AMTAtmospheric Measurement TechniquesAMTAtmos. Meas. Tech.1867-8548Copernicus PublicationsGöttingen, Germany10.5194/amt-9-2147-2016Quantification and parametrization of non-linearity effects by higher-order
sensitivity terms in scattered light differential optical absorption spectroscopyPuķīteJānisjanis.pukite@mpic.deWagnerThomasMax Planck Institute for Chemistry, Mainz, GermanyJānis Puķīte (janis.pukite@mpic.de)13May2016952147217715October201519January20166April201625April2016This work is licensed under a Creative Commons Attribution 3.0 Unported License. To view a copy of this license, visit http://creativecommons.org/licenses/by/3.0/This article is available from https://amt.copernicus.org/articles/9/2147/2016/amt-9-2147-2016.htmlThe full text article is available as a PDF file from https://amt.copernicus.org/articles/9/2147/2016/amt-9-2147-2016.pdf
We address the application of differential optical absorption spectroscopy
(DOAS) of scattered light observations in the presence of strong absorbers
(in particular ozone), for which the absorption optical depth is a non-linear
function of the trace gas concentration. This is the case because
Beer–Lambert law generally does not hold for scattered light measurements
due to many light paths contributing to the measurement. While in many cases
linear approximation can be made, for scenarios with strong absorptions
non-linear effects cannot always be neglected. This is especially the case
for observation geometries, for which the light contributing to the
measurement is crossing the atmosphere under spatially well-separated paths
differing strongly in length and location, like in limb geometry. In
these cases, often full retrieval algorithms are applied to address the
non-linearities, requiring iterative forward modelling of absorption spectra
involving time-consuming wavelength-by-wavelength radiative transfer
modelling. In this study, we propose to describe the non-linear effects by
additional sensitivity parameters that can be used e.g. to build up a lookup
table. Together with widely used box air mass factors (effective light paths)
describing the linear response to the increase in the trace gas amount, the
higher-order sensitivity parameters eliminate the need for repeating the
radiative transfer modelling when modifying the absorption scenario even in
the
presence of a strong absorption background. While the higher-order absorption
structures can be described as separate fit parameters in the spectral
analysis (so-called DOAS fit), in practice their quantitative evaluation
requires good measurement quality (typically better than that available from
current measurements). Therefore, we introduce an iterative retrieval
algorithm correcting for the higher-order absorption structures not yet
considered in the DOAS fit as well as the absorption dependence on
temperature and scattering processes.
Introduction
Differential optical absorption spectroscopy (DOAS)
is a well-established remote sensing method to
retrieve trace gas abundances in the atmosphere, either from direct or
scattered light observations
e.g..
Based on Beer–Lambert law, it assumes a linear relation between the trace
gas concentration (i.e. number density) and the optical depth (OD, logarithm
of the intensity; in this paper we specifically define OD as the logarithm of
the Sun normalized radiance). The assumption of linearity, however, generally
does not hold for scattered light observations since light takes various
paths before observed by the instrument . The longer
light paths will become more unlikely than the shorter ones due to the higher
absorption probability with increasing concentration. Therefore, the OD has a
complex, non-linear relation with respect to the concentration of atmospheric
absorbers, with higher non-linearity for strong absorbers (in particular
ozone). Since the scattering probability due to molecules and aerosols
changes with wavelength, this relation also varies with wavelength. Also
interferences between different absorbers and especially the impact of strong
on weak absorbers need to be accounted for to not under- or overestimate
their amount . Due to the strong ozone absorption,
retrievals in the UV and in the mid-visible (green/yellow) range can be
substantially affected. The non-linearity effect is especially strong for
observation geometries where the light contributing to the measurement is
crossing the atmosphere (in particular the regions where the trace gas of
interest is located) under spatially largely distributed paths differing
strongly in length and location, like observations of stratospheric
trace gases in limb geometry or at high solar zenith angles (SZAs). However,
besides observations in these extreme viewing geometries, also nadir observations of
tropospheric trace gases might be substantially affected by the
non-linearity, especially if the tropospheric trace gas absorption is strong
and the probability of multiple scattering is high (e.g. with increased
aerosol loads present). Usually, iterative approaches like weighting function
DOAS
or the so-called full retrieval approach, developed especially for limb
measurements e.g., requiring
extensive radiative transfer model (RTM) calculations wavelength by
wavelength in the spectral region of interest, are employed to provide an
exact treatment of non-linearity. Besides that, linear retrieval methods for
scattered light observations exist, e.g. air mass factor (AMF) modified DOAS
or the Taylor series approach
that considers the non-linearity by parametrizing it
by the fit components. In particular the Taylor series approach, which
parametrizes the non-linearity by including first-order Taylor series terms
on absorption and wavelength as additional fit components, benefits from
being more time efficient because RTM calculations, similarly as for the
standard DOAS, are necessary only for the spatial evaluation (trace gas
profile or vertical column retrieval) and can be limited to one wavelength.
However, to be not only more time efficient (by performing RTM calculations at
only one or a limited number of wavelengths) but also more independent from
a priori constraints, one might wish to quantitatively characterize the
influence on the retrieval caused by strong absorption. One possibility would
be to introduce additional functionals quantifying the non-linearity effects,
besides the already well-known conventional box AMFs. The
box AMFs quantify the linear influence of the trace gas absorption
coefficient on the logarithm of the measured intensity; they describe the
effective light paths through a given atmospheric region (also called box or
voxel) normalized by the box spatial extent. Here, we parametrize the
contribution to OD by non-linearity effects by introducing higher-order
box AMFs or higher-order effective light paths. The mth-order effective
light path of a set of given m boxes is defined as the weighted mean
product of the individual light paths through the m boxes. When normalized
by the product of the spatial extents of the considered boxes, the mth-order box AMF is obtained. The higher-order effective light paths can help to
approximate the forward model in an optimum way: first, compared to the
standard DOAS approach and even to the Taylor series approach
, non-linearities will be much better taken into account;
second, with respect to both standard DOAS and Taylor series approach, the
impact of a priori constraints is minimized; third, compared to the exact
formulation of the forward model, it is much more time efficient because no
full RTM employment for forward model calculations is
needed. We limit most of the descriptions to the second- and third-order
effective light paths due to their practical relevance since the effect of
even higher-order contribution to OD is practically negligible.
The consideration of the higher-order contributions allows us to implement a two-step algorithm (separating the spectral retrieval part from the spatial
retrieval part) even for strong absorption scenarios similarly as
in. In addition, the quantitative consideration of these
contributions allows us to implement an iterative Gauss–Newton-like retrieval
scheme iteratively correcting for the higher-order OD terms not yet
considered in the DOAS fit. The concept of higher-order effective light paths
allows us to perform the necessary forward modelling quickly (i.e. without
employment of RTM calculations). During the iterative
process the absorption dependence on temperature and scattering processes are
considered as well.
The article is structured as follows: in Sect. we introduce an
approximation for the radiative transfer equation considering higher-order OD
terms while Sect. generalizes DOAS considering these terms.
Section introduces an approach to approximate the effective
light paths of any order as function of wavelength and trace gas absorption,
which is practical to quickly calculate absorption spectra and the first-order effective light paths. Section discusses the application
of the higher-order OD terms in the retrieval algorithms. First, the
application of Taylor series DOAS is studied, and the
method is extended by considering the higher-order OD terms of different
trace gases. In addition, an iterative two-step approach is described and
investigated in sensitivity studies. Section investigates the
application of the new retrieval scheme for the retrieval of BrO vertical
concentration profiles from satellite limb measurements by SCIAMACHY
and compares the retrieved profiles with
correlative balloon measurements
. Finally,
Sect. draws some conclusions.
Characterization of absorption effects for scattered light observations
The intensity (in our study the Sun normalized spectral radiance) as observed
by a detector depends on the paths the photons from the Sun took in the
atmosphere until reaching the detector, each with a certain probability to be
observed by the detector. The intensity is basically the mean of the
probability of the light paths after entering the atmosphere per element of
the area, element of time, and element of wavelength to enter an element of
the detector's aperture solid angle. The path integral formulation of the
radiative transfer equation compare in a discrete
summation form is expressed as follows:
I=limN→∞1N∑i=1NwTi.
The weight wTi expresses the probability of such an individual light
path i to reach the element of the detector's aperture solid angle. The
weight depends on the path's geometry and the optical properties along it
(e.g. a path with many scattering events or with high trace gas absorption
along it will have low weight). For the sake of simplicity, the limit
expression and the summation boundaries along i will be skipped in the
following.
One can modify the weight wTi, describing it as a product of the weight
without absorption wi and the absorption probability along the path i
(this is possible because Beer–Lambert law holds along one single path). The
equation becomes
I=1N∑iwie-∑jlij(∑kβkj),
where lij describes the path length of the ith trajectory through the
box j (if the box is not crossed by a path, the length for the path there
is 0). The summation is performed over all boxes belonging to the
atmosphere. To correspond to reality, each box has to have an infinitesimally
small 3-D volume. βkj is the absorption coefficient of absorber k
in box j, i.e.
βkj=ckjσkj,
where ckj and σkj are number density and absorption
cross section of the absorber respectively.
Since the Beer–Lambert law does not hold for a multiple light path ensemble,
only its approximation can be applied for scattered light observations.
Standard DOAS applications use its approximation up to the first order.
However, for strong absorption scenarios the first-order approximation is
usually not applicable, which motivates us to investigate the use of higher-order terms.
Expanding the OD τ (i.e. the logarithm of the intensity of
Eq. ) in a Taylor series similarly as it is
done by with respect to the absorption coefficients
βkj up to the third order (at zero absorption background), one
obtains
Here, τ0, referred to in the following as the scattering term, is the
contribution to the OD that can be explained by scattering processes only
because the weights wi do not contain any absorption. Because of its
broadband variation with wavelength, it can be approximated by a polynomial
in DOAS. It equals the logarithm of the intensity I0 which would be
obtained for the atmosphere without any absorbers present (note that I0 in
this notation is not the solar irradiance). The rest, i.e. the difference
between the observed OD τ and the scattering term, is the OD
caused by absorption. τ1, τ2, and τ3 approximate the absorption
OD of the first-, second-, and third-order contributions to the observed
absorption OD by k absorbers present in the atmosphere. We refer to them in
the following as first-, second-, and third-order OD terms respectively. The
first-order OD terms can be also called linear OD terms while the second- and
third-order (and even larger-order) OD terms are generally referred to as
higher-order OD terms in the following. The terms of a respective order are
always a sum of the OD contributions of different trace gases. The absorbers
k, K, and k‾ index all trace gases belonging to the same
atmospheric scenario, and also j, J, and J‾ index all boxes
belonging to the same scenario spanning the whole atmosphere; therefore the
summation order, performed over all trace gases and all boxes in the
atmosphere, is arbitrary. While the linear OD term is a sum of the first-order or linear kth absorber OD contributions, the higher-order OD terms
contain contributions arising from the same trace gas (if k=K for the
second
order, or k=K=k‾ for the third-order OD terms, we call these
contributions self-correlative OD terms of absorber k) and from the
combination of different trace gases (if k≠K for the second order, or any
k, K, and k‾ is not equal to another one, we call these
contributions cross-correlative OD terms e.g. of absorbers k and K).
Lj is the effective light path (of the first order) through box j
describing the sensitivity of the measurement to a certain atmospheric
volume, in this case for an atmosphere without absorbers:
Lj=∑iwilij∑iwi.
A sketch of light paths (with corresponding weights) crossing two
boxes (layers) in limb geometry for the illustration of the variables in
Eqs. () and (). w1…w4 indicate weights of
various paths photons may take travelling from the Sun and reaching the
detector. j and J represent two arbitrarily selected boxes (layers);
l1…4,j…J are the lengths of individual paths through the
boxes. The red arrows indicate that the path is obtained by summation of the
two segments through the box. In the case the path is not crossing any box,
its length is 0, as indicated.
Normalized by the box dimensions (usually by the height) a dimensionless
functional is obtained: the well-known box AMF e.g.,
Aj=1hjLj.
However, the higher-order OD terms of Eq. () introduce the new
sensitivity functionals: L2jJ and L3jJJ‾. The second-order effective light paths are
L2jJ=∑iwilijliJ∑iwi.
