The instrumental spectral response function (ISRF) is a key quantity in DOAS analysis, as it is needed for wavelength calibration and for the convolution of trace gas cross sections to instrumental resolution. While it can generally be measured using monochromatic stimuli, it is often parameterized in order to merge different calibration measurements and to plainly account for its wavelength dependency. For some instruments, the ISRF can be described appropriately by a Gaussian function, while for others, dedicated, complex parameterizations with several parameters have been developed.

Here we propose to parameterize the ISRF as a “super-Gaussian”, which can reproduce a variety of shapes, from point-hat to boxcar shape, by just adding one parameter to the “classical” Gaussian. The super-Gaussian turned out to describe the ISRF of various DOAS instruments well, including the satellite instruments GOME-2, OMI, and TROPOMI.

In addition, the super-Gaussian allows for a straightforward parameterization of the effect of ISRF changes, which can occur on long-term scales as well as, for example, during one satellite orbit and impair the spectral analysis if ignored. In order to account for such changes, spectral structures are derived from the derivatives of the super-Gaussian, which are afterwards just scaled during spectral calibration or DOAS analysis. This approach significantly improves the fit quality compared to setups with fixed ISRF, without drawbacks on computation time due to the applied linearization. In addition, the wavelength dependency of the ISRF can be accounted for by accordingly derived spectral structures in an easy, fast, and robust way.

The instrumental spectral response function (ISRF), also denoted as
“instrument transfer function” or just “slit function”, describes the
spectral response to a monochromatic stimulus and is thus a key quantity in
spectroscopy. Within differential optical absorption spectroscopy (DOAS)

The ISRF is determined by the optical properties of the instrument (entry and exit slits, gratings, detector properties etc.) but is typically too
complex to be accurately reproduced by a physical model. It can, however,
generally be measured accurately in the laboratory using quasi-monochromatic
stimuli

For the instruments considered in this study with moderate spectral resolutions (some 0.1 nm), the line width of the stimuli is usually negligible.

like spectral light sources (SLSs) combining different atomic emission lines, light that has passed a monochromator, or a (tunable) laser.The measured ISRF might be directly applied to the convolution of high-resolution trace gas cross sections. Often, however, the ISRF is parameterized by an appropriate function in order to (a) merge different calibration measurements, (b) describe the wavelength dependency of the ISRF by wavelength-dependent parameters, and (c) determine the ISRF directly from measurements of direct or scattered sunlight, making use of the highly structured Fraunhofer lines.

One of the simplest possible parameterizations of the ISRF is a Gaussian,
which often describes the measured line shapes fairly well by only one free
parameter,

Such tailored parameterizations have been demonstrated to reproduce the measured line shapes well. However, parameterizations using many parameters generally introduce ambiguities, leading to parameters being (anti-)correlated (in the sense that changes of different parameters can cause very similar responses of the ISRF). Thus, while these complex parameterizations can be fitted well to a measured monochromatic stimulus, a fit within wavelength calibration based on measured Fraunhofer lines is generally challenging, as ambiguities result in slow and often unstable fits.

In this study, we propose to use a modification of the Gaussian, the
so-called “super-Gaussian” (SG), for the parameterization of the ISRF. The
SG has long been in use in laser physics to describe flat-topped beam
distributions

The SG can reflect a wide variety of shapes by adding just one parameter
compared to the classical Gaussian, and thus it is still a rather simple, but
powerful, extension. Within the DOAS community, the parameterization of the
ISRF by an SG is already implemented in Pandora

In the second part of this study, we focus on ISRF changes. While the ISRF can usually be measured with high accuracy in the lab, the ISRF might change over time, in particular if the instrument is moved or if conditions like temperature change.

In particular, for GOME-2 the ISRF has changed significantly over time, as
shown in

Recently, also the impact of changing ISRF over one satellite orbit has been
addressed.

Here we formally extend this approach and deduce the spectral effects caused
by ISRF changes by a Taylor expansion. This allows for a linearization within
the spectral analysis by inclusion of spectral structures which are just
scaled within the fit procedure. Linearization leads to more stable fits, as
it excludes local side minima of the residual function, and allows for a much
faster DOAS analysis, as soon as all relevant effects such as spectral shifts
are linearized

While the presented Taylor expansion is given in general form, it is applied for the SG parameterization, which is particularly suited due to the limited number of parameters which are uncorrelated and have a descriptive meaning (width and shape).

In this study we

introduce the SG and its properties in Sect.

demonstrate how far the SG is capable of reproducing ISRFs (Sect.

give examples of applications of the linearized treatment of ISRF changes and the
benefit for wavelength calibration and trace gas retrievals in Sect.

