Experience of differential atmospheric absorption spectroscopy (DOAS) shows that a spectral shift between measurement spectra and reference spectra is frequently required in order to achieve optimal fit results, while the straightforward calculation of the optical density proves inferior. The shift is often attributed to temporal instabilities of the instrument but implicitly solved the problem of the tilt effect discussed/explained in this paper.

Spectral positions of Fraunhofer and molecular absorption lines are systematically shifted for different measurement geometries due to an overall slope – or tilt – of the intensity spectrum. The phenomenon has become known as the tilt effect for limb satellite observations, where it is corrected for in a first-order approximation, whereas the remaining community is less aware of its cause and consequences.

It is caused by the measurement process, because atmospheric absorption and convolution in the spectrometer do not commute. Highly resolved spectral structures in the spectrum will first be modified by absorption and scattering processes in the atmosphere before they are recorded with a spectrometer, which convolves them with a specific instrument function. In the DOAS spectral evaluation process, however, the polynomial (or other function used for this purpose) accounting for broadband absorption is applied after the convolution is performed.

In this paper, we derive that changing the order of the two modifications of
the spectra leads to different results. Assuming typical geometries for the
observations of scattered sunlight and a spectral resolution of 0.6 nm, this
effect can be interpreted as a spectral shift of up to 1.5 pm, which is
confirmed in the actual analysis of the ground-based measurements of
scattered sunlight as well as in numerical radiative transfer simulations. If
no spectral shift is allowed by the fitting routine, residual structures of
up to 2.5

For a measured structured spectrum

The tilt effect was previously described and is explicitly corrected for by

We will derive the tilt correction as an interpretation of a spectral shift for Gaussian instrument functions, which can, however, also vary with wavelength. We show that the calculated spectral shifts due to the tilt effect agree with the observed shifts from DOAS analyses of ground-based measurements.

We observed that for Multi Axis Differential Optical Absorption
Spectroscopy (MAX-DOAS) evaluations

MAX-DOAS instruments typically contain thermally stabilized spectrometers in
order to avoid changes in their pixel-to-wavelength calibration. For such
instruments, the spectral stability within 1 day often has a similar
magnitude to the tilt effect (often less than a few picometres). So-called Fraunhofer
reference spectra are recorded regularly at zenith-viewing direction: these
are used as reference for the spectral analysis. If Fraunhofer reference
spectra are recorded 10–15 min each, then the change in spectral
calibration of the instrument for the measurement spectrum relative to the
Fraunhofer reference spectrum becomes small (typically

When a measured spectrum is evaluated relative to another spectrum of the
same set-up, instrumental effects on the tilt are expected to cancel out, as both
spectra are influenced in the same way, e.g. by the efficiency of the grating
and the detector. However, if a measured spectrum is evaluated relative to a
so-called Kurucz Sun spectrum

Another interesting aspect is that correction of the measured shifts
including the tilt effect will allow the spectral stability of
passive DOAS instruments to be estimated more precisely as shown in
Sect.

In Sect.

A sketch of the principle of the tilt effect is shown in
Fig.

Two emission lines

The instrument response function or instrument slit function

Illustration of the tilt effect: an explanation is found in
Sect.

Let

Finally

A low-resolution Sun spectrum

The optical density which is typically fitted in DOAS applications

The tilt effect in the current literature describes the fact that
absorption structures in

Apart from the shift

Without restriction of generality, a Gaussian instrument function is used in
the following, as it has some useful analytical properties. Here

Note that any instrument function can be represented by a sum of Gaussian
functions (Sect.