Figure illustrates the terms used in Eqs. ()
and () assuming a finite-sized 1-D layered atmosphere in limb
geometry (as it is used in the studies later in the paper).
L2jJ (Eq. ) are weighted products of the individual light
paths through two boxes characterizing how dependent the sensitivity is
between two different places in the atmosphere; i.e. they account accounts
for all those light paths which cross both selected boxes. If, however, both or
any of the boxes are not crossed, the individual contribution of the light
path is 0. In the case of complete correlation, which occurs if only one light path exists (like for direct light observations), the second-order effective light paths are equal to the product of the corresponding
first-order effective light paths:
L2jJ=LjLJ.
In this case the contributions to the observed intensity by the second-order
terms like in Eq. () (and even higher-order terms) become 0 and
the DOAS equation becomes linear. In Sect. the properties of
second-order contribution are investigated in more detail.
The third-order effective light paths are
L3jJJ‾=∑iwilijliJliJ‾∑iwi.
They are weighted products of the individual light paths through three boxes
and quantify the dependence of the sensitivity between three locations in
atmosphere; i.e. they account for all light paths which cross the given three
boxes.
In general, any order effective light paths can be defined, if the Taylor
series expansion of Eq. () is extended to higher
orders. The mth-order term for any M boxes is
LM1…M=∑iwi∏m=1Mlim∑iwi.
Note that similar to the effective light paths Lj, the higher-order
effective light paths are equal for any trace gas considered in a certain
atmospheric scenario.
In analogy with Eq. (), also higher-order box AMFs can be defined
when normalizing effective light paths by the extension (height) of the
involved boxes:
AM1…M=1∏m=1MhmLM1…M.
Generalization of DOAS for scattered light observations
In the next subsections the validity of different approximations (linear or
higher-order terms) in DOAS is assessed.
Linear approximation in the standard DOAS
In order to get a DOAS-like equation as in for
scattered light observations, one basically needs to consider the Taylor
series expansion of the logarithm of the intensity in Eq. () up to
the first order as a function of wavelength within a certain wavelength
interval (fit window):
τ(λ)=-ln(I(λ))=τ0(λ)+τ1(λ).
In the following we skip explicitly mentioning the dependence of wavelength
for better readability.
The wavelength dependence of the scattering term τ0 is usually
approximated by a low-order polynomial, i.e.
τ0=∑p=0Pλpap,
where λ is wavelength, P indicates the polynomial order, and ap
is the coefficient of the different polynomial orders obtained by the fit.
τ1 in Eq. () approximates the absorption OD. Considering
wavelength-dependent absorption cross sections but neglecting their spatial
variation (assuming them being constant with temperature and pressure), the
linear OD term τ1 can be rewritten:
τ1=∑kσkSk,
with the slant column density (SCD) Sk in the DOAS literature referred to
as the concentration of absorber k integrated along the effective light
path.
Sk=∑jckjLj
Since the Taylor series expansion is performed at the absorber concentration
of 0, the SCD in the formula represents the effective light path for an
atmosphere without absorption. Here it is important to note that for strong
absorption scenarios the retrieved SCDs as well as ODs are
systematically underestimated by this linear model derived at a zero
absorption background, because longer light paths contributing to the
measurement become less probable. This leads to the fact that a fit by a
linear model will distribute the higher-order absorption structures (with a
mostly negative contribution by the second-order terms, see also the
discussion later in Sect. ) to the existing fit parameters
(polynomial and cross-section terms). This effect can be addressed in a
simple way by calculating effective light paths, including a background
absorption scenario like it is done in,
when inverting SCDs into spatial concentration distributions. This is,
however,
only useful for minor absorbers because the relationship between the OD and
their concentration is still almost linear. Applied to the spatial
interpretation only, this simple approach, however, still does not consider
spectral effects, i.e. systematic biases in the fitted SCDs due to ignoring
higher-order absorption structures.
Taylor series approach
In order to correct for both the systematic underestimation of the SCDs and
to minimize the systematic biases due to higher-order absorption structures,
the Taylor series approach in suggested considering the
first-order dependency of the SCD in Eq. () on
absorption and scattering as
Sk=Sk,0+Sk,λλ+Sk,σσk.
Thus the OD terms for absorber k are given as
τ1,k=Sk,0σk+Sk,λλσk+Sk,σσk2,
whereSk,λ, describing the first-order variation with wavelength, and Sk,σ,
providing the first-order effect on trace gas absorption, are directly
obtained by the DOAS fit in addition to Sk,0, parametrizing the constant part of the SCD. In this way the linear dependence between the
logarithm of the intensity and the absorption parameters is kept and the
retrieval problem can be solved by the usual least squares technique.
, however, have not yet provided a quantitative
interpretation of these terms in the framework of radiative transfer theory
(see next subsection).
This modification for a strong absorber (ozone) in the UV largely decreased
the fit residual structures. However, at the same time a systematic bias for
weak absorbers still remained (see Figs. 12 and 13 therein) because the
method does not account for cross-correlative terms (see also Sect. 5.1 where
we evaluate the method in sensitivity studies).
Besides the significant improvements, the method relays on a priori
considerations when performing radiative transfer simulations for the spatial
evaluation. Effective light paths are evaluated at a background a priori
scenario at an empirically selected wavelength (where the discrepancy with
the true profile as determined in sensitivity studies is minimum).
Another possibility to address the problem of scattered light observations
with a linear method is the so-called AMF modified DOAS
, where the product of a
wavelength-dependent AMF and cross section is used instead of a simple
cross section. This method, however, relies on the approximate knowledge or
assumption of atmospheric properties, e.g. trace gas and aerosol profiles,
for AMF calculation which needs to be performed at all wavelengths used for
the spectral analysis.
Higher-order DOAS
Instead of limiting the DOAS fit to the linear approximation (standard DOAS)
or a qualitative consideration of the variation of the SCD due to scattering
and second-order self-correlative absorption effects
Sect. ,, all higher-order terms of
Eq. () can generally be considered in a quantitative manner.
Like for the approximation of the wavelength dependence of τ0 in the
linear case (Eq. ), the wavelength dependence of the effective
light paths can also be approximated by polynomial functions in Eq. (),
in particular because the weights wi in Eqs. (), (),
and () are broadband functions of wavelength since they consider
only scattering:
τ=∑p=0P0λpap︷τ0+∑k∑p=0PkλpσkSpk︷τ1-12∑k∑K∑p=0PkKλpσkσKSpkK︷τ2+16∑k∑K∑k‾∑p=0PkKk‾λpσkσKσk‾SpkKk‾︷τ3.
Here λp, their products with the cross sections, as well as their
product with the products of the cross sections are fit components describing
the wavelength dependence of the OD due to scattering. The retrieved
quantities from the fit are the polynomial coefficients ap, SCD
coefficients Spk that parametrize the wavelength dependence due to the
scattering, as well as the higher-order SCD coefficients SpkK and
SpkKk‾.
Considering only the τ0 and τ1 terms with Pk=0 the equation
corresponding to standard DOAS is obtained (compare
Eqs. –). Considering also terms with Pk=1 (at least
for strong absorbers) and including also τ2 terms with k=K (for
strong absorbers) and PkK=0 the approximation by the Taylor series
approach (compare Eq. ) is obtained.
The retrieved SCD coefficients are quantitatively associated with the trace
gas concentration by
∑jLjckj=∑p=0PkλpSpk.
The equation allows to investigate the relationship between the SCD and the
spatial distribution of an absorber at any wavelength. Additionally, it is
worth noting that the SCDs correspond to an atmosphere without absorption,
so theoretically it is possible to calculate the effective light paths
Lj for such an atmosphere. This eliminates the need for a priori profile
assumptions in the calculation of effective light paths and avoids possible
subsequent iterative steps. For the second- and the third-order SCD
coefficients the relations can be written in a similar manner:
∑j∑JL2jJ-LjLJckjcKJ=∑p=0PkKλpSpkK
and
∑j∑J∑J‾L3jJJ‾-3L2jJLJ‾+2LjLJLJ‾ckjcKJck‾J‾=∑p=0PkKk‾λpSpkKk‾.
From these two equations it can be basically concluded that additional
information for the spatial evaluation can be derived by fitting higher-order
terms, because different-order effective light paths have different spatial
distributions, as discussed later. Note, however, that in this paper the
content of this additional information is not explicitly investigated because
(a) a non-linear inversion algorithm would be needed and (b) the high
correlations between the derived SCDs of different orders, especially when
they correspond to the same trace gas. For accurate fit results, in general a
very good measurement quality (e.g. signal-to-noise ratio) would be needed
and, depending on the trace gases of interest and the measurement quality, fit
components with a significant contribution to the fit results should be
considered.
Terms for second-order approximation
In order to account for the effects caused by strong absorption scenarios,
let us consider the higher-order Taylor series terms of the expansion in
Eq. (). The considerations are generally valid for combinations
between any absorbers, both strong or weak. However, the impact of
combinations between weak absorbers only is expected to be negligible.
The second-order Taylor series term for any two absorbers X and Y can be
split in three terms: two for the respective absorbers (important for strong
absorbers only) and one for an interdependence between the absorbers (can be
important also for combinations between a strong and a weak absorber).
τ2=τXX+τYY+τXY=12∑j∑JL2jJ-LjLJβXjβXJ+βYjβYJ+2βXjβYJ
(a) Contribution (absolute values) to the OD by different
Taylor series terms in the UV spectral region with three absorbers (ozone,
NO2, and BrO) for a simulated scenario characteristic for SCIAMACHY limb
observations at high northern latitudes in March (TH = 19.83 km). All OD
terms up to the second-order and the largest third-order terms are shown.
(b) Simulated (true) absorption OD in comparison with the OD
calculated from the effective light paths considering absorption up to a
certain order. (c, d) Same as (a) and (b) but in
the visible spectral range with two absorbers (ozone and NO2).
Self-correlative term
Assuming that the cross section is invariant in space (i.e. neglecting its
variation due to temperature and pressure), the single trace gas term for any
of the trace gases is
τXX=-σX22∑j∑JcXjcXJ(L2jJ-LjLJ).
Including the squared cross section in the DOAS fit ,
the second-order contribution of the absorber is accounted for.
Cross-correlative term
The second-order Taylor series expansion term for the interdependence between
two absorbers X and Y is
τXY=-σXσY∑j∑JcXjcYJ(L2jJ-LjLJ).
Note the similar structure with Eq. (). Since the term consists of
the product of the absorption cross sections of two absorbers, it will be
important if absorption structures of two absorbers overlap. However, it is
left open how much this term contributes to the SCDs of different trace
gases; i.e. it describes the contribution to both trace gases in a similar
way. It seems clear that the error on the SCD fitted by the linear DOAS model
will be largest for weak absorbers since the ratio between the OD associated
with the cross-correlative term and the OD attributed to the trace gas will
be larger for a weak absorber:
τXYτX+τXX>τXYτY+τYY if τX+τXX<τY+τYY.
As an example, the Taylor series terms of different orders for a typical
scenario with ozone (strong absorber, vertical column ∼ 460 DU),
NO2, and BrO (weak absorbers) characteristic for subarctic latitudes in
March see Table 4.1 infor more details are
calculated by applying the effective light paths of different orders. The
wavelength-dependent effective light paths for the atmosphere without
absorption are derived from trajectory ensembles generated by McArtim
(see Appendix
for calculation details) at distinct wavelengths and approximated (see
Sect. for details) by a broadband function for wavelengths in
between. The obtained Taylor series terms for UV and visible (VIS) spectral ranges
are plotted in Fig. a and c, for a limb viewing geometry,
tangent height (TH) of 19.83 km, and solar zenith and azimuth angles of 75 and
60∘ respectively. First of all, non-negligible values of the ozone
self-correlative term and the inter-correlative terms between ozone and the
minor absorbers with respect to their first-order terms can be seen. For
ozone, also the third-order self-correlative term shows rather high values. The
importance of these and other higher-order terms should be evaluated also
taking into account the detection limit of a particular detector. The
consideration of the higher-order effects for measurements in limb geometry
is most important at wavelength regions with strong ozone absorption bands
for an instrument able to detect ODs below 10-2 to 10-3.
(a) Conventional (first-order) effective light paths at
545 nm calculated for the same scenario as in Fig. .
