CALCULATION OF UNDERSAMPLING CORRECTION SPECTRA FOR DOAS SPECTRAL FITTING

Transcription

1 CALCULATION OF UNDERSAMPLING CORRECTION SPECTRA FOR DOAS SPECTRAL FITTING Sander Slijkhuis 1, Albrecht von Bargen 1, Werner Thomas 1, and Kelly Chance 2 1 Deutsches Zentrum für Luft- und Raumfahrt e.v., Deutsches Fernerkundungsdatenzentrum (DFD), Oberpfaffenhofen, D Weßling, Germany 2 Harvard-Smithsonian Center for Astrophysics, 60 Garden Street, Cambridge, MA 02138, USA Sander.Slijkhuis@dlr.de, Albrecht.von-Bargen@dlr.de, Werner.Thomas@dlr.de, kchance@cfa.harvard.edu ABSTRACT We describe a method to improve the retrieval of minor trace gases from observations of the GOME instrument onboard ERS-2. This method uses a correction spectrum which is fitted to the observed Earthshine spectrum simultaneously with all other trace gas reference spectra, using the standard DOAS fitting. The correction spectrum accounts for undersampling effects which occur when the GOME Solar calibration spectrum is fitted to a GOME Earthshine spectrum. We show that this undersampling correction spectrum can be calculated from first principles using the wavelength calibration of the Earthshine spectrum, the wavelength shift between Earthshine spectrum and Solar calibration spectrum, and the instrument s slit function. A sensitivity study on these basic parameters shows that especially the latter must be known with high accuracy for the method to give satisfactory results. Using the theoretical undersampling correction spectrum, our r.m.s. errors on the DOAS fit can be reduced by almost a factor of 3. A test run of simultaneous fitting of BrO and other trace gases on the GDP yielded an improvement on chi-square of better than a factor of 5, but this may partly be due to compensation of errors in the Ring spectrum used. Higher reductions may be possible using better knowledge of the slit function. 1. INTRODUCTION Residuals from the spectral fitting of trace gases with very small absorption depths show a fixed pattern of high-frequency oscillations, which is very stable over one GOME orbit of data. This was first noted by K. Chance (Ref. 1) in the fitting of BrO from GOME data. He noted that the residuals did not have the character of either a known (or likely) molecular absorption or of a potential contribution from the Ring effect. The effects were attributed to spectral undersampling ; it was speculated that a different filling of the instrument slit in Solar and Earthshine observations might cause the problem. The fit residuals could be largely reproduced as the difference between a solar spectrum sampled from a highly oversampled solar reference and a solar spectrum sampled at two phases from a representation of this solar spectrum taken only at the GOME wavelength grid. This two-phase undersampling correction was succesfully applied in the retrieval of BrO by several authors (e.g. Chance, Ref. 1, and Richter et al., Ref. 4). However, the selection of the appropriate phase is rather ad-hoc, based on finding the best match to BrO fit residuals without a full understanding of the physical background. In this paper we show that most of the BrO fit residual can be simply explained by the fact that the GOME solar calibration spectrum is Doppler-shifted with respect to the Earthshine spectra: due to undersampling in the instrument, systematic interpolation errors occur when in the spectral fitting the Doppler-shifted solar spectrum is shifted back to the Earthshine spectra. Knowing basic instrument parameters, these systematic interpolation errors can be calculated. This enables the a priori calculation of an undersampling correction spectrum for any spectral window in the GOME instrument. 2. METHOD The method is based on the following consideration. The Earthshine spectrum is recorded in the GOME instrument rest frame, because the field-ofview direction is perpendicular to the spacecraft motion. The spectral calibration parameters as derived from the internal Pt/Cr/Ne calibration lamp are also measured in this instrument rest frame. The Sun is recorded with a negative Doppler shift, because during Solar calibration its light enters the instrument through a viewport which makes an angle of 22.5 to the spacecraft motion. Therefore, relative to the position of its Fraunhofer lines, the Solar calibration spectrum is sampled by the detector pixels on a wavelength grid which is offset from the Eartshine spectrum with a positive Doppler shift.