We use

The average

For measurements,

With Eq. (

Taylor expansion of

which is, with the shift from Eq. (

This is the more general case of Eq. (

Previously the tilt effect was also associated with spectral undersampling

The colour index CI(

Instead of analysing the DOAS polynomial, it is often sufficient to look at
the difference in colour indices of measurement spectra and reference spectra, or in
other words, at the tilt of the spectrum (or part of a spectrum) as in

This means that the difference in colour indices between different spectra is
proportional to the derivative of the DOAS polynomial at a certain point

For a spectral resolution of

The Multi Axis DOAS – Comparison campaign for Aerosols and Trace gases
(MAD-CAT) in Mainz, Germany took place on the roof of the Max Planck Institute
for Chemistry (MPIC) during June and July
2013

Measured shift

We apply data obtained by an EnviMeS

now continued by Airyx GmbH,
Eppelheim, Germany,

Mercury discharge lamp spectra recorded at different spectrometer
temperatures yield a shift of the spectral calibration of this spectrometer
type of about 4.5 pm K

During laboratory test measurements, a change of the spectral calibration of
the instrument over time was found to be proportional to the temperature
difference outside the thermally insulated spectrometer box and the
spectrometer temperature and is therefore attributed to the residual
temperature differences due to heat flux from the Peltier element through the
spectrometer and the thermal insulation. We assume that this is the main
reason for the variation of the inferred instrumental spectral shift in
Fig.

Mercury discharge lamp spectra used to obtain the instrument slit function

Retrieval wavelength intervals and reference spectra for the
MAX-DOAS.

The 1-D-telescope unit measures its elevation angle constantly using a MEMS
acceleration sensor to determine the true vertical direction and corrects
the elevation angle when it deviates from the nominal elevation angle. It
has a vertical and horizontal field of view (FOV) of 0.2 and 0.8

Even though the tilt effect is a general effect and not restricted to a
certain wavelength range, here we adapted the HONO retrieval settings
suggested by

The analysis of measured and synthetic spectra (see Sect.

The shift

This is parameterized typically in the following way:

The choice of

The variation of the observed spectral shifts of the measurement spectrum
during 16 June 2013 is less than 4 pm, as can be seen from
Fig.

A correlation coefficient

DOAS fit results from 16 June 2013 at 04:46 UTC for a spectrum at
2

Having shown that the shifts are mostly caused by the tilt effect, this
allows the measured shift of the reference spectrum for the shift to be
corrected by the tilt effect to obtain the instrumental shift at higher precision, also
during unsupervised field measurements and without the need for calibration
lamps. This is also shown in Fig.

Corresponding plots of fit residuals to the cases 1, 3 and 5 from
Table

Correlation of the shift determined from the DOAS polynomial according to the tilt effect and measured relative shift of the measurement spectrum to the following zenith sky spectrum (in order to minimize the influence of instrumental instabilities). To reduce the scatter of the data points further, four subsequent elevation angle sequences were co-added.

For an individual spectrum recorded at an elevation angle of 2

Additionally the tilt effect is demonstrated for synthetic spectra in order to exclude any instrumental influences.

All simulations were conducted with the radiative transfer model SCIATRAN

Absorption by the trace gases, ozone (O

The simulated observation geometry was similar to the measurement sequences
as described in Sect.

Water vapour absorption according to

The spectral analysis was performed in analogy to
Sect.

The synthetic spectra represent measurements of an ideal instrument without
any changes of the wavelength calibration due to external influences.
Therefore, the initial expectation of the analysis of the synthetic spectra
was that no shift is needed in the spectral analysis between reference
spectrum and measurement spectrum. However, as described in
Sect.

Fits with an rms of more than

The small discrepancy could be also caused by the fact that the influence of
rotational Raman scattering is calculated differently in DOASIS (according to

Overall the very good agreement of theoretically expected and calculated spectral shifts also shows the validity of the derivation of the tilt effect.

Even for a perfect MAX- or zenith sky DOAS instrument (as shown in
Sect.

The spectral shift depends on the spectral resolution of the instrument (see
Eq.

Another approach is to calculate the effective shift spectrum from the
explicit calculation of the commutator of polynomial and convolution, or in
other words the difference between

The apparent change in the wavelength determination due to the tilt effect
can be determined from the DOAS polynomial using Eq. (

For a known DOAS polynomial

Peak-to-peak optical density caused by shift, squeeze and higher-order
squeeze due to the tilt effect, using the data set from
Fig.

