Introduction
For a measured structured spectrum s(λ) (e.g. scattered sunlight),
the tilt effect emerges, because structures do not cancel out completely in
the ratio of a measured spectrum t(λ) relative to another spectrum
with a different colour, which denotes the broadband spectral dependence. We
show later that this can be interpreted as a spectral shift. This is due to
the fact that the broadband shape of the atmospheric transmission and the
convolution with the instrument function do not commute .
The tilt effect was previously described and is explicitly corrected for by
,
, , ,
, and by including one additional
tilt-effect pseudo absorber in the spectral analyses, the magnitude of which was
determined from the spectral fit. In their pioneering work,
named this effect the “tilt effect” and
corrected its impact by including a correction spectrum calculated using a
radiative transfer model in the spectral analysis. However,
do not provide a mathematical derivation and
instead estimates the effect's magnitude. It remains unclear whether this effect is
related to spectral undersampling and if it is also significant for other
observations of scattered sunlight. also provide
formulae for the correction spectrum, state that the effect is directly
related to spectral undersampling and note that the effect is stronger for
NO2 retrievals in the blue spectral range than for O3 in the green
spectral range due to smaller Fraunhofer lines. provide
a mathematical derivation for the tilt effect correction spectrum (in
their Appendix B) and state that the tilt effect
can be interpreted as a spectral shift. However, also here, only one fixed
correction spectrum is used, which is scaled accordingly with the fitting
routine.
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
e.g., which allow for a spectral shift
of the measurement spectrum relative to a reference spectrum, systematic
spectral shifts of up to 2 pm at a spectral resolution of the instrument of
0.6 nm are observed as shown in Fig. with the exact
magnitude depending on the observation geometry. If no spectral shift is
allowed in the fitting routine, residual structures of up to 2.5×10-3 peak-to-peak are observed (see
Fig. ). Thus, this effect needs to be
considered for any DOAS application using a structured light source (such as
the Sun), aiming at an rms of the residual of 10-3 and below. This is done
implicitly in many DOAS retrieval codes by allowing for a spectral shift
between measurement spectra and reference spectra. This option was originally
introduced into the different analysis software to account for real shifts
caused by instrumental instabilities compare e.g..
In fact, the observed shifts derived from the spectral analysis were
usually attributed to such instrumental stabilities only.
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 < 0.1 pm, cf.
Fig. ) and no larger spectral shifts in the DOAS
analysis can be explained by instrumental instabilities any more. However,
significant spectral shifts are still observed and are furthermore related to
the telescope elevation angle of the MAX-DOAS observation. These can be
explained in such cases by the tilt effect as shown in
Sect. .
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 as e.g. in,
the instrumentally induced tilt change can also lead to an apparent relative
spectral shift.
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. , we mathematically derive the expected spectral shift
for a simplified instrument function. The expected spectral shifts are
compared in Sect. to field measurements. For these, as
in Sect. for the case of synthetic spectra, good
agreement is found. Finally in Sect. we discuss
different ways of how the tilt effect can be corrected. We provide examples
and estimate its impact on the spectral retrieval.
Mathematical derivation
Principle
A sketch of the principle of the tilt effect is shown in
Fig. where two individual δ-shaped emission lines
are used instead of a Sun spectrum.
Two emission lines δ1 and δ2 at 349.8 and 350.2 nm are
observed using a (virtual) spectrometer with a spectral resolution of
0.6 nm. These are shown as Gaussian peaks (p1 and p2) around each of
the lines (grey areas). If both lines have the same intensity, the resulting
total observed intensity (blue) has its maximum in the middle of the two
lines at 350 nm. If the lines are attenuated by the polynomial p(λ)
(drawn in red, in intensity space), the resulting total observed intensity
(green) appears to be shifted in wavelength by Δλ=-0.08 nm.
The unrealistically steep broadband slope in p(λ) was chosen to
illustrate the effect: typically the slope of the polynomial in intensity
space in DOAS observations is 2 orders of magnitudes smaller, as is the
spectral shift due to the tilt effect (see Sect. ).