2 In the spectral analysis the Solar spectrum is shifted back to the Earthshine spectrum. This process causes interpolation errors. In the case of GOME, these interpolation errors are enhanced by the fact that the detector pixels undersample the signal: according to the Nyquist criterion we should at least have two sampling points per resolution element, but especially in the BrO fitting window we have significantly less. The detector pixels are separated by 0.11 nm, whereas the FWHM of the slit function is nm in that region. The interpolation errors can be directly calculated if the Solar spectrum is known with a resolution much better that the instrument resolution. We do have an accurate high-resolution solar spectrum at our disposal; this spectrum and its calibration are described in Ref. 2. Under the assumptions that the Sun remains constant, and that the instrument response is constant on the detector pixel scale, this spectrum approximates the true input spectrum (the intensity scaling is irrelevant) to the instrument. Neither of these assumptions is quite true of course, but in first order the interpolation errors on the difference between true and synthetic Solar spectrum can be neglected. In the first step of the calculation of undersampling errors, a synthetic solar spectrum is calculated by convolving the high-resolution solar spectrum from Ref. 2 with the GOME instrument slit function. The convolved spectrum is sampled at the centre wavelengths of the detector pixels in the instrument rest frame. Denote this spectrum by [r]. The convolved spectrum is also sampled at the centre wavelengths of the detector pixels in the Dopplershifted frame. The resulting spectrum is shifted back by cubic spline interpolation to the instrument rest frame. Denote this spectrum by [d]s. In the spectral fitting process, the errors made by the interpolation can now be compensated by multplying the shifted, observed Solar calibration spectrum by the ratio [r]/[d]s. In practice we follow a slightly different way of correcting, because the above direct multiplication requires that the instrument parameters are known with very high accuracy. It turns out that for small errors on wavelength calibration or Doppler shift, the amplitude of the correction is more affected than the shape of the correction spectrum. Therefore, we treat the correction spectrum the same way as the trace gas reference spectra. It is simultaneously fitted with the trace gas spectra, which enables a bestfit scaling of its amplitude. In the DOAS spectral fitting method (see Ref. 3 and references therein) the Earthshine and Solar reference are logarithmised before fitting. In this case, the undersampling correction spectrum (UC) is calculated as: UC = log([r]/[d]s) In the direct fitting method used in Ref. 1 we obtain the difference between synthetic spectrum without and with interpolation error as: UC = [r] [d]s Notes: (1) It is essential that the (cubic spline) interpolation here is identical to the interpolation method used in the spectral fitting program, since the interpolation is the source of error. (2) GOME does not really sample the spectrum, but integrates this over the pixel width. It therefore may seem appropriate that we do not simply convolve our high-resolution Solar spectrum and take the values at the wavelength grid points, but that we integrate the convolved highresolution spectrum over the pixel width. We have tried both and obtain nearly identical results. The exact shape and width of the convolution funtion is essential here (see also Sect. 3.2). We use a Gaussian slit function and, for the sampling method, its width can be derived by a non-linear fit of the convolved high-resolution spectrum to the measured GOME spectrum. However, this width cannot be used in the pixel binning method, because this width already incorporates the degradation in resolution due to pixel binning. In this case the width of the convolution function before pixel binning is needed. This needs fitting of an instrument model which takes effects of pixel binning and pixel-to-pixel charge transfer into account. Since the results are nearly identical, we use the more straightforward sampling method. 3. RESULTS 3.1 Direct Fitting We first compare the results of our method with the results obtained from BrO fitting in Ref. 1. To this end we compare our calculated UC spectrum with the residual structures found in Ref. 1. Figure 1a shows as a histogram the BrO fit residue for each GOME pixel wavelength [nm], with the UC overplotted (solid line). The UC is calculated using the wavelength calibrations of the Earthshine and the Solar reference, using a Gaussian slit function with 0.16 nm FWHM, as determined in Ref. 1. Its amplitude is scaled such that it attains the same r.m.s. as the BrO fit residue spectrum. Figure 1b shows the difference between the BrO fit residue and the scaled UC. The remaining structure has an r.m.s. which is a factor of 2.4 smaller than the r.m.s. of the BrO fit residue. For comparison, using the two-phase undersampling correction from Ref. 1, we obtain similarly a reduction by a

