Undersampling correction for arraydetector-based satellite spectrometers

Kelly Chance, Thomas P. Kurosu, and Christopher E. Sioris

Array detector-based instruments are now fundamental to measurements of ozone and other atmospherictrace gases from space in the ultraviolet, visible, and infrared. The present generation of such instru-ments suffers, to a greater or lesser degree, from undersampling of the spectra, leading to difficulties inthe analysis of atmospheric radiances. We provide extended analysis of the undersampling suffered bymodern satellite spectrometers, which include the Global Ozone Monitoring Experiment, ScanningImaging Absorption Spectrometer for Atmospheric Chartography, Ozone Monitoring Instrument, andOzone Mapping and Profiler Suite. The analysis includes basic undersampling, the effects of binning intoseparate detector pixels, and the application of high-resolution Fraunhofer spectral data to correct forundersampling in many useful cases. © 2005 Optical Society of America

OCIS codes: 010.1280, 120.4570, 280.1120.

1. Introduction

Array-based spectrometers used in atmospheric re-mote sensing can suffer substantially from spectralundersampling, with negative consequences to thequality of data retrieved from the measurements. Thiscircumstance was first obvious in measurements fromthe Global Ozone Monitoring Experiment (GOME),1 inwhich scientists fitting trace gases from GOME spec-tra found large systematic fitting residuals, and largefitting errors, even after correcting for Doppler shiftsbetween radiances and irradiances. In a previousstudy it was recognized that this result was mainly dueto spectral undersampling, and a technique was pre-sented for correcting most (�90%) of the undersam-pling error in spectral regions where atmosphericabsorption effects are small.2 For GOME, this includesfitting regions used for nitrogen dioxide (NO2), bro-mine monoxide (BrO), chlorine dioxide (OClO), andformaldehyde (HCHO). If used with caution, the tech-nique can also be applied successfully to ozone (O3) andsulfur dioxide (SO2). (Caution is required because theabsorption optical depths for O3 are higher than for theother trace species listed; correction assumes that, to

first order, the radiance spectrum consists mostly ofbackscattered Fraunhofer structure. The SO2 absorp-tion occurs in the region where O3 also absorbsstrongly.) The technique consists of comparing fullysampled and undersampled versions of a high-resolution Fraunhofer reference spectrum,3 with thedifference being the effect of undersampling. Slijkhuiset al.4 suggested that the observed residuals are in-duced by the resampling needed to compare Earthradiance and solar irradiance spectra in the fitting,because they are measured with different Dopplershifts of the ERS-2 satellite with respect to the sun. Itis demonstrated here that this is not entirely the case:Wavelength shifts between GOME radiances and ir-radiances are larger than can be accounted for byDoppler shifts. Slijkhus et al.4 also presented a versionof the technique (see Ref. 2) implemented in the Scan-ning Imaging Absorption Spectrometer for Atmo-spheric Chartography (SCIAMACHY) operationalprocessor. The technique is now also used for measure-ment of HCHO5 and NO2

6 and has been implementedin the operational processor for measurements of BrO,OClO, and HCHO by the Ozone Monitoring Instru-ment (OMI).7

The present analysis improves on the previous un-derstanding of correction for undersampling becausethe sampling theorem in conjunction with the instru-ment slit function is fully considered. The analysis isapplied to measurements by GOME and OMI, sincethey represent the two major instrument types: diodearray detectors (GOME) and CCD detectors (OMI).The development applies as well to other instruments,including SCIAMACHY (diode array detectors) and

K. Chance (kchance@cfa.harvard.edu), T. P. Kurosu, and C. E.Sioris are with the Harvard-Smithsonian Center for Astrophysics,Mail Stop 50, 60 Garden Street, Cambridge, Massachusetts,02138-1596.

Received 7 January 2004; revised manuscript received 12 Au-gust 2004; accepted 27 September 2004.

0003-6935/05/071296-09$15.00/0© 2005 Optical Society of America

1296 APPLIED OPTICS � Vol. 44, No. 7 � 1 March 2005

the Ozone Mapping and Profiler Suite (OMPS), part ofthe National Polar Orbiting Environmental SatelliteSystem, which uses CCD detectors.