Definitions
The instrument response function or instrument slit function h(λ0,λ) describes the response of the spectrometer for incoming radiation
of wavelength λ0 at the response wavelength λ on the
detector.
Illustration of the tilt effect: an explanation is found in
Sect. .
Let p(λ) be a polynomial in intensity space describing the broadband
change in the shape of the spectrum due to scattering processes and broadband
absorption in the atmosphere.
Finally k(λ) is a high-resolution Sun spectrum, e.g. from
.
A low-resolution Sun spectrum s(λ) can be calculated from these
quantities, where ⊗ denotes the convolution operator. For
simplicity, s(λ) is assumed to be direct sunlight with neither
extinction nor absorption.
s(λ)=k(λ)⊗h(λ0,λ)=∫dλ0h(λ0,λ)k(λ0),
and with the wavelength-dependent attenuation p(λ), we obtain
t(λ)=k(λ)p(λ)⊗h(λ0,λ)=∫dλ0h(λ0,λ)p(λ0)k(λ0).
The optical density which is typically fitted in DOAS applications
is then
τ(λ)=lnt(λ)s(λ).
The tilt effect in the current literature describes the fact that
absorption structures in s(λ) and t(λ) (Fraunhofer lines and
atmospheric absorbers on Earth) do not cancel out completely when calculating the
optical depth τ, even if p(λ) is smooth. We will show that it
produces an apparent shift Δλ of t(λ) with respect to
s(λ). It is caused by the broadband spectral variation p(λ),
typically approximated by a polynomial in optical density, which does not
commute with the convolution with the instrument function
p(λ)s(λ)-t(λ)≠0. In the next subsection, we
therefore want to show the following equation:
p(λ)s(λ)≈!t(λ-Δλ)=t(λ)-Δλ∂t(λ)∂λ+O(Δλ2).
Apart from the shift Δλ, higher orders O(Δλ2) are neglected here. Note that the right-hand side of
Eq. () is similar to the tilt definition in
their Eq. 20, even
though it is not directly connected to a spectral shift there.
Derivation
Without restriction of generality, a Gaussian instrument function is used in
the following, as it has some useful analytical properties. Here σ is
the standard deviation and λ0 the centre wavelength.
g(λ0,λ)=1σ2πe-(λ-λ0)22σ2
Note that any instrument function can be represented by a sum of Gaussian
functions (Sect. ) and that many
instrument functions are indeed close to Gaussian shape, as in
. To show a useful relation which is needed later, we set
λ0=0 for simplicity and we use a first-order polynomial
q(λ)=1-wλ.
We use ∂g(λ)∂λ=-λσ2g(λ) to reformulate
q(λ)g(λ)=g(λ)-wλg(λ)=g(λ)+wσ2∂g(λ)∂λ,
which then is g(λ+Δλ)+O(Δλ2)
with Δλ=wσ2. We find that the spectral shift Δλ is indeed proportional to the product of the tilt w of the
spectrum and the square of the width of the instrument function.
O(Δλ2) represents second-order effects. This can
also change of the effective shape of the instrument function (see
Sect. ).
The average w from Eq. () (or later ∂∂λd(λ) with the DOAS polynomial d(λ) from
Eq. ) was found within 0.025–0.01 nm-1 averaged over
the fit interval using a fixed Fraunhofer reference spectrum for the MAD-CAT
campaign described in Sect. .
For measurements, p(λ) is normally not linear in λ due to the
characteristics of Mie and Rayleigh scattering. Therefore the derivative
∂∂λp(λ) is not constant and the
spectral shift Δλ also depends on the wavelength λ
itself.
With Eq. () we can calculate t(λ) from
Eq. ():
t(λ)=k(λ)p(λ)⊗g(λ0,λ)=∫dλ0g(λ0,λ)p(λ0)k(λ0).
Taylor expansion of p(λ0) around λ yields
t(λ)=∫dλ0g(λ0,λ)[p(λ)+(λ0-λ)∂∂λ′p(λ′)|λ′=λ+O(λ0-λ)2]k(λ0),
which is, with the shift from Eq. () at
wavelength λ and neglecting higher-order terms:
≈p(λ)∫dλ0gλ0,λ+Δλ(λ)k(λ0)=p(λ)s(λ+Δλ(λ)),
with
Δλ(λ)=σ21p(λ)∂∂λp(λ)=σ2∂∂λln(p(λ)).
p(λ) is defined in intensity space and is related to the
DOAS polynomial d(λ) in optical density space (logarithm of
intensity) via
p(λ)=e-d(λ).