The normalized Gaussian function,

For application of

Figure

Illustration of the “super-Gaussian” as defined in
Eq. (

The SG as defined in Eq. (

The full width at half maximum (FWHM), which is often used as measure for the
width of a distribution, is

Changes of the ISRF cause a spectral response to the measured spectra. In particular for direct or scattered sunlight, this response is highly structured due to the Fraunhofer lines. Such spectral structures impair spectral analyses like DOAS, resulting in larger fit residuals, larger statistical errors of fitted column densities, and possibly also systematic biases if not appropriately accounted for.

In this section we show that the spectral structures caused by a change of the ISRF can be linearized with respect to the parameter change and thus can be accounted for by adding correction spectra to the spectral analysis. This generally makes the fit more stable, as local side minima are excluded, and significantly faster.

Illustration of the effect of ISRF changes on spectra in intensity
and optical depth space. Left: super-Gaussian

Be

Thus, the change of

The ISRF describes the response to a monochromatic input. For a
high-resolution input spectrum

In case of a wavelength-dependent ISRF,

If RCS are included in the spectral calibration procedure (see Appendix

Within DOAS analysis, slant column densities (SCDs)

Respective PAs can be defined in order to include the effects of ISRF changes
in DOAS analysis in a linearized way: In optical depth space, the respective
change caused by

Thus, the PA is defined as

The respective factors

The above formalism allows for a unique definition of PAs by
Eqs. (

In this section, the impact of ISRF changes is derived generally for any ISRF
parameterization

In this section, we briefly describe the datasets and instruments used in this study. Further details are provided in the given references.

A solar spectrum with high accuracy and high spectral
resolution is required for the calculation of RCS and PAs (previous section)
and the wavelength calibration as described in Appendix

In order to limit computational costs (e.g., for the convolution with the
ISRF), the original data were pre-convolved with a Gaussian of

We exemplarily illustrate the ISRF parameterization
for a MAX-DOAS instrument based on an Avantes ultra-low stray-light
spectrometer (AvaSpec-ULS2048x64) using a back-thinned Hamamatsu S11071-1106
detector. The instrument is similar to that described in

The spectrometer is temperature stabilized (

GOME-2 on MetOp-A was launched in 2006 as first of
three GOME-2 instruments, providing a multi-annual time series of spectral
measurements of the light reflected by the Earth's surface and atmosphere.
The instrumental characteristics of GOME-2 are described in

GOME-2 spectral measurements are provided by EUMETSAT. In this study, we investigate the daily solar measurements from GOME-2 on MetOp-A for the years 2007–2014, and Earth's backscattered radiance for one arbitrarily chosen orbit on 1 April 2009.

OMI on AURA was launched in 2004 as part of the “A-train”

OMI is operated in push-broom mode, i.e., the across-track dimension is measured simultaneously by a CCD instead of scanned consecutively by a mirror, as for GOME-2. This implies that different viewing angles have different instrument properties, i.e., ISRFs.

For OMI, the ISRF is significantly different from a simple Gaussian, being
more flat-topped

In this study, we use the solar irradiance climatology compiled from daily OMI measurements in 2005.

TROPOMI will be launched in 2017 within the S-5P
mission

The TROPOMI ISRF has been extensively measured on ground before launch based on various SLSs. Generally it was found to be extremely flat-topped for the UV below 310 nm, Gaussian to triangular for the UVIS (310–500 nm), flat-topped for the NIR, and slightly flat-topped for the SWIR.

Here we investigate the performance of the SG parameterization for sample TROPOMI ISRFs for each spectral channel. The respective calibration measurements for UV, UVIS, and NIR are based on a slit function stimulus (SFS) constructed by a monochromator using a rotating grating, and have been provided by Antje Ludewig and Joost Smeets from KNMI (personal communication, 2016). The SWIR calibration measurements were performed with an optical parametric oscillator (OPO) and have been provided by Paul Tol from SRON (personal communication, 2016).

ISRF fit results for Avantes. Left: measured (crosses) line shape of
the Hg line at 404.66 nm and best matching parameterizations

In this section we investigate the performance of a super-Gaussian parameterization of the ISRF for different detectors and demonstrate its benefits compared to a simple Gaussian parameterization.

Figure

The simple Gaussian roughly reproduces the width of the measured Hg line
(Fig.

