The DOAS technique is based on Beer–Lambert law, which describes the attenuation of light passing through a medium. Here, it is applied to measurements of scattered sunlight. The spectral attenuation caused by scattering is smooth in wavelength (e.g. λ-4-dependence for Rayleigh scattering) while molecular absorption often has structured spectra. In DOAS, the total spectral attenuation is therefore split into a high-frequency part consisting of the (high-frequency components of) trace gas absorptions and a low-frequency part that accounts for elastic scattering, which is described by a low-order polynomial, also compensating for intensity changes, e.g. caused by clouds. In addition, the effect of inelastic scattering, which is predominantly due to rotational Raman scattering, known as the Ring effect is accounted for by a pseudo cross section e.g.. Similarly, intensity offsets mostly resulting from stray light within the spectrometer are accounted for by pseudo cross sections σoff (see Sect.  for more details). In total, the optical depth τ is approximated by τ=ln⁡I0I=∑iσi⋅SCi+σRing⋅SCRing+σoff⋅SCoff+∑papλp+r, where the first sum is over all i absorbers having cross section σi, the polynomial degree is p, and the residual term r contains the remaining (uncompensated) optical depth. An important quantity used within this study to identify and evaluate differences between DOAS retrieval codes is the root mean square of the fit residual r (for simplicity denoted rms in the context of this study), which is a measure of the fit quality. In Eq. (), known as the DOAS equation, all quantities with the exception of ap depend on wavelength λ. For tropospheric absorbers, the spectrum I is normally taken at small elevation angles above the horizon where the tropospheric light path is large. The reference spectrum I0 is normally a zenith spectrum either at small solar zenith angle (SZA), which is called a noon reference fit in the following, or close in time to the measured spectrum I, in the following called a sequential reference fit.

The DOAS equation is usually an overdetermined problem (τ is measured at more wavelengths λ than unknowns exist in Eq. ) and solved by means of a least-squares fit (see also Sect. ), i.e. minimizing the residual term. The resulting fit factors are the polynomial coefficients ap and the so-called slant columns SCi, which are the quantities of interest. The slant column is the integrated concentration ρi of absorber i along the effective light path s (for simplicity we use the SC also for the fit coefficients of the Ring and offset spectra): SCi=∫ρids. Note that this is a simplification as normally an ensemble of different light paths contribute to the measurement. In extreme cases, the absorption optical depth can become a non-linear function of the trace gas concentration. However, this is usually only of importance for satellite limb measurements and is discussed in detail, for example, in . A comprehensive discussion of the DOAS technique can be found, for example, in .

The Multi-Axis DOAS Comparison campaign for Aerosols and Trace gases (MAD-CAT) was carried out in Mainz, Germany, in summer 2013. During the intensive phase of the campaign (7 June to 6 July 2013), 11 groups deployed MAX-DOAS instruments on the roof of the MAX-Planck Institute for Chemistry at the Mainz University campus. Being located in the densely populated Rhine-Main region, the observations are dominated by anthropogenic pollution, predominantly by NO2 (in the visible). The main azimuthal viewing direction was 51∘ (from the north), pointing roughly in the direction of the city of Frankfurt ≈ 30 km away. Series of vertical scans comprising elevation angles of 0, 1, 2, 3, 4, 5, 6, 8, 10 and 30∘ were performed. In each direction, single measurements of varying exposure time depending on illumination were integrated for 20 s (for the IUPB instrument, other instruments used different integration times). Between vertical scans, multiple zenith measurements were taken (see Fig.  for an example of resulting NO2 slant columns). Detailed information about MAD-CAT can be also found at http://joseba.mpch-mainz.mpg.de/mad_cat.htm and the first campaign data focusing on range-resolved distributions of NO2 measured at different wavelengths were published by . Furthermore, demonstrated the presence of vibrational Raman scattering on N2 molecules in spectra measured during MAD-CAT. In addition, publications based on MAD-CAT data are in preparation focusing on HCHO , HONO and CHOCHO .

The IUPB MAX-DOAS instrument deployed in Mainz during the MAD-CAT campaign is a two-channel CCD-spectrometer system measuring in the UV and visible. Within this exercise, only data from the visible spectrometer are used as NO2 is best retrieved in this spectral range. The spectrometer is an ANDOR Shamrock 303i, covering a spectral range from 399 to 536 nm at a resolution of ≈ 0.7 nm. The spectrometer was actively temperature stabilized to 35 ∘C. Spectra were recorded with a charge-coupled device (CCD) of the ANDOR iDUS 420 type with 1024 × 255 pixels (26 × 26 µm each), leading to a spectral sampling of 7–8 pixels nm-1.