We get
Δλ(λ)=-σ2∂∂λd(λ).
This is the more general case of Eq. () for non-linear DOAS
polynomials.
Relation to undersampling
Previously the tilt effect was also associated with spectral undersampling
: as, for example, described in
for spectral data from satellite, a spectral
shift between the observed measurement and reference spectra was introduced
in order to correct for Doppler shifts between them. As these shifts
(typically < 5 pm) are small compared to the spectral resolution of the
instrument (typically ≈ 0.5 nm), the spectral shift can be
linearized and directly calculated from a high-resolution Sun spectrum (such
as ) in order to also include artefacts of spectral
undersampling. In the DOAS fit finally contained
this linearized shift as well as the non-linear shift and squeeze parameters
of the measurement spectrum relative to the reference spectrum. Also, the
tilt effect introduces a spectral shift of similar magnitude and is
corrected (in first-order approximation) in the same way. This potentially
led to confusion in the available literature. The derivation of the
tilt effect shown above does, however, not depend at all on the properties of
the spectral binning of the instrument and can therefore be considered
independent of the undersampling effects.
Relation to the colour index
The colour index CI(λa, λb) is defined by the ratio of
intensities Ia and Ib at two distinct wavelengths λa and
λb see e.g. and can be used
to describe the tilt of a spectrum in first-order approximation.
CI(λa,λb)=IaIb
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
. For a measurement spectrum I′ with a DOAS
polynomial d(λ) relative to the reference spectrum I, we get
CI′(λa,λb)=Iae-d(λa)Ibe-d(λb),
and thus we obtain using Eq. ()
CI′-CI=Ia(e-d(λa)-e-d(λb))Ibe-d(λb)=IaIbλa-λbe-d(λb)∂∂λe-d(λ)|λ=λc;λc∈λa,λb≈-Ia(λa-λb)Ib∂∂λd(λ)|λ=λc;λc∈λa,λb.
This means that the difference in colour indices between different spectra is
proportional to the derivative of the DOAS polynomial at a certain point
λc within the fit interval. The derivative of the DOAS polynomial is
again proportional to the apparent spectral shift due to the tilt effect
(Eq. ). As an example the colour indices for 1 day of
MAX-DOAS measurements are shown in Fig. b.
Measurements
For a spectral resolution of ≈ 0.6 nm and typical DOAS
polynomials, shifts of up to around 1 pm are expected using
Eq. (). In this section, we will set the expected spectral
shift due to the tilt effect in relation to the spectral shift of the
measurement spectrum in DOAS fits.
Measurement site
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.
The measurement site is located on the outskirts of Mainz and is close to
Frankfurt as well as several smaller towns. Eleven research groups participated
with the MAX-DOAS instruments. The intercomparison is aimed primarily at the
spectral retrieval of nitrogen dioxide (NO2), formaldehyde (HCHO),
nitrous acid (HONO) and glyoxal (CHOCHO), their azimuthal
distributions and the retrieval of their respective vertical concentration
profiles. Data from this campaign have been already published, e.g. in
, , and
.
Measured shift (a) and colour index (340,
370 nm) (b) as a function of time and observation elevation (colour
coded) for 1 day (16 June 2013) during the MAD-CAT campaign relative to a
Fraunhofer reference spectrum recorded close to local noon (thin blue
vertical line). The thick blue line in the upper panel represents the pure
instrumental shift after the shift introduced by the tilt effect was removed
(see text).
