The sum over states (or partitioning function) Z=ZrotZvib is given by the product of the rotational state sum Zrot and the vibrational state sum Zvib: Zrot=∑J∞gJ(2J+1)e-Erot(J)/kTZvib=∑ν∞e-Evib(ν)/kT.

Zvib can be calculated explicitly using a geometric series: Zvib=e-hcν̃2kT1-e-hcν̃kT.

The geometric factors fij(Θ) in Eq. () are determined by the relative orientations of the incident light polarization, the polarizability tensor and the direction of observation, given by the scattering angle Θ. Using Table  the non-normalized phase function p′(Θ) for each of the scattering processes can be calculated. p′(Θ)VRS depends thus also on the ratio of b′2=γ′2a′2 and shows more dominant scattering into the forward and backward direction and differs for N2 and O2, compare Table . The phase function of VRS is similar to the Rayleigh phase function. p′(Θ)RRS=p′(Θ)VRRS=13+cos⁡2(Θ)p′(Θ)VRS=1+745b′2+1+345b′2⋅cos⁡2(Θ)

From the general expression for the total cross-section σtotal=∑ν,Jσν,J→ν′,J′1ZgJ(2J+1)e-E(ν,J)/kT the total cross-section of the Q branch of the vibrational transition can now be calculated e.g. by summing over all possible transitions of the Q branch (ΔJ=0;Δν=1) and averaging the respective phase functions over the full solid angle: σQ,total=128π5(ν̃in+ν̃)49(1-e-hcν̃/kT)3a′2+23γ′2SΔJ=0h8cπ2ν̃ using SΔJ=0=∑J1ZrotgJ(2J+1)bJ,Je-Erot(J)/kT.

σQ,total depends only slightly on the temperature, since also SΔJ=0 varies by less than 1 % between T=230 K and T=298 K and the term 1-e-hcν̃/kT less than 10-4. The numerical values for the resulting overall cross-section are given in Table . The important information for passive DOAS applications is the (ν̃in+ν̃)4-dependency from Eq. (), together with the spectral shift ν̃ (Table ) itself. This information already allows the calculation of correction spectra in a first approximation. The remaining calculations are necessary to obtain the absolute magnitude of the effect. The resulting apparent optical densities are shown in Fig. . Equation () was used to calculate the spectra provided in the Supplement. Figure  was calculated using Eq. ().

The effect of VRS of N2/O2 is shown for two measurement campaigns, using two different spectrometers and telescopes at different locations. The M91 campaign data set shows low NO2 concentrations at a high spectral resolution using an Acton spectrometer. The measurements took place on the Pacific. To exclude possible interferences from liquid water absorption and VRS of liquid water, the MAD-CAT data set was analysed: this data set was recorded during a MAX-DOAS intercomparison campaign in Mainz. However, this data set was recorded with a compact spectrometer at lower spectral resolution and NO2 and glyoxal concentrations were significantly larger.

The aim of the spectral retrieval was to identify the contribution of inelastic scattering due to VRS(N2/O2) to measured intensities in MAX-DOAS measurements. The most pronounced signal is expected between 430 and 440 nm, as the calcium Fraunhofer line ghosts due to VRS(N2) are expected to be found here. As a number of absorbers show absorption structures in this spectral range, these had to be included in the fit or their detection limits had to be determined. An overview over the respective settings in shown in Table .

The recorded scattered sunlight spectra were analyzed for absorptions using the software package DOASIS . A measured Fraunhofer reference spectrum (FRS) I0 from the current elevation sequence at an elevation angle of 90∘ for MAD-CAT was chosen. For M91 40∘ spectra were used to avoid direct sunlight into the telescope in the southern part of the cruise track. The measured OD τObs was then calculated for each spectrum I (Eq. ).