2. Definitions and Assumptions

The spectrometers considered here use array detec-tors, which respond to the incoming light over finiteregions, with defined response profiles. We will use theterm instrument line shape (ILS) for the response of aspectrometer to a monochromatic source up to where itenters the array detector and the term instrumenttransfer function (ITF) for the ILS convolved with thedetector pixel response. The following properties areassumed to be valid over the region of the array detec-tor that is necessary to consider for ITFs and under-sampling corrections at a given pixel, as developedhere.

1. Linear response. Each pixel responds lin-early to input light intensity.

2. Equal pixel response. Pixels respond equallyto photons of different wavelength. Pixels respond (orcan be calibrated to respond) equally in output signalfor equal input light intensity.

3. Linear dispersion. Pixel wavelengths are atequal increments.

4. No end-point issues. The region under con-sideration is sufficiently far from the array end forwavelengths beyond the array wavelength range tocontribute to the signal. (Proximity to the array endwould require recalculation of the ILS from deconvo-lution of the ITF, as discussed below.)

5. Calibration light source issues. ITFs aredetermined using either

Y lines with negligible spectral width orY a tunable source with negligible spectral

width and equal intensity as it is tunedover the ITF.

6. Slit width variation is negligible over theITF centered at each particular wavelength.

Higher-order corrections will likely be needed in thefuture to account for the breakdown of some of theseassumptions.

3. Spectral Undersampling

Consider a wavelength range extending from �min to�max that is fully or partly sampled by an array ofdetectors, spaced at wavelength increment ��. Anycontinuous incoming signal spectrum extending overthe wavelength range can be represented fully byexpansion in spatial (i.e., wavelength) frequency in aFourier series

S(�) � a0 ��k�1

�

[ak cos kt � bk sin kt]

� a0 ��k�1

�

{ak cos[�k(� � �min)]

� bk sin[�k(� � �min)]}, (1)

where

t � 2� � �min

�max � �min� 2

� � �min

�, (2)

and the spatial frequencies are

�k �2k�

. (3)

Suppose that the signal is band limited, that is, lim-ited in content of spatial frequency components to amaximum spatial frequency �max. Then, by the sam-pling theorem,8,9 the information content of the spec-trum is fully known if the spectrum is sampled overthe full range �min to �max to twice this maximumspatial frequency, 2�max (the Nyquist-sampling fre-quency):

��

2�

12�max

. (4)

In this case, the signal expansion includes only theterms necessary to measure spatial frequencies��max:

S(�) � a0 ��k�1

N

[ak cos kt � bk sin kt], N ��

2��.

(5)

The spectrum is completely determined by this ex-pansion. Its values at points other than the sampledpoints can be determined by the fact that the value ateach sampled point �n represents the intensity cn of asampling function, sinc�2��n � ������, centered atthat point,8

sinc[2(�n � �)���] �sin[2(�n � �)���]

[2(�n � �)��] , (6)

plus the constant offset a0:

cn � S(�n) � a0 ��k�1

N

[ak cos ktn � bk sin ktn],

tn �2(�n � �min)

�. (7)

The sampling function is written here in this way toemphasize the pixel dependence and orientation ofthe following discussion. The sampling function pre-sented here is actually an approximation to a fullerand more complex form,9,10 but the differenceamounts to a completely negligible correction exceptwithin several sample points of the �min and �max endpoints.

If the signal is not band limited to �max but is stillsampled to only 2�max, spatial frequencies greaterthan �max are aliased into the band 0 �� � �max, withsignal information for �max � � � 2�max appearing

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at �max � � and information for 2�max � � � 3�maxappearing at � � 2�max, etc. (Ref. 9, pp. 16–18). It willbe demonstrated in Section 4 that informationaliased from the spatial frequency band �max � �� 2�max represents the most important sources ofproblems for GOME and the main target of correc-tions. Problems from aliasing in the spectra becomemost evident when it becomes necessary to resamplea spectrum in wavelength, for example, when com-paring an atmospheric radiance spectrum with a so-lar irradiance spectrum to determine atmosphericcomposition from molecular absorption lines, but theinterference from aliasing is present in any casewhen the spectrum is undersampled (i.e., the spec-trum is not fully Nyquist sampled). For example, syn-thetic spectra calculated during the fitting process todetermine abundances of atmospheric gases wouldnot normally include aliasing, whereas the measuredspectra would.