Instrument description
We apply data obtained by an EnviMeS MAX-DOAS instrument during the
MAD-CAT campaign. It is based on two Avantes ultra-low stray-light
AvaSpec-ULS2048x64 spectrometers (f=75 mm) using a back-thinned
Hamamatsu S11071-1106 detector. The spectrometer is temperature stabilized at
20 ∘C with deviations of ΔT < 0.02 ∘C
at the temperature sensor. The UV spectrometer covered a spectral range of
294–458 nm at a FWHM spectral resolution of ≈ 0.6 nm or
≈ 7 pixels. The spectral stability was determined from the position
of the Ca lines at around 393 and 397 nm and was typically better than
±2 pm per day and better than ±5 pm for the duration of the
measurements from 6 June 2013 to 31 July 2013.
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-1. The maximum deviation of the spectrometer temperature from the
nominal temperature was ΔT < 0.02 K; thus less than
0.1 pm spectral shift can be attributed to temperature instability close to
thermal equilibrium under ideal conditions.
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. , which was already corrected for the shift
introduced by the tilt effect. This observation later led to an improved
mechanical set-up of the spectrometer box to reduce these internal temperature
differences.
Mercury discharge lamp spectra used to obtain the instrument slit function
h(λ0,λ) were recorded manually. No significant change of the
instrument slit function shape was observed during the campaign.
Retrieval wavelength intervals and reference spectra for the
MAX-DOAS. S0 denotes the SCD used for the I0 correction during
convolution, if applicable.
T
S0
HONO
Wavelength interval (nm)
Start
335
End
373
H2O vapour
298 K
×
∗
,
O4
293 K
×
O3
223 K
1×1018 molec cm-2
×
243K
1×1018 molec cm-2
×
HCHO
×
HONO
×
BrO
×
NO2
293 K
1×1016 molec cm-2
×
×
Linear and square terms according to
220 K
1×1016 molec cm-2
×
Ring spectrum at
273 K
×
DOASIS
243 K
×
based on
Ring spectrum ⋅λ4
×
Polynomial degree
5
Additive polynomial degree
1
e.g.
∗ Water vapour absorption around 363 nm was not
considered for the calculation and analysis of synthetic spectra.
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∘. During
daylight, spectra were recorded for 1 min each at 11 elevation angles of
90∘ (zenith), 30, 15, 10, 8, 6–1∘ as long as
solar zenith angles (SZA) were smaller than 87∘. Until a SZA of
100∘ zenith sky spectra were recorded at 90∘ telescope
elevation. The exposure time was adjusted within the DOASIS
measurement script to obtain spectra at a typical saturation of 50 %.
Analysis
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 for the spectral analysis (see
Table ). Similar results were obtained
in other wavelength intervals (e.g. a glyoxal retrieval window from 432 to
458 nm).
The analysis of measured and synthetic spectra (see Sect. )
was done using the DOASIS software using a noon Fraunhofer reference
spectrum. The literature cross sections were convolved using the measured
instrument slit function at 334 nm.
Shift and squeeze parameters
The shift a and squeeze b (also called stretch) allow the DOAS fit to
shift and squeeze cross sections in order to minimize the fit rms and
compensate for instrumental instabilities and other factors which can
influence the spectral calibration of the instrument.
This is parameterized typically in the following way:
Δλshift(λ)=a+b(λ-λ0)+c(λ-λ0)2+O(λ-λ0)3.
Higher orders such as c are often not used and set to zero.
The choice of λ0 depends on the implementation. It is the minimum
wavelength of the fit interval in DOASIS and the middle of the fit interval
in the QDOAS software package . Choosing λ0 in the
middle of the fit range has the advantage that the corresponding base
functions for shift and squeeze are linearly independent, which can be
favourable in terms of numerical stability.
Results
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. . This translates to a spectral shift of less than
0.3 pm h-1 or less than 0.06 pm per elevation angle sequence. This
accuracy allows the shift to be distinguished due to the tilt effect (up to 2 pm)
within each elevation angle sequence from instrumental instabilities. The
resulting correlation of measured shift and calculated shift determined from
the DOAS polynomial is shown for the complete data set (June and July 2013) in
Fig. evaluated relative to the next zenith
Fraunhofer reference spectrum. The shift due to the tilt effect was
calculated from the DOAS polynomial using Eq. (). As the
shift varies with wavelength, we used the average shift calculated on an
equidistant grid of 0.1 nm within the fit interval.