In order to reduce noise of both data sets, several elevation sequences were co-added. When adding 16 elevation sequences for M91 (corresponding to 2 h measurement time), a mean RMS (root mean square) of the residual of 7.5×10-5 is observed. Example fits are shown in Fig.  for both campaigns. Adding more than 16 elevation sequences is typically not possible due to instrumental instabilities and changing atmospheric conditions and radiative transfer. For residual spectra dominated by photon shot noise, the RMS depends on the integrated intensity I as I according to Poisson statistics. When increasing the number of co-added spectra, the residual will be eventually dominated by systematic structures. This behaviour can be visualized in a so-called Allan plot (plotting RMS vs. intensity on a log-log scale) and was used to find a compromise between time resolution and low RMS. This led to summing 16 elevation sequences for the M91 data set. For the MAD-CAT data set more elevation angles were recorded, and therefore only eight elevation sequences were co-added to obtain a similar time resolution of 1.5 h.

The pixel to wavelength mapping for the recorded spectra was performed by using mercury emission lines. The correctness of the pixel to wavelength mapping was tested by convoluting a high-resolution sunlight spectrum by and comparing it to a measured spectrum.

The broad-band contributions to the observed optical depth were compensated by a polynomial fitted together with the trace gases. The (rotational) Ring spectrum was taken from DOASIS where it is calculated according to for each Fraunhofer reference spectrum, agreeing with the expression given in Eq. (), without vibrational transitions. A second Ring spectrum, the original one scaled by λ4/λ04-1 to account for radiative transfer effects discussed in was also included in the fit and was clearly detected in the measured OD.

The second Ring spectrum compensates for additional Ring spectrum structures which appear with a changing colour index within one elevation sequence, for example, due to particle scattering at lower elevation angles. The first Ring spectrum is calculated according to Eqs. () and (). However, if in Eq. () the colour index of I' changes (e.g. due to scattering on particles), this translates directly to the Ring correction spectrum. The second Ring spectrum compensates for this effect.

The spectrum was calculated by multiplication of the original Ring spectrum by (λ4/λ04-1). By construction, this Ring correction spectrum vanishes at the wavelength λ0, which is used then as the wavelength of the RRS signal to be compared to the VRS signal. It is almost orthogonal to the Ring spectrum in the interval from 420 to 440 nm: by comparison with an explicitly orthogonalized second Ring spectrum (within the complete fit interval) the change in Ring dSCD was found to be less than 1×1024moleccm-2 for both data sets used. This is less than 2 % of the range of observed Ring signals, as shown in Fig. ().

The cross-sections listed in Table  were used after being convolved with the instrument slit function. The instrument slit function was measured as the response of the spectrometer to the mercury emission line at 404.7 nm. It was assumed to be constant, in Sect.  an error estimate is given for this assumption for the case of strong NO2 absorption. A parameterization by a Gaussian function with a wavelength dependent width would have also introduced uncertainties, since the slit function shape of both spectrometers is not exactly of Gaussian shape.

It was not necessary to incorporate tropospheric ozone absorption cross-sections in the fit, since the expected OD due to stratospheric ozone is τ≤1×10-4. However, the resulting stratospheric light-path change during morning and evening made it necessary to include an ozone cross-section measured at 223 K by . The zeroth order intensity offset correction was realized by including an inverse reference spectrum in the DOAS fit according to Eq. ().

For water vapour absorptions based on HITEMP , the relative sizes of the absorption bands in the fit interval were adjusted according to . This correction was not crucial for the detection of VRS, but yields reasonable water vapour dSCDs even for the weak water vapour absorption band at 426 nm.

While for the Ring spectrum neither shift nor squeeze was allowed, the remaining cross-sections' shift and squeeze parameter were linked together and determined by fitting the measurement spectrum against the Kitt Peak Flux Atlas 2010 by . No significant differences were found when determining the shift and squeeze of all cross-sections together using the Levenberg Marquard implementation of the DOAS-fit within DOASIS as shown in Table .

The VRS correction spectra τVRS for N2 and O2 were calculated according to Eq. () from a solar atlas and then separately included in the spectral evaluation of the measurements in the spectral range of 420–440 nm. This wavelength range has been chosen to avoid the main water vapour absorptions at 416 and 442 nm and to include the main Fraunhofer ghosts at 433 and 436 nm due to VRS(N2).

The results for M91 of the spectral analysis of 16 co-added elevation sequences (2 h time resolution) were filtered: only spectra with a low fit residual with a root mean square (RMS) of less than 1.5×10-4 were used. This ensures that spectra during twilight and/or high NO2 concentrations, which can increase the overall residual of the fit, are not included in the further analysis.