To use it in the DOAS fit as a pseudo-absorber (PA), it can be converted (in
a first-order approximation) to optical density space by division with

This correction spectrum, introduced in the fit results shown in
Table

The DOAS polynomial can be determined with sufficient precision without
correcting the tilt effect, as a small spectral shift can be represented via
Taylor expansion by an individual spectrum, which is dominated by narrowband
contributions

Note that this approach might need to also consider strong absorbers present
in the observed spectra, which are not present in the solar atlas spectrum.
This can play a role in ozone and sulfur dioxide absorption in the UV range
and for strong absorbers such as H

Apart from the tilt-effect-induced shift, the correction spectra calculated using the DOAS polynomial also includes the effect of the squeeze parameter
and higher orders. Therefore a correction spectrum needs to be calculated
corresponding to each fit. As seen from Fig.

As shown in Eq. (

As summing and convolution are interchangeable, Eq. (

As already pointed out for Eq. (

As the tilt effect will influence all spectra recorded at low resolution, it
will also have an effect on the spectral calibration of scattered sunlight
spectra, if done by fitting it to a high-resolution solar atlas, as e.g.

Note that also other calibration methods, as e.g. the calibration using line
emission spectra, have uncertainties: If the position of the emission lines
is determined by fitting Gaussian peaks, the fit error of the centre of the
peak also typically amounts to 2–3 pm, as the shape of the observed
emission line is rarely Gaussian

The centre of mass of an emission peak can be more accurately determined when the emission peaks are not undersampled.

The impact of the tilt effect on the spectral retrieval of trace gases is two-fold: if the tilt effect is not corrected for, the remaining residual structures can cause deviations for retrieved trace gases. The shift induced on the measurement spectrum is the same as for the absorbers, as similar considerations apply to the convolution of trace gases as to the convolution of the Fraunhofer spectrum. However, if the shift of the trace gases is not determined from the fit but from a fit of the Fraunhofer reference spectrum to a solar atlas (typically with a different tilt or colour indices), small shifts of the order of a few picometres can occur, which are not the same for the absorbers.

Using a pseudo-absorber for the spectral shift of NO

For the case of HONO and a spectrum with an apparent shift due to the
tilt effect of 1 pm, the results are shown in Table

Based on a theoretical analysis as well as on measured and simulated
scattered sunlight spectra, we have shown that the tilt effect can cause
artificial shifts and enhanced residuals, which are introduced by the
fact that any modification of the broadband spectral variation of a spectrum
(e.g. caused by atmospheric scattering processes) does not commute with the
convolution with the instrument slit function. Thus an effective shift
between measurement and reference spectra can be observed. This effect is
called the tilt effect according to

A shift between measurement spectra and reference spectra is typically allowed for in DOAS retrievals and motivated by instrumental instabilities. We show that the shift caused by the tilt effect is significantly larger than typical instrument shifts within one elevation sequence and that the main reason to allow for this shift is eventually the tilt effect. For measured as well as for simulated spectra a good correlation between fitted and calculated shifts is found due to the tilt effect.

For ground-based passive DOAS instruments with a spectral resolution of
0.6 nm, we find apparent spectral shifts of more than 1 pm due to the
tilt effect. This shift can result in residual optical depths of

Alternatively, using the known instrument function, correction spectra can be explicitly calculated for a given DOAS polynomial or approximated from a given difference in colour indices between measurement spectra and reference spectra, similarly to suggestions in previous publications.

The effect is generally present for spectroscopic measurements at medium spectral resolution with wavelength-dependent attenuation. Therefore the same effect can be expected for active measurements (e.g. cavity-enhanced or long-path DOAS measurements).

The spectra used for the data evaluation in this paper (about 420 MB) can be obtained on request from the authors.

The authors declare that they have no conflict of interest.

We thank the MAD-CAT team for support during the MAD-CAT campaign. We thank Klaus Pfeilsticker and Alexey Rozanov for helpful comments during the preparation of the manuscript. We thank EnviMeS/Airyx for providing the MAX-DOAS instrument during the MAD-CAT campaign. The article processing charges for this open-access publication were covered by the Max Planck Society. Edited by: Michel Van Roozendael Reviewed by: three anonymous referees