A correlation coefficient R2=0.83 and slope of 0.95±0.02 was
observed. The y axis intercept of the fitted polynomial was small and
amounted to 0.05 pm, which is less than 1/1000 of the spectral width of a
detector pixel. The small deviation of the fitted slope of the correlation
from unity can result from a slightly varying instrument function width
within the fit interval (< 2 %, estimated from widths of recorded
mercury emission line spectra) and effective weighting of the shift at
different wavelengths due to variable depth of the Fraunhofer lines
(estimated from tilt effect calculations of spectra with and without
weighting due to Fraunhofer lines to be less than 3 %). The average
measurement error of the shift (estimated by twice the fit error following
) amounts to 0.03 pm (< 1.5 %). Furthermore
the instrument function of the spectrometer used here is not exactly
Gaussian.
DOAS fit results from 16 June 2013 at 04:46 UTC for a spectrum at
2∘ elevation for different settings of the spectral shift and squeeze
of the reference spectrum (see second row) and with and without a tilt effect
correction spectrum marked by crosses in the first row
(Sect. ). To minimize photon shot noise, four
subsequent elevation angle sequences were co-added. Values in round brackets
denote fixed values for shift and squeeze. Fit residuals and a fit of the
correction spectrum are shown in Fig. . The average
shift within the fit interval due to the tilt effect calculated from the DOAS
polynomial itself amounts to 1.14 pm and a squeeze of 1+8×10-7,
which could not be resolved from the measurement data. The row named
“σfit” lists the respective fit errors of the obtained
differential slant column densities (dSCDs) from the DOAS fit (squeeze
definition from DOASIS, λ0=λmin; see
Eq. ).
Case
1
2
3
4
5
6
Free shift parameters
shift, squeeze
shift
none
corr. spectrum
corr. spectrum
none
and shift
Tilt effect correction spectrum
×
×
×
Shift (pm)
1.1±0.1
0.99±0.04
(0)
(0)
0.019±0.04
(0)
Squeeze
1.00±3.4×10-6
(1)
(1)
(1)
(1)
(1)
Rms [10-4]
2.83
2.84
4.30
2.79
2.79
2.82
dSCD HONO [1×1014 molec cm-2]
2.64
2.48
0.18
1.99
2.01
2.03
σfit HONO [1×1014 molec cm-2]
2.46
2.47
3.67
2.39
2.44
2.46
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. . The resulting
instrumental shift is stable until about 09:00 UTC (with a standard
deviation of less than 0.1 pm), a time after which the room temperature
changed, probably as the door was opened and the temperature outside the
instrument started changing. The gap in measurement data around noon is
caused by a restart of the measurement routine. As the temperature
stabilization routine was also restarted, the gap is followed by a shift in the
spectral calibration of 0.4 pm, as the temperature control needed a few
minutes to stabilize. This effect would not have been as clearly visible
without correction for the tilt effect.
Corresponding plots of fit residuals to the cases 1, 3 and 5 from
Table and the tilt-effect correction spectrum for
case 5. The tilt-effect correction spectrum is shown in red; the sum of it
and the residual are shown in grey.
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∘ the
fit results are shown in Table using a reference
spectrum recorded in the same elevation angle sequence at an elevation angle
of 90∘. Here six cases are distinguished with different numbers of
free parameters for shift, squeeze and the explicitly calculated tilt-effect
correction spectrum according to Eq. (). For the calculation
of the tilt-effect correction spectrum we used a DOAS polynomial obtained
from a fit without considering the tilt effect (see also
Sect. ).
The tilt effect in synthetic spectra
Additionally the tilt effect is demonstrated for synthetic spectra in order
to exclude any instrumental influences.
Calculation of synthetic spectra
All simulations were conducted with the radiative transfer model SCIATRAN
, version 3.6.0 (3 December 2015). SCIATRAN was
operated in raman mode to simulate intensities of scattered sunlight
in Mainz, Germany (49.99∘ N, 8.23∘ E), including the effect
of rotational Raman scattering in the Earth's atmosphere. The scalar
radiative transfer problem was solved in a pseudo-spherical atmosphere (i.e.
the solar beam is treated in spherical geometry, while the scattered or
reflected beam is treated in plane-parallel geometry) using the discrete
ordinate method. The simulations from 330 to 395 nm were conducted with
0.01nm spectral sampling, and Raman lines were calculated using the
forward-adjoint approach and binned to the 0.01 nm wavelength grid
.