The results for MAD-CAT of the spectral analysis of eight co-added elevation sequences (1.5 h time resolution) were filtered additionally to use only measurements with a NO2 dSCD of less than 7×1016moleccm-2 to avoid an influence of large NO2 absorptions on the detection of the weak VRS signal. This filter removed 28 % of all measurements.

The absolute cross-section of VRS is small and the mean free path for VRS and RRS is significantly larger than the scale height of the atmosphere (See cross-sections in Table ). The light paths in Eq. () are thus the same for both effects and cancel out. This then allows for the estimation of the size of the cross-section of VRS relative to the cross-section of RRS.

A comparison of the VRS and RRS signal is shown for each campaign in Fig. . For M91, a clear correlation of the contribution of rotational Raman scattering with the contribution from vibrational Raman scattering on N2 is observed (R2=0.8); on is also observed for MAD-CAT (R2=0.7). The contribution from O2 is also positively correlated with the Raman signal, but the slope is nearly zero within the measurement error (R2=0.1). For MAD-CAT this correlation is stronger (R2=0.17). For both campaigns, the contribution from rotational-vibrational inelastic scattering was not identified in measurement spectra. The relative magnitude of the cross-sections of RRS and VRS is calculated from the slope of the linear fits shown in Fig. . In Table  the results are listed and compared to calculations.

The effective contribution to the measured intensity was estimated by using the Fraunhofer line at λ=430 nm: The intensity caused by each of the processes, RRS, VRS N2 and VRS O2 at this wavelength will create a maximum in the respective pseudo cross-section IRamani(λ)/I0(λ), which then allows for the estimation of constant intensity offsets due to these processes.

A perfect correlation of both effects is not expected, because the phase function for VRS and RRS differ, as shown in Sect. .

Alternatively, the optical depth due to VRS can be extracted from residual spectra of a DOAS fit using only the cross-sections listed in Table  and correction spectra calculated according to Eq. (). A schematic of both approaches is shown in Fig.  and is here applied to the M91 data set. From the underlying properties of VRS described in Sect.  a linear relationship between the observed RRS and VRS signal is expected. This relationship is found for actual MAX-DOAS measurements as shown in Fig. . Parameters which can modify this relation are discussed in Sect. .

In order to linearly decompose the residual spectra based on column densities obtained from the DOAS fit, a system of linear equations was set up to determine contributions vi to the residual spectra R that are correlated with the column densities Si. The residual spectrum vRing, which is found to correlate with the Ring signal SRing, is then expected to contain the residual structures caused by VRS, independently of potential influences of other absorbers on the residual spectra. Insufficiently modelled absorption cross-sections of the absorber i are found in vi if the respective dSCDs are not strongly correlated.

This approach is based on the following procedure

The complete M91 MAX-DOAS data set was fitted from 420 to 440 nm to avoid the main water vapour absorptions at 416 and 442 nm. The retrieval was the same as in the first analysis (see Table ), but without the VRS correction spectra and based on individual spectra with an exposure time of 1 min.

AT⋅V=(SRingSH2O…)T⋅(vRingvH2O…)=R

The overdetermined system of linear equations (Eq. ) was solved for V using a least-squares approach, minimizing |AT⋅V-R|2. This yields V=(AAT)-1AR. The resulting correction vRing corresponding to the Ring dSCD SRing was then fitted with the already included absorbers from the original fit scenario and additionally the pseudo cross-section for vibrational and vibrational–rotational Raman scattering of N2 and O2. This fit is shown in Fig. . Adding the other absorbers to the fit is necessary due to possible compensation of residual structures due to VRS by other absorbers within the first fits to obtain the residuals. In other vectors corresponding to other absorbers or parameters a significant contribution of the structure associated with VRS(N2/O2) was not found. The vector corresponding to the intensity offset correction could have contained some contribution of VRS(N2/O2), as a part of it can be compensated by the intensity offset correction; however, no correlation of the magnitude of intensity offset correction and Ring signal was observed. We assume that the intensity offset correction is mainly affected by instrumental stray light, which, for example, depends on the colour index of the measured spectrum.

