5.1      Introduction

          The bandwidth, band shape and centre wavelength of the Dobson instrument's bands depend on the mechanical adjustment of the instrument's slits and other optical components. Absolute centre wavelengths can be determined by interpolation between measured lines of spectral line sources. For each of these adjustments and calibrations, and also for each operational wavelength pair selection, there be some uncertainty and the resulting uncertainty in the wavelength bands will propagate through to the instrument's ozone measurement as it is defined by equation (1.1). It is only through the spectrally very variable solar extraterrestrial irradiance and ozone absorption coefficient terms in equation (1.1) that any significant error arises.

          Dobson (1957b) was well aware of the problem. He made spectral intensity measurements with the instrument to help guide his choice of band centre wavelengths and bandwidths and his recommendations for calibration precisions. However, as far as is known, the only specific study of band uncertainty propagation is that summarised by Basher (1981), and it is this study to which the remainder of this Section 5 is devoted.

          The study is based on a Taylor expansion of the measurement equation in terms of varying band centre wavelength and band shape. Only the first linear term in the expansion is used and only the variations of extraterrestrial irradiance and ozone absorption coefficient are considered. The ozone uncertainties depend non-linearly on airmass and ozone and so the calculations are repeated for a range of likely conditions of airmass and ozone.

          There were three components to the study. Firstly, high resolution spectra of solar irradiance and ozone absorption were used to find band-integrated linear dependences, called here sensitivities, to changes in band position and band shape about mean model bands. Secondly, uncertainties in band position and shape were deduced from the recommended adjustment and operating procedures. Thirdly, the above sensitivities and uncertainties were combined to calculate the propagated uncertainties in band ozone absorption coefficients, in extraterrestrial constants, in standard lamp calibrations, and in column ozone estimations.

          The accuracy of the calculations is necessarily poor, owing to the uncertainty in the input sensitivities and the deduced band uncertainties, but the calculations do clearly indicate the general magnitude and behaviour of the various uncertainties and the general quality requirements for the relevant adjustments and calibrations. They also give some insight into the difficulties of using standard lamps and the relative merits of the various methods for obtaining extraterrestrial constants.

5.2      Basis of calculations

          The very high resolution extraterrestrial flux spectrum of Furukawa et al. (1967), which is a composite of the experimental spectra available at the time, was integrated to provide a spectrum at 0.05 nm resolution, and from this the irradiance sensitivities were calculated. The Furukawa et al. spectrum has only modest accuracy in absolute flux, but is notable for its considerable wavelength detail and high wavelength accuracy, which makes it well suited to this sensitivity study. A section of the original spectrum is shown in Figure 5.1 and helps illustrate this point.

Figure 5.1
Figure 5.1   Section of the Furukawa et al. (1967) solar spectrum illustrating
             its wavelength detail and variability.

          The ozone absorption data of Vigroux (1953 and 1967) for -44°C were plotted and a continuous spectrum was hand drawn to give realistic band shapes, especially in the data-sparse regions (see Figure 1.1). This was then integrated to provide a 0.05 nm resolution spectrum from which the ozone absorption coefficient sensitivities were calculated. The spectrum used has its greatest concentrations of original data in the vicinity of the Dobson bands.

          The transmittance bands of the instrument were modelled as simple symmetrical triangular or trapezoidal shapes, unaffected by lens aberrations or other optical effects. Each band was thus assumed to be defined by, firstly, a centre wavelength, as given by the standard wavelengths in Table 1 of Dobson (1957a), secondly, a bandwidth of 1.0 nm for the short wavelength of a bandpair and 3.0 nm for the long wavelength, and thirdly, a "slopewidth", the spectral width of the bands' sloping sides, of 0.9 nm for all bands. Actually, the bands are smoothed to some extent by optical aberrations (see Figure 2.1) and the spectral widths of the bands increase slightly as wavelength increases owing to the decreasing prism dispersion with wavelength, but these factors will not substantially alter the conclusion of the study.