Absorption by the trace gases, ozone (O3), nitrogen dioxide (NO2),
formaldehyde (HCHO), bromine oxide (BrO), nitrous acid (HONO), and by the
O2–O2 collision complex (O4) was considered. The respective
cross-section references are the ones also used for analysis in
Table . Aerosols were assumed to be
mostly scattering, having an optical depth (AOD) of 0.135, an asymmetry
factor of 0.68, and a single scattering albedo (SSA) of 0.94. All aerosol
parameters were assumed to be constant over the whole wavelength range.
The simulated observation geometry was similar to the measurement sequences
as described in Sect. . A more detailed
description, also of the concentration height profiles, can be found in
.
Water vapour absorption according to and
was not considered for the synthetic spectra but was
compensated in the measured data. A detailed analysis can be found in
.
Results
The spectral analysis was performed in analogy to
Sect. . The absorption of water vapour in the UV was
not considered for the calculation of the spectra and thus also not in the
spectral analysis.
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. , some spectral shift was found and needed to be
compensated for.
Fits with an rms of more than 4×10-4 during twilight were filtered
out, as saturation and radiative transfer effects of stratospheric ozone
absorption increased the residuals of the fits significantly and have the
potential to modify the calculated shift values. The correlation of
calculated and fitted shift for the remaining 120 spectra due to the
tilt effect was very good, with R2=0.9993. Shifts of up to 1.2 pm due to
the tilt effect were found. The shift from the DOAS fit was about 2.1 %
larger than from the calculation of the tilt effect. However, the average
measurement error of the shift amounts to 0.02 pm and is thus of similar
magnitude.
The small discrepancy could be also caused by the fact that the influence of
rotational Raman scattering is calculated differently in DOASIS (according to
from the convolved, synthetic spectrum
itself) and SCIATRAN (according to at the
higher spectral resolution of 0.01 nm before convolution).
Overall the very good agreement of theoretically expected and calculated
spectral shifts also shows the validity of the derivation of the tilt effect.
Discussion – correction of the tilt effect
Even for a perfect MAX- or zenith sky DOAS instrument (as shown in
Sect. ), the tilt effect needs to be considered and corrected.
Typically it is corrected by allowing a shift between measurement spectra and reference spectra. As the shift at each wavelength depends on the derivative
of the broadband spectral dependence, which is usually corrected by the DOAS
polynomial, an additional squeeze (and higher orders of the spectral shift)
of the measurement spectrum can be necessary, depending on the desired
magnitude of the residual. This is discussed in
Sect. .
The spectral shift depends on the spectral resolution of the instrument (see
Eq. ). In fact it is proportional to the square of the
spectral resolution.
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 p(λ)s(λ) and t(λ).
This approach is discussed in Sect. .
Shift and squeeze
The apparent change in the wavelength determination due to the tilt effect
can be determined from the DOAS polynomial using Eq. ().
The shift, squeeze and higher-order parameters can then be determined by a
polynomial fit of Δλ(λ) using the QDOAS definition of
λ0 of squeeze and higher orders (see Eq. ). For
each of the parameters of the polynomial (corresponding to shift, squeeze,
quadratic squeeze etc.), the maximum shift inside the fit range can be
determined and can then be used for estimating the residual structure which
is caused by the tilt effect. This shift, converted to the corresponding
optical depth, is shown in Fig. . For typical
applications (FWHM = 0.6 nm, rms > 1 × 10-4), it is
therefore sufficient to allow shift and squeeze between measurement spectra and reference spectra in order to correct for this effect. The conversion factor
αOD from shift to peak-to-peak optical density within the
fitting interval was determined from the pseudo-absorber of the spectral
shift within the fit interval for the given spectral resolution of the
instrument and amounted in this case to 1.5 nm-1.