3 Figure 1: a. Undersampling correction spectrum (solid) and fit residual from Ref. 1 (histogram) b. Difference between fit residual and undersampling correction spectrum Figure 2: a. As Fig. 1, but correction calculated for GDP using DOAS (solid); fit residual from GDP (histogram) b. Difference between GDP fit residual and undersampling correction spectrum

4 Figure 3: As Fig. 2b, but undersampling correction calculated for less than optimal parameters to show the sensitivity of the method: a. Using an instrument resolution of 0.20 nm instead of 0.15 nm b. Using a wavelength calibration shifted by 0.02 nm c. Using a single hyperbolic slit function instead of a Gaussian of the same FWHM

5 factor of 2.3. In both cases the differences of (fit residual U C) show basically the same pattern; only in the region longward of nm does the shape of our UC differ somewhat from the two phase UC. 3.2 DOAS Fitting The DOAS fitting was performed using a test version of the operational GOME data processor (GDP) at DLR. The logarithmised Earthshine spectrum was fitted with the logarithmised Solar calibration spectrum, reference cross-sections of O 3, NO 2, O 4, BrO, a theoretical Ring spectrum from Ref. 2, and a polynomial closure term. For BrO the cross-sections from Ref. 5 were used, with a wavelength correction of 0.17 nm (see Ref. 4). Except for the Solar reference, no shifts or squeezes in wavelength were allowed. The UC is calculated here using the wavelength calibration of the GOME level 1 data product. For the convolution of the high-resolution solar spectrum we used a Gaussian slit function with 0.15 nm FWHM, which gives us a slightly better fit than the 0.16 nm from Ref. 1. Figure 2a shows the theoretical undersampling correction spectrum calculated for the DOAS fitting method (solid line). Similar to Fig. 1a, the UC is scaled to the fit residues of the GDP fit (histogram). The same GOME observation as for Fig. 1 is used. Figure 2b shows the difference between the GDP fit residue and the scaled UC. The remaining structure has an r.m.s. which is a factor of 2.9 smaller than the r.m.s. of the GDP fit residue. A comparison with the results in Fig. 1 is not direct possible since the fitting methods are too different. Sensitivity analysis In order to demonstrate the sensitivity of the UC to the instrument parameters, we provide a few calculations where basic parameters are varied. The results presented in Figure 3 are to be compared to the result shown in Fig. 2b. Figure 3a shows a calculation where the nominal GOME resolution of 0.20 nm is used instead of the resolution of 0.15 nm. Figure 3b shows the effect of shifting the instrument rest frame by 0.02 nm (this is the expected shift in wavelength for the nominal instrument lifetime). In both cases a substantial degradation (60-70% r.m.s.) of the fit result can be expected. This shows that for operational processing one cannot simply use one standard undersampling correction spectrum, but on the basis of accurately fitted instrument parameters one has to calculate (or select a pre-calculated) one. Calculations show that up to a wavelength shift of nm, the UC is not significantly changed. This implies that the variation of UC over one orbit can be neglected, in line with the BrO fit results obtained in Ref. 1. A further illustration of how sensitive the UC is to the instrument properties is given in Figure 3c. Here the Gaussian slit function for convolution of the high-resolution spectrum is replaced by a single hyperbolic function ( F = a 2 / [a 2 + λ 2 ] ) of the same FWHM (a different FWHM did not improve the result). Also here the result is significantly worse. Tests with a top-hat slit function showed even worse results. Improvement in BrO fitting Figures 4 and 5 illustrate the improvement in BrO fit quality which can be made using the current test version of the GDP. The calculations were performed for a GOME ground-pixel with a relatively large BrO content (ground pixel 1957 in orbit 7355); the slant column density for BrO is molec/cm 2. Fig. 4a shows the residual BrO absorption in the GOME observation after removal of all interfering trace gas species O 3, NO 2, O 4, solar spectrum, and Ring effect. Fig. 4b shows the same but with the calculated undersampling correction spectrum taken into account in the fitting. No shifts or squeezes in wavelength were allowed for the reference spectra, except for the Solar reference and the BrO spectrum. Figure 5 is similar to Fig. 4; here shifts and squeezes were allowed for the Ring spectrum. The value of chi-square on the simultaneous fit is reduced by more than a factor of seven for Fig. 4, and more than a factor of five for Fig. 5. The DOAS fit was performed using the Ring spectrum for the nominal resolution of 0.20 nm. This is not quite consistent since the Ring should also be calculated for the fitted resolution to obtain a consistent calculation. This may be the reason why allowing shifts/sqeezes on Ring yields a better result in this case. The difference between Fig. 4a and Fig. 5a indicates that in the absence of an UC spectrum in the fit, a shift/squeeze of Ring may partially compensate undersampling effects. For instance, the feature near is corrected although it clearly shows in our UC spectrum and therefore is unlikely an error in Ring. The purpose of this exercise was not to obtain the best possible scientific result, but to understand the undersampling effect and to show its impact on BrO fitting in the context of the current version of the GDP. The examples presented here give an indication of the performance which can be expected before any further optimisations are made. The factor of 5-7 in the improvement of chi-square may be a bit too optimistic, since the inclusion of the UC in the fit may also correct some errors on the Ring spectrum