A Nyquist-sampled spectrum S��� is fully describedby summing over the contributions from the m indi-vidual sample points

S(�) � c0 ��i�1

m

ci sinc[2(�i � �)���], (8)

where m � 2N � ����. It is now possible to inves-tigate the case in which the spectrum is not fullyNyquist sampled. Consider a spectrum input to theinstrument Sinp���. If the spectrum is completelyknown a priori, it can be expanded as before in aFourier series

Sinp(�) � a0 ��k�1

�

[ak cos kt � bk sin kt]. (9)

Sinp can be separated into a Nyquist-sampled part,SNyq, containing only spatial frequencies � �max, andan undersampled part, Sund

SNyq(�) � a0 ��k�1

N

[ak cos kt � bk sin kt], (10a)

Sund(�) � �k�N�1

�

[ak cos kt � bk sin kt]. (10b)

The undersampled part of the spectrum is thus

Sund(�) � Sinp(�) ��i�1

m

ci sinc[2(�i � �)���] � c0,

(11)

where m runs over the sampled points.

4. Slit Functions (Instrument Transfer Functions)

An ITF serves as a low-pass filter to limit the spatialfrequency content of the spectrum. Ideally, an ITFwould limit the spectral information to frequencies

� �max. The ITFs for satellite-based spectrometers aremeasured in one of three manners:

1. By use of a reference line lamp (often a Pt-NeCr lamp11). This method has the advantage thatlamp lines are much narrower than the ITF. Thedisadvantages are that lines are not always com-pletely separated, that spectral coverage may beinadequate in some regions, and that a set of points(one per detector pixel) is mapped out, rather thana continuous ITF. This method was used for GOMEand SCIAMACHY.

2. By use of a tunable source consisting of abroadband light source and a monochromator. Thismethod has the advantage of mapping out a contin-uous ITF but the disadvantage of being a spectrallybroader source than the PtNeCr line source. Thismethod was used for OMI, where the full-width athalf-maximum (FWHM) of the source was �0.1 OMIdetector pixel.12 The residual effects from finitesource width could, in principle, be reduced by use ofthe deconvolution techniques introduced in Section 5.

3. By fitting of flight spectral irradiance data to ahigh-resolution solar reference spectrum,3 for whichthe fitting includes wavelength adjustment and si-multaneous fitting to a parameterized ITF.2,13

An ITF, ���, can be expanded to include Nyquist-sampled and undersampled portions, if one ignores(for now) detector pixel binning:

(�) � �0 ��i�a

b

�i sinc[2(�i � �)���] � und(�).

(12)

The limits a and b are selected to include portions ofthe detector array where the slit function contributessignificantly. As examples, we show the GaussianGOME ITF we normally use for wavelength calibra-tion purposes. A more complex, compound hypergeo-metric ITF was determined during the instrumentcharacterization.14 We find that the Gaussian spec-trum provides better wavelength calibration forGOME spectra, and we use it routinely in our dataanalyses. It also provides much better undersam-pling correction for GOME spectral fitting. We showan OMI ITF (see Ref. 12) for this same wavelengthregion and for the NO2 fitting region (405–465 nm;the ITF determined at 432 nm is used here). The OMIITFs are selected for CCD row 150, corresponding toa viewing azimuth angle of 29.5°, approximately half-way from the nadir view to the extremity of the OMIswath. Additional types of ITFs for OMI may be con-sidered once there is flight data for comparison. Forall ITFs shown in this section, the complication ofbinning over the detector pixel response function isnot yet included. This complication will be discussedin Section 5.

Figure 1 shows the GOME Gaussian ITF, deter-mined from ERS-2 orbit 81003031 (3 October 1998),used here as a test orbit for determination of under-sampling correction, appropriate to the fitting win-

1298 APPLIED OPTICS � Vol. 44, No. 7 � 1 March 2005

dow used for BrO retrieval in GOME.2 The FWHM ofthe ITF is 0.160 nm, and there are 1.4 samples perFWHM in this region of GOME spectra. The ITF isadditionally decomposed into fully sampled and un-dersampled components. Figure 2 shows this ITFwith the hypothetical sampling to twice the GOMEsampling frequency. Figure 3 shows the OMI ITF forthis same wavelength region (FWHM � 0.421 nm,2.8 samples per FWHM) and its decomposition intofully sampled and undersampled components. Figure4 shows this ITF with the hypothetical sampling totwice the OMI sampling frequency. Figures 5 and 6show the OMI ITFs and decompositions for the NO2fitting wavelength range (FWHM � 0.639 nm, 3.0samples per FWHM). Note the asymmetry in themeasured OMI ITFs and their decompositions.