(b) Product of the first-order effective light paths at two
different altitudes. (c) Second-order effective light paths as
function of two different altitudes. (d) Difference between the
second-order effective light paths and the products of the first-order effective
light paths.
The lines in Fig. b and d, show the sum of the absorption OD
terms up to a certain order, while the dots indicate the true total absorption
OD (logarithmic ratio between intensities simulated by the RTM without and
with absorption). The overestimation of the absorption is clearly seen if
only the first-order (linear) terms are considered. Second- and third-order
terms improve the characterization converging towards the truth.
Second-order effective light paths
The second-order effective light paths (Eq. ), as well as higher-order
terms, similarly to the classical first-order effective light paths
(Eq. ), depend on the specific composition of light paths for the
observation geometry and are the same for any trace gas; i.e. the effective
light paths only quantify the effect to the observed OD an absorber will have
if it is “added” to the atmosphere.
Let us analyse the difference between the second-order effective light paths
and the products of the corresponding (first-order) effective light paths (as
in Eqs. , , , or ) in more detail.
This difference can in principle be used as a measure for the complexity of
the observation geometry. Indeed, this difference in mathematical terms is
the covariance between light paths through the two boxes. For the same box
(if j=J), the difference (i.e. the variance) will always be ≥0
because the mean of the squared light paths is larger than the squared mean
of the light paths if their lengths and/or weights are not equal. For
different boxes (j≠J) the difference can be both positive or negative
but not smaller than the negative product of the effective light paths. A
negative difference will appear when the two boxes are more likely to be
crossed by different than by similar light paths (e.g. different parallel
light paths from the Sun towards the line of sight of the instrument). In
practice, the difference will be larger if boxes j and J are located
close to each other (with maximum if j=J) and will decrease and even become
negative with increasing distance between the boxes j and J. Negative
values are more characteristic for 2-D or 3-D simulations of the atmospheric
radiative transfer where it is more likely that different light paths cross a
different set of boxes; for the 1-D atmosphere considered here mostly
positive values occur.
For illustration, first-order effective light paths and their product between
different boxes are compared with second-order effective light paths in
Fig. . The same scenario as in Fig. is used.
Similar to first-order effective light paths, as well as for the second-order
effective light paths, the highest sensitivity is found around the tangent
point. Interestingly there are also the largest differences between the second-order effective light paths and the products of the first-order effective light
paths are observed. This finding is caused by the increased scattering
probability around the tangent point, which causes photons to change the
direction. This also means that higher-order contributions to the OD increase
for absorbers located around the tangent point. The different sensitivity
distributions of the different-order terms could in principle allow us to obtain
additional information about the spatial distribution of the absorber.
However,
this possibility is not further exploited in this study.
Third-order contribution
Continuing analysing the Taylor series expansion of
Eq. () one can realize that the sensitivity at a
certain region can depend not just on the sensitivity at another region but
also on a third (Eq. ), fourth, etc. region (there can be also
higher-order self-correlation terms for the same region as well). A
quantitative description of the third-order terms as being products of
cross sections is provided in Appendix ; also these third-order products of cross sections can be included in the DOAS fit as fit
parameters.
From Fig. a and c one can see that, for the chosen
scenario, even the third-order self-correlative ozone term might be important
enough
to be considered in the fit. It can be seen however that the third-order
terms generally show a smaller contribution than the second-order terms.
Since the measurement geometry along with the radiative transfer introduces
strong limits for the possible trajectories, it is likely that higher-order
contributions decrease rapidly with increasing order. Besides the Taylor
series nature that diminishes the importance of even higher terms according
to 1/n!, it is less likely that the light path that had crossed some
atmospheric box/boxes will cross another one in a completely different way,
because the probability of scattering events per trajectory decreases
rapidly. The cases where many scattering events are still probable (e.g.
highly polluted regions with high aerosol load or fog) would be worth
investigating in a separate study.
Limitation of the classical absorption optical depth definition
In this section we discuss the applicability of fundamental concepts of DOAS
for scattered light observations. We show that these fundamental concepts are
not valid in a strict sense.
While the basic consequence of these findings is a possible bias of the
results of the retrieval algorithms, in practice these aspects are
specifically relevant only for scenarios with strong non-linearities due to
absorption. For most of the current DOAS applications these considerations
are not relevant since they are performed in the (almost) linear range.
The fundamental assumption by the standard DOAS is that it is possible to
distinguish between absorption features of different trace gases; i.e. the
total atmospheric absorption τT (being the logarithm of the ratio of
the intensity without and with absorption) can be written as the sum of the
ODs of the individual absorbers:
τT=lnI0I=∑kτk.
In order to improve the fit results sometimes differential cross sections are
applied to better differentiate between narrow and broadband spectral
features, but it is always assumed that each trace gas has its own
“fingerprint” completely originating from the contribution of this
absorber.
Looking at the higher-order terms discussed before, it becomes clear that
this assumption (Eq. ) does not hold for scattered light
observations; i.e. there are “fingerprints” that cannot be attributed to
one trace gas alone but are products of several absorbers.
It can in particular be shown that the definitions for optical quantities
(OD, SCD, and AMF) used in DOAS are affected. In the following subsections we
discuss two commonly used definitions: the classical definition and the light
path integral definition. They are introduced in the following subsections.
Classical definition
The classical definition is used by e.g. and
defines OD as the logarithmic ratio of intensities with and without an
absorber of interest.
The OD τc according to this definition (c stays for
“classical”) for absorber X in terms of Eq. () is
expressed as
τcX=lnII-X=ln∑iwie-∑jlij(∑kβkj)∑iwie-∑jlij(∑kβkj-βXj),
where I-X means intensity without absorber X and βXj is the
absorption coefficient of the selected absorber. One might wish to
investigate the contribution of each absorber to τT in
Eq. () separately. However, looking at Eq. ()
one can immediately realize that for two absorbers X and Y:
τcX+Y≠τcX+τcY,
since
ln∑iwie-∑jlij(∑kβkj-βXj-βYj)≠ln∑iwie-∑jlij(∑kβkj-βXj)+ln∑iwie-∑jlij(∑kβkj-βYj).
This means that the OD of two absorbers (or any number of absorbers) is not equal to the
sum of the individual ODs of these absorbers, since the cross-correlation
between different absorbers is not considered. Consequently also the
definitions of the AMF ,
AcX=τXVXσX,
and SCDs,
ScX=AcXVX=τXσX,
are fundamentally inaccurate (VX is the vertical column density (VCD) of
absorber X). Hence not only standard DOAS but also “extended”
applications like the AMF modified DOAS are
affected.
Difference between the sum of the ODs of the considered absorbers,
each calculated by the classical formula, and the OD calculated for the whole
absorption (red crosses), in comparison with the sum of the inter-correlative
terms (blue line: sum of second-order terms only; green line: sum of second- and
third-order terms). Top panel: the result in the UV for the limb observation
scenario with TH = 19.83 km and absorption scenario containing ozone,
NO2 and BrO. Bottom panel: same but in visible range and for two
absorbers, ozone and NO2.
Figure , top panel, shows the difference between the sum of the
ODs of the considered absorbers in the UV, each calculated by the classical
formula, and the OD calculated for the whole absorption for the limb
observation scenario with TH = 19.83 km and absorptions of ozone,
NO2,
and BrO at distinct wavelengths. The rest OD is compared with the sum of the
inter-correlative terms estimated by Eqs. (), (), and
() which is approximated by the approach in
Sect. for all wavelengths within the spectral region
similarly as for Fig. . Not accounting for the
inter-correlation between the different absorbers represents the approach of
simply adding the individual ODs obtained by the classical definition. While
perfect agreement is found between the rest OD and the sum of the
inter-correlative terms of up to the third order, already the sum of the second-order
terms gives good agreement for most wavelengths. While the difference
is shown for wavelengths where the full radiative transfer simulations are
performed, the wavelength-dependent effective light paths are approximated
for all wavelengths in the spectral interval similarly as for Fig. 1 (see
Sect. 4.1 for details). The bottom panel shows the corresponding results for
an absorption scenario containing ozone and NO2 in VIS. Here
almost perfect agreement is found already by the second-order absorption
contribution alone.
The result in Fig. is confirmed when the OD according to the
classical definition is expressed including higher-order terms. The exact
intensities with and without absorbers in Eq. () are replaced
with the derivation according to Eq. (). Considering terms up to
the second order, the OD is
τcX=∑jLjβXj-12∑j∑J(L2jJ-LjLJ)βXjβXJ-∑j∑J(L2jJ-LjLJ)βXj∑k,k≠XβkJ,
or, assuming spatially constant cross sections,
τcX=σX∑jLjcXj-12σX2∑j∑J(L2jJ-LjLJ)cXjcXJ-σX∑k,k≠Xσk∑j∑J(L2jJ-LjLJ)cXjckJ.
The classical OD thus consists of the absorption terms attributed to the
absorber X (the first two terms on the right side of the equation) and the
cross-correlative terms between X and all remaining absorbers. Summing the
ODs for the individual absorbers, the cross-correlative terms are considered
twice as much as in the OD calculation for the whole absorption together
while the self-correlative terms are considered correctly.
Applying Eq. (), SCD is expressed
ScX=∑jLjcXj-12σX∑j∑J(L2jJ-LjLJ)cXjcXJ-∑k,k≠Xσk∑j∑J(L2jJ-LjLJ)cXjckJ.
Light path integral (LPI) definition
Alternatively, according to the LPI definition, the SCD
is defined as concentration integrated along the effective light path
e.g.. In the discrete notation it can be written as
SlX=∑jLTjcXj,
with LT the effective light path obtained for the given absorption
scenario:
LTj=∑iwie-∑JliJ(∑kβkJ)lij∑iwie-∑JliJ(∑kβkJ).
The corresponding definition of the OD would be the integration of the
absorption coefficient along the effective light path:
τlX=∑jLTjcXjσXj=∑jLTjβXj.
Please note that τlX≠τcX (as defined in
Eq. ), indicating that even for a scenario with only one absorber
present in the atmosphere (where the classical definition in Eq.
is valid), LPI definition (Eq. ) is not accurate.
Also for the LPI definition the optical quantities can be expressed in terms
of the Taylor series expansion. Expanding effective light paths in
Eq. () in Taylor series on absorption (for more details see
Sect. ) up to the second order, the SCD in
Eq. () becomes
SlX=∑jLjcXj-∑kσk∑j∑J(L2jJ-LjLJ)cXjckJ.
Please notice that the summation over k here also contains the
self-correlative term, i.e. X∈k. In a similar way the OD can be
approximated:
τlX=σX∑jLjcXj-∑kσk∑j∑J(L2jJ-LjLJ)cXjckJ.
While the application of the LPI definition has been limited for retrievals
of weak absorbers , Eq. () allows us to
quantify the disagreement for strong absorbers which arises due to
overestimating the self-correlative term by 2 times in comparison to the
classical definition (Eq. , see its approximation in
Eqs. and ). The disagreement in principle can
now be addressed in a quantitative way. The cross-correlative terms in the
LPI definition are considered in a similar way as in the classical formula.
Taylor series definition
In summary, both the classical and LPI definitions of the OD are not
applicable to describe the total OD as the sum of the ODs of individual
absorbers. In order to obtain an agreement between the sum of the
approximations of the ODs τk of individual absorbers and the Taylor
series approximation of the total OD in the form of Eq. (),
one can use arbitrary factors aX,k instead to assign part of a
cross-correlative OD to absorber X and factors ak,X to assign another
part of it to absorbers k, so that
aX,k+ak,X=1,
for any k≠X and with aX,k=X=0.5. The OD definition for absorber
X can then be written as
τX=σX∑jLjcXj-σX∑kaX,kσk∑j∑J(L2jJ-LjLJ)cXjckJ.
Therefore, the approximation of the OD in Eq. () can be
used to define more appropriate quantities for SCD and AMF in
Eqs. () and () for applications e.g. in forward
modelling for profile retrievals (e.g. satellite limb or DOAS observations)
like in AMF modified DOAS. The selection of factors aX,k is arbitrary
and could depend on empirical considerations like the correlation properties
between different fit parameters; e.g. for an absorber with more pronounced
differential structures one could assign a larger factor, to account for the
fact that cross-correlative structures can become assigned to the strong
absorber in the fit. In other words, the (self-correlative) absorption
structure would largely correlate with the cross-correlative structure.