The flat-topped shape of the measured Hg line is much better reflected by the
super-Gaussian parameterization with a shape parameter

Fit results of ISRF parameterized as

The fitted

ISRF fit results for the GOME-2 direct sun measurement on 23 January
2007. Left: least-squares fit of the ISRF during wavelength calibration in
the 420–440 nm interval (top) and corresponding residual (bottom), with
ISRF parameterized as

For GOME-2, the ISRF is usually parameterized by a
Gaussian

Fit results of ISRF parameterized as

Figure

Symmetric and asymmetric parameterizations yield basically the same results,
and the fitted asymmetry parameters are close to zero. But still, allowing for asymmetry significantly improves the fit quality (this effect is much
larger for the UV spectral range). For the asymmetric parameterization, the
fitted widths

Fit results of ISRF parameterized as

Figure

ISRF fit results for the OMI direct sun measurement climatology for
cross-track pixel 2 (0-based). Left: least-squares fit of the ISRF during
wavelength calibration in the 420–440 nm interval (top) and corresponding
residual (bottom), with ISRF parameterized as

Obviously, a parameterization of the ISRF by a Gaussian is not appropriate for OMI and results in a highly structured residual with 5.64 ‰ RMS. With the super-Gaussian parameterization, residuals are significantly smaller (0.85 ‰ RMS).

Fit results of ISRF parameterized as

Exemplary TROPOMI ISRF for the UV

The operational OMI ISRF has been found to be asymmetric

We apply the super-Gaussian parameterization exemplarily to one set of SFS
measurements for each TROPOMI detector around the center row and column of
the detector, corresponding to the central wavelength and nadir-viewing
geometry. The ISRFs and best matching parameterizations

The TROPOMI ISRF is different for the four detectors. In the UVIS, it is similar to a Gaussian, while it is more flat-topped in the NIR and almost approaching a boxcar shape in the UV. However, the SG parameterization is capable of reproducing the measured ISRFs well with shape parameters of 7.4, 2.4, 3.0, and 2.7 for the UV, UVIS, NIR, and SWIR, respectively.

Fit results of ISRF parameterized as

The official ISRF parameterizations (Antje Ludewig and Paul Tol, personal
communication, 2016) are based on

advanced sigmoids, involving nine parameters, for the UV and NIR;

a “generalized exponential” with eight parameters for the UVIS; and

the convolution of a skew-normal and a block distribution, plus a Pearson VII distribution, for the SWIR, involving eight parameters in total.

In the second part of the paper we present applications of the
linearisation of ISRF

In Sect.

For OMI (not shown), we could not find indications for a significant change of the ISRF over time.

In Sect.

We investigate the temporal evolution of the GOME-2 ISRF width around 429 nm
by performing wavelength calibration fits for the daily solar spectra for
four different fit settings:

The ISRF is fitted as Gaussian, as in

The ISRF is fitted as super-Gaussian.

The ISRF is fixed to the results of setting 2 for the first day of the time series.

As in setting 3 but, in addition, the RCS

Temporal evolution of the GOME-2 ISRF fitted for daily solar
measurements (420–440 nm) for four fit settings, i.e., a Gaussian
parameterization
(green), a super-Gaussian parameterization (orange), a fixed ISRF matching the super-Gaussian from the first day (grey), and a fixed ISRF plus the RCS

The first evaluation reproduces the findings shown in

The results of the SG fit have already been discussed in Sect.

If wavelength calibration and ISRF fit are done in the beginning of the
time series and the ISRF is kept constant afterwards, the resulting fit
residual is almost as good as for setting 2 within 2007 but starts to
increase significantly later on. In 2010, when the change of

In setting 4, the ISRF is kept constant as well, as for setting 3, but the effect
of ISRF changes is accounted for by including the RCS

Thus, while the application of a fixed ISRF for the GOME-2 time series begins to become suboptimal after 2 years, the additional inclusion of RCS actually accounts for the spectral changes caused by the ISRF changes over time.

The case study shown above illustrates that the linearisation of ISRF changes generally works; however, a full ISRF fit might easily be performed for each daily measured sun reference instead. This is different if the ISRF changes along orbit: due to the high number of spectra, a full fit of the ISRF is not feasible any more. Thus, in the case of trace gas retrievals, the concept of linearisation by accounting for changes by a PA, which can be included in a linear fit setup, is highly beneficial.

We have investigated the benefit of the PA

Temporal change of the GOME-2 ISRF along orbit on 1 April 2009 as
derived from a linear fit including the PAs

The systematic change of ISRF width along orbit is closely related to the
temperature of the pre-disperser prism (Fig.

The respective change of the fitted shape parameter

Figure

Illustration of the generation of a synthetic spectrum in order to
investigate wavelength-dependent ISRF changes. In black, the result of

The ISRF generally depends on wavelength. In
Sect.

We construct a synthetic spectrum by
convolution of

In Fig.