It is clear from all three examples that higher sam-pling by a factor of 2 greatly decreases the under-sampled portion of the ITF, almost eliminating

undersampling. Since for GOME this sampling corre-sponds to 2.8 samples per FWHM, this result demon-strates that information aliased from the spatialfrequency band �max � � � 2�max represents themost important source of undersampling in this spec-tral region.

5. Binning into Detector Pixels

GOME and SCIAMACHY use Reticon-S RL-1024SRU-type linear diode array detectors, which are de-signed to have the sensitivity profiles shown in Fig. 7for the spectral resolution and sampling of GOME.These detectors have 1024 photodiode elementsspaced at 25 �m . The center 13 �m of each elementis n doped, with the bulk of the material p doped, togenerate the response profile shown (the newerReticon-L detectors are also spaced at 25 �m but with19��m n-doped regions and 6��m interdiode gaps).

Characterization of the actual pixel responsesmust take into account that the input light source

Fig. 1. GOME Gaussian ITF in the spectral region used for BrOdeterminations, and its decomposition into Nyquist-sampled andundersampled components.

Fig. 2. GOME Gaussian ITF and the Nyquist-sampled and un-dersampled portions for the hypothetical case in which the slitfunction is sampled to twice the GOME spatial frequency.

Fig. 3. OMI ITF for the BrO fitting region and the Nyquist-sampled and undersampled portions.

Fig. 4. OMI ITF for the BrO fitting region and the Nyquist-sampled and undersampled portions for the hypothetical case inwhich the slit function is sampled to twice the OMI spatialfrequency.

1 March 2005 � Vol. 44, No. 7 � APPLIED OPTICS 1299

(mostly Fraunhofer spectrum, convolved with theILS) varies significantly across the detector pixel pro-file. The final response is the integral across the pixelprofile of the convolution of the input spectrum withthe ILS

Rpix(�) ��profile

S(��) � (� � ��)d��, (13)

where Rpix is the pixel response, S is the input spec-trum, � is the ILS, and R denotes convolution. Tocalculate the response, it is first necessary to obtainthe ILS by deconvolution, since the measured ITF isthe convolution of the ILS with the pixel responseprofile. This response is accomplished with the Jans-son method, as described by Blass and Halsey,15 forGOME, and a nonlinear least-squares fitting to aparameterized profile shape for OMI. The result forthe GOME slit function is shown in Fig. 8. The de-convolved slit is the best match to the ILS that couldbe obtained in the iterative deconvolution process.The difference between the initial and the recon-volved slit is an indicator of the goodness-of-fit forthis procedure. The response functions for the OMICCD detectors are Gaussian with a FWHM of 25 �mand separated by 22.5 �m.12 Figures 9 and 10 show

Fig. 5. OMI ITF for the NO2 fitting region and the Nyquist-sampled and undersampled portions.

Fig. 6. OMI instrument transfer function for the NO2 fitting re-gion and the Nyquist-sampled and undersampled portions for thehypothetical case where the slit function is sampled to twice theOMI spatial frequency.

Fig. 7. Response of three Reticon-S detector pixels as a functionof location in GOME detector channel 2.

Fig. 8. Deconvolution of the GOME ITF from the pixel responsefunction to determine the ILS in the BrO fitting region.

Fig. 9. Deconvolution of the OMI ITF from the pixel responsefunction to determine the ILS in the BrO fitting region.

1300 APPLIED OPTICS � Vol. 44, No. 7 � 1 March 2005

these along with the decomposition to determine theILS for the BrO and NO2 fitting regions of OMI.

