardle (1969), Khrgian (1975) and Basher (1977). The effective ozone absorption coefficient of a band at any time is the integral over wavelength of the ozone absorption spectrum weighted by not only the slit transmittance function, but also the spectral irradiance incident on the instrument at that time, and, less importantly, the detector spectral sensitivity and the spectral transmittance of any filters present. It can be shown that this effective ozone absorption coefficient, α may be written as:
-1 ∫P0(λ)10-μhXα(λ)-mΒ(λ)-secZδ(λ)dλ α = ----- log ------------------------------------ (6.1) μhX ∫P0(λ)10-mΒ(λ)-secZδ(λ)dλwhere P0(λ) is the electrical response of the instrument per unit wavelength outside the atmosphere (this is indicated by the subscript zero) and is the product of solar spectral irradiance, slit spectral transmittance, filter spectral transmittance and detector spectral sensitivity. The other terms are defined in Section 1.2. (Note that here in equation (6.1) wavelength λ is a continuous variable, whereas in equations (1.1) to (1.7), the convention of λ being used to label the centre wavelength of a band is followed. This results in some inconsistency, for example, P(λ) here is a spectral quantity whereas in equation (1.5) it is the band integral of this spectral quantity.) Expressions similar to equation (6.1) can be written for an effective Rayleigh scattering coefficient, Β, and an effective aerosol scattering coefficient, δ, but in practice this is not necessary as the variation in these coefficients is negligible. The aerosol scattering term in equation (6.1) can be safely neglected (Basher, 1977).
Vanier and Wardle (1969) calculated α for the Dobson instrument's A, B, C and D bandpairs and gave the range of variation for the A bandpair coefficient as 0.003, which is equal to about 0.2% in α. They state that the ranges of variation for the other bandpairs are also small and conclude that the bandwidth effect is negligible. Basher's (1977) results showed some small differences to these results but this was probably due to differences between the absorption spectra, solar spectra and slit spectra used by the authors.
Some new and more extensive calculations of the bandwidth effect are presented below in Table 6.1. The method and basic data used are essentially the same as those described by Basher (1977), but the resolution of the calculations was increased from 0.2 nm to 0.05 nm and an improved and more detailed ozone absorption spectrum was constructed and used (see Section 5.2 for more details). Of course it must be remembered that no amount of improving can overcome the relatively high uncertainty in the available absorption data (see Section 9). The calculations assume triangular and trapezoidal slit transmittance functions of 1.0 and 3.0 nm bandwidth for the S2 and S3 bands respectively, and of 0.9 nm slopewidth (the spectral width of the sloping sides). The results are given in Table 6.1 as the corrections in percent that need to be applied to the absorption coefficients calculated for zero airmass.
The tabulated corrections are positive (except for some double bandpair combinations) and increase approximately linearly with ozone amount and airmass. This reflects the expected decline in α with increasing absorber amount. The linearity of the dependences allows their representation by simple equations for data correction by computer, though as far as is known, Dobson data are never corrected for the bandwidth effect. Note that the second decimal place is given in the table's data to show trends, but is otherwise not significant. The corrections appear to be quite dependent on band centre wavelength, which implies that the values calculated will be rather dependent on the accuracy of the absorption data and slit data used. The accuracy of the corrections listed is probably no better than 0.2% of the relevant coefficient.
The bandwidth effect errors tend to be smallest for the shorter wavelength bands, and this is due to the smaller spectral variability of α(λ) at the shorter wavelengths (see Figure 1.1). This is a fortunate thing since it is the shorter wavelength bands which dominate the bandpair absorption coefficients. The magnitudes of the errors for the recommended AD bandpair combination are smaller than their uncertainty for all conditions and are therefore negligible. In contrast, the errors for the often-used C and CD methods are the largest of the usual band combinations, and, while mostly they are less than 1%, under the conditions of high ozone and airmass when these methods are most important, they will rise to 2%.