Variability of effective light paths on absorption and on wavelength
The effective light paths defined earlier (Eqs. , ,
, ) are calculated at a single wavelength and at no
absorption while the DOAS fit is performed over a selected wavelength range,
over which the effective light paths change due to both variability of the
trace gas absorption and the atmospheric scattering properties.
Therefore it can be worth it to parametrize the different effective light paths
to account for such variations in the retrieval process without the need to
perform RTM simulations for every wavelength within the fit window if
atmospheric trace gas composition changes.
Effective light paths for limb geometry (TH = 18 km, 1 km
layer from 18 to 19 km) modelled by RTM at five distinct wavelengths and
approximated afterwards by a fit with a Rayleigh cross section, its square
plus an offset.
Wavelength dependence due to scattering
If the Taylor series expansion of Eq. () is made and
effective light paths are calculated at zero absorption, the wavelength
dependence of the weights wi is caused only by changes of the scattering
probability. So basically, light paths and their weighs wi as in
Eq. () are different at different wavelengths. Since
the variation is broadband, one can approximate it by a broadband feature.
For example, one can simulate effective light paths (box AMFs) of different
orders at selected wavelengths by RTM and interpolate them or fit the
variation by a low-order polynomial or some other characteristic function.
For illustration, box AMFs as function on wavelength are shown in
Fig. , where RTM simulations are performed at five different
wavelengths and approximated afterwards.
Dependence on absorption
By performing the Taylor series expansion at a zero absorption background,
effective light paths at an arbitrary absorption background can be expressed
as a function of higher-order effective light paths (simulated for the zero
background scenario) and the trace gas distributions. Assuming spatially
constant cross sections, the first-order effective light path (as in
Eq. ) for box J considering terms up to the third order can be
written as
LBJ=LJ-∑kσk∑jckjL2jJ-LjLJ+∑k∑KσkσK∑j∑J‾ckjcKJ‾12L3jJJ‾-LJ12L2jJ‾-LjLJ‾-LjL2J‾J.
Here, trace gas concentrations ck and cK, with k and K belonging to
the same set, are arbitrary background distributions of trace gases.
Such a parametrization is practical, since it can be used for any wavelength
chosen for the procedure described in Sect. and for any trace
gas cross section as well as for any trace gas profile. This allows us to adjust
the effective light paths with small calculation effort, which can be useful
for iterative Gauss–Newton-like retrievals, for which a linearization of the
forward model is used and is updated after each iteration. An iterative
algorithm can be implemented without the need to redo (much slower) RTM
simulations. With small modification (see next subsection) the spatial
variability of cross sections can also be accounted for.
Figure a shows an example for first-order effective light
paths calculated for a certain absorption background from the effective light
paths obtained for an atmosphere without absorption at 545 nm. Good
agreement with the effective light paths explicitly simulated for the given
absorption is obtained. Already if only the terms up to the second order are
considered, the discrepancies are hardly visible by eye; the largest
discrepancies are located around the tangent point where most non-linearities
come from (compare Fig. d). The absolute contributions of the
different-order terms are shown in panel b. Panels c and d show the relative
differences between the approximated and explicitly simulated effective
light paths for the absorption background at different wavelengths in the
mid-visible spectral range (520–570 nm). The largest discrepancies occur
for altitudes below the TH, which usually only have a small effect on the
measured signal. The largest contribution to the measurement is from and
above the TH. At TH they exceed 2–3 % only at 570 nm where the
absorption OD is extremely strong (around 0.9, see Fig. b);
above the TH the discrepancy strongly decreases and converges towards 0 a
few kilometres above the TH.
(a) Effective light paths at 545 nm explicitly simulated
by RTM for an absorption background (true) in comparison with first-order
effective light paths approximated for this background from the effective
light paths simulated for an atmosphere without absorption. Calculations
considering effective light paths up to either second or third order are
performed. (b) Contributions from terms of different orders to the
effective light paths in (a). (c ,d) Relative differences
between the approximated effective light paths considering up to second- or third-order terms respectively and the explicitly simulated effective light paths
at the absorption background for different wavelengths in the mid-visible
spectral ranges.
Spatial variability of cross sections
If the true cross-section σTkj at a certain box j differs from
the originally considered cross-section σk, the impact on the
absorption will be the ratio of both Rkj=σTkj/σk. In
this way one can apply modifications by scaling the concentration of absorber
k by the cross-section ratio in the respective formulas. For example, the SCD in
Eq. () can be rewritten in order to account for the spatial
variability of the cross section and/or the cross-section variation on
wavelength:
Sk=∑jRkjckjLj.
DOAS evaluation of a simulated limb spectrum at TH = 19.83 km.
The simulated spectrum is shown on top left, the fit residual on top right.
The remaining plots show fit results for the individual OD terms listed in
Table , third column.
In a similar way, i.e. by considering the cross-section ratio, one can also
include modifications in all other formulas containing products of
cross section and concentration.
Application in retrieval algorithmsImplications for Taylor series DOAS
In comparison to standard DOAS, Taylor series modified DOAS
suggested adding higher-order terms for strong
absorbers (e.g. ozone) directly in the fit. Also terms to account for the
wavelength dependence can be fitted. The originally not considered
cross-correlation between strong and weak absorbers can be considered by
adding the cross-section product of both in the fit to improve it. Although
easy to implement, several reasons complicate the use of these
cross-correlative terms for a non-iterative retrieval algorithm. They
include (1) considerations about the optical quantity definitions (they are
now clarified in Sect. ); (2) the spectral structures of
these terms showing spectral features similar to the cross sections of the
absorbers involved – thus, they provide the potential for high correlations
between the different terms included in the DOAS fit; (3) these terms are a
product of contributions of different absorbers and their interpretation
(e.g. for the retrieval of spatial distributions) requires a non-linear
algorithm.
The OD for two absorbers considering terms up to the second order can be
written as
τX+Y=σX∑jcXjLj1+σY∑jcYjLj2+σX212∑j∑JcXjcXJ(L2jJ-LjLJ)3+σY212∑j∑JcYjcYJ(L2jJ-LjLJ)4+σXσY∑j∑JcXjcYJ(L2jJ-LjLJ)5.
Standard DOAS accounts only for the terms 1 and 2. Taylor series DOAS as
discussed in also added term 3 (X≡ O3).
(Terms considering the wavelength dependence caused by scattering are also
added but are not shown here.) Term 4 can be neglected for weak absorbers,
while term 5 shows a rather strong impact if it is ignored see e.g.
Fig. 12, bottom panel, in Appendix C in. However, adding it,
assuming it as fully contributing to the OD of the weak absorber, leads to
errors by overestimating the impact of the cross-correlative terms
(Sect. ). The effect is not negligible with respect to the
OD of the weak absorber (see Eq. ). If, however, the correlative
term is added to the fit but not assumed to contribute to the OD of the weak
absorber, the fit coefficient of term 2 of the weak absorber is basically
an approximation for the SCD, which would be measured if the strong absorber
X were not present in the atmosphere. So one could in principle
retrieve profile information from the retrieved SCD by performing forward
modelling for an atmosphere with no absorption. In practice, however, this
first-order term has a rather high fit error because of the correlation with
the inter-correlative term and consequently the profile retrieval will give
rather poor results (the retrieval error is high).
Fit parameters included in the DOAS evaluation of synthetic spectra
in the spectral range 520–570 nm.
Optical depth associated withApproximationTerms fittedOrthogon-alizationCalculation from simulated termsScattering, broadband wavelength (λ) functionFirst-order polynomial1, λyeslog(I0(λ))OzoneUp to second-order Taylor series terms w.r.t. ozone absorption and λσO3, σO3σO3, λσO3yesσO3∑jcO3jLj(λ)+σO322∑j∑JcXjcXJ(L2jJ(λ)-LjLJ(λ))NO2First-order Taylor series termw.r.t. NO2 absorptionσNO2–σNO2∑jcNO2jLj(λ)Ozone and NO2Second-order cross-correlative term between ozone and NO2 absorptionσO3σNO2–σO3σNO2∑j∑JcXjcXJ(L2jJ(λ)-LjLJ(λ))
(a) Results for the same evaluation as in
Fig. , but wherein the fitted ODs are summed according to their
attribution to certain trace gases or scattering in Table in
comparison with a calculation from the simulated terms at zero absorption
(formulae in the right column in the table). (b) The same
as (a) but the retrieval is performed with random Gaussian noise
added to the simulated spectra. The noise leads to an increase of the fit
residual by 10 times when compared to the fit in Fig. . The
results are shown for 10 different random noise spectra.
As an example, a DOAS evaluation for a simulated limb spectrum at
TH = 19.83 km by applying the Taylor series approach is shown in
Fig. for the wavelength range 518–570 nm, where the ozone
absorption is especially strong. The fit window in the green spectral region
was selected for demonstration purposes primarily because of the strong O3
absorption. However, the NO2 absorption is in general relatively well
structured in this spectral range and has been investigated before (see e.g.
). For limb measurements this wavelength could be
beneficial due to the decreased Rayleigh scattering probability (in
comparison to the blue spectral region where the NO2 retrieval usually is
performed), thus causing an increased sensitivity to the lower stratosphere.
The fit parameters are listed in Table . To improve the
condition of the forward model matrix, the polynomial parameters are
orthogonalized. The same is done for the ozone parameters; they are all
labelled as O3 in the figure. Ten million light path trajectories are used
to simulate the spectrum, reducing the noise below 2×10-4 and allowing
a good distinction also for the second-order cross-correlative term between
ozone and NO2. Figure a compares the fitted ODs summed
according to their attribution to certain trace gases or scattering in
Table with the corresponding values calculated according to
the formulas in the right column of the table. Good agreement is obtained. It
can be seen that the OD of the second-order cross-correlative term between
ozone and NO2 is approximately 10 % of the OD of the first-order
NO2 term – a potential error source if neglected. Additionally, the
effect of spectral noise on the retrieval is investigated by adding random
Gaussian noise to the simulated spectrum. The noise level was selected in a
way that the fit residual increased ∼ 10 times (i.e. to 2×10-3) in comparison to the residual for the original simulation.
Figure b, however, reveals even higher variation (up to 5×10-3) in the fitted NO2 and cross-correlative terms. The
variation can be explained by the large covariance between both terms as
shown in Fig. where the statistical properties (mean and
standard deviation) of the fitted NO2 ODs are plotted in comparison with
the calculated OD from the effective light paths. For the statistical
analysis, the spectrum with 1000 different noise spectra added is
evaluated. It can be seen that (1) the random noise does not introduce
obvious systematic error structures in the fit result and (2) the standard
deviations of the two fit terms are very similar in magnitude, but the
standard deviation of their sum is reduced almost 3 times. In practice,
therefore, it might be necessary to utilize both terms together for further
retrieval steps (e.g. spatial evaluation).
Mean and standard deviation of the fitted OD of the first-order
NO2 term (top), second-order cross-correlative term between ozone and
NO2 (middle), and the sum of both terms (bottom). For a statistical
evaluation, the analysis is performed for the simulated limb spectrum at
TH = 19.83 km with 1000 different random Gaussian noise spectra
added (similar as for Fig. ). Additionally, the respective
terms as calculated from the simulated effective light paths are included for
comparison.
To illustrate the consequences if the cross-correlative term between ozone
NO2 is not considered, Fig. compares the resulting
NO2 OD when the cross-correlative term is either included or excluded from
the fit with the simulated OD according to the classical formula
(Eq. ). Also the respective second-order approximation for the
classical definition calculated from the effective light paths, i.e. the sum
of the first-order NO2 and the cross-correlative term between ozone and
NO2 (compare to Eq. ), are plotted. Again a good agreement
between the simulated classical OD and the approximated OD can be seen
(around 1 %). It is also found that the agreement with the evaluation,
where both the first-order NO2 and the cross-correlative terms are
included, is varying only slowly with wavelength and is less than 2 %. In
addition to the higher-order effects, the variation might also be caused by
broadband (i.e. scattering) features, the RTM behaviour (e.g. Monte Carlo
(random) noise in the simulated spectrum), as well as numerical effects in
the fitting algorithm. If however the first-order NO2 term is left alone in
the fit, the relative difference has a similar shape as the ozone
cross section with the amplitude 1 order of magnitude larger reaching
15 %.