6. Undersampling Correction

A. Wavelength Calibration Issues

Satellite radiances and irradiances can be calibratedin wavelength with high absolute accuracy (typically� 0.0004 nm for the GOME BrO fitting region) by useof cross correlation to the solar spectrum described byChance and Spurr,3 with simultaneous fitting of theITF.2,13 Previously it was suggested that the appar-ent wavelength shift between GOME radiances andirradiances was due to instrumental effects, influenc-ing the way the detectors are illuminated in the dif-ferent measuring geometries.2 Slijkhus et al.4 statedthat the shift is due to the Doppler effect, since irra-diances are obtained during the portion of the orbitwhen the satellite is moving toward the sun. ForERS-2 orbit 81003031, the maximum velocity towardthe sun is 7.46 km s�1, whereas the average wave-length shift (irradiance � radiance) for the BrO fit-ting window is equivalent to 7.865 � 0.025 km s�1

(0.000923 � 0.00003 nm). The satellite Doppler shiftcontributes 6.89 km s�1 (0.0081 nm), rather than thefull 7.46 km s�1, given the GOME solar measurementprocedure.4 There is an additional component of0.50 km s�1 at this season from the ellipticity of theEarth’s orbit, for a total Doppler shift of 7.39 km s�1,leaving a significant instrument, a spectral compo-nent, or both (see the discussion in Ref. 16 for possiblespectral tilt contributions). For the other spectrome-ters considered here, the relative contributions arenot yet determined. For OMI in particular, since itsundersampling is calculated here, a relative shiftequal to the full Doppler shift is assumed; this as-sumption will be modified when flight data becomeavailable.

B. Undersampling Calculations

Calculation of the undersampling correction for eachcase is accomplished by convolution of the high spec-

tral resolution (0.01 nm) solar reference spectrum,described by Chance and Spurr,3 with the ITF deter-mined for the instrument and wavelength region andby means of differencing fully sampled and under-sampled representations of this convolved solar spec-trum at the sampling grid of the satellite radiances.

1. Convolve the high-resolution solar referencespectrum Eref with the satellite instrument ITF tocreate a lower spectral resolution, but a highly over-sampled, solar reference spectrum, Eover,

Eover(�) ��profile

Eref(��) � ITF(� � ��)d��. (14)

The portion of Eref used in fitting BrO for GOME isshown in Fig. 11, top panel; Eover for this spectralregion is shown in Fig. 11, middle panel.

2. Determine from direct cross correlation (forGOME) or by an estimate (for OMI) the wavelengthgrids for irradiance and radiance spectra girr and grad.

3. Sample Eover at the wavelength grid girr, to giveEirr, and at the grid grad, to give Erad, using cubic splineinterpolation17 to determine values at each exact gridpoint. These are now undersampled representationsof the solar reference spectrum, although each is cor-

Fig. 10. Deconvolution of the OMI ITF from the pixel responsefunction to determine the ILS in the NO2 fitting region.

Fig. 11. Portion of the high-resolution solar reference spectrumEref used in fitting BrO for GOME (top); Eref convolved with theGOME ITF to create the lower spectral resolution but highly over-sampled, solar reference spectrum, Eover (middle); Eover sampled atthe wavelength grid of the GOME irradiance (Eirr) (bottom). Eirr

and Erad (not plotted here; see text) are undersampled representa-tions of the solar reference spectrum.

1 March 2005 � Vol. 44, No. 7 � APPLIED OPTICS 1301

rect at the points on its sampling grid. Erad is shownin Fig. 11, bottom panel; Erad is virtually indistin-guishable when overplotted.

4. Resample Eirr to the wavelength grid grad, usingcubic spline interpolation, giving E�rad.

5. The undersampling correction Cu, in opticalthickness units, is the difference between Erad andE�rad, normalized to the average of Erad over the fittingwindow:

Cu(�) �Erad(�) � E�rad(�)

E� rad

. (15)