Despite the significantly better detection of the VRS signal of O2 and N2, the contribution of VRRS of both species could not be disentangled from the intensity offset correction.

The fit of the Ring-correlated residual structure shown in Fig.  yields more information than only the contribution of VRS to the observed OD; it also yields approximations for the changes of other involved trace gases in the respective spectral region, a factor with which the Ring signal can be multiplied to obtain the averaged effect on NO2; for example, the fit yields a NO2 fit coefficient of 7.65×10-12. For a large Ring signal of 5×1025moleccm-2 this results in a change of NO2 dSCDs of (3.8±0.4)×1014moleccm-2. This is larger than the estimate from individual fits shown in Fig. , because Fig.  combines all measurements with different Ring dSCDs.

The main advantage of this approach is that it allows for the obtainment of an average over the residual structures associated with a certain absorber over the whole period of a campaign, e.g. a month or a year for a sufficiently stable instrument. This can reduce the influence of photon shot noise to a minimum. Furthermore an identification of the VRS signature from spectral data is possible for N2 as well as for the weaker O2 signal, which was not found to be significant in Fig.  for individual elevation sequences.

Ignoring the potentially significant impact of VRS can lead to systematic biases in the spectral evaluation. Since IO, NO2, water vapour and glyoxal are trace gases typically absorbing in a spectral range which is mainly affected by VRS, its effect is exemplary studied for these trace gases for our M91 data set. Also here 16 elevation sequences were co-added in order to reduce the overall fit errors. Fit results with an RMS larger than 2×10-4 were not used. We observed an agreement of the results when not co-adding elevation sequences.

We found that correcting for the effect of VRS will reduce the total RMS of the fit residual for a large Ring signal and thus in most cases also the fit error. It does not significantly lower the RMS for spectra with small Ring signals. Whether the correction has an impact on the retrieved column densities has to be tested for each trace gas and also for different spectral resolutions of the respective MAX-DOAS instruments, which can have an influence on the way in which the neglected apparent OD was compensated for. The difference of the squares of the RMS χ2 of the evaluation with and without correcting for VRS is expected to have a linear relationship with the square of the Ring signal, R: Δχ2=a⋅R2.

This relation is found in MAX-DOAS data, for example, from M91, with a=(0.25±0.05)×10-8/(5×1025molec cm-2)2 for the wavelength interval 420–438 nm, effectively reducing the RMS of a fit residual of 1×10-4 by about 15 % for a Ring signal of 5×1025moleccm-2. In other words, if no VRS spectrum is included, the minimum possible RMS for a DOAS fit showing a Ring signal of this magnitude is larger than 5×10-5. In this case, 30 % of the VRS signal is already compensated for by other absorbers and the intensity offset correction.

The following subsections show the impact on the spectral retrieval of different trace gases when neglecting this effect. It may vary depending on fit settings and the spectral resolution of the instrument. Within each of the comparisons the same retrieval settings and intervals were used, once without and once with the correction for VRS of N2 and O2. From the resulting dSCDs of these two fits the impact of VRS was calculated.

The influence of VRS on the observed differential OD at 433 and 436 nm is directly visible due to the high OD of the Ca-II-Fraunhofer lines, combined with overall small residual sizes due to large amounts of available light.

Towards shorter wavelengths, the amount of Raman scattered light scales with 1/λ4 according to Eq. (). This is compensated, however, by the fact that Fraunhofer lines below 390 nm are less pronounced. In the wavelength range of 332–380 nm it was not possible to clearly identify the contribution of VRS of N2/O2. The residual spectra were dominated by systematic residual structures correlated with the dSCD of O4 with typical ODs of (4-8)×10-4 for light-path lengths of ≈10 km. The structures were observed independently of the employed O4 literature cross-section. We expect an apparent differential OD due to the contribution by VRS of N2 to measured ODs of about 3×10-4 at 360 nm (see also Fig. ) for a large Ring signal of 5×1025moleccm-2.

If the incident light was attenuated by any strong absorber before being scattered inelastically, then strong absorbers such as ozone, water vapour or NO2 need to be considered.

But even without strong Fraunhofer or absorption lines, this effect contributes to the observed differential OD according to Eq. (). However, constant or polynomial intensity offset corrections are included in typical spectral evaluations of DOAS measurements, compensating in this case for most of the contribution of VRS and VRRS.