Light was collected by a telescope unit mounted on a commercial ENEO VPT-501 pan-tilt head allowing pointing in any viewing direction. Photons entering the telescope through a fused silica window were focused by a lens on an optical fibre bundle. The instrument's field of view (FOV) was ≈1.2∘. The Y-shaped optical fibre bundle (length 20 m) connecting the telescope to both spectrometers consists of 2 × 38 = 76 single fibres, minimizing polarization effects. A video camera inside the telescope housing takes snapshots of every recorded spectrum for scene documentation, and a mercury-cadmium (HgCd) line lamp allows for wavelength calibration measurements. Dark current and slit function measurements are taken every night. The same instrumental set-up has been used in previous campaigns, e.g. CINDI and TransBrom .

Spectra recorded by the IUPB MAX-DOAS instrument on 18 June 2013 during MAD-CAT were distributed to partners. It is worth mentioning that this intercomparison exercise was not restricted to groups participating in MAD-CAT, as common observations were provided. 18 June 2013 was selected as having good viewing and weather conditions. However, temperature stabilization of the IUPB spectrometer was problematic due to unusually hot weather which, due to a lack of air conditioning on that day, led to the instrument overheating. As a result, spectral stability was not as good as usual, providing the opportunity to investigate how different retrieval codes deal with potential spectral shifts during the day. The provided data comprised observations at several elevation angles in the main azimuthal viewing direction of 51∘ (see Fig.  for an example). In addition, common trace gas cross sections as well as DOAS fit settings were distributed as summarized in Table , which are the common fit settings agreed on during the MAD-CAT campaign. All groups then performed DOAS fits using these prescribed settings on IUPB spectra using their own retrieval software. The influence of different retrievals was then analysed, focusing on the impact on NO2 columns. A brief description of retrieval codes is given in the following. The participating groups (with abbreviations used within this study) and retrieval codes they used are summarized in Table .

Differences between groups for the 425–490 nm fit using a noon reference (v1 fit settings; see Table ) are shown in Fig.  for individual measurements at an elevation angle of 2∘. Small elevation angles above the horizon are associated with long tropospheric light paths and are therefore important for the detection of tropospheric absorbers. Differences shown in Fig.  are relative to IUPB results. For the objective of identifying retrieval-code-specific effects, the use of a single retrieval code as a reference seems advantageous in comparison to using the mean of all retrieval codes which would average over all retrieval-specific features. Note that this does not exclude IUPB from the intercomparison as problems with the IUPB retrieval would be easily detected, leading to the same systematic patterns in all lines shown in Fig. .

Absolute differences (institute-IUPB) and relative differences (absolute difference/IUPB) of NO2 slant columns and fit rms are shown in Fig. a–d. In general, NO2 slant column differences are in the range of ±2–3 ×1015 molec cm-2 or < 2 %. This is about a factor of 2–3 larger than NO2 slant column errors, which are typically <1×1015 molec cm-2 or < 0.6 % for 2∘ elevation. A clearly enhanced disagreement is observed for the first data point in the INTA time series as well as for MPIC_WD. The latter could be linked to the reference spectrum, which in this case is not the one with the smallest sun zenith angle (SZA), while the outlier in the INTA time series was identified to arise from different implementations of the intensity offset correction (see Sect. ). Note that these NO2 differences for individual measurements are much smaller than the variability (diurnal cycle) of NO2 and thus almost invisible in Fig. e where no differences but absolute NO2 slant columns from each groups (including IUPB in black) are plotted.

Interestingly, many groups show a smooth behaviour (constant offset) in absolute NO2 differences (Fig. a), which is mostly an effect of the choice of the reference (see Sect. ), while relative differences (Fig. c) reflect the shape of NO2 slant columns in Fig. e (smaller slant columns lead to larger relative differences and vice versa). However, some groups show a smooth line, not for absolute differences but for relative differences. Thus, two types of disagreements are observed, (1) constant in absolute differences and (2) constant in relative differences. These two types are linked to differences in retrieval codes, which are investigated in detail in Sect. .

Absolute rms differences (with respect to IUPB) in Fig. b show the same shape as NO2 slant columns. This is because at small elevations the rms itself reflects the shape of the NO2, which was already demonstrated and discussed in Fig. . A better measure for the identification of differences between retrieval codes is thus the relative rms disagreement shown in Fig. d. Interestingly, the first data point for INTA, showing a large disagreement with IUPB and other groups in NO2 slant columns is prominent in absolute rms differences as well, but not in relative rms differences. The reason is that the rms in the morning is very large (Fig. f, compare also to Fig. ) and thus decreases the relative difference. Remarkably, relative rms differences are found up to 80–100 %, which is substantially more than NO2 slant column differences (only a few percent).