Cu corresponds to differencing fully sampled and un-dersampled representations of the convolved solarspectrum since only Eirr is resampled, thus inducingundersampling error on the radiance wavelengthgrid grad. This undersampling correction assumesthat, to first order, the radiance spectrum consistsmostly of backscattered Fraunhofer structure.Higher-order corrections could be made to account foratmospheric absorption and the Ring effect. Figure12 shows the residuals from fitting for BrO (withoutthe use of an undersampling spectrum as a basisfunction) in ERS-2 orbit 81003031 for a measurementpixel midway through the orbit (pixel 800) and as anorbit average (top panel); the bottom panel shows theundersampling Cu calculated here for the averagerelative wavelength shift between irradiance and ra-diance fitted for this orbit (0.0092 nm). The calcu-lated undersampling correction accounts for morethan 90% of the fitting residuals. Figure 13, toppanel, shows the undersampling calculated for theOMI baseline BrO fitting window, assuming a rela-tive radiance–irradiance shift corresponding to thefull Doppler shift for the Aura orbit (7.5 km s�1). Thebottom panel shows the absorption for a typical BrO

slant column density of 1 � 1014 cm�2. The units (op-tical thickness and transmission) are equivalent inscale for these small interferences and absorptions.The undersampling correction is roughly an order ofmagnitude smaller than for GOME but is still largerthan the BrO absorption and needs to be carefullyincluded in the fitting of the satellite data. Figure 14,top panel, shows the undersampling calculated forthe OMI baseline NO2 fitting window, assuming arelative radiance–irradiance shift corresponding tothe full Doppler shift for the Aura orbit. The bottompanel shows the absorption for a typical NO2 NorthAmerican summer slant column density of1.25 � 1016 cm�2. The contribution is less severe but,at ca. 20% of the nominal NO2 signal, it is still by no

Fig. 12. Residuals from fitting GOME spectra for BrO in ERS-2orbit 81003031, for a single spectrum and as an average for allspectra in the orbit (top); Synthetic undersampling, Cu, calculatedhere for the relative wavelength shift between the GOME irradi-ance and the GOME radiance (bottom).

Fig. 13. Synthetic OMI undersampling, Cu, in the BrO fittingregion, calculated here for a relative wavelength shift of the irra-diance and radiance corresponding to 7.5 km s�1 (top); absorptionfor a typical BrO slant column density of 1 � 1014 cm�2 (bottom).

Fig. 14. Synthetic OMI undersampling, Cu, in the NO2 fittingregion, calculated here for a relative wavelength shift of the irra-diance and radiance corresponding to 7.5 km s�1 (top); absorptionfor a typical NO2 slant column density of 1.25 � 1016 cm�2 (bot-tom).

1302 APPLIED OPTICS � Vol. 44, No. 7 � 1 March 2005

means negligible. Undersampling correction will verylikely need to be included to derive meaningful tro-pospheric NO2 abundances from OMI measurements.Under heavily polluted conditions over North Amer-ica in summertime, for example, the troposphericcontribution to the NO2 slant column usually does notexceed 40%.6 Effective monitoring of moderate pollu-tion requires correction to substantially better thanthat level.

An attempt was made to improve the undersam-pling correction for GOME by calculation of the under-sampling spectrum for the deconvolved ILS, followedby convolution of the resulting spectrum with the Reti-con response function. The result of this procedure wasexpected to be an improved undersampling correctionthat would correspond even more closely to the GOMEfitting residual, as shown in Fig. 12, top panel. Theactual result was substantially worse. The magnitudewas approximately correct, but the spectral detailswere not. We think that the reason is that the actualdetector response function deviates considerably fromthe trapezoidal shape shown in Fig. 7. For the present,at least, the best undersampling corrections continueto be those determined with the entire ITF. These areshown in Figs. 12–14. When OMI flight data becomeavailable, this procedure will be attempted to seewhether it provides improved undersampling correc-tion for this case.

7. Discussion and Conclusions

We provide a method that may be used to simply andreliably estimate the degree of undersampling an in-strument configuration will have. For instruments inwhich the ITF may be approximated as Gaussian(GOME), we show that sampling at 2.8 pixels�FWHM will almost completely eliminate undersam-pling. For more complex ITFs (OMI), significantundersampling can exist at 3.0 pixels�FWHM.

Undersampling for ground-based, zenith-sky spec-trometers has been discussed in Ref. 18 in the contextof measuring O2, NO2, and NO3. On the basis of nu-merical experiments, it is recommend to use sam-pling ratios between 4.5 and 6.5 pixels�FWHM, forGaussian ITFs, to avoid undersampling. This recom-mendation is consistent with our finding that for OMINO2 measurements undersampling will be signifi-cant and will highly affect tropospheric NO2 mea-surements at 3.0 pixels�FWHM, but that at 6.0pixels�FWHM, it becomes negligible. The improve-ment in undersampling of OMI over that of GOME isdue to the higher sampling rates (2.8 and 3.0 samplesper FWHM for the OMI UV and visible examplesversus 1.4 for the GOME UV example). AsymmetricITFs such as those of OMI and perhaps other imagingspectrometers may require higher sampling ratiosthan symmetric ITFs.