Fit results for the synthetic spectrum based on an ISRF fit with the
spectral structure

The fit of a wavelength-independent super-Gaussian yields the average

If the RCS

In this section, we apply the concept of RCS
for describing the ISRF wavelength dependency for GOME-2 measurements in the
UV. We have determined the wavelength dependency of the ISRF, parameterized as

Wavelength dependency of the GOME-2 ISRF width

In a second step, we have performed the wavelength calibration over the full
fit window (325–375 nm) at once, with wavelength dependencies accounted for
by including the RCS

Figure

Thus, the wavelength dependency of the ISRF can be accounted for by including
RCS in the wavelength calibration procedure, while the actual convolution

The super-Gaussian is a powerful extension of the Gaussian which allows
to represent a variety of different shapes by adding just one free shape
parameter

Changes of the ISRF over time or wavelength can be accounted for by including spectral structures derived from the linear term of a Taylor expansion. In intensity and OD space, RCS and PAs are defined to be included in spectral calibration and DOAS analysis, respectively. The linearization makes the spectral analysis robust and fast; thus the inclusion of RCS and PA comes without notable performance loss. While this approach is possible for any ISRF parameterization, the SG is particularly suited due to the low number of parameters and the illustrative meaning of its parameters.

For GOME-2, the inclusion of PAs significantly improves the fit quality and removes a systematic component of the residual along orbit, as it appropriately accounts for the effects of ISRF broadening along orbit. The fitted change of ISRF width directly corresponds to temperature. Generally, including RCS and PAs allows for easy monitoring of the long-term stability of an instrument by straightforward fit parameters.

Fit result of the ISRF excluding (orange) or including (cyan) RCS for the solar spectrum of GOME-2 on 23 January 2007 in the UV.

Accounting for the wavelength dependency of the ISRF by the proposed
linearisation allows for considering wide fitting windows during spectral
calibration and is thus a fast and robust alternative for the “subwindow”
approach as implemented in QDOAS

The solar atlas (Kurucz et al., 1984) is available at

The super-Gaussian as defined in Eq. (

Illustration of the asymmetric super-Gaussian

ISRFs based on the

Note that by this implementation of asymmetry, the FWEM of

Exemplary TROPOMI ISRF for the UV

For an asymmetric function, the first moment (“center of mass”, COM) is
generally not 0 any more. Consequently, the application of such an asymmetric
ISRF would cause a net spectral shift in the measured spectrum. However, the
effect of a possible spectral shift is usually accounted for during spectral
calibration and should not interfere with the asymmetry of the ISRF. In order
to separate both effects, we demand that the ISRF does not cause a shift; i.e., after calculating the ASG according to Eq. (

In Fig.

The combined variation of

Fit results of ISRF parameterized as

The wavelength calibration of a spectrometer can be performed based on monochromatic stimuli with known wavelength, such as SLSs. However, as the instrument characteristics generally slightly changes during operation, an a posteriori wavelength calibration might be necessary. Within DOAS analysis, wavelength calibration is thus often done based on measured spectra of direct or scattered sunlight, making use of the highly structured Fraunhofer lines. Within this procedure, both the wavelength grid and a parameterized ISRF of the detector can be determined simultaneously.

In the following, we indicate spectral data with high resolution with the
tilde symbol. We model a high-resolution spectrum of direct or scattered
sunlight by the function

The respective spectrum on the instrument's wavelength grid

Fitted parameters are as follows:

parameters of

column densities

intensity of Raman scattered light

polynomial coefficients for

the RCS fit coefficients

shift and stretch of the wavelength grid transformation.

The resulting calibrated wavelength grid and best-matching ISRF are used to
provide the relevant cross sections, necessary for a subsequent DOAS
analysis, on the instrument's spectral resolution:

Based on the ISRF change

In Sect.

Figure

Errors induced by the linearization as investigated for synthetic
spectra with width

For small changes of the ISRF width, the linearization works well. For

The authors declare that they have no conflict of interest.

We would like to thank Andreas Richter (IUP Bremen, Germany), Michel van Roozendael (BIRA Brussels, Belgium), Alexander Cede (LuftBlick, Austria), Antje Ludewig (KNMI, de Bilt, the Netherlands), Paul Tol (SRON, Utrecht, the Netherlands), and Kang Sun (Smithsonian Center for Astrophysics, Harvard, Cambridge, MA, USA) for helpful discussions on the topic of ISRF and the parameterization of its change.

Steffen Dörner and Jan Zörner from MPIC Mainz are acknowledged for helpful discussions on Python.

We thank Mingxi Yang from Plymouth Marine Laboratory for setting up and operating the EnviMeS/Avantes MAX-DOAS instrument at Penlee Point.

R. L. Kurucz is acknowledged for providing the solar atlas. GOME-2 spectral measurements are provided by EUMETSAT, Darmstadt. OMI spectral measurements are provided by NASA. TROPOMI ISRF sample measurements are provided by Antje Ludewig and Joost F. C. Smeets (KNMI, de Bilt, the Netherlands) for UV/UVIS/NIR, and by Paul Tol (SRON, Utrecht, the Netherlands) for SWIR. This research was supported by the FP7 project QA4ECV, grant no. 607405. The article processing charges for this open-access publication were covered by the Max Planck Society. Edited by: M. Weber Reviewed by: two anonymous referees