The IUPB spectra provided were wavelength precalibrated using nightly HgCd line lamp measurements as explained in Sect. . However, the DOAS fit quality can be improved (rms reduced) by applying a postcalibration. In addition, the night-time calibration can change during the day as a result of temperature drifts. This is accounted for by DOAS retrieval codes in terms of a non-linear shift fit (for a more detailed discussion see Sect. ). Figure g and h (being a zoomed-in image of g) show the reported shift resulting from the wavelength calibration in participating retrieval codes. Note that absolute values for the shift are shown here and the IUPB result is explicitly included. For an ideal spectrometer without drifts, only a very small shift would be expected and it would be caused by a non-commutivity of convolution and DOAS polynomial, which is known as the tilt effect and typically in the order of less than 1–2 pm, depending on the instrument resolution . The shift retrieved here is larger than that and driven by the overheating of the system on this day. In general, time series of shifts shown in Fig. g and h agree well in shape. A small offset is observed for NIWA, who successfully found and corrected a fault in their retrieval code (related to the intensity offset calculation) in the course of this exercise. The DOAS fit was then repeated using the corrected retrieval code and the result, called NIWA2, is included in Fig. . Apparently, NIWA2 shows a much better comparison than other groups (most obvious in the fitted spectral shift) and performs better because the rms in Fig. d is much smaller than before. Similarly, INTA fixed a problem in their retrieval code related to the wavelength calibration. The reanalysed INTA data are denoted INTA2 in Fig. . The INTA2 shift is comparable to other groups, while large disagreements were seen before, and the rms is largely reduced in comparison to the first INTA submission. Notably, almost no differences between INTA and INTA2 are observed in terms of NO2 slant columns; in particular the first data point is still outlying. This indicates (1) that the wavelength calibration has a large impact on the fit rms but only a minor impact on the NO2 slant columns, and (2) that the disagreement of the first data point is not related to the wavelength calibration. Despite INTA and NIWA, only KNMI is fitting a rather different shift (and provides the shift only in 0.01 nm resolution which leads to the displayed discrete steps). This is caused by different definitions of the shift (see Sect. ): KNMI is fitting the shift of the optical depth relative to the cross sections while the other groups are fitting the shift of I relative to the reference spectrum I0. As a result, the shift of KNMI and other groups cannot be expected to match.

The agreement between groups, including other elevation angles, has been quantified by linear regression analyses to correlation plots (of each group relative to IUPB results, not shown). Resulting slopes, offsets and correlation coefficients are summarized in Fig.  and colour-coded for different elevation angles. As expected, the correlation coefficient is > 99.98 % with the only exception of the INTA (and INTA2) 2∘ elevation which is predominantly caused by the outlier already seen in Fig. . The slope ranges between 0.985 and 1.01, the offset between -4 and 2.5 ×1015 molec cm-2. Apart from USTC and INTA, no large separation of the slope and offset with elevation angle is observed. An important observation is that groups using the same retrieval code (QDOAS) do not necessarily show the same systematic behaviour in Fig. , implying that the influence of remaining fit parameters that are different from the harmonized general settings in Table  is still larger than the effect of the specific retrieval software used. For example, the best agreement in terms of slope, offset and correlation coefficient is found between IUPB, AUTH, IAP and CU Boulder, which all use different retrieval codes.

The agreement between groups was investigated in the same way when using a sequential reference (denoted v1a fit setting; see Table ) instead of a noon reference spectrum (v1 fit). A sequential reference is often preferred if tropospheric absorbers are of interest as stratospheric effects are removed to a large extent.

Figure  is similar to Fig. , but some groups are missing here (e.g. IAP) as they only provided noon reference fits. The range of disagreements for individual NO2 slant columns is up to ≈ 8 % and therefore larger than NO2 disagreements using a noon reference (v1 fit). In addition, neither absolute nor relative differences between groups are smooth (Fig. a and c). The reason is that, in contrast to the noon reference, a different (zenith) I0 spectrum is used for every scan and thus the impact of details of the implementation of the reference changes from scan to scan. Different implementations of the reference spectrum are further investigated in Sect.  and were found to be the major reason for NO2 disagreements between groups. The outlier (first data point) seen for v1 (noon) fit settings is present for v1a as well but it is not prominent here as fluctuations for the above-mentioned reason are of same magnitude.

In terms of rms, differences between groups are comparable to v1 results and as large as ≈ 80 %.

Compared to the noon reference fit, correlations from linear regressions (not shown) are smaller, especially for the 30∘ elevation angle, mainly resulting from the larger disagreements as explained above (but also partly expected as slant columns are smaller using a sequential reference). Correlations are still > 99.2 % for 30∘ elevation and even > 99.8 % for smaller elevations.

In addition to the 425–490 nm fit (v1 and v1a), a smaller fit window used within the MAD-CAT campaign was intercompared (v2 and v2a using a spectral range of 411–445 nm; see Table ). Results of these fits are not explicitly shown as mainly confirming observations and findings above. Typical correlation coefficients (as well as offsets and slopes) are summarized in Table  for all fits. A very small tendency of better agreement between groups if using the larger fit window is seen. Although this should not be overinterpreted, it could be caused by more information being present in the large fit window and therefore more accurate results (fewer statistical fluctuations).