Suggested algorithms
Due to the high cross-correlation between the different-order terms, the
retrieval quality, among others, depends on the way this correlation is
accounted for. There are several possibilities, including the following.
Results of the Taylor series DOAS evaluation either with or without
the cross-correlative ozone and NO2 term included in comparison with the
NO2 OD simulated according to the classical definition (Eq. )
and calculated from the simulated effective light paths (up to the second
order). Top: absolute values of the simulated, calculated, and retrieved OD
for NO2; bottom: relative difference with respect to the classical
definition.
A quasi-full retrieval approach: the absorption OD is calculated for
each considered spectral point according to the higher-order Taylor series
expansion, and the forward model is linearized at a certain a priori
absorption scenario first. Then an iterative least squares fitting technique
is applied, improving the knowledge about the trace gas distributions and
adjusting the forward model according to the improved knowledge using the
effective light paths of different orders. This method must be the most
accurate but also the most time consuming as the calculations are performed
for every spectral point and the inversion is performed for all spectral
points and measurement geometries together. However, it will be still faster than
the classical full retrieval approach like e.g.
because no new full RTM calculations are necessary for each iterative step.
A quasi-AMF modified DOAS: other than the method by
where full RTM simulations are performed for every wavelength, here AMFs are
calculated utilizing pre-calculated effective light paths of different
orders. Also the limitation of the classical AMF definition for a multiple
absorber scenario can be considered, e.g. by applying the OD definition
according to Eq. () when calculating AMFs e.g. with
Eq. (). AMFs for the retrieved trace gases are calculated for an
a priori profile and are applied in a DOAS fit for each measured spectrum
separately. Then the trace gas profile is retrieved and the procedure is
repeated, adjusting the AMFs e.g. with respect to the spatial variation of
the cross sections. This method is also quite accurate since it considers
effects between different absorbers in the approximation of the AMFs. Similar
to the previous method, calculations (in this case of the AMF) for every
spectral point are required, but the spatial retrieval is done separately
from the spectral analysis (which is used to retrieve the SCDs, the product
of the fitted VCD and AMF), so the inversion requires less computational
resources.
A two-step approach: in the first step, a DOAS fit is performed
including all relevant higher-order terms, especially the
inter-correlative terms. In the second step (spatial retrieval) a retrieval
procedure is performed at a selected wavelength (e.g. a wavelength in the
centre of the spectral region or where the differential absorption structures
are largest or the absorption effects smallest; with a compromise between
all these aspects). Further, there are two possibilities: (a) to use only the
first-order terms which describe the absorption as there would be no
influence of absorption by the strong absorber (the absorption is sorted out
by the cross-correlative term) or (b) to perform a non-linear spatial retrieval
including the cross-correlative terms of the absorber of interest and the
strong absorber. The possible correlation between the different-order terms
of the same absorber, however, needs to be approached iteratively and at the
end a balance between (a) and (b) should be found. This method would be
faster than the first two, because the effective light path calculations are
limited to one or few spectral points. At the same time, the algorithm would
be less accurate due to the correlations between the first-order and second-order terms in the DOAS fit, or, in other words, it requires better
measurement quality.
An iterative two-step approach
While the implementation of the algorithms mentioned in
Sect. is not a subject of this study, for the following
sections a fourth retrieval approach is suggested and implemented. Here the
effective light paths up to the second order are applied to retrieve vertical
profiles. The principle is similar to the two-step approach in
. First, a DOAS fit is performed to the measured
spectra; second, the spatial information is obtained by inverting the fit
results to vertical profiles. The substantial change here is the
implementation of an iterative process where the retrieved profiles of the
relevant trace gases are used to recalculate the absorption spectra. In the
subsequent iterations, the measured spectrum is corrected by the calculated
spectra, the DOAS fit is performed for the difference, and the retrieved
profile is adjusted by eventually arising corrections (a Gauss–Newton
approach is implemented).
Spectral evaluation
First, a DOAS fit is performed for the simulated spectra similarly as it is
done by the non-iterative algorithm described in .
Afterwards, the obtained SCDs Sf parameters (f means fitted) are used to
calculate the OD contribution for absorber X:
τfX=∑iSfXiσiX,
where σiX are the fit components of trace gas X. For weak
absorbers they are just the corresponding cross sections of the trace gases
(i=1). For ozone, the additional Taylor series terms are also considered as
in . The calculation of the OD by Eq. () is
performed at the selected wavelength at which the spatial evaluation is later
performed. A wavelength of 342 nm (located between strong ozone absorption
bands while close to the most pronounced BrO absorption band) is used for the
UV fit window of 337–357 nm. For the visible fit window (519–570 nm), the
selected wavelength is 545 nm. The trace gases X considered in the
calculation and in the profile retrieval (see next subsection) are BrO,
NO2 and O3, and NO2 and O3 for the respective fit windows.
Settings of the iterative two-step algorithm for simulation studies.
Retrieval in the UVRetrieval in the visibleFit window (nm)338–357 (or extended to 332–357)519–570Spectral fit parametersPolynomial (second-order); cross sections for trace gases (O3, NO2, and BrO as included in spectra simulation); Taylor series terms for ozone (product of cross section and wavelength and squared cross section)Polynomial (first-order); cross-sections for trace gases (O3, NO2, as included in spectra simulation); Taylor series terms for ozone (product of cross section and wavelength and squared cross section)Tangent height range considered (km)12–3612–36RTM first- and second-order effective light paths simulated at wavelengths (nm)332.0, 335.6, 338.0, 344.2, 357.0514.3, 519.9, 545.2, 570.0, 576.4Approximation of the sensitivity variation on wavelengthLeast squares fit with Rayleigh cross sections , with their square and offset as fit parametersSame as for the UV fit windowForward modelling, initial retrieval iteration stepCalculation of ODs of individual absorbers at spatial retrieval wavelength from a priori profiles according to the second-order approximation of the classical definition (Eq. ); first-order effective light paths for a priori scenario are obtained from first- and second-order effective light paths simulated for an atmosphere without absorptionSame as in the UVForward modelling, following iterationsSame as above but for trace gas profiles at current iteration; also absorption spectra are calculatedSame as in the UVWavelength at which spatial retrieval is performed (nm)342545 nmRetrieved trace gasesSame as included it the spectral fitSame as included it the spectral fitRetrieval altitude range (km)10–3810–38Altitude resolution (km)11A priori informationA priori profiles as plotted in Fig. (dashed lines), constant a priori uncertainty at all altitudes (for ozone: 10×1012; NO2: 1×109; BrO: 2×107 molecules cm-3); smoothing constraint: correlation length of 3 km introduced in the a priori covariance matrixSame as in the UV (for ozone and NO2)Spatial evaluation
Effective light paths of the first- and second-order pre-calculated for zero
absorption are used to calculate first-order effective light paths for the
profiles retrieved in the previous iteration (for the initial iteration for
the a priori data) at the selected wavelength according to Eq. ().
In order to allow the use for the selected wavelength, they are approximated
by a fit with broadband wavelength function (see Sect. ).
The profile is retrieved according to the Gauss–Newton optimal estimation
scheme :
cXi+1=cXi+(EXa-1+KXiTEXϵ-1KXi)-1KXiTEXϵ-1ΔτX-EXa-1(cXi-cXa),
starting with the a priori information as a first guess, i.e. cXi=cXa (suffixes a, i, and i+1 indicate quantities corresponding to
a priori, existing retrieved knowledge, and updated knowledge during the
iteration respectively). ΔτX is the difference between
τfX and τXa for the retrieval of the first guess.
τXa, which is applied only for the initial inversion, is the
absorption OD calculated according to the approximation of the classical
definition (Eq. ) for the a priori scenario at the selected
wavelength. The elements of the weighting function matrix KXi
are the products of the first-order effective light paths and the
cross sections. Weighting functions and ODs are used instead of effective
light paths and SCDs in order to enable the possibility to account for the
temperature dependence of the cross sections (important when the retrieval is
applied on real measurements). EXe is the error covariance
matrix with the DOAS fit variances as its diagonal elements; the off-diagonal
elements are 0. EXa is the a priori covariance matrix. The
applied retrieval settings for the sensitivity studies are summarized in
Table .
Iteration process
In further iterations, absorption spectra based on the first- and second-order
absorption terms according to Eq. () are calculated from the
profiles retrieved in the previous iteration for all wavelengths within the
fit window considering the broadband approximation and possible temperature
dependencies of the cross sections. Then the DOAS evaluation is performed
again for each measured spectrum corrected by subtracting the calculated
absorption spectrum (referred to as corrected spectrum in the following). The
spatial retrieval is performed again, and the difference Δτ in
Eq. () is now the newly obtained τfX. Since the retrieval
is converging fast, a fixed small number of iterations (2–3) is selected.
Sensitivity studies of the retrieved profiles
We apply the iterative two-step algorithm to absorption spectra simulated at
different THs to investigate its ability to reproduce the true profiles used
for the simulated spectra and compare it with the linear algorithm where
(1) no iteration for spectral correction is applied and (2) a linear forward
relationship between the a priori and OD is assumed. The simulation is
performed for the same scenario (limb observation in the subarctic atmosphere
in spring) as for previous studies; i.e. while the spatial retrieval is
performed similarly as the initial step of the iterative algorithm applying
Eq. (), τXa (as introduced in Sect. )
is calculated according to the light path integral definition in
Eq. ().
Residual and fitted ODs of the absorbers obtained by the iterative
two-step approach in the UV. The spectra are simulated for the same scenario
(limb observation in subarctic atmosphere in spring) as for previous figures.
Left panel: initial DOAS fit for a spectrum at TH = 19.83 km; right
panel: fit for the corrected spectrum (original spectrum corrected by
subtracting the spectrum calculated from profiles retrieved in the last
iteration).
Comparison of the iterative two-step approach with the linear two-step approach
Figures and illustrate the
effectiveness of the iterative two-step approach for considering spectral
features of the involved absorbers in both the UV and VIS spectral range.
In both cases the ODs of the minor absorbers (as derived from the fit to the
corrected spectra) are reduced to 0 (for the given simulation precision)
when the iterative approach is applied. It means that the forward model
considers all spectral features correlated with the fitted terms of these
trace gases. Here especially the wavelength dependence and cross-correlation
between different absorbers is accounted for. Some remaining absorption can
still be seen for ozone comparable with the magnitude of the contribution of
its third-order terms (compare with Fig. ). Also an almost
complete removal of systematic residual structures is observed for the UV fit
window; the rest is most likely attributed to higher-order effects or
simulation errors. For the VIS spectral range, where the residual is
dominated by simulation noise (although the same number of photon
trajectories is simulated by RTM as in the UV, less scattering events are
obtained that reduces the statistics), small changes can be observed
especially at longer wavelengths with higher ozone absorption.
Same as Fig. but for the visible fit window.
Odd plots (from left for ozone, NO2, and BrO): vertical
concentration profiles retrieved by different approaches (blue: linear
approach with effective light paths simulated for the true profiles; green:
same but with effective light paths calculated for the a priori profiles from
the first- and second-order absorption terms simulated at zero absorption
background; red: profile retrieved at the initial step of the iterative
algorithm; magenta solid line: final profile obtained by the iterative
approach; magenta dotted line: final profile but without consideration of the
cross-correlative terms between different absorbers) in comparison with the
true profile (black dotted line: original profile; black solid line: original
profile smoothed by averaging kernels). The a priori is shown as a dashed line.
Even plots: relative difference to the true profiles smoothed by the
averaging kernels. Top: results for the UV; bottom: results for the visible.
