A critical step in satellite retrievals of trace gas columns is the
calculation of the air mass factor (AMF) used to convert observed slant
columns to vertical columns. This calculation requires a priori information on the
shape of the vertical profile. As a result, comparisons between
satellite-retrieved and model-simulated column abundances are influenced by
the a priori profile shape. We examine how differences between the shape of the simulated and a priori profiles can impact the interpretation of satellite retrievals
by performing an adjoint-based four-dimensional
variational (4D-Var) assimilation of synthetic

Satellite observations provide a wealth of information on the abundance of
trace gases in the troposphere (Fishman et al., 2008). The
next generation of satellite instruments, including the upcoming
geostationary constellation of TEMPO (Chance
et al., 2013; Zoogman et al., 2017), Sentinel-4 (Bazalgette
Courrèges-Lacoste et al., 2011; Ingmann et al., 2012), and GEMS
(Bak et al., 2013;
Kim, 2012), will provide information on

There are three main stages in retrieving trace gas abundances from
ultraviolet and visible solar backscatter radiance measurements: calculating
a light-path slant column by fitting observed spectra to known spectral
signatures of trace gases, removing the stratospheric portion of the column,
and converting the slant column to a vertical column density using an air
mass factor (AMF). AMFs are calculated using a radiative transfer model and
are a function of viewing geometry, surface reflectance, clouds, and
radiative transfer properties of the atmosphere. AMF calculations also
require an a priori estimate of the trace gas vertical profile and are sensitive to
the profile shape (Eskes and Boersma, 2003; Palmer et al.,
2001). Uncertainties in AMF calculations are the dominant source of
uncertainty in satellite

Boersma et al. (2016) highlighted the issue of representativeness errors in comparing model-simulated values with UV–Vis satellite-retrieved columns. Vertical representativeness errors arise from the satellite's altitude-dependent sensitivity due to atmospheric scattering and can degrade the quality of model–measurement comparisons beyond errors that arise from either modeling or measurements alone. A consistent accounting of the altitude-dependent sensitivity is necessary to limit these errors.

Two common methods are used to account for vertical representativeness. In
one method, observed slant columns are converted to vertical columns using
an air mass factor calculated with scattering weights to represent
instrument vertical sensitivity and shape factors to represent the vertical
profile (Palmer et al., 2001). Another commonly used method
employs an AMF provided with the retrieval to convert slant columns to
vertical columns and then applies an averaging kernel to the simulated
profile to resample the simulated profile in a manner that mimics the
satellite vertical sensitivity (Eskes and Boersma, 2003). In this
method both the averaging kernel and the retrieval AMF are calculated using
an a priori

A common application of comparisons between satellite-observed columns and
model simulations is to constrain

In this work we examine how a priori profile assumptions impact satellite–model comparisons and use the GEOS-Chem adjoint as a case study to assess how this impact can affect the interpretation of satellite observations. Section 2 provides the mathematical framework for AMF calculations and satellite–model comparisons. Section 3 describes the adjoint model and synthetic observations for the case study. Section 4 discusses the results.

The air mass factor translates the line-of-sight slant column abundances
(

In the method described by Palmer et al. (2001), a radiative
transfer model is used calculate scattering weights

In an alternative formulation, the air mass factor is represented as part of
an averaging kernel. As formulated by Rodgers and Connor (2003), the averaging kernel (

A lexicon is given in Table 1 as notation used to describe these treatments
has varied across the literature. We choose

Lexicon comparing notation used in this paper to that used in previous studies.

Figure 1 shows examples of typical shape factor, scattering weight, and
averaging kernel profiles for a range of atmospheric conditions.

The following section expresses mathematically how satellite–model comparisons are made using various a priori profiles.

Following Palmer et al. (2001), a retrieved vertical column (

The difference

Equation (11) describes how this comparison is used in practice. However, we
can rearrange this expression in terms of model (

Comparison of simulated and retrieved columns using the averaging kernel is
described by Eskes and Boersma (2003) and in the retrieval
documentation in Boersma et al. (2011). The averaging
kernel is applied to the simulated profile in order to sample the simulated
column in a manner that reflects the retrieval sensitivity:

The resampled simulated column is then compared to the retrieved vertical
column (

Equation (16) describes how this method is used in practice. To facilitate
the comparison with Eq. (13), Eq. (16) can be rewritten using an alternative
formulation relating averaging kernels to scattering weights:

