In order to avoid problems connected with the content of a priori information in volume mixing ratio vertical profiles measured with the Michelson Interferometer for Passive Atmospheric Sounding (MIPAS), a user-friendly representation of the data has been developed which will be made available in addition to the regular data product. In this representation, the data will be provided on a fixed pressure grid coarse enough to allow a virtually unconstrained retrieval. To avoid data interpolation, the grid is chosen to be a subset of the pressure grids used by the Chemistry–Climate Model Initiative and the Data Initiative within the Stratosphere-troposphere Processes And their Role in Climate (SPARC) project as well as the Intergovernmental Panel of Climate Change climatologies and model calculations. For representation, the profiles have been transformed to boxcar base functions, which means that volume mixing ratios are constant within a layer. This representation is thought to be more adequate for comparison with model data. While this method is applicable also to vertical profiles of other species, the method is discussed using ozone as an example.

The often ill-posed nature of inverse problems in remote sensing of the
atmosphere is typically fought by formal regularization, i.e. by inclusion
of prior information in a Bayesian or related sense. The most common of
such methods is optimal estimation

As a test case for the method suggested and for purposes of demonstration,
measurements obtained with the Michelson Interferometer for Passive Atmospheric
Sounding are used. MIPAS, a limb-viewing mid-infrared Fourier
transform spectrometer

The IMK–IAA data processor relies on multi-parameter non-linear least-squares
fitting of measured and modelled spectra within a framework of Newtonian
iteration

“True” (whatever this is) vertical profiles of atmospheric state variables are
regarded as continuous. This issue is critically discussed in

Constant values within layers or linear interpolation between adjacent levels
are two common options to implement prior assumptions in the ML case. The
ML estimator

The estimator using a formal constraint is

The advantage of ML representation of vertical profiles is that the averaging
kernel matrix, which includes the partial derivatives of the retrieved
profile values at altitude

This master vertical grid, however, is still too fine to allow stable
unregularized retrievals. In order to avoid any unnecessary interpolation,
the actual retrieval grid is chosen to be a subset of the master grid. The
approximate allowed minimum gridwidth is determined with the method presented
by

For reasons of computational accuracy, the forward model used for the retrieval
solves (i.e. numerically integrates) the radiative transfer equation on an
altitude grid which is finer than the retrieval grid. For this purpose,
the constituent and temperature profiles are interpolated to the forward model
grid points linearly with altitude. This means that the profile is represented
by triangular base functions. The larger peak values of the ML profiles
compared to the regular representation where the peaks are more rounded is a
typical feature of triangular base functions: their peak is slimmer, which is
compensated by a larger peak value to be consistent with the same amount of
molecules (Fig.

A MIPAS ozone profile from the regular
processing (black) and the same profile in a maximum likelihood representation
where linear interpolation of vmr between altitude grid points was assumed
(red). This particular example is a measurement at 17.75

Often profiles based on triangular base functions are not directly
comparable to modelled profiles, because the latter frequently are layer models,
which assume constant atmospheric conditions within a layer around the
vertical grid point. In other words, modelled profiles often have rectangular
base functions; i.e. they are “staircase profiles”. Thus the retrieved
ML profiles are converted in a postprocessing step into staircase profiles
containing the same amount of molecules. The transformation is reported in
Appendix A, and results are shown in Fig.

The staircase profiles could easily be obtained without the postprocessing described above simply by using rectangular base functions in the retrieval and the underlying forward model. We think, however, that linearly interpolated profiles are more adequate because they represent the true atmosphere better than the staircase profiles with their discontinuities at the layer boundaries. To avoid related inaccuracies in the radiative transfer calculations, we consider the triangular base functions as more appropriate and accept the additional effort implied by the transformation to the rectangular base functions.

An ozone profile represented in triangular (black) and rectangular (red) base functions.

In cases when the ML data are sampled exactly on the model grid, direct comparison of observations and model data without any transformation is possible and adequate. It is sufficient that the grids match in the altitude range considered for the intercomparison. Occasionally, the observations will, for reasons discussed above, be represented in a subset of the master grid. In these cases a mass-weighted mean of the respective two or more contiguous model layers will be the adequate quantity to be compared with the observation. This approach, while still simple and intuitive, is roughly equivalent with the application of the averaging kernel evaluated on the master grid, assuming linearity in a sense that the instrument is equally sensitive to the target gas regardless of its actual distribution within the combined retrieval layer.

Here we compare the differences between the regular and the ML representation
of typical ozone retrievals on the basis of zonal mean volume mixing ratio
cross sections (Fig.

Zonal mean ozone distributions from the regular retrieval on a fine grid (top panel) and the maximum likelihood staircase representation on a coarse grid (bottom panel).

Admittedly the statement that the averaging kernel of the ML retrieval is
unity can be challenged: while algebraically this is certainly true when the
averaging kernel is calculated on the coarse ML retrieval grid, the correct
sensitivity of the retrieval with respect to the true atmospheric state
would require consideration of atmospheric variability on a finer scale

For each species the altitude grid is chosen time-invariant. Thus, the altitude
resolution, which in an ML retrieval is determined solely by the vertical
grid chosen, does not vary with time either. Variations of the information
content of the measurements with time thus change only the error bars, while
the vertical resolution remains constant. Variable error bars, however, can be
handled by any advanced statistical toolbox, as opposed to varying averaging
kernels. In consequence, problems in time series analysis encountered by

In the upcoming version of MIPAS data, both representations of the data will be
made available: the regular retrievals will be available along with the usual
diagnostics including averaging kernels and error estimates. The easy-to-use
ML data product will be made available as an alternative, for
applications where related problems cannot easily be solved by application of
the averaging kernels (amplitudes of annual variation; trends) or for data
users who have no experience in working with data containing a priori knowledge.
It must be mentioned that for some species the ML representation is inadequate.
Abundances of species like CO, NO, or NO

Our method is initially targeted at trace gas distributions measured with the
limb-sounding technique and should be applicable to a wide range of instruments
measuring limb emission or solar/stellar/lunar occultation. Application to
nadir-viewing instruments could be more difficult because of their limited
vertical resolution. Nevertheless, similar efforts were made in the
context of the nadir-looking Tropospheric Emission Spectrometer (TES) by

The column density of gas g in a layer between altitudes

This work was partly funded by BMBF under DLR contract 50 EE 0901. J. Plieninger was supported by Helmholtz within the framework of their climate initiative REKLIM. The article processing charges for this open-access publication were covered by a Research Centre of the Helmholtz Association. Edited by: H. Worden