TABLE 6.1 Bandwidth effect corrections for the ozone absorption coefficients of Dobson instrument bands and band combinations, in percent, and as a function of ozone amount in atm cm and airmass. OZONE μh 305. 309. 311. 318. 325. 329. 332. 340. A B C D AB AC AD BC BD CD 0.200 1.0 0.06 0.12 0.15 0.14 0.48 0.66 0.43 0.06 0.04 0.09 0.13 0.14 -0.09 -0.05 0.01 0.02 0.07 0.12 2.0 0.12 0.25 0.29 0.28 0.95 1.31 0.86 0.12 0.07 0.19 0.26 0.28 -0.20 -0.10 0.01 0.04 0.15 0.25 3.0 0.18 0.37 0.44 0.42 1.43 1.95 1.28 0.18 0.10 0.29 0.40 0.43 -0.33 -0.16 0.02 0.07 0.23 0.38 4.0 0.24 0.51 0.60 0.57 1.90 2.57 1.69 0.25 0.13 0.40 0.54 0.58 -0.47 -0.22 0.02 0.11 0.32 0.51 5.0 0.29 0.64 0.75 0.71 2.38 3.19 2.11 0.32 0.16 0.51 0.68 0.73 -0.63 -0.29 0.01 0.16 0.42 0.65 0.250 1.0 0.07 0.14 0.17 0.16 0.54 0.75 0.48 0.07 0.04 0.11 0.16 0.17 -0.11 -0.05 0.01 0.02 0.09 0.15 2.0 0.14 0.29 0.35 0.33 1.09 1.49 0.95 0.15 0.08 0.23 0.32 0.34 -0.25 -0.12 0.02 0.06 0.19 0.30 3.0 0.21 0.45 0.53 0.50 1.63 2.21 1.42 0.22 0.12 0.36 0.48 0.51 -0.41 -0.19 0.02 0.10 0.29 0.46 4.0 0.28 0.61 0.72 0.67 2.17 2.91 1.88 0.30 0.16 0.49 0.65 0.69 -0.60 -0.27 0.02 0.16 0.41 0.63 5.0 0.34 0.77 0.91 0.85 2.72 3.60 2.33 0.38 0.19 0.63 0.83 0.87 -0.81 -0.37 0.01 0.22 0.53 0.80 0.300 1.0 0.08 0.17 0.20 0.18 0.61 0.84 0.53 0.08 0.05 0.13 0.18 0.19 -0.13 -0.06 0.01 0.03 0.11 0.18 2.0 0.17 0.34 0.41 0.38 1.22 1.66 1.05 0.17 0.10 0.27 0.37 0.39 -0.30 -0.14 0.02 0.08 0.23 0.36 3.0 0.24 0.53 0.62 0.58 1.83 2.46 1.56 0.26 0.14 0.42 0.57 0.59 -0.50 -0.23 0.02 0.13 0.35 0.55 4.0 0.31 0.72 0.84 0.75 2.44 3.25 2.06 0.35 0.18 0.58 0.77 0.80 -0.73 -0.33 0.02 0.20 0.49 0.75 5.0 0.38 0.91 1.06 0.99 3.05 4.01 2.56 0.45 0.21 0.75 0.98 1.01 -0.99 -0.44 0.01 0.29 0.64 0.95 0.350 1.0 0.10 0.19 0.23 0.22 0.68 0.93 0.57 0.10 0.06 0.15 0.21 0.22 -0.16 -0.07 0.02 0.04 0.13 0.20 2.0 0.19 0.39 0.47 0.44 1.36 1.83 1.14 0.20 0.11 0.32 0.43 0.44 -0.36 -0.16 0.03 0.09 0.26 0.42 3.0 0.27 0.60 0.71 0.66 2.04 2.72 1.70 0.30 0.16 0.49 0.66 0.68 -0.60 -0.27 0.03 0.17 0.42 0.64 4.0 0.35 0.82 0.96 0.90 2.71 3.58 2.25 0.40 0.20 0.68 0.89 0.91 -0.88 -0.39 0.02 0.26 0.58 0.87 5.0 0.43 1.05 1.22 1.13 3.38 4.41 2.79 0.51 0.24 0.88 1.13 1.15 -1.20 -0.53 0.01 0.37 0.76 1.11 0.400 1.0 0.11 0.22 0.26 0.24 0.75 1.02 0.62 0.11 0.07 0.17 0.24 0.25 -0.18 -0.08 0.02 0.05 0.14 0.23 2.0 0.21 0.44 0.52 0.40 1.50 2.01 1.23 0.22 0.12 0.36 0.49 0.50 -0.41 -0.19 0.03 0.11 0.30 0.47 3.0 0.30 0.68 0.80 0.78 2.24 2.97 1.84 0.34 0.18 0.56 0.74 0.76 -0.70 -0.31 0.03 0.20 0.48 0.73 4.0 0.39 0.94 1.08 1.01 2.98 3.90 2.43 0.46 0.22 0.78 1.01 1.03 -1.03 -0.46 0.02 0.31 0.68 1.00 5.0 0.47 1.20 1.38 1.28 3.71 4.81 3.01 0.58 0.27 1.01 1.29 1.30 -1.42 -0.62 -0.00 0.45 0.89 1.28 0.450 1.0 0.12 0.24 0.29 0.27 0.82 1.11 0.67 0.12 0.07 0.19 0.26 0.27 -0.20 -0.09 0.02 0.05 0.16 0.26 2.0 0.23 0.49 0.58 0.54 1.63 2.18 1.33 0.25 0.14 0.41 0.54 0.55 -0.47 -0.21 0.03 0.13 0.34 0.53 3.0 0.33 0.76 0.89 0.83 2.44 3.22 1.97 0.38 0.19 0.64 0.83 0.84 -0.80 -0.36 0.03 0.24 0.55 0.82 4.0 0.42 1.05 1.21 1.12 3.24 4.22 2.61 0.51 0.24 0.88 1.13 1.14 -1.20 -0.52 0.01 0.38 0.77 1.13 5.0 0.51 1.35 1.54 1.42 4.04 5.20 3.23 0.64 0.29 1.14 1.45 1.45 -1.65 -0.71 -0.01 0.54 1.02 1.45 0.500 1.0 0.13 0.26 0.31 0.20 0.89 1.19 0.72 0.14 0.08 0.22 0.29 0.30 -0.23 -0.10 0.02 0.06 0.18 0.29 2.0 0.25 0.55 0.64 0.60 1.77 2.35 1.42 0.27 0.15 0.45 0.60 0.61 -0.53 -0.24 0.03 0.15 0.39 0.59 3.0 0.36 0.85 0.98 0.91 2.64 3.47 2.11 0.42 0.21 0.71 0.92 0.93 -0.91 -0.40 0.03 0.28 0.62 0.92 4.0 0.46 1.16 1.34 1.23 3.50 4.54 2.79 0.56 0.26 0.99 1.26 1.26 -1.37 -0.59 0.01 0.44 0.87 1.26 5.0 0.55 1.50 1.70 1.57 4.36 5.58 3.46 0.70 0.31 1.28 1.61 1.60 -1.90 -0.81 -0.02 0.64 4.15 1.626.3 Summary
(i) The effective ozone absorption coefficient for a band of finite bandwidth decreases with increasing absorber amount, owing to the spectral variation of absorption across the band. The customary use of fixed ozone absorption coefficients therefore gives rise to an error which depends on airmass and ozone amount.(ii) Bandwidth effect errors calculated for the Dobson instrument are neglibily small for the AD band combination. The errors for the C and CD band combinations are among the largest found, and while these are mostly less than 1%, they do rise to 1.5% when the airmass and ozone amount are very high.
(iii) Bandwidth effect errors are approximately linear with airmass and ozone amount and are easily corrected for. The uncertainty in the calculated errors is possibly 0.2% for the relevant coefficient.
(iv) The Dobson slit transmittance functions are not well known. Optical effects, such as lens aberrations will smear and broaden the functions and generally will increase the bandwidth effect errors.
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