          The sensitivities to change of centre wavelength, calculated as the mean in just the 0.1 nm interval surrounding the band centre wavelength, are listed in Table 5.1 below. The calculated sensitivities to change in bandwidth and slopewidth are relatively small and are not listed, but their effects are generally included in subsequent calculations. For bandwidth changes, the sensitivity of flux per unit bandwidth was calculated, in order to avoid the linear flux increase with bandwidth. The effect of this increase is eliminated in normal calibration techniques.

                                 TABLE 5.1                                 

      Sensitivities of band-integrated extraterrestrial irradiance and      
      ozone absorption coefficient to change in band centre wavelength.     

     λ    305.5    308.8    311.45    317.6    325.4    329.1    332.4    339.8 nm    

dI/Idλ    -11.4    -11.4     -4.0     -22.2      8.4      0.9     -3.0      1.3 % nm-1

dα/αdλ    -8.0    -13.2    -16.8     -17.1    -10.7    -20.3    -20.1    -12.9 % nm-1

          The predominance of negative values in Table 5.1 for both irradiance and absorption implies that, to some extent, the effect of one will compensate the effect of the other, since a decrease in absorption will result in an increase in transmitted irradiance. This compensation is by no means perfect and will vary with airmass and ozone. Figure 5.2 shows the calculated sensitivities of irradiance and absorption as a function of wavelength between 303 and 308 nm for the 1.0 nm triangular passband. Some very large sensitivities are evident. The spectra suggest that the original choice of the 305.5 nm short wavelength band of the A bandpair was partly dictated by the need to minimise the nett effect of the sensitivities.

          In the calculation of ozone uncertainties, the ozone absorption coefficient sensitivities are weighted by the coefficients themselves, and therefore it is the shorter wavelength sensitivites which are most significant. The irradiance sensitivities combine with equal weight. It is worth noting that the sensitivities to centre wavelength change are approximately inversely proportional to bandwidth, and that this sets fundamental limits to the narrowness of the bands of any UV ozone measuring instrument. Practical bandwidths are 0.5 to 1.0 nm.

          The uncertainties of adjustment, calibration and operation, and their mutual dependences, were deduced largely from the recommended procedures in Dobson (1957b). There is a wide variety of possibilities, especially for the bandwidths and slopewidths and their interdependences, but for the purposes of this study, these have been condensed to the following. Firstly, calibration uncertainties are assumed to comprise: independent centre wavelength uncertainties of 0.025 nm, mutually dependent bandwidth uncertainties of 0.025 nm for the four shorter wavelength bands, mutually dependent bandwidth uncertainties of 0.05 nm for the four longer wavelength bands, and mutually dependent slopewidth uncertainties of 0.025 nm for all bands. (It is thus somewhat arbitrarily assumed that the bandwidths are defined by the slit S2 and S3, and that all the slopewidths are defined by the slit S1.) Secondly, routine operational uncertainties are assumed to comprise uncertainties of 0.025 nm in centre wavelength only, which are mutually dependent within each bandpair, but independent between bandpairs.

Figure 5.2
Figure 5.2   Calculated sensitivities of band-integrated extraterrestrial
             irradiance Po and absorption coefficient α as a function of
             wavelength, for a short wavelength Dobson band.

          Instruments for which the above calibration uncertainties hold true are termed here "well calibrated", while instruments for which the above operational uncertainties hold true are termed here "well operated". These standards of calibration and operation should he attainable by a conscientious, skillful technician, knowledgeable about the instrument's principles of design.

          In the following calculations, the above model uncertainties are taken to represent the standard errors of a normal distribution of uncertainties which is assumed to be present in the population of Dobson instruments. Dependent uncertainties are combined by simple addition, and independent uncertainties are combined by summing their variances.

          The calculations have been made separately for routine operation and for each of the three methods for establishing an instrument's extraterrestrial constants. These methods are:

firstly, the traditional Langley method, which is the ultimate means of calibrating reference instruments, but which in general can be very inaccurate if atmospheric conditions are poor or the instrument has a stray light problem;

secondly, the transfer of a reference instrument's constants by means of transportable incandescent lamps, which is a very attractive option, but which in the past has been plagued by large errors of uncertain origin;

and thirdly, by direct intercomparison with a reference instrument, which, although a rather expensive and tedious option, is reliable and has become the recommended standard practice.