In the IUPB spectra provided to intercomparison partners, two different zenith spectra at the same smallest SZA were reported, which is of course non-physical. Actually, the second zenith spectrum is the one with the smallest SZA, but for rounding reasons (the SZA was a 4-digit number in the input data provided), both spectra had the same SZA. Consequently, for the noon reference fits v1 and v2 within this intercomparison exercise, four options exist: (1) taking the first zenith spectrum of smallest SZA, (2) taking the second one, (3) taking the first for the morning and the second for the afternoon (i.e. always taking the closest in time) and (4) taking the average of both spectra. Similarly, different options exist for calculating the sequential references for fits v1a and v2a: (1) always taking the zenith spectrum closest in time, (2) always taking the last zenith spectrum before the actual measurement, (3) always taking the next zenith spectrum after the measurement, (4) taking the average of the two (before and after) and (5) interpolating the two zenith spectra to the time of the actual measurement.

All different options for noon and sequential reference fits were evaluated using the IUPB retrieval code NLIN (Table ). Figure  shows the resulting absolute and relative slant column differences (top and middle) as well as relative rms differences (bottom) for noon reference (left) and sequential reference (right) with respect to v1 or v1a fit results.

Taking another noon reference spectrum results in a constant offset in absolute NO2 differences (Fig. , top left). Test TR0 (using the first spectrum) yields the same results as the IUPB v1 fit from Sect. , which uses the first zenith spectrum as well. In contrast, using the second zenith spectrum as a reference (TR1) results in a constant offset of 1.5 × 1015 molec cm-2 (0.5–2.5 % in relative differences, depending on the actual NO2 slant column), because the second zenith spectrum apparently had a smaller NO2 content. The change of the NO2 content could be related to changes of the atmospheric NO2 amount or the atmospheric light path. In terms of rms, TR1 is up to 3 % larger (Fig. , bottom left). The main reason for this is probably the larger NO2 slant columns as associated effects like the wavelength dependence of the NO2 slant column were not compensated in this intercomparison exercise as discussed in Sect. .

Test TR2 yields results which are identical to TR0 am and TR1 pm values. This is not seen for any groups in Fig.  above; i.e. this option is apparently not present in any retrieval code. Not surprisingly, TR3 (averaging both zenith spectra) yields results which are between TR0 and TR2.

In contrast to noon reference tests, sequential references show no smooth behaviour, neither for absolute nor for relative differences (Fig. , right). The reason is that for each vertical scanning sequence (from which only the 2∘ elevation is shown here and in Fig. ) another reference spectrum is used and consequently reference-related differences between groups are also changing from scan to scan. Note that almost no difference can be seen in Fig.  between TR4 and TR5 as the 2∘ elevation angle is shown here and consequently the closest zenith measurement in time is normally the one before the scan as IUPB measurements proceed from low to large elevations. TR8 (interpolation to the measurement time) resembles the sequential reference treatment normally implemented in the IUPB code; thus the TR8 line is zero. For TR6 and TR7, absolute and relative differences (up to 8 %) are remarkably similar to observed differences between groups using v1a fit settings, both in shape and in absolute values (compare to Fig. ).

To conclude, the exact treatment of reference spectra is the major reason for observed NO2 discrepancies between groups in Sect. , causing differences of up to 8 %. Unfortunately, no clear recommendation can be derived from relative rms differences in Fig.  (bottom right) as all lines scatter around zero. However, the relative difference in rms can be as large as 6 % and in general, using a single zenith reference spectrum before or after the scan tends to produce larger rms. However, while the reference treatment explains the majority of NO2 disagreements, it cannot explain the large rms differences (up to 100 %) between groups.

The slit function distributed to intercomparison participants originated from HgCd line lamp measurements made in the night before 18 June 2013. It was preprocessed in terms of subtraction of the (also measured) dark signal, centred and provided on an equidistant 0.1 nm grid.

While some groups used this slit function as is, other retrieval codes include a further online processing of the slit function, e.g. by fitting line parameters. In order to evaluate resulting differences, trace gas cross sections have been convolved offline (before the fit) using different treatments of the slit function as summarized in Table . Again, the IUPB retrieval code NLIN has been used then performing the DOAS fit.

Examples of different treatments of the slit function are shown in Fig. . The original slit function is displayed in blue. If fitting a Gaussian shape to it, the maximum is not exactly centred around zero, presumably caused by a small asymmetry. The slit function after fitting a Gaussian shape and forcing centring (i.e. μ=0) is shown in red (as used in TS7 and TS10). Furthermore, performing a discrete convolution of cross sections requires the same wavelength sampling; i.e. the slit function has to be interpolated to the cross section grid, which was 0.01 nm. Here, an Akima interpolation has been applied (green line). However, if the original slit function is first linearly interpolated to the required 0.01 nm grid and then a Gaussian shape is fitted, a slightly different result is obtained, shown in dark green (as used in TS8 and TS11). Note that often not a discrete convolution but a (faster) convolution using Fourier transformation is implemented in DOAS retrieval codes (see below).