By comparing Eqs. (13) to (18), it is evident that the underlying difference between the two approaches is the choice of a priori profile information used to calculate the AMF, as the averaging kernel method is not independent of a priori profile assumptions. This bias could be addressed by replacing the a priori-based AMF in Eq. (18) with a simulation-based AMF using the following relationship (Boersma et al., 2016; Lamsal et al., 2010):

It should be noted that both the averaging kernel and scattering weight
methods are equivalent for comparisons that examine ratios of retrieved and
modeled columns:

For ratios, both methods are dependent on geophysical assumptions used to calculate scattering weights but are independent of a priori profile information. Lastly, some studies (e.g., Qu et al., 2017) may directly assimilate slant column densities rather than vertical column densities using

This approach is also still dependent upon the scattering weights but not
upon external a priori profile information. Overall, the choice of approach may be
influenced by whether or not scattering weights are available from either
the

The GEOS-Chem chemical transport model (

The GEOS-Chem adjoint (Henze et al., 2007, 2009) is
used here to perform a 4D-Var data assimilation. The adjoint seeks to
iteratively minimize a cost function generally defined by the difference
between satellite-retrieved and simulated columns (

In this study we perform 4D-Var data assimilation experiments to infer
surface

Synthetic observations (Obs

For these tests, we use one observation per hour per

To test the impact of a priori profile information, seven different tests are performed using seven different

Case SF

Case SF

Case SF

Case SF

Case SF

Case SF

Case SF

An advantage of using scattering weights and the simulated shape factor in a
4D-Var framework is that it allows for the shape factor, and thus the AMF,
to be updated at each iteration. When a priori profiles from an external source are
used it is not possible for them to update during the inversion. The
SF

Global mean air mass factors and synthetic observation vertical column density for shape factors tested here.

Figure 3 shows root mean square errors (RMSEs) for the a posteriori emissions estimated
by the 4D-Var assimilations of Obs

Global root mean square error (RMSE) values for 4D-Var estimates
of

Difference between root mean square error (RMSE) of adjoint tests
for Obs

Figure 4 shows maps of the difference in RMSE between the SF

Table 3 summarizes additional error statistics focused on grid boxes with
significant emission sources. Errors in a posteriori emission estimates are correlated
with the true emissions in the SF

Summary of error statistics for adjoint tests. Values marked with asterisks (

Tests using Obs

Accounting for the vertical profile dependence of satellite observations is essential to accurately interpret those observations. This work examines how the choice of shape factor affects differences between simulated and satellite-retrieved quantities in a data assimilation framework. Examination of the mathematical frameworks behind two common methods for comparing simulated and retrieved columns highlights how the method introduced by Palmer et al. (2001) facilitates separation of observation sensitivity (scattering weights) from the profile shape (shape factor) enabling the model–retrieval comparison to be independent of a priori profile assumptions.

Scatterplot of adjoint test results.

In these case studies, vertical representativeness errors were best reduced
by using a shape factor that was consistent with the model simulation. This
was especially true in polluted regions where the AMF errors dominate
observation uncertainties, as deviations between the tests were largest in
these regions. The further the shape factor deviated from the model state
the larger the inversion errors became, as indicated by Fig. 5. The
SF

The case study presented here demonstrates that the shape factor source can
have a strong influence on adjoint inversion results. However, the magnitude
of this influence can vary. Inversion tests performed using synthetic
observations based on random 30 % perturbations to emissions were
insensitive to the AMF, despite large differences in a priori vertical column
densities. In these tests, the cost function was more sensitive to the
larger difference between the observed and simulated slant columns (i.e.,

As it is beneficial for a consistent shape factor to be used when comparing satellite-retrieved values to model-simulated results, it will be useful for data products to provide the information required for this method to the user community. This is most straightforward when scattering weights (rather than averaging kernels) are provided alongside retrieved column data, as scattering weights and shape factors are independently calculated; however, simulation-based air mass factors can be calculated using the averaging kernel and a priori-based air mass factor via Eq. (19).

In summary, when comparing a model simulation to a satellite-retrieved

The GEOS-Chem chemical transport model and its adjoint are available at

MJC and RVM designed the overall study. MJC designed and carried out the case studies and their analysis. All coauthors provided guidance in analyzing results. MJC prepared the manuscript with contributions from all coauthors.

The authors declare that they have no conflict of interest.

This research has been supported by the Canadian Space Agency. Daven K. Henze was supported by NASA (grant no. NNX17AF63G).

This paper was edited by Ronald Cohen and reviewed by two anonymous referees.