Correction spectra for VRS of N2 and O2 need to be included especially in the spectral retrievals of NO2, IO and glyoxal, but also water vapour in the blue spectral range, whenever wavelengths between 430 and 440 nm are included and systematic residual structures are smaller than the optical depths attributed to VRS of N2 and O2. For a residual RMS of 1×10-4 an improvement of the RMS of up to 15 % was observed.

Since the atmospheric O2/N2 ratio is constant (at the precision relevant to correcting spectra for VRS contributions) , it is advisable as a first approximation to include only a single combined N2+O2-spectrum compensating for VRS contributions to measured intensities. The contribution by VRRS is mostly proportional to the VRS signal and can therefore be added to the VRS correction spectrum. On the other hand, the contribution of VRRS is relatively smooth (compare Fig. ) due to the rotational shifts of the transitions and a large part is already compensated in fit settings by the intensity offset correction. Given a Ring signal with an OD of 2 %, for example, this translates to an additionally measured intensity due to VRRS of 0.8×10-4. Due to the Raman remapped Ca lines the largest variations are expected for the spectral region between 414 and 440 nm. The effect is already corrected by the intensity offset correction except for a remaining part of 30–50 %, which translates to total differential ODs of <0.4×10-4. This is currently negligible for most ground-based measurements, but this effect might need to be considered for geometries where a strong filling-in of Fraunhofer lines is observed

All calculations for a spectral resolution of the instrument of 0.5 nm.

Due to the observed correlation of the spectral signature of VRS scattering on N2 and the Ring dSCD shown in Fig. , a direct correction of the Ring spectrum together with the VRS signal seems reasonable, at least in the spectral ranges where an effect has been observed. The actual values of the cross-sections can be found, for example, in Table . This would allow furthermore to correct for VRRS scattering, which is also expected to behave in the same way as the Ring spectrum, but is often (by coincidence) corrected by the intensity offset correction (see Eq. ), as estimated above. This correction of the Ring spectrum would therefore improve how the intensity offset correction can be interpreted, which usually has been introduced in the first place to compensate for instrumental stray light. The contribution of the VRRS on O2 and N2 can then be removed from the intensity offset correction polynomial, because the dominating factor for determining the Ring spectrum dSCD is the actual RRS contribution, which is about 50 times larger.

However, the following points need to be considered:

Vibrational Raman scattering leads to a red-shift of solar radiation, thus the ratio of vibrational Raman scattered and elastically scattered light is influenced by the wavelength dependency of the aerosol extinction and Rayleigh scattering. For the case of rotational Raman scattering (the Ring effect) this is inherently compensated for, since the frequency shift due to RRS is not significant for the wavelength dependence of the aerosol extinction. For the shift of 40 nm of N2 VRS from the Ca-II K Fraunhofer line at 393 nm, significant relative intensity differences at 393/433 nm can be observed for different measurement geometries and/or aerosol loads. During M91, relative intensities at 390 and 432 nm (the so-called colour-index CI(390, 432 nm)) were observed between 0.3 and 0.8 for zenith sky measurements, even more if the complete elevation angle sequence is considered. This relation translates directly to the size of VRS correction spectrum via Eq. (). Therefore, a direct correction of the Ring spectrum is not possible in general, if not considering these effects explicitly. The same argumentation can be applied for separating the influence of VRS(N2) and VRS(O2), since the wavelength shifts are different. However, the influence of VRS(O2) is hardly detected in current measurement spectra. A potential dependence of VRS(N2)-OD on the colour-index was not observed in measurements.

A way to avoid these points is to use a measured spectrum to calculate the correction spectra, which includes the effects due to changes in radiative transfer, such as aerosol load and measurement geometry. When using measured spectra, this would require the same quantum efficiency of the instrument at both wavelengths or a radiometrically calibrated instrument, as well as a constant instrument slit function. This is typically not the case.

It is therefore recommended to include the correction spectrum for VRS of N2 without including the contribution from VRRS, calculated based on a solar atlas. Correction for VRS of O2 is only needed if the RMS of the residual is significantly below 10-4.