The total error in any case is the combination of the calculated standard errors arising from the calibration and from routine operation.

          A general point, applicable in all three cases, is that uncertainties in ozone absorption coefficients contribute ozone errors which are essentially constant with airmass, whereas uncertainties in extraterrestrial irradiance, and therefore extraterrestrial constants, contribute ozone errors which are inversely proportional to airmass and therefore are most important at low airmass, i.e., high solar elevation.

5.3      Routine operation

          During routine operation, the Dobson instrument's wavelength bandpairs are manually selected and the settings are periodically adjusted to account for temperature changes. The resulting band uncertainty gives rise to uncertainty in the appropriateness of the standard ozone absorption coefficients and in the appropriateness of the extraterrestrial constants. The calculated standard errors in ozone for "well operated" instruments due solely to operational uncertainties are given in Table 5.2. This shows, among other things, standard errors for the AD and CD combinations which are generally less than 0.5% and 1% respectively, but which rise to 1% and 2.5% respectively at an airmass of 1 and an ozone amount of 0.200 atm cm. The standard error for the C bandpair is less than 0.5% for all conditions. (Note that this nominal minimum value of 0.200 atm cm is an extreme minimum. Typical minima, found in the tropics, are near 0.230 atm cm).

          The random component of the errors can be reduced by averaging a set of observations, though of course operator and temperature errors will remain. The standard error of the differences between two instruments will be √2 times the values shown.


Standard errors in column ozone measurements
due solely to wavelength-pair selection uncertainty in routine
operation for "well operated" instruments.

               A       B     C      D      AB     AC     AD     BC     BD     CD     
X=.200 atm cm                                                                        
      μ = 1    .41    .23    .35   3.10   1.45    .83    .95    .98   1.34   2.54   %
      μ = 2    .11    .05    .38   1.34    .37    .39    .37    .78    .56   1.26   %
      μ = 3    .01    .14    .39    .75    .31    .34    .19    .89    .37    .92   %
      μ = 4    .04    .18    .40    .45    .44    .35    .13    .97    .32    .80   %

X=.300 atm cm                                                                        
      μ = 1    .21    .04    .37   1.92    .70    .51    .56    .76    .81   1.67   %
      μ = 2    .01    .14    .39    .75    .31    .34    .19    .89    .37    .92   %
      μ = 3    .06    .20    .40    .35    .49    .36    .12   1.00    .32    .77   %
      μ = 4    .09    .23    .40    .16    .60    .39    .12   1.06    .33    .73   %

X=.400 atm cm                                                                        
      μ = 1    .11    .05    .38   1.34    .37    .39    .37    .78    .56   1.26   %
      μ = 2    .04    .18    .40    .45    .44    .35    .13    .97    .32    .80   %
      μ = 3    .09    .23    .40    .16    .60    .39    .12   1.06    .33    .73   %
      μ = 4    .12    .25    .40    .01    .69    .41    .15   1.11    .36    .73   %

X=.500 atm cm                                                                        
      μ = 1    .05    .10    .39    .98    .28    .35    .26    .84    .44   1.05   %
      μ = 2    .07    .21    .40    .28    .53    .37    .12   1.02    .32    .75   %
      μ = 3    .11    .25    .40    .04    .67    .41    .14   1.10    .35    .73   %
      μ = 4    .13    .26    .41    .08    .74    .43    .17   1.13    .38    .73   %

          A comparison of the calculated standard errors with experimental data may be made. If from the intercomparison of ten instruments at Belsk (Dziewulska-Losiowa and Walshaw, 1975) we take 1/√2 times the standard deviation of the log Pi/Pj differences between each instrument and a reference instrument (no. 83), with the bias and the airmass dependence removed, and at an approximate airmass of 1.7 and approximate ozone amount of 0.375 atm cm, we find that for the A, C and D bandpairs, the ozone standard errors are 0.4, 0.7 and 1.7% respectively, as compared to calculated values of 0.1, 0.4 and 0.8%, respectively, from Table 5.2. The experimental standard errors are about twice the calculated values, and although other error sources will be involved, there remains the suggestion that at the time of the intercomparison the operational uncertainties were greater than the "well operated" criteria assumed here.