αOD=2×∂s(λ)∂λ1s(λ)max
Tilt-effect correction spectrum
For a known DOAS polynomial d(λ), a correction spectrum c(λ)
can be calculated to compensate for the tilt effect. This implies that an
iterative fit process is performed and thus means higher computational costs.
The correction spectrum c(λ) is the difference between two synthetic
sun spectra calculated from a highly resolved solar atlas, one where the
attenuation with the p(λ)=e-d(λ) in intensity space is
applied before the convolution operation and one where it is applied after:
c(λ)=p(λ)s(λ)-t(λ).
Peak-to-peak optical density caused by shift, squeeze and higher-order
squeeze due to the tilt effect, using the data set from
Fig. . The shift due to the tilt effect was calculated
from the DOAS polynomial for the corresponding mean wavelength of each pixel
within the fit interval. Then a third-order polynomial was fitted to this
data to calculate the corresponding shift, squeeze and higher-order terms and
thus the corresponding peak-to-peak ODs caused by the tilt effect. It can be
seen that shift and squeeze already compensate for most of the effect. The
colour scale is the same as in 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
s(λ)
cPA(λ)=c(λ)s(λ).
This correction spectrum, introduced in the fit results shown in
Table , was indeed found in the spectral fit and reduced
the shift of the reference spectrum from 1.1±0.1 pm (case 2) to 0.019±0.04 pm (case 5). As the calculation from Eq. () also
provides the absolute magnitude of the effect, this correction spectrum does
not even need to be fitted as in previous publications but can be applied
directly (case 6). This can, if the instrument itself is stable, potentially
reduce the degrees of freedom of the fit and thus result in lower detection
limits. This could, however, not be observed here for measurement data.
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 as it is defined via the derivative with
respect to wavelength of the respective spectrum. To test this, the DOAS
polynomial was determined for the spectrum shown in
Table . This polynomial was used to calculate the
correction spectrum. The absolute magnitude of the resulting DOAS polynomials
with and without correcting for the tilt effect differed relative to each
other by up to 3 %. Calculating the correction spectrum from the second
DOAS polynomial results in a second correction spectrum, which differs
absolutely with an OD of 6×10-5. Therefore further iterations of
the fitting process are not needed in this case.
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 H2O and O2 in the red and near-IR
spectral range. A potential disadvantage is that this calculation requires
knowledge of the exact instrument slit function, which is implicitly included
in the first approach (Sect. and see also
Sect. ). As the spectral shift of the
instrument also needs to be accounted for often, shift and squeeze need to be
implemented in any case, which can make the calculation of explicit
correction spectra in most cases obsolete. This choice depends on the desired
precision of the result for a very small rms (compare
Fig. ).
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.
applying shift and squeeze is sufficient for most applications, but
calculation of the correction spectrum can reduce the impact of the
tilt effect even further, as seen in Table . Here the rms
of the default fit using shift and squeeze of the reference spectrum (1) is
reduced by using an explicitly calculated correction spectrum slightly by
1 % (4), even though the number of degrees of freedom of the fit stayed
constant (cases 1 + 5). When only the correction spectrum was used, and
the shift fixed to zero, assuming no shift between Fraunhofer reference and
measurement spectrum (cases 1 + 4), the rms is the same, but the HONO fit
error is reduced.
The influence of the instrument slit function
As shown in Eq. () for the case of a Gaussian instrument
response function, the spectral shift depends on the spectral resolution of
the instrument, in fact it is proportional to the square to the spectral
resolution. A real instrument function h(λ0,λ) is in general
not a Gaussian function but can be approximated by N Gaussian functions of
different widths σi shifted by Δλ0i and weighted by
wi, as it is typically also measured at finite spectral resolution.
h(λ0,λ)=∑iNwigσiλ0+Δλ0i,λ
As summing and convolution are interchangeable, Eq. ()
can be written as a sum over different si(λ)=k(λ)⊗gσi(λ0+Δλ0i,λ) using
Eq. (). To these Gaussian instrument functions the
derivation of the tilt effect applies individually. However, as the
derivative with respect to λ in the Taylor series for the spectral
shift in Eq. () also commutes with the sum, the shift
calculated from s(λ) also correctly compensates for the tilt effect
for non-Gaussian instrument functions.