Figure  shows resulting absolute and relative differences in NO2 slant columns (top and middle) as well as relative differences of the fit rms (bottom) with respect to the IUPB v1 fit (without I0-correction as this was not applied to the slit function test fits). The reference IUPB v1 fit uses an online convolution of cross sections using FFT and a further geometrical centring of the slit function is applied.

No difference between the reference fit, TS0 and TS5 is observed, neither for NO2 nor for the rms. In TS0, the same slit function treatment is applied as in the reference fit, but using discrete convolution instead of FFT, meaning that no difference arises due to the method of convolution. In addition, TS5 and TS0 are identical tests except that TS0 applies an explicit geometrical centring (i.e. centring the area) of the slit function. As this has no visible effect on NO2 slant columns and rms, the centring during preprocessing was already sufficient (or the non-linear shift fit was compensating for small deficits in the slit function centring).

The largest impact on NO2 (up to -1.3 %) is seen for TS2 and TS4 which both subtract the smallest value from the slit function (i.e. forcing the slit function to be zero for the smallest value measured). This is usually not performed in DOAS fits and only advisable if instrumental stray light is a large problem or if the dark signal drifts which is normally not the case in state-of-the-art CCD detectors that are cooled and temperature-stabilized using Peltier elements.

In contrast, the largest effect in terms of rms (up to 6.5 %) occurs not for TS2 and TS4, but for all tests using a fitted Gaussian. Interestingly, the Gaussian tests (TS6–TS11) show almost no difference among them in terms of rms in the morning, but split up towards the evening. This might have to do with decreasing NO2 slant columns in the afternoon or changes of the slit function during the day. In terms of NO2, all tests using Gaussian slit functions yield smaller NO2 slant columns of 0.2–0.7 %.

In addition to the basic Gaussian line parameters fitted here, some retrieval codes offer the possibility of fitting more sophisticated line shape parameters while taking into account potential asymmetry much better (the slit function in Fig.  shows indeed slight asymmetry). However, the exact implementations differ and were therefore not reproduced in the tests performed here, which can then be regarded as two extreme scenarios, namely (1) using the (original, slightly asymmetric) slit function as it is, or (2) fitting basic line parameters (a more sophisticated fit taking into account the slit function's slight asymmetry would clearly lead to a result in between these scenarios).

An important finding is that all performed tests lead to constant offsets in relative (and not absolute) NO2 differences. In addition, all tests yield larger rms than the reference fit using the measured slit function.

Photons may hit the CCD detector at locations not corresponding to their wavelength (e.g. through scattering on mirrors, surfaces etc. inside the spectrometer) which produces an intensity offset, also called stray light. In addition, other effects such as changes in the dark current can lead to intensity offsets and the vibrational Raman scattering (VRS) is known to produce spectral effects that are very similar to intensity offsets . In the DOAS fit, pseudo cross sections are usually included in order to compensate for intensity offsets. If the measured spectrum I is superimposed by a constant intensity C (which is the most simple assumption), the optical depth reads -τ=ln⁡I+CI0=ln⁡II0+ln⁡1+CI≈ln⁡II0+CI, with the Taylor expansion ln⁡(1+x)≈x. Thus, in a first approximation the intensity offset causes an additive term of optical depth that is proportional to 1/I, which is often used as a pseudo absorber (and showing large similarities to the Ring cross section as this compensates a filling-in of Fraunhofer lines).

However, often more sophisticated approaches are used. For example, the IUPB retrieval code NLIN allows either the simple implementation of σoff=1/I or σoff=ln⁡I+C⋅ImaxI, omitting the Taylor expansion in Eq. () and superimposing I by a certain constant C of the maximum intensity within the fit interval. In addition, higher correction terms are also sometimes used, assuming that, not only does a constant superimposes the spectrum, but a contribution also changes with wavelength (in which case often λ/I is included in the DOAS fit). Furthermore, sometimes the offset is not included in the linear DOAS fit, but fitted non-linearly (this is not included in tests performed here).

Different implementations summarized in Table  were tested in order to evaluate the influence of the intensity offset correction. Figure  shows resulting absolute and relative differences of NO2 slant columns and rms values. Again, the reference for these differences is the IUPB v1 (noon) fit.

No difference in NO2 or rms is seen between TI0 and the v1 fit as both fits use the same simple approach of σoff=1/I. In contrast, TI1 uses 1/I0. In terms of NO2 differences, the TI1 line slightly follows the shape of total fit rms and NO2 slant columns (compare to Fig. ). Interestingly, while the disagreement of TI1 with respect to the reference fit is on average ≈2 % for NO2, the first data point is clearly off by almost 10 % for NO2 and 60 % for rms. This agrees perfectly with the observed outlier in the INTA analysis (compare to Fig. ); i.e. the reason for this disagreement could be identified as a different offset implementation (which has been verified by INTA). The SZA and the sun azimuth angle (SAA) of this measurement are ≈ 90 and 58∘ (from the north), respectively, while the instrument's elevation angle is 2∘ and the azimuthal viewing direction 51∘; i.e. the instrument was pointing close to the rising sun. Enhanced stray light in the spectrometer (caused by the large contribution of photons at longer wavelengths while observing the red sky during sunrise) seems plausible and using 1/I is a better choice for compensation. It was verified (not shown) that the fit coefficient of the offset (also called the offset slant column) is particularly large not only in this but also in adjacent measurements and a colour index indicated that these spectra are indeed more reddish. However, these measurements could also be affected by direct sunlight in the instrument, which is known to increase rms (e.g. due to polarization issues). Figure f demonstrates that the respective measurement is affected by a very large rms (potentially caused by a combination of direct light and stray light).