5.4      Case where extraterrestrial constants are determined independently

          Uncertainty in the calibration of the wavelength bands in this case implies uncertainty only in the appropriateness of the standard ozone absorption coefficients used. The calculated standard errors for "well calibrated" instruments are given in Table 5.3.


Standard errors in the appropriateness of the standard ozone
absorption coefficients for "well calibrated" instruments.

 A      B      C      D      AB     AC     AD     BC     BD     CD     

0.22   0.35   0.45   0.48   1.09   0.58   0.31   1.38   0.53   0.87   %

Of note is that the AD standard error is the smallest of those for double bandpairs and that the CD standard error is about three times the AD standard error. The total standard error for instruments calibrated independently is the RMS addition of these standard errors and the operational standard errors of Table 5.2. Again, when two instruments are compared, the standard errors of the differences are √2 times these values.

          The independently determined extraterrestrial constants are appropriate to the bands irrespective of any band uncertainty, (although in practice the constants often have large errors). However, the small wavelength differences between any two instruments will result in different standard lamp readings for the instruments. This arises because the measured intensity ratios for sunlight, and hence the extraterrestrial constants, depend on the sensitivities shown in Table 5.1, which are generally negative, whereas the intensity ratios for the lamp light depend on the lamp sensitivites, which are positive, with values of about +3.0% nm-1 being found for the 3000 K blackbody assumed here. Table 5.4 shows, for a comparison of two instruments, the calculated standard error in the difference of their lamp log intensity ratios when their solar log intensity ratios are equal, or vice versa. The listed values incorporate an arbitrary increase by a factor √2 to provide some account of the operational measurement uncertainties. These data go some way toward explaining the otherwise puzzling discrepancies in standard lamp readings between instruments.


Standard errors in the difference of lamp log intensity
ratios for two instruments whose measurements of solar
log intensity ratios agree or vice versa.

  A       B       C       D       AB      AC      AD      BC      BD      CD 

.0025   .0014   .0020   .0042   .0028   .0030   .0052   .0024   .0044   .0047

          A comparison with experimental data from the Belsk intercomparison (Dziewulska-Losiowa and Walshaw 1977, Table IIA) is possible here too. The standard deviations of the differences in lamp log intensity ratios between nine instruments and a reference instrument were 0.0100, 0.0074 and 0.0106 respectively for the A, C and D bandpairs. These are approximately three times the calculated values, which suggests that the calibration standard of the instruments concerned were not up to the "well calibrated" criteria assumed here. By contrast, the corresponding standard deviations for the 1977 Boulder intercomparison (Komhyr et al., 1981a) were 0.0052, 0.0039 and 0.0060 respectively for the A, C and D bandpairs. These represent a considerable improvement on the Belsk results, though they are still about twice the calculated values.

5.5      Case where extraterrestrial constants are established by standard lamp transfer

          In this case there will be uncertainty in the secondary instrument's extraterrestrial constant owing to band uncertainty in both the reference instrument and the secondary instrument and the different sensitivities of sunlight and lamplight, as described in the previous section. Also, there will be uncertainty in the appropriateness of the fixed ozone absorption coefficients for the secondary instrument, and there will be operational uncertainties.

          The combined effect of these uncertainties was calculated but the results are not listed since the lamp transfer method is not often used now. The ozone standard errors are very approximately 1.6 times those of Table 5.2, and range from about 0.5% to 0.6% for the C pair, from 0.4% to 1.5% for the AD combination, and from about 1.1% to 4.1% for the CD combination. The largest values occur for low airmass, low ozone situations, and are appreciable then, especially for the CD combination. It must be noted that these calculated values do not include any account of the uncertainty in the reference instrument's extraterrestrial constants or the large uncertainty found in practical lamp transfer calibrations.