The previously developed undersampling correc-tion2,4 is now commonly used in GOME scientificanalyses and has been implemented operationally forSCIAMACHY. It has been demonstrated here thatcorrection will be required for OMI BrO and tropo-spheric NO2 measurements, at least. Complete im-

plementation will not be possible until the ITFs andirradiance–radiance wavelength shifts are character-ized in flight versus CCD row. A similar conclusionalmost certainly applies to the OMPS instruments.

Further refinement of the undersampling correc-tion for OMI, to include full averaging of the solarspectrum over the ILS and convolution with the pixelresponse, is currently in progress, to be ready forapplication to flight spectra. For GOME, this furtherrefinement has been shown not to be an improve-ment, likely because of incorrect characterization ofthe Reticon response function. Convolution with theITF followed by the sampling procedure describedabove in Subsection 6.B provides very accurate cor-rection.

Higher-order corrections for instrumental effects,such as uneven sampling in wavelength space, andspectroscopic effects, such as atmospheric absorptionand the Ring effect, may eventually be developed ifanalysis of flight data indicates that they are war-ranted.

This research was funded by the National Aeronau-tical and Space Administration and the SmithsonianInstitution. We thank Marcel Dobber and Ruud Dirk-sen of the Koninklijk Nederlands Meteorologisch In-stituut (Royal Dutch Meteorological Institute) andRuud Hoogeveen of the Stichting Ruimte OnderzoekNederland (Space Research Organisation of the Neth-erlands) for help with information on GOME and OMIdetector responses and ITF determinations. It is al-ways a pleasure to acknowledge the European SpaceAgency and the German Aerospace Center for theirongoing cooperation in GOME and SCIAMACHY.

References1. European Space Agency, The GOME Users Manual, European

Space Agency Publication SP-1182, F. Bednarz, ed. (EuropeanSpace Agency, Noordwijk, The Netherlands, 1995).

2. K. Chance, “Analysis of BrO measurements from the GlobalOzone Monitoring Experiment,” Geophys. Res. Lett. 25, 3335–3338 (1998).

3. K. V. Chance and R. J. D. Spurr, “Ring effect studies: Rayleighscattering, including molecular parameters for rotational Ra-man scattering, and the Fraunhofer spectrum,” Appl. Opt. 36,5224–5230 (1997).

4. S. Slijkhuis, A. von Bargen, W. Thomas, and K. Chance, “Cal-culation of undersampling correction spectra for DOAS spec-tral fitting,” in Proceedings of the European Symposium onAtmospheric Measurements from Space (1999), pp. 563–569.

5. K. Chance, P. I. Palmer, R. J. D. Spurr, R. V. Martin, T. P.Kurosu, and D. J. Jacob, “Satellite observations of formalde-hyde over North America from GOME,” Geophys. Res. Lett. 27,3461–3464 (2000).

6. R. V. Martin, K. Chance, D. J. Jacob, T. P. Kurosu, R. J. D.Spurr, E. Bucsela, J. F. Gleason, P. I. Palmer, I. Bey, A. M.Fiore, Q. Li, and R. M. Yantosca, “An improved retrieval oftropospheric nitrogen dioxide from GOME,” J. Geophys. Res.107, 4437, doi:10.1029/2001JD0010127 (2002).

7. NASA, OMI Algorithm Theoretical Basis Document Volume 4:Trace Gas Algorithms, K. Chance, ed., (2002), http://eospso.gsfc.nasa.gov/eos_homepage/for_scientists/atbd/.

8. S. Goldman, Information Theory (Prentice-Hall, N.Y., 1953).9. R. W. Hamming, Digital Filters (Prentice-Hall, Englewood

Cliffs, N.J., 1977).

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10. J. C. Brasunas, Jr., “Far-infrared spectroscopy of gaseous neb-ulae,” Ph.D. dissertation (Department of Physics, HarvardUniversity, 1981).

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