The obtained vertical concentration profiles are plotted together with the
true profiles in Fig. for ozone, NO2 and BrO (odd plots
from left to right respectively). For the linear retrieval, two results are
shown: one, for which the effective light paths were simulated for the true
scenario (blue lines), and another, for which the effective light paths are
calculated for the a priori profiles from the first- and second-order
absorption terms simulated at zero absorption background (green lines). For
the iterative two-step retrieval, results for the initial retrieval step (red
lines) and the final results after iterative process (solid magenta) are
shown. For this retrieval, additional results are shown for which the
cross-correlative terms are skipped in the calculation of the correction
spectra (dashed magenta line), as discussed in detail in
Sect. . The top panel shows results for the retrieval in
the UV and the bottom for the retrieval in the visible. To minimize possible
effects due to spatial under-sampling and a priori constraints, true
profiles smoothed by the averaging kernels according
to are also included in the figure. Even plots show the relative
difference between the retrieved profiles and the smoothed true profile. It
should be noted that the differences to the true (non-smoothed) profiles
could in principle be caused by a contribution of the a priori profiles to
the retrieved profiles (if the effect of the a priori on the retrieval is
large). However, here the measurement response (i.e. the sum of the rows of
the averaging kernels) is close to unity above the altitude of
∼ 12–16 km (depending on involved trace gases), indicating that this
effect is small. Hence the differences mentioned above are mainly caused by
the smoothing alone. The ozone profile retrieved in the UV (see two plots on
the top, left) shows an overestimation of less than 4 % for all retrieval
methods, indicating that there is only a small non-linearity effect on the
retrieval of the ozone profile itself. This finding can be explained by the
fact that (a) the spatial retrieval is performed at a wavelength outside the
strong ozone absorption band, (b) the non-linearities are largely accounted
for by the fitted Taylor series ozone terms, and (c) by a good representation
of OD at the selected wavelength by the fitted ozone OD. Interestingly, the
iterative retrievals (even without the necessity for iterations) show good
agreement (better than 1 %) for altitudes between 17 and 33 km, while
the linear retrievals show discrepancies of up to 2 % for practically all
altitudes. For altitudes below 15 km, however, the iterative retrieval shows
larger discrepancies of ∼ 3 %. The application of effective light
paths simulated for the true profiles or approximated for the a priori
profiles does not give significantly different results: the results agree
within ∼ 1 % range with respect to each other.
In the VIS spectral range, much larger ozone absorption occurs, increasing
gradually with wavelength. Thus it is not possible to select a wavelength
with low ozone absorption for the spatial evaluation. Therefore an
overestimation of the retrieved profiles (see the two plots on the left in
the bottom panel) by the linear retrieval of up to 16 % is observed
because the effective light path integral formalism for the OD used in the
retrieval set-up overestimates the higher-order contributions to the
absorption OD (see Sect. ). At the same time, in the
initial step of the iterative approach the classical OD definition
(Eq. ) is used. Thus, higher-order self-correlative terms are
correctly approximated in τXa in Eq. () (while the
contribution of the cross-correlative terms with weak absorbers can be
ignored). Good agreement (within 1 %) is achieved above the ozone peak
between 20 and 28 km altitude. Below and above this altitude range the
concentration is under- and overestimated respectively by up to 4 %.
The iteration process leads to an improvement also for these altitudes,
reaching an overall agreement within 1 %.
From left: ozone, NO2, and BrO: relative differences between the
trace gas profiles retrieved at different wavelengths by different approaches
with respect to the true profile smoothed by averaging kernels. Rows from top
to bottom: retrieval in the UV by the linear approach (with effective light
paths calculated for the a priori), retrieval in the UV by the iterative
approach, retrieval in the visible by the linear approach (with effective
light paths calculated for the a priori), and retrieval in the visible by the
iterative approach.
For weak absorbers like NO2, the higher-order self-correlative terms are
negligible, explaining that almost no difference between the linear retrieval
approach and the initial step of the iterative approach (see upper third and
fourth plots) is found (because both OD definitions consider cross-correlative
terms similarly). The difference with respect to the truth of up to 25 %
for altitudes just below the ozone peak in the UV could be caused by the
incorrect accounting for the cross-correlative effect with ozone (either
caused by the lower ozone a priori profile in comparison with the truth or
the incorrect OD definition; see discussion in Sect. ).
Since, however, the retrieval utilizing the effective light paths simulated
for the true profiles agrees with the retrieval utilizing the effective light
paths approximated for the a priori profiles, the discrepancy is not caused
by the difference between the true and a priori ozone profile. The most
likely reason for the disagreement is the broadband variation of the NO2
OD with wavelength (due to scattering), since other than for ozone, no
variation of the NO2 OD due to dependence of scattering with wavelength is
assumed in the DOAS fit. In addition, in contrast to BrO, the NO2 absorption
features are distributed rather equally along the fit window, while the
selected wavelength for the spatial retrieval is more at the beginning of the
fit window. The profile obtained after the iteration process, however, shows
good agreement (better than ∼ 1 %) with the truth for altitudes
above 16 km. Deviations below this altitude for the retrieved NO2 profiles
are caused by the a priori profile according to our investigation. The
results for VIS (bottom, middle plots) reveal disagreement with the
truth for the linear retrievals when the effective light paths are calculated
for the a priori profiles. While above the ozone profile peak an
overestimation of up to 6 % is found, below the ozone profile peak an
underestimation of up to 8 % occurs. Much better agreement with the truth
is obtained for the linear retrieval with effective light paths calculated
for the true profiles and the iterative retrieval (better than 2–3 % for
altitudes above 20 km). Below this altitude, an overestimation of up to
10 % for the linear and 8 % for the iterative retrieval is obtained,
which does not depend on the a priori and is potentially caused by the
cross-correlation of the fit parameters or even higher-order Taylor series
terms not considered in the retrieval. Due to the almost perfect agreement
between the iterative retrieval and the linear retrieval (when the effective
light paths are calculated for the true profiles) the assumption of an
influence of possibly incorrect cross-correlative terms in the OD definition
can be excluded. Similarly to in the UV, the NO2 profiles below 14 km are
affected by the a priori.
The BrO retrieval in the UV (right two plots on the top) shows similar
findings as the NO2 retrieval in this spectral range: an agreement between
the linear retrievals and the initial step of the iterative retrieval is
found, but with a smaller discrepancy to the truth. This finding is explained
by the selection of the wavelength for the spatial evaluation close to the
strongest BrO absorption band: the BrO profiles are slightly underestimated
by up to 4 % for altitudes up to 30 km. The iterative retrieval improves
the agreement to 2–3 % for altitudes below 25 km, while at higher
altitudes the agreement is similar to the linear approaches.
Same as upper panel of Fig. but for an extended
UV fit window (332–357 nm) including an additional BrO absorption band at
shorter wavelengths.
Effect of cross-correlative terms between different trace gases
As stressed in Sect. and demonstrated by the application of
the Taylor series approach (Sect. ) the consideration of
cross-correlative higher-order terms is especially important for the
retrieval of the minor absorbers. We investigate their importance in the
application of the iterative algorithm by performing an iterative retrieval
without considering the cross-correlative terms in the calculation of the
correction spectra. The resulting profiles (see magenta lines in
Fig. ) confirm the aforementioned arguments and results:
substantial discrepancies with respect to the true profiles are observed for
both minor absorbers (BrO and NO2) while they are negligible for ozone.
The discrepancies are largest for the NO2 results in the VIS spectral
range (more than 20 % around 20 km altitude) due to the larger ozone
absorption, while in the UV the difference to the truth does not exceed
5 %.
Same as Fig. but for a standard DOAS fit, i.e.
without considering the higher-order ozone absorption terms.
Dependence on the wavelength selection for the spatial retrieval
We investigate the stability of the iterative retrieval if the spatial
retrieval is performed at different wavelengths. Figure
displays the relative difference with respect to the true profiles when the
spatial evaluation step of the retrieval is performed at different
wavelengths within the fit window. While for the linear retrieval the
discrepancies strongly depend on the selected wavelength, the iterative
retrieval shows similar results for all wavelengths. For the linear
retrieval, the discrepancies for the ozone profile are larger for wavelengths
at the strongest ozone absorption bands in the UV as well as smaller at
shorter wavelengths in the visible where smaller ozone absorptions occur. For
NO2, the agreement is better at longer wavelengths in the UV but even then
some discrepancies are left. In the visible, the profile is overestimated at
higher altitudes (above 20–25 km depending on wavelength) increasing with
wavelength, while it is underestimated at lower altitudes. For BrO, however,
good agreement for the linear retrieval is obtained for the lower wavelength
part of fit window (where the strongest BrO absorption peak occurs) but close
to the ozone absorption bands. In short, the application of the linear
approach requires detailed sensitivity studies to find the best retrieval
wavelength and the agreement may even depend on the involved fit parameters.
At the same time, the iterative approach shows similar performance at all
wavelengths for all involved trace gases.
Same as Fig. but for the visible fit window.
Same as Fig. but for the application of standard
DOAS. The final profile but without consideration of the cross-correlative
terms between different absorbers (magenta dotted line in the original
figure) is not shown here.
Sensitivity to the fit window selection
Figure compares the retrieved profiles with the true
profiles for the UV fit window extended towards shorter wavelengths
(332–357 nm) including in addition two stronger ozone bands and one BrO
absorption band. Compared to the findings for the standard fit window
(Fig. , upper panel) similar but larger discrepancies with
respect to the true profiles are obtained both for the linear retrieval and
the iterative retrieval. For ozone they reach 5–8 % below 17 km
altitude for all the retrieval approaches, while for BrO the underestimation
increases for the linear retrievals at all altitudes being now 3–6 %
below 30 km. For the iterative retrieval, the retrieved BrO profile varies
more around the true profile being slightly more underestimated at low
altitudes (by 4 % at 15 km) and overestimated by 2 % at around
21 km. For NO2, the profiles retrieved by the linear approach show a
shift towards lower altitudes, while for the iterative retrieval approach an
underestimation around 20 km altitude is observed (which is not present for
the standard fit window). The agreement, however, improves (differences are
less than 0.5–1 %) for all trace gases compared to the agreement
obtained for the standard fit window if the third-order absorption effects of
ozone are included in the DOAS fit (not shown here due to almost one to one
matching with Fig. , upper panel). We also found out that
to obtain this good agreement it is only necessary to include the third-order
absorption effects in the DOAS fit, but it is not necessary to consider the
third-order terms in the calculation of the effective light paths for the
spatial evaluation and the calculation for the correction spectra in the
iterative process.
Effect of including the Taylor series terms in the fit
For strongly non-linear problems, and if the a priori is too far away from
the true profiles, the results during the iterative retrieval might converge
towards profiles that differ from the truth. Applying the standard DOAS fit
(i.e. skipping higher-order terms), much larger discrepancies are found even
after the iterative process (Figs.
and for the UV and VIS respectively) when compared
with the retrieval when the higher-order terms of ozone are included
(Figs. and ). The residual values are of
the magnitude of the third-order ozone absorption OD (compare with
Fig. ), indicating the impact from the higher-order spectral
features: these features (1) are not considered by the standard DOAS fit so
well as by ozone Taylor series terms and (2) consequently the retrieved
profiles become systematically biased, which affects the calculated correction
spectra. The retrieved profiles from different approaches are shown in
Fig. . While the significantly disagreeing results of the
linear retrievals are corrected to a large extent by the iterative approach,
the uncertainties in the UV are by a factor of 2–4 larger compared with the
retrieval including the additional ozone terms. The iterative retrieval in
the VIS spectral range, which is characterized by much stronger absolute
but smaller differential ozone absorption, also results in larger
uncertainties (by a factor 2–4) above the ozone profile peak, while the
ozone concentration is disagreeing by 7 % and the NO2 concentration by
more that 30 % at 20 km altitude.
Settings of the iterative two-step algorithm for the application to
SCIAMACHY limb observations in the UV.