TI2 and TI3 are more sophisticated approaches (Eq. ) based on either I or I0. However, the resulting lines largely follow TI0 and TI1 (simple approaches). Unexpectedly, both TI2 and TI3 lead to larger rms values for the sunrise measurement in the morning where the simple approach performs better.

The total effect of using an intensity offset compensation is evaluated by TI4 which includes no offset correction. The resulting effect on NO2 is almost the same as TI1 and TI3 (based on I0) leading to the clear recommendation of using an offset compensation based on I instead of I0, which is supported by largely increased rms values of TI4 (Fig.  bottom).

An intensity offset correction of 1st-order (i.e. a term varying linearly with λ in addition to a constant term) was tested in TI5, which is in practice often used not only for intensity offsets but also for compensation of the wavelength-dependence of the Ring slant column. The resulting NO2 differences with respect to the reference fit (or TL0) are small with the exception of the first data point that is slightly off. In terms of rms, TI5 leads to improvements of ≈ 20 % in the morning. Interestingly, the TI5 rms line shows some similarities to the IUPHD-IUPB line in Fig. d. However, no 1st-order offset was included in the IUPHD fit; i.e. the offset implementation is not causing the observed similar shape in the morning (and the reason remains unclear). This is supported by increasing IUPHD rms values in the evening in Fig. d, which are not present in TI5.

Fit TI6 includes again only a 0th-order intensity offset, but a pseudo cross section accounting for the wavelength-dependence of the Ring slant column was added (the Ring cross section was multiplied by λ and orthogonalized against the original cross section). The resulting rms is indeed almost identical to TI5 (using a 1st-order intensity offset) while the NO2 is identical to TI0; i.e. the outlying first data point which is still partially present in TI5 disappeared in TI6. Thus, using a pseudo-cross section for compensation of the Ring wavelength-dependence seems preferable compared to using a 1st-order intensity offset correction.

In addition to the intensity offset tests, the effect of inclusion of an I0-correction was evaluated in TI7 (which is identical to TI0 except for the I0-correction; see Table ). The respective line is shown in addition to the intensity offset investigations in Fig. . The I0-effect addresses the problem that the limited instrument's resolution can cause an incomplete removal of Fraunhofer structures in the vicinity of strong narrow-banded absorption bands .

Only a very small constant offset in relative NO2 slant column differences is obtained in TI7 (≈ 0.25 %, which is almost invisible in Fig. , middle). In terms of rms, exclusion of the I0-correction leads to an increased rms of up to 20 %, which is comparable to different treatments of the intensity offset correction. Thus, including an I0-correction is recommended in polluted environments such as the MAD-CAT site. It should be noted that the first data point is not an outlier in TI7; i.e. it is not sensitive to the I0-correction.

The DOAS equation (Eq. ) is a linear inverse problem Ax=b, with the vector x (size n) containing the n trace gas slant columns and polynomial coefficients of interest, the vector b=ln⁡(I0I) (size m) containing the measured optical depths at m wavelengths and the m×n DOAS matrix A with columns consisting of absorption cross sections and polynomial terms (1, λ, λ2, etc.) for the m wavelengths.

Different numerical methods exist to solve Eq. () for x. As A is non-square, no inverse exists. However, most retrieval codes calculate a pseudo-inverse A-1 (almost) fulfilling A-1A=I (identity matrix) using a singular value decomposition (SVD) and obtain the slant columns of interest by x=A-1b. This method is frequently recommended for solving overdetermined linear inverse problems in terms of least squares see e.g..

However, after multiplying Eq. () with AT, the matrix ATA is quadratic and can be decomposed into an upper and a lower triangular matrix, L and U. The linear inverse problem ATAx=LUx=ATb can then be solved by a forward substitution, obtaining y from Ly=ATb, and a backward substitution, obtaining x from Ux=y.

Furthermore, ATA can also be directly inverted, normally by LU decomposition as well. The vector of slant columns is then obtained by x=(ATA)-1ATb. As the inversion normally takes many more computational steps, this method is known to be subject to round-off errors and is therefore not recommended .