5.6      Case where extraterrestrial constants are established by direct intercomparison

          In Dobson instrument intercomparisons, it is assumed that the standard ozone absorption coefficients apply to both instruments. The extraterrestrial constants of the secondary instrument are then adjusted to ensure the best agreement in total ozone over the normally used range of airmass. However, because of band uncertainties, the absorption coefficients appropriate to the two instruments are in fact different. The resulting small difference in the instruments' absolute ozone scale is automatically accomodated by the adjustment of the secondary instrument's extraterrestrial constants. However, the compensation is only perfect at one condition of airmass and ozone, and as the airmass or ozone departs from this condition there develops an increasing residual ozone error.

          The calculated calibration standard errors in ozone for this situation are not given here on their own, but instead are combined with 1/√2 of the operational errors listed in Table 5.2, in order to give an idea of the minimum ozone error, or best performance, among a group of instruments calibrated by direct intercomparison. The 1/√2 factor is intended to represent a possible reduction in random error by the taking of means. An airmass of 1.5 and an ozone amount of 0.300 atm cm has been assumed as the point of perfect agreement. The results are shown in Table 5.5. Briefly, the standard errors are generally less than 0.5%, 0.4%, and 1.3% for C, AD and CD band combinations respectively, but at an airmass of 1.0 and the extreme minimum ozone amount of 0.200 atm cm they rise to 0.9%, 1.1% and 3.0% respectively.


Standard errors in ozone estimates due to band uncertainty arising from
both calibration and operation, for a "well calibrated", "well operated"
instrument whose extraterrestrial constants are gained by direct

             A     B     C     D     AB    AC    AD    BC    BD    CD   

X=.200 atm cm                                                           
     μ = 1   .57   .66   .87  3.21  2.41  1.32  1.10  2.62  1.64  2.97  %
     μ = 2   .12   .08   .39  1.34   .42   .40   .37   .82   .57  1.27  %
     μ = 3   .08   .18   .42   .77   .50   .39   .22  1.02   .42   .97  %
     μ = 4   .15   .28   .49   .54   .80   .50   .23  1.29   .46   .97  %
X=.300 atm cm                                                           
     μ = 1   .26   .25   .49  1.95   1.04  .65   .60  1.23   .89  1.78  %
     μ = 2   .08   .18   .42   .77    .50  .39   .22  1.02   .42   .97  %
     μ = 3   .17   .32   .51   .49    .91  .55   .25  1.39   .49   .99  %
     μ = 4   .21   .39   .57   .45   1.13  .64   .30  1.61   .57  1.06  %
X=.400 atm cm                                                           
     μ = 1   .12   .08   .39  1.34    .42  .40   .37   .82   .57  1.27  %
     μ = 2   .15   .28   .49   .54    .80  .50   .23  1.29   .46   .97  %
     μ = 3   .21   .39   .57   .45   1.13  .64   .30  1.61   .57  1.06  %
     μ = 4   .25   .43   .60   .48   1.30  .71   .35  1.77   .65  1.14  %
X=.500 atm cm                                                           
     μ = 1   .06   .11   .39   .98    .32  .36   .26   .86   .45  1.06  %
     μ = 2   .18   .34   .53   .46   1.00  .58   .27  1.48   .52  1.01  %
     μ = 3   .25   .43   .59   .47   1.26  .70   .34  1.74   .63  1.13  %
     μ = 4   .27   .46   .64   .53   1.39  .76   .38  1.87   .69  1.20  %

5.7      General remarks

          Estimates of uncertainty in the above calculations were made by recalculating the tables using as input, not the sensitivities, but the uncertainties in the sensitivities as defined by the change in the sensitivity for a 0.05 nm change in the centre wavelength. The uncertainty in the lamp sensitivity was taken as the difference in sensitivity between 3000 K and 3200 K blackbodies. The resulting uncertainties vary greatly among the calculations, but generally they are best represented as one part in five, except for the C bandpair whose uncertainties sometimes rise to one part in two. The additional uncertainty arising from inaccuracy in the spectral modelling of solar irradiance, ozone absorption and band transmittance is difficult to assess, but may perhaps raise the uncertainty to one part in three.