6 Figure 4: a. Atmospheric spectrum processed by GDP after removal of all components except BrO; the fitted BrO reference spectrum is overplotted (solid line) b. As a) but with undersampling correction spectrum fitted as additional component Figure 5: As Fig. 4, but wavelength shifts and squeezed allowed for (only) the Ring spectrum

7 or on other reference spectra. 4. DISCUSSION Using the theoretical undersampling correction spectrum, our r.m.s. fit errors on the DOAS fit can be reduced by almost a factor of 3 if all other fits of reference spectra remain the same. By allowing the simultaneous fit of UC spectrum and reference spectra, an experimental version of the GDP fitting software achieved an improvement of chi-square of a factor of 5. The present results brings the r.m.s. fit residuals down to a factor of 2.3 above the level of the measurement precision given in the level 1 product. We have not yet investigated the stability of the fit residuals over the orbit, but results from Ref. 1 suggest that most of the residual is systematic. In view of the sensitivity of the results to the shape of the slit function, it may well be that a better characterisation of the slit function would further reduce the fit residuals. Other instrument effects may also contribute to systematic errors: 1. One of the reasons of using a GOME-measured solar reference has always been that instrument calibration errors on e.g. pixel-to-pixel gain or etalon correction would divide out if GOME-measured Earthshine spectra are fitted with the corresponding solar spectrum. In case of a (Doppler) shift between these two spectra this is not longer true: the shift will partly transfer errors on one detector pixel to the neighbouring pixel. For this reason it might be justifiable to take the average fit residue over the orbit and treat this as additional instrument correction spectrum, as has been done in some work using the undersampling correction spectrum described in Ref. 1. However, if pixel transfer effects are playing a part, one would expect that the largest fit residues correspond to the largest gradients in the radiance spectrum. This is not readily apparent in our fit residues. 2. Another effect which will apparently provide systematic errors is instrument noise on the solar calibration spectrum. However, this is only systematic as long as the same solar reference is used on time scales shorter than one day. 3. Other potential sources of error might arise from the baseline correction (e.g. dark signal, straylight). These would induce errors with structures somewhat similar to the Ring spectrum; possibly the shift/sqeeze on Ring partly compensates for these errors. ACKNOWLEDGEMENTS The authors wish to thank Ernst Hegels at DLR- DFD for stimulating discussions. REFERENCES 1. Chance, K. Analysis of BrO Measurements from the Global Ozone Monitoring Experiment, 1998, Geophys. Res. Letters, 25, p Chance, K. and R.J.D. Spurr, Ring effect studies: Rayleigh scattering, including molecular parameters for rotational Raman scattering, and the Fraunhofer spectrum, Applied Optics 36, p.5224, GOME Interim Science Report, edited by T. D. Guyenne and C. J. Readings, SP-1151, ESA publications Division, ESTEC, Noordwijk, The Netherlands, ISBN , 1993, and references therein 4. Richter, A., et al., GOME observations of tropospheric BrO, this issue, Wahner, A., et al., Absorption cross sections of BrO between 312 and 385 nm at 298 K and 223 K, Chem. Phys. Lett., 152, p.507, 1988