The influence of these different numerical methods on resulting slant columns could not be easily tested with the IUPB retrieval code and was thus evaluated in a Python script solving the DOAS equation using the same input (spectra, cross sections). Python (which is a well-established programming language in scientific computing) provides numerous different routines within its numpy and scipy packages (based on different subroutines from the LAPACK package, http://www.netlib.org/lapack/) that were tested for solving the DOAS equation. All performed tests are summarized in Table . Again, differences of NO2 slant columns and fit rms have been calculated with respect to the IUPB v1 fit results (i.e. using NLIN). In order to restrict differences to the influence of numerical approaches only, the same slit function treatment as the IUPB retrieval code NLIN was applied in the Python script and the same (noon) reference spectrum was used. In addition, no further wavelength calibration, i.e. no non-linear shift and squeeze fit, was performed (neither in the Python script nor in the NLIN reference fit used here) and no I0-correction was included. The test results are shown in Fig. , again for the 2∘ elevation angle.

Results of test TL0 appear to be identical to the reference fit of the IUPB retrieval code, both using a SVD for inversion of the DOAS matrix. However, very small differences exist between TL0 and the reference fits (too small to be seen on the scale of Fig. ), which are < 0.006 % for NO2 slant columns and < 0.07 % for rms. These tiny disagreements can be attributed to numerical differences in programming languages.

TL1 and TL2 yield identical NO2 slant columns (both are using an LU decomposition) which differ from SVD results by up to 0.7 %. However, the rms from TL2 was found to be an order of magnitude larger compared to the other tests and is therefore not shown in Fig.  (bottom). This is most likely due to problems mentioned above; i.e. this finding is in agreement with common recommendations in textbooks. Interestingly, both TL1 (and TL2) NO2 and rms lines (which are differences with respect to IUPB) are similar in shape to the total fit rms (compare to Fig. f). Thus, when the rms increases, SVD inversion and LU decomposition lead to larger disagreements, both in rms and NO2. As the SVD yields smaller rms values, it seems to be preferable, although the obtained improvement is only about 2.5 %.

Numerical differences may be obtained when performing the linear DOAS fit (Eq. ) on another wavelength grid. Changes of the grid potentially arise from the wavelength calibration. Some retrieval codes (e.g. NIWA) also use an internal, equidistant wavelength grid. To test the effect of changes in the wavelength grid, the TL0 fit was repeated on an equidistant 0.01 nm grid (i.e. I, I0 and cross sections were interpolated to 0.01 nm before solving Eq. ). TL3 and TL4 are identical to TL0, but a linear interpolation was applied in TL3, while TL4 uses a cubic spline interpolation to 0.01 nm. Apparently, this results in a constant offset in relative NO2 differences, which is seen most clearly in the TL3 line in Fig.  (middle). The resulting constant shift is ≈0.4 % for TL3, but only ≈ 0.02 % for TL4 meaning that the type of interpolation to the equidistant grid is of importance and the spline interpolation (not surprisingly) seems to resemble the spectrum better than a linear interpolation. However, using different wavelength grids might, for example, explain some of the observed differences between IUPB and NIWA, which were found to be constant in relative differences as well. In terms of rms, the computation on an equidistant 0.01 nm grid using linear interpolation behaves even a bit better on average (up to 1 %). However, no recommendation can be drawn from this, as discussed above.

As mentioned in Sect. , the spectra provided from the IUPB instrument were precalibrated using nightly HgCd line lamp measurements, which provide accuracies better than 0.1 nm. However, usually a postcalibration is included in DOAS retrieval codes in order to increase the fit quality (reducing rms). This wavelength calibration is implemented in different ways in participating retrieval codes. Most groups calibrate the reference spectrum I0 to a high-resolution Fraunhofer atlas, apply the resulting calibration to all measured spectra and allow in addition a shift and squeeze between I and I0 in order to compensate spectral shifts of the spectrometer during the day, e.g. caused by temperature changes (KNMI uses a slightly different definition of the shift as discussed below). This non-linear shift and squeeze fit of the wavelength axis is mostly implemented in an iterative scheme together with the linear DOAS fit on ln⁡(I0/I). However, codes differ, for example in whether trace gases are included in the shift and squeeze fit of I0 to the high-resolution Fraunhofer atlas. Sometimes also a higher-order calibration is allowed or several subwindows are used in order to differently characterize different parts of the spectra.

Table  summarizes tests performed to investigate the impact of different wavelength calibration approaches using the IUPB retrieval code NLIN. In addition, some tests were performed with the Python script form Sect. , which has therefore been extended to perform the non-linear shift fit as not all tests could be easily implemented in the comprehensive NLIN software. Note that, in contrast to the shift, the squeeze has been excluded from the intercomparison as it was found to always be 1.0 for measurements shown here. Also, QDOAS-specific implementations were not tested here.

Figure  shows the resulting impact on NO2 and rms as well as the fitted shift between I and I0 (not present in all tests). NO2 and rms are again differences relative to the IUPB v1 fit results (without I0-correction as this was not implemented in the Python routine). As in Fig. , the shift in Fig. d is no difference, and for comparison the shift from the reference fit is shown explicitly in black. The (fixed) shift retrieved from the non-linear fit of I0 to the Fraunhofer atlas is -0.035 nm. This is roughly a factor of 10 larger than the fitted shift between I and I0 shown in Fig. d. It is interesting to note that the shift is not zero around noon (time of the reference spectrum), indicating correlations between shift fit and other effects (predominantly intensity offset correction) and also indicating the presence of the tilt effect .