          Bearing in mind the assumptions made and the uncertainties involved, the calculated standard errors should be used as a guide to instrument performance rather than as a specification of accuracy. Also, it should be remembered that the calculations describe the standard errors of a large group of instruments, and hence that the performance of any individual instrument can be considerably different to these statistics.

          All of the standard errors calculated here will vary in inverse proportion to the degree that instruments are "well calibrated" and "well operated". The standards of calibration and operation assumed are not exceptionally high and would appear to be readily attainable by reasonably dedicated scientifically trained staff, such as those who take part in international comparisons. The 1974 Belsk intercomparison data suggests that the standards were not met at that time, but the better results of more recent intercomparisons (Komhyr et al., 1981a) indicate a much improved standard, of calibration in particular.

          Overall, the calculations show that for instruments which are properly adjusted and operated and which are calibrated by the direct intercomparison method, the standard error in AD ozone estimates due to wavelength band uncertainty can be less than 0.5% usually, and less than 1% for low ozone, low airmass situations. The corresponding standard errors for the CD ozone estimates are about 1.5% and 3%. However, there is some suggestion from past intercomparisons that these calculated values may underestimate the true standard errors, by up to a factor of two. An important point is that, although often small, the standard errors are not negligible, especially for the CD band combination, and that every effort should be made to meet, and preferably exceed the recommended precisions for wavelength band calibration and for operational bandpair selection.

5.8      Summary

(i)      Uncertainty in the spectral transmittance of the Dobson instrument's bands, particularly in centre wavelengths, contributes uncertainty to the ozone measurements via the spectrally very variable extraterrestrial irradiance and ozone absorption coefficient, whose band-integrated sensitivities to centre wavelength change are typically 10 to 20% nm-1.

(ii)     Calculated ozone standard errors due to band uncertainty among instruments calibrated and operated to the recommended standards depend on the band combination, on airmass and ozone amount, and on the method by which the instrument's extraterrestrial constants are determined.

(iii)    Band uncertainty explains at least part of the differences in measured standard lamp log intensity ratios between instruments, and, similarly, the errors in extraterrestrial constants transferred by standard lamps. Calculated standard errors in these quantities lie between 0.002 and 0.005. The experimental values for the A, C and D bandpairs found at intercomparisons are about twice the calculated values.

(iv)     Calculated ozone standard errors due solely to operational bandpair selection are generally less than 0.5%, 0.5% and 1% respectively for the C, AD and CD ozone estimates, though the latter two rise to nearly twice these values when the airmass is 1 and the ozone amount is at the nominal extreme minimum of 0.200 atm cm. Values for the A, C and D bandpairs found at the Belsk intercomparison are about twice the calculated values.

(v)      Calculated ozone standard errors due to both calibration and operational uncertainty for the C, AD and CD band combinations are generally less than 1%, 1% and 2% respectively among instruments whose extraterrestrial constants are obtained independently or by lamp transfer, and less than 0.5%, 0.5% and 1.5% respectively where the constants are obtained by direct intercomparison. Maximum values, of about twice these values, generally occur when the airmass is very low and the ozone amount is very low.

(vi)     The calculations indicate deficiencies in the adjustment or operation of instruments at the Belsk intercomparison. Subsequent intercomparisons have shown rather better performances, probably as a result of the greater attention given to prior laboratory adjustment and calibration.

(vii)    Overall, the standard error due to band uncertainty among properly calibrated and operated, directly intercompared instruments can be less than 0.5% for the standard AD ozone estimate in most circumstances. Greater standard errors will occur for other situations and other band combinations. To obtain such levels of uncertainty, assiduous efforts are required to meet, or if possible exceed, the recommended standards of calibration and operation.

Return to Table of Contents

Forward to The Bandwidth Effect