Instrument slit function changes due to tilt effect
As already pointed out for Eq. (), the squeeze and second-order
effects of the tilt effect also lead to a slight modification of the
effective instrument slit function's shape, apart from the spectral shift.
Based on the DOAS polynomials obtained from the fits of measurements from
16 June (compare Fig. ) and using an initial Gaussian
instrument slit function, the effective instrument slit function was
numerically calculated and fitted again with a Gaussian function. The first-order
tilt-effect shift was reproduced within 2×10-8 nm. The
relative width of the instrument function varied by up to 5×10-4 %. For an absorber with a differential OD of unity, this results
in an OD of less than 5×10-6 and is therefore negligible.
Pixel-wavelength calibration of spectra
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.
. As will be shown in Sect. ,
the effect on retrieved trace gases is typically negligible, as the expected
shifts due to the tilt effect are also here of the order of less than a few
pm.
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 e.g.. The
width of a single pixel for the measurements shown above is typically 60 pm
or larger. The variation between different mercury emission lamps is about
0.07 pm and thus significantly smaller than the tilt effect itself
.
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
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 NO2 ∂/∂λσNO2(λ), we obtain a residual OD for a
shift of 2 pm due to the tilt effect of the NO2 absorption
cross section of 0.2 %. Thus a 1.5 % differential absorption by
NO2, which corresponds to a differential slant column density (dSCD) of
1 × 1017 molec cm-2, can result in a systematic residual
structure due to the tilt effect of 3×10-4 (2 % of the
original absorption), which is often acceptable.
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 . It
becomes clear that the overall influence of the tilt effect on the retrieved
HONO dSCDs is small and within the measurement error in this case, for this
absorber and for this instrument. However, as the residual rms and thus the
fit error are significantly reduced, the correction of this effect is crucial
for a correct determination of measurement errors and detection limits (cf.
e.g. ). If the shape of the structures caused by the
tilt effect shows more similarities with an absorber, the changes in its
dSCDs might, however, be larger. This depends on the fitting interval, spectral
resolution and the respective absorber and cannot be answered in general.
Conclusions
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 . In the
context of limb satellite observations, this effect was mathematically
described by . We showed that the spectral shift due to
this effect is proportional to the square of the instrument resolution
σ and the slope of the broadband spectral shape. It can be
described by the so-called DOAS polynomial, which accounts for broadband
spectral differences between the measured spectrum and the Fraunhofer
reference spectrum (e.g. caused by Mie and Rayleigh scattering and broadband
absorptions). In contrast to previous publications
e.g., it is not directly connected to
spectral undersampling and is not restricted
to a certain wavelength range. It affects any medium-resolution spectroscopic
application where the spectral evaluation involves a step in which the
convolution and effects like scattering are commuted, which leads to a broadband variation
of the shape of the spectrum. Lab measurements of trace gas
absorptions are, however, often done at higher spectral resolution, which
minimizes the apparent shift of the tilt effect due to the relation shown in
Eq. ().
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 2.5×10-3 if not corrected for. This will increase the calculated fit and
measurement errors and can also lead to deviations of retrieved dSCDs,
depending on the settings of the spectral retrieval and the instrument's
properties. For DOAS fits with a residual rms of more than 10-4, we
estimate that the tilt effect can be compensated for by allowing for a shift
and squeeze term. For DOAS fits with a residual rms of less than 10-4,
which can be obtained by co-adding a large number of spectra, higher-order
terms for the parameterization of the wavelength shift might be necessary. The
shift due to the tilt effect is typically not constant with respect to
wavelength λ within the fit intervals, as it is proportional to the
derivative of the so-called DOAS polynomial (Eq. ). For
observation geometries which show larger differences in colour indices, such
as satellite limb observations, such corrections might even be necessary if
the requirements on the magnitude of the residual are less strict. As the
spectral shift due to the tilt effect can be calculated from the DOAS
polynomial, the remaining observed spectral shift can be attributed to
instrumental properties and it can thus be used for monitoring purposes.
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).