The most extreme test is TW0, which excludes both the calibration of I0 to the Fraunhofer atlas as well as the shift between I and I0. When omitting both calibration steps, the rms is largely enhanced by up to 80 %, peaking in the morning and decreasing towards noon with a second smaller maximum around 12:00 UT. Interestingly, a very similar shape is seen in the rms differences of KNMI, NUST, UNAM, USTC and INTA (but not in INTA2 where a fault in the wavelength registration module was corrected) in Fig. d. Although all of these groups are performing a wavelength calibration, TW0 indicates that differences in the calibration procedure are causing most of the disagreements between groups in terms of rms. This is in contrast to NO2 where changes of only 0.4 % are obtained from TW0.

TW1 still excludes the absolute calibration to the Fraunhofer atlas, but includes the shift fit between I and I0. As seen before, the impact on NO2 is very small (≈ 0.4 %), but absolute NO2 differences of TW1 reflect the shape of total rms and NO2 slant columns (compare to Fig. ); i.e. the relative differences are smooth in shape. The rms is similar in shape to TW0, but the morning maximum is slightly later at 07:00 UT; the noon maximum is around 11:00 UT and a small maximum in the evening occurs at 18:00 UT. This shape is similar to the relative rms of NUST, UNAM and NIWA in Fig. , but the absolute numbers are different. However, this behaviour indicates that differences in the fitted Fraunhofer shift are partially responsible for observed differences between these and other groups. Interestingly, the fitted shift between I and I0 of TW1 in Fig. d is very similar to corresponding values of the reference fit, which is because the missing Fraunhofer shift is a different effect than the shift between I and I0.

In contrast to TW1, TW2 includes the Fraunhofer shift fit, but excludes the shift between I and I0. As this is the larger effect (-0.035 nm compared to only ≈ 0.004 nm), the rms is much smaller than in TW1 with a single maximum (up to 50 %) in the early morning at 06:00 UT. The rms time series shape is similar to the KNMI line in Fig. . The reason is a different definition of the shift in the KNMI retrieval: while in most retrievals I is shifted relative to I0, KNMI calculates the optical depth τ=ln⁡(I0/I) without any shifts but then allows a shift of all cross sections relative to τ. This is in first order compensating the effect of the Fraunhofer shift but neglecting potential shifts between I and I0 (in this case Fraunhofer lines would not cancel out completely in the optical depth τ). As a result, the KNMI approach is similar (but not identical) to TW2. The fit quality following the KNMI approach is expected to be better using a sequential reference as the temperature drift of the spectrometer is much smaller then. It has to be mentioned that the change of NO2 in TW2 is small (0.3 %) compared to the change in fit rms (50 %).

TW3 uses the same wavelength calibration treatment as the IUPB reference fit, but was run in another programming code (Python) to evaluate how much difference is caused by use of another programming code and numerical issues. The fitted shift is mostly identical to the reference fit except for the early morning and late evening when the NO2 also shows very slight differences. The largest disagreement of NO2 is 0.2 % for the first measurement of the day that was affected by large stray light effects (and potentially direct light) and thus most likely indicating cross-correlations between shift fit and intensity offset correction. The resulting rms is almost identical to the reference fit (Fig. c).

TW4 is the same as TW3 but the spline interpolation (calculating I at spectral points of I0 during the non-linear shift fit) is replaced by a simple linear interpolation. The fitted shift is changed slightly and NO2 differences are of almost the same order than when not performing any shift fit at all, demonstrating that the shift fit has a negligible impact on NO2. In contrast, it has a large impact on rms, where the marginally different methods between TW3 and TW4 produce the same rms while the effect of excluding the shift completely leads to largely enhanced rms.

In the Fraunhofer calibration of the reference fit using NLIN (non-linear shift fit of I0) as well as in TW3 all trace gas absorptions are omitted; i.e. an iterative scheme between shift fit and DOAS fit comprising only a polynomial of order 4 is applied. In contrast, all trace gas absorptions are included in the Fraunhofer calibration in TW5. As a result, the rms of the DOAS fit between the Fraunhofer spectrum and I0 is reduced by a factor of 2 and the non-linearly fitted shift is -0.031 nm instead of -0.035 nm in the reference fit. However, this only has a marginal influence on NO2, rms and fitted shift between I and I0 in Fig.  and consequently explains none of the observed differences between groups.

Notably, all shifts (even TW1) in Fig. d show the same general shapes that are retrieved by most groups in Fig. . Only shifts of KNMI clearly differ, for the reasons mentioned above. Rms shapes in Fig. c suggest that differences in the wavelength calibration are the major reason for observed rms disagreements between groups in Fig. .