Fit window (nm)338–357Spectral calibrationWavelength calibration for a selected Sun spectrum by WinDOASSpectral fit parametersPolynomial (second-order); absorption cross sections: O3 at 193, 213, and 233 K (I0 corrected), NO2 at 223 K, and BrO at 223 K; Taylor series terms for O3 (product of cross section at 213 K and wavelength and squared cross section) and NO2 (product of cross section and wavelength), functions to account for polarization (eta, zeta), offset (1/I0, λ/I0), Ring, shiftTangent height range considered (DOAS)Spectra interpolated on 9–48 km grid with 3 km resolution; measurements across track are averagedTangent height range (spatial evaluation)Fit results at 9–36 (BrO, NO2), 12–39 (O3) km are further evaluatedReference spectraSun spectrum (A0 as in SCIAMACHY level 1c files) to account for possible systematic offset; fitted SCDs are corrected by fit result at high TH (BrO: by average SCD from 39 and 42 km, NO2: SCD at 39 km, O3: 42 km)RTM first- and second-order effective light paths simulated at wavelengths (nm)332.0, 335.6, 338.0, 344.2, 357.0, 363.4, 380.0, 391.1, 399.9, 419.8, 435.1, 450.2, 519.9, 545.0, 570.2Approximation of the effective light path variation on wavelengthFit with Rayleigh cross sections ; their square and offset as fit parametersForward modelling, initial retrieval iteration stepCalculation of ODs of individual absorbers at spatial retrieval wavelength from a priori profiles according to the second-order approximation of the classical definition (Eq. ); first-order effective light paths for a priori scenario are obtained from first- and second-order effective light paths simulated for atmosphere without absorption. Cross-section dependence on temperature is accounted for by using temperature-dependent cross sections ( (O3), (NO2), and (BrO) are used)Forward modelling, following iterationsSame as above but the calculation is performed for trace gas profiles obtained in the previous iteration; also correction spectra are calculatedWavelength at which the spatial retrieval is performed (nm)342Retrieved trace gases, altitude range (km)10–38 (BrO, NO2), 10–41 (ozone)Altitude grid resolution (km)1Regularization settingsOptimal estimation scheme with constant a priori uncertainty at all altitudes (for ozone: 10×1012; NO2: 1×109; BrO: 2×107 molecules cm-3); smoothing constraint: correlation length of 3 km introduced in the a priori covariance matrixApplication to SCIAMACHY measurements
In the following we apply the two-step iterative algorithm introduced in
Sect. 5.3 for the retrieval of trace gas profiles from SCIAMACHY limb
measurements. We investigate the response to different retrieval settings
(like the cross-section dependence on temperature or the consideration of
the cross-correlative terms). We also compare the retrieved profiles with
correlated balloon measurements provided by
and . Note that the real
measurements and retrieval parameters are subject to substantial errors
including a relatively small signal-to-noise ratio and other instrumental
problems, as well as uncertainties in the cross sections, temperature
profiles, spectral calibration, and consideration of the Ring effect. For the
comparison of the profiles from different instruments, the selection of
different cross sections, fit windows, trajectory modelling, and photochemical
correction for balloon measurements are also additional error sources
.
Instrument description
The SCIAMACHY instrument was flown on
the ENVISAT satellite operated in a near-polar Sun synchronous orbit with an
inclination from the equatorial plane of ∼ 98.5∘ and probed the
atmosphere at the day side of Earth in alternating sequences of nadir and
limb measurements from August 2002 to April 2012. Limb scans in one scanning
sequence were performed with approximately 3.3 km elevation steps at the tangent point
in flight direction. The cross-track swath was 960 km at the tangent point and
consisted of up to 4 pixels for the UV/VIS spectral range. The field of view
was 0.045∘ in elevation and 1.8∘ in azimuth. This
corresponds to approximately 2.5 km in vertical direction and 110 km in
horizontal direction at the tangent point, respectively. Measurements were performed in
the UV/VIS/NIR spectral range from 240 to 2380 nm with a spectral resolution
of approximately 0.25 to 0.55 nm in the UV/VIS range. More instrumental
details can be found in .
Comparison of the current retrieval algorithm implementation for BrO
with the algorithm in .
Current retrievalRetrieval in AlgorithmIterative two-step approachTwo-step approachSpectral range338–357 nm Tangent height range∼ 9–40 km∼-3–32 kmTangent height griddingRadiance linearly interpolated on regular 3 km gridOriginal measurement grid is usedReference spectraSun reference spectra in DOAS analysis; fitted SCDs are offset by fit result at high TH afterwards to account for possible systematic offset (for BrO, average SCD obtained at 39 and 42 km is used)Measurement at TH of ∼ 35 km is used as reference spectrumInformation for scan across trackRadiance is averaged across track before spectral analysisFitted SCDs are averagedRadiative transfer modelMcArtim3 + tool for higher-order effective light path calculation from light path trajectory ensemblesMcArtim1Calculated sensitivity parameters by RTMFirst- and second-order effective light paths at selected wavelengths (see Table )First-order box AMFs at one wavelength (344.2 nm)Forward modelling includesCalculation of absorption spectra and first-order effective light paths for trace gas profiles at current iteration from first- and second-order effective light paths obtained for an atmosphere without absorptionConsideration of first-order effective light paths obtained at a priori scenarioConsidered wavelengths in forward modellingAbsorption spectra are calculated for all wavelengths within the fit window from first- and second-order effective light paths (the wavelength dependence of effective light paths is approximated from the calculated effective light paths by RTM at the selected wavelengths by a fit to scale them by a constant + Rayleigh cross section); spatial evaluation is performed at 342 nm (outside ozone absorption band)No spectral modelling; spatial evaluation is performed at 344.2 nm (at ozone absorption band but showing good agreement for BrO in sensitivity studies)Considered atmospheric species in forward modellingBrO, O3, NO2 (in all retrieval steps and iterations)BrO, O3, NO2 (considered in DOAS fit), their a priori distribution is considered in spatial forward modelling (box AMF calculation), vertical inversion is performed for BrO onlyTemperature dependence of cross sectionsFully accounted for all considered species in forward modelling and spatial inversion. For the spatial inversion, fitted ODs of absorber are inverted (instead of SCDs), effective light paths are accordingly multiplied by cross section interpolated at temperature at given altitude; in spectroscopic analysis temperature and profile weighted tangent height-dependent cross sections are usedIn spectroscopic retrieval part O3 cross sections at 223 and 243 K are used, NO2 and BrO cross sections at 223 KPolynomialSecond orderFourth orderSpectral corrections (polarization sensitivity)Logarithm of all eta, zeta functions from SCIAMACHY key data are orthogonalized and two functions with the largest eigenvalues are included in the analysisOne eta and one zeta function from SCIAMACHY key data are included according to Spectral corrections (shift)Spectral shift is fitted according to approach in to allow shift between analysed spectra and the reference (spectra is adjusted by the fitted shift and iterated to account for non-linearity in shift)Non-linear method is used as in ; see e.g.
Continued.
Current retrievalRetrieval in Spectral corrections (various)Ring, offset (1/I0, λ/I0), I0 correction for O3 cross sections Accounting for non-linearityeffects in absorptionIn spectral analysis Taylor series approach is used for ozone to account for absorption variation on wavelength (by a broadband and an ozone absorption terms) to improve spectral fit quality and by the iterative algorithm non-linearity in absorption is considered up to second-orderIn spectral analysis Taylor series approach is used for ozone to account for absorption variation on wavelength (by a broadband and an ozone absorption terms); in the spatial inversion, global linearity in BrO absorption is considered at the a priori stateModel (retrieval) grid1 km layers1 km layers (for studies with simulated spectra); grid aligned with measurement (∼ 3.3 km) for retrieval from SCIAMACHY measurementsRetrieval settings
The retrieval of vertical profiles from SCIAMACHY measurements is performed
in a similar way as for the studies with simulated spectra above. The
retrieval scheme is based on the linear two-step retrieval algorithm
developed in our group but
with the addition of the iterative scheme. In order to account for real
measurement conditions and to address instrumental effects, additional fit
parameters for the spectral analysis are introduced in addition to those in
the sensitivity studies: Ring effect, polarization, intensity offset, shift,
and temperature dependence of cross sections are accounted for. The algorithm
parameters as applied for the SCIAMACHY measurements are summarized in
Table . Besides the iterative retrieval scheme, the
parameters considered are similar as in , with some
modifications as indicated in Table . The essential
modifications in the spectral fit include
use of the more recent temperature-dependent ozone cross sections from
; we include cross sections at 193, 213, and 233 K
in the spectral fit;
the measured spectra are gridded to a TH grid of 3 km;
spectra at the same TH (across track) are averaged;
a Sun spectrum (A0 spectrum in the SCIAMACHY data set) is used as a
reference spectrum to improve the signal to noise ratio in the fit – in order
to correct for possible offsets in the retrieved quantities caused by
unexplained spectral structures in the Sun spectrum, the absorber ODs are corrected (by subtraction) by the ODs retrieved at high
TH;
the spectral shift is linearized and the shift between the reference and
the analysed spectra is fitted as additional fit parameter according to the
method in (during the iterative process, the wavelength
grid is adjusted to account for non-linearities in the shift);
to account for the spectrally varying polarization sensitivity of the
SCIAMACHY instrument eta and zeta spectra are orthogonalized and the spectra
corresponding to the largest eigenvalues are considered in the fit.
The settings for the spatial evaluation and the calculation of the correction
spectra are similar to for the sensitivity studies, but in addition
cross sections at all available temperatures are considered for the
calculation of correction spectra and effective light paths.
The most important modifications in the spatial retrieval step include the
following (see Table for more details):
The vertical profiles are now obtained on a 1 km retrieval grid.
Instead of the SCD, the OD is used for the
inversion allowing the consideration of the temperature dependence of the
cross sections.
Weighting functions (i.e. the product of the effective light paths and the
cross sections) are calculated at 342 nm (outside the strong ozone
absorption band) while in box AMFs at 344.2 nm (at an
ozone absorption band but providing good agreement according to sensitivity
studies) are used.
Effective light paths are approximated by a broadband function from the
first- and second-order effective light paths obtained at zero absorption and
at selected wavelengths. The absorption contribution is approximated up to
the second order, while in box AMFs are calculated at a
given wavelength and for a priori absorption scenario.
Although the modifications are discussed in the scope of the iterative
algorithm, the retrieval with the linear approach used in the comparison plots
in the next subsection also includes all respective modifications while
omitting the iterative process. Consequently the results are not the same as
presented in or .
Evidence of the higher-order absorption structures in measured spectra
Before discussing the retrieved profiles by the new iterative algorithm (next
subsection) we provide evidence for higher-order absorption structures in the
measured spectrum. Figure shows a difference between the
fit residuals for a fit without including higher-order ozone terms (standard
DOAS) and with including them (initial fit of the iterative approach). The
comparison is performed for a measurement with the same observation geometry
as used for the simulation studies (compare e.g. Fig. ).
Because all other residual structures are removed by the subtraction, a
similar pattern as in the upper left plot in Fig.
(initial fit without higher-order terms added to the fit) can be seen. The
neglected higher-order structures can be seen.
Difference between residuals of the standard DOAS fit and the fit
with higher-order ozone terms included according to .
The fit is performed for a SCIAMACHY measurement with the same
observation geometry as for the sensitivity studies. Please note that a
similar pattern is obtained for the simulated spectrum in the upper left plot
in Fig. illustrating the residual for the initial DOAS
fit where the higher-order terms are omitted.
Results for BrO
We retrieve vertical concentration profiles by the iterative algorithm and
the linear algorithm and analyse the effect of considering the temperature
dependence of the cross sections or the second-order cross-correlative terms.
We compare the vertical profiles retrieved from SCIAMACHY measurements with
collocated LPMA/DOAS balloon observations of direct sunlight
. For a detailed
description of the balloon DOAS instrument, the retrieval algorithm, and the
collocation criteria, please refer to these publications. In
Figs. and , results are shown for the
available balloon measurements for Kiruna (67.9∘ N,
21.1∘ E) in March 2003, Aire-sur-l'Adour (43.7∘ N,
0.3∘ E) in October 2003, Kiruna in March 2004, Teresina
(5.1∘ S, 42.9∘ W) in June 2005, and Kiruna in September
2009. The SCIAMACHY retrieval results from the linear algorithm (green line),
the iterative approach (red line), the iterative approach without considering
the temperature dependence of the cross sections (i.e. assuming a constant
temperature of 223 K in the forward modelling) (blue line), or the second-order
cross-correlative terms (cyan line) are included in the plot. To not
overburden the plots we do not include profiles showing the temperature
effect for the linear retrieval. This effect is caused by the temperature
dependence of the cross sections on the calculation of the higher-order
effective light paths, but this effect is negligible. Also for minor
absorbers, the results of the linear algorithm are practically identical with
the result of the first iteration of the iterative algorithm, and therefore they
are not plotted.
Comparison of SCIAMACHY BrO profiles retrieved by the linear
approach (green), iterative approach (red), iterative approach without
considering the temperature dependence of the cross sections (blue), and
iterative approach without considering the cross-correlative terms in the
calculation of the correction spectra (cyan) with collocated balloon
observations. In order to match the SZA of the SCIAMACHY measurement the
balloon results are photochemically corrected
. Grey areas illustrate the uncertainty of the
balloon results (1σ). The altitude range where the sounded air masses
of both instruments match is indicated by a yellow shading.
Same as Fig. but showing the relative differences
for the SCIAMACHY results with respect to the balloon measurements.
An agreement between SCIAMACHY and the balloon profiles (within the error
bars) can be seen for most of the profiles and altitudes, which fulfil the
collocation criterion (indicated by the yellow shading). The profiles
obtained by the linear retrieval are systematically lower than presented in
the retrieval comparison by where they were found
systematically too high in comparison to retrievals by other groups. This
decrease compared to the previous version can largely be explained by the use
of another wavelength (342 nm) for the spatial retrieval. Also the different
treatment of the Fraunhofer reference might contribute to the change (we
found a systematic bias caused by TH reference spectrum applied in the
previous algorithm). The profiles from the iterative retrieval without yet
considering the temperature dependence of the cross sections shows an
increase compared to the linear retrieval. The difference is increasing with
altitude (from around 0 % below 20 km to 10 % around 30 km). The
tendency is comparable to the findings from the simulated spectra where
a positive offset is also obtained for the altitude range around 20 km between
the linear and iterative retrieval. However, for the real measurements the
difference is smaller by 5 %. The consideration of the temperature
dependence of the cross sections leads to lower values in comparison to the
retrieval assuming a constant temperature. A seasonal variation is found
similar to the findings by . They investigated the
effect of the temperature-dependent BrO cross sections and found a typical
effect of 10 % with a strong seasonal variation with altitude. Keeping in
mind that also the temperature dependence of the ozone cross section in the
current study is considered, a larger effect (up to 15–20 %) for high
latitudes in spring is obtained, caused by both the temperature-dependent BrO
and ozone absorption features. Exclusion of the cross-correlative terms in
the correction spectra (which is subtracted from the measured spectra in the
upcoming iterations of the iterative retrieval) shows only an effect of a few
percent, which is in accord with the simulation studies.
Conclusions
For scattered light observations the Beer–Lambert law generally does not
hold due to multiple light paths contributing to the measurement. In the
presence of strong absorption and a strong wavelength dependence of the light
scattering probability (characteristic for complex viewing geometries like
satellite limb measurements) the commonly used linear approximation
between the trace gas concentrations and the logarithm of the measured
intensity does not hold anymore. The non-linearity is especially strong when
large wavelength intervals are used in the spectral analysis. Therefore
algorithms based on the DOAS technique are especially affected. In essence,
basic quantities like the OD or the SCD cannot be unequivocally defined anymore. Even with modifications, such as
including additional fit terms accounting for the effects of strong
absorbers,
this problem cannot be solved completely because of remaining
cross-correlative structures between weak and strong absorbers or remaining
structures due to the broadband variability on wavelength. While we show
that, in principle, the higher-order cross-correlative structures can be
considered in the DOAS fit as additional fit parameters, to obtain meaningful
results a good signal-to-noise ratio is required to minimize the
cross-correlation between the fitted parameters.
To avoid the need to apply a full retrieval algorithm requiring
time-consuming, iterative wavelength-by-wavelength forward modelling of
absorption spectra, we introduce an iterative two-step retrieval algorithm
separating the spectroscopic and spatial retrieval steps. From the profiles
retrieved in a previous step, corrections for the measured spectra are
calculated. After they are applied to the measured spectra, the next
iteration is performed. Usually, convergence is found after 2–3 iterations.
The correction spectra are calculated based on the a priori profiles or the
profiles derived in the respective iteration. The calculation is based on RTM
calculations at only a few selected wavelengths for an atmosphere with zero
absorption, which strongly reduces the calculation time. Also higher-order
absorption effects and the temperature dependence of the cross sections is
considered. The quantitative definition of higher-order absorption effects
can be obtained by expanding the radiative transfer equation into higher-order Taylor series. Higher-order effective light paths are calculated as
weighted mean products of the light paths through several boxes. By
approximating the wavelength dependence of the first- and higher-order
effective light paths derived from pre-calculated RTM simulations at several
wavelengths by a broadband function, we calculate absorption spectra and
first-order effective light paths for any atmospheric trace gas composition
and for any wavelength within the fit range, thus eliminating the need for RTM
simulations during the retrieval process.
We find a strong influence by second-order absorption structures for
retrievals from measurements in limb geometry, which often exceed the
contributions by minor absorbers (ODs ∼ 10-3–10-1). Even
third-order terms show potential significance: for atmospheric scenarios
typical for Arctic spring (with the stratospheric ozone concentration larger
than the yearly average), we found an OD for the third-order ozone absorption
term of up to 10-2 at the absorption peak at ∼ 333 nm and
exceeding 10-4 even at ∼ 344 nm, while also in the visible (at
550 nm) it is ∼ 10-3. Also cross-correlative terms between ozone
and other trace gases can have significant ODs and thus have the potential to
cause retrieval errors if not considered by retrieval. Since higher-order
contributions to the OD structures are proportional to products of the trace
gas concentrations, these terms will increase or decrease proportionally to
changes in the respective products; e.g. second-order self-correlative terms
are proportional to the squared concentration.
By applying the iterative two-step algorithm for simulated measurements, the
residual structures caused by absorption are reduced: sensitivity studies
reveal good agreement within 1 % for retrieved ozone profiles with
respect to the truth. This good agreement is obtained for most of the
altitudes for retrievals both in the UV and the VIS spectral range even
if the effective light paths up to just the second-order are considered. The
agreement practically does not depend on the selected wavelength for the
spatial evaluation. Also for the minor absorbers NO2 and BrO the retrieved
profiles are wavelength independent with an agreement within a few (1–3)
percent for altitudes above 17–20 km. At the same time, linear retrieval
algorithms (without an iterative process) without complete correction for
higher-order effects show partly significant discrepancies and the retrieved
values are strongly dependent on the wavelength selected for the spatial
evaluation. Good agreement is also found between the retrieval results when
the second-order approximation for the effective light paths or their direct
calculation by RTM are used.
The pre-calculation of higher-order sensitivity parameters by RTM allows to
enhance the performance speed of iterative retrieval algorithms. The
consideration of only second-order absorption terms in the simulation of the
correction spectra in the iterative two-step approach accounts for most of
the non-linear absorption effects, reducing the systematic residual structures
to the order of ∼ 10-4 for scenarios even with strong ozone
absorption. An agreement within a few percent for simulated trace gas
profiles typical for SCIAMACHY limb geometry is obtained. Also the
application of the iterative two-step algorithm on real SCIAMACHY
measurements confirms the findings of the sensitivity studies based on
synthetic spectra and shows good agreement with the collocated balloon
measurements. The algorithm profits from the additional possibility to
correct the temperature dependence of the cross sections. The improved time
efficiency when applying the pre-calculated higher-order effective light
paths instead of online RTM could help for applications for future
instruments with better quality and higher data amount. Additionally, it is
not necessary to consider the trace gas profile variability in the lookup
tables like in applications relying on weighting function DOAS as in
e.g. because the algorithm is
fully flexible to derive the trace gas absorption from the effective light
paths. Together with the flexibility for the wavelength selection, the method
is, in principle, easily implementable for different fit windows, observation
geometries, and instruments.
Although our studies are limited to the limb geometry which is in many
aspects the most sophisticated one, MAX-DOAS and even nadir geometry under
extreme conditions (high SZA and/or high pollution) can benefit from the
approach, but this needs to be investigated in separate studies. The
obtainable improvement depends on the precision at which one is measuring
(uncertainties of the spectral retrieval) and performing the retrieval (e.g.
uncertainty of the radiative transfer simulations).
Calculation of effective light paths from trajectory ensembles generated by RTM McArtim
We use the 3-D full spherical Monte Carlo RTM McArtim
to simulate light trajectory
ensembles necessary to calculate effective light paths of second and third order
according to the definitions provided in Eqs. ()
and (),
with the possibility to extend them to even higher orders according to
Eq. (). A short introduction about the calculation of the weight
and intensity can be found also in Appendix A of
(performed for the RTM model Tracy-II, the predecessor of McArtim). Although
Tracy-II does not assume absorption in the light trajectory simulation
process (rather it adds it afterwards), the description is still valid (with a
minor modification), for McArtim for which absorption is directly considered
in the path generation process. For McArtim, in case of an absorption event
along the trajectory, the contribution (weight) of the radiative tracing
event is set to 0 and the trajectory simulation stops; otherwise the
trajectory is simulated until it leaves the atmosphere.
and describe the calculation of the
intensity and first-order effective light paths while performing the
trajectory simulation. Therefore we need only to introduce the application of
the simulated trajectory ensembles to obtain the higher-order effective light
paths which are not calculated by McArtim directly.
Information provided by McArtim about the simulated light trajectories
include, among others, the coordinates of the scattering/absorption/ground
reflection/box boundary crossing events, information about the event type,
and,
in case of scattering, the weights characterizing the probability that the
trajectory from the scattering position will reach the Sun (in a
probabilistic sense representing the weights used in
Eq. i.a.)
According to the dependent sampling formalism used in McArtim, each
trajectory contains as many light paths as radiative transfer events involved
(technical events like box boundary crossing are not counted as an event in
this sense). Therefore, considering Eqs. () and (), one
needs to consider as many different light paths as the sum of all scatter
events and not just the sum of the simulated trajectories. Individual light
paths extend from detector until a particular scatter event on the simulated
trajectory and then continue towards the Sun. The part of light path from
the scatter event towards the Sun (i.e. forcing to the Sun) is not directly
provided by the McArtim output and must be calculated separately. For that an
algebraic equation system through a spherical layer system corresponding to
the different altitudes is solved according to the Sun coordinates, giving the
respective intersection points (for 3-D atmosphere also cones corresponding
to latitudes and planes for longitudes can be considered). The light path
lengths through separate boxes are calculated as distances between the box
intersection points via radiative events (if any) by a simple trigonometry.
Afterwards the second-order effective light paths for boxes j and J are
obtained as
L2jJ=∑i=1N∑is=1Mwi,isli,isjli,isJ∑i=1N∑is=1Mwi,it,
and the third-order effective light paths for boxes j, J, and J‾
are
L3jJJ‾=∑i=1N∑is=1Mwi,isli,isjli,isJli,isJ‾∑i=1N∑is=1Mwi,is,
where wi,is is the weight and li,isj, li,isJ, and
li,isJ‾ are the light paths for the isth scattering event
of the ith trajectory. i and is here correspond to i in the notation of
Eqs. () and (), while the notation of j, J, and
J‾ is the same. Also here light paths not crossing some or all
two or three involved boxes are considered by a length product being 0,
with their weight contributing to the sum in the denominator.
Third-order contributions
Considering an absorption cross section invariant in space (an assumption
used in DOAS), a third-order impact of the absorption of one trace gas
(compare Eq. ), between two or even three trace gases, can be
quantitatively described.
The self-correlation for trace gas X is defined as
τXXX=σX3∑j∑J∑J‾cXjcXJcXJ‾16L3jJJ‾-12L2jJLJ‾+13LjLJLJ‾.
The third-order cross-correlative term between two trace gases X and Y
is
τXYY=σXσYY2∑j∑J∑J‾cXjcYJcYJ‾12L3jJJ‾-L2jJLJ‾+LjLJLJ‾-12LjL2JJ‾.
Since there are three combinations of XYY, the summation is performed three
times, which leads to the corresponding expression of the sensitivity part.
The third-order cross-correlative term between three trace gases X, Y, and
Z is
τXYZ=σXσYσZ∑j∑J∑J‾cXjcYJcYJ‾L3jJJ‾+2LjLJLJ‾-LJ‾L2jJ-LjL2JJ‾-LJL2jJ‾
There are six combinations of XYZ; the summation therefore is performed
three times, which leads to the corresponding expression of the sensitivity
part.
It can be seen that the expressions are third-order products of
cross sections and these cross-section products can in principle be
considered in a DOAS fit.
Acknowledgements
We want especially to thank Tim Deutschmann for providing the RTM McArtim and
his knowledge about radiative transfer. We thank also ESA and DLR for
providing the SCIAMACHY level 1 data. J. Puķīte is funded by the
Deutsche Forschungsgemeinschaft (PU518/1-1).
The article processing charges for this open-access
publication were covered by the Max Planck Society.Edited by: Jochen Stutz
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