This work presents a new iterative method for optimally selecting a vertical
retrieval grid based on the location of the information while accounting for
inter-level correlations. Sample atmospheres initially created to parametrise
the Radiative Transfer Model for the Television Infrared Observation
Satellite Operational Vertical Sounder (RTTOV) forward model are used to compare the presented iterative selection
method with two other common approaches, which are using levels of equal
vertical spacing and selecting levels based on the cumulative trace of the
averaging kernel matrix (AKM). This new method is shown to outperform
compared methods for simulated profile retrievals of temperature, H

Retrieved profiles of temperature and composition from nadir-viewing
instruments are often presented on a grid much finer than can be justified
given the actual vertical resolution of the measurements. Therefore, we
propose a method to determine the optimal subset of vertical levels from
a fine vertical grid by selecting levels according to their contribution to
the degrees of freedom that come from the signal (DFS) rather than the a
priori

When designing a retrieval scheme, it is useful first to determine the subset of coarse vertical levels that efficiently contribute to the estimate. By reducing the number of attempted estimates, the retrieval relies less on formal prior knowledge and becomes more sensitive to the true state. There are also computational benefits during the retrieval due to the improved conditioning of the problem, possibly faster convergence and greater tolerance of ad hoc assumptions in the a priori.

Consider, for example, the Infrared Atmospheric Sounding Interferometer
(IASI) level two (L2) product; where temperature, water vapour, and ozone
profiles are presented on a vertical grid of 100 pressure levels ranging from
surface pressure up to

Post-processing methods are developed to reduce the reliance upon a priori
information. Possible a priori sources include Numerical Weather Prediction
(NWP) data and chemical transport models such as the Goddard Earth Observing
System Chemical transport model (GEOS-Chem;

Previous work by

The presented work is applied to profile retrievals of temperature,

Section

Atmospheric profile retrievals with a nadir viewing satellite tend to be
significantly ill-conditioned. In other words, the attempted number of
estimated parameters (

This section reviews inverse theory, applied to ill-conditioned atmospheric
sounding. While there are at least two separate notations commonly used, we
adopt the notation consistent with

When the radiative transfer function is sufficiently linear about a reference state vector (

Solutions to Eq. (

The term a priori is meant to include both a mean state,

Diagnostic information about the retrieval is succinctly contained in a unitless

Repeated analysis of

Solutions to ill-conditioned inverse problems can be improved by simply reducing the number of estimated parameters. However, information content may be lost if the retrieved state is reduced too much. Therefore, as the original parameter space is reduced, the information content should be monitored in a consistent mathematical way. This is done by defining operators that map the retrieval between the original and reduced state space.

Consider two vertical grids for the problem of retrieving atmospheric
profiles. First, a fine grid from the discretisation of the full state
vector,

For convenience we apply a linear mapping from the retrieval to the fine grid so that

To transform the a priori to the retrieval state space, the following averaging operation is required,

More rigorous derivations and discussions of mapping between states can be
found in

AKM columns are plotted for a sample temperature retrieval with IASI using a fine vertical grid (black) compared to
a coarser grid (grey) selected using the iterative vertical selection method. Note that the pressure axis changes from
linear to logarithmic above

From inspection of Eqs. (

When determining a coarse retrieval grid the number and location of profile
levels can be chosen in an ad hoc manner or decided based upon the
distribution of information in the profile. The DFS is a natural scalar
metric of information to use when constructing and comparing different
vertical grids, because it can be directly compared to the number of
attempted retrieval levels. When the DFS is approximately equal to the number
of levels, then little prior knowledge appears in the estimate. Other
possible scalar metrics of information include the Shannon information
content and the trace of the Fisher information matrix

DFS for a temperature retrieval with IASI vs. both ranked atmospheric pressure levels and ranked spectral channels
from the

The simplest possible selection method is to segment the atmosphere into layers of equal thickness. Levels of equal pressure may be used for better tropospheric sensitivity, or levels of equal height for stratospheric sensitivity. Nadir-viewing instruments such as IASI are typically more sensitive to the troposphere, so equal pressure spacing will be assumed for these comparisons, even though it is clearly inappropriate for species with stratospheric concentrations such as ozone.

Alternatively, a vertical selection method proposed by

While using the diagonal of

Simulated IASI spectrum showing the spectral ranges considered in this study, which are typical for temperature and trace gas profile retrievals with this instrument.

The proposed vertical selection method is outlined as follows.

Calculate the DFS on the fine grid by making

Next, a single level is removed by modifying

The resulting DFS from removing that level are determined from Eqs. (

This process is repeated to find the second-least important level until all vertical levels have been ranked and discarded down to the two levels that contribute the most to the DFS.

To visualise the effect, Fig.

Sampling locations of the 80 atmospheres comprising the RTTOV training ensemble colour coded according to the atmospheric surface temperatures of the profiles.

Clearly, the downside to this iterative selection method is the added
complexity and computational cost of checking each available vertical level
during the merging process, as compared to the method of interpolating from
the cumulative trace of

Choosing the number of retrieval levels ultimately depends upon the tolerance
for a priori appearing in the estimate. For an effectively a priori free retrieval, the number of estimates should be at most equal to the DFS on the
fine grid rounded down

Figure

The DFS from the ensemble for temperature and trace gases as a function of their surface values. The points are
colour coded by atmospheric surface temperature using the colour bar shown in Fig.

Notice that the DFS increases nearly linearly with pressure level number
initially, but quickly reaches a plateau once

Figure

In this section the three different selection methods are compared, using
a variety of atmospheric conditions for simulated IASI retrievals of
temperature,

Radiances measured by IASI are accurately reproduced by configuring
a radiative transfer model to the specifics of IASI's orbital geometry and
instrument response

Instrument noise was provided by the Centre National d'Etudes Spatiales
(CNES) 2008 post-launch estimate

Continuing work from the Thermodynamic Initial Guess Retrieval (TIGR)
database

Temperature and gas profiles are presented on a fixed fine grid of 101
pressure levels ranging from

Contours of AKMs for a mid-latitude summer retrieval of temperature,

The original purpose of the TIGR database was to sample representative a priori information for accurate modelling of forward or inverse radiative transfer problems. As such, a prior covariance matrix was created for each constituent studied here by calculating the sample covariance from the 80-atmosphere ensemble. Therefore, the prior covariances used represent global statistics and include full off-diagonal correlations often neglected in retrieval schemes. This choice was made primarily to highlight the effect off-diagonal sensitivities in the AKMs have upon the level selection results. While it is unlikely these particular prior covariances will be used in an operational retrieval, they may provide the most interesting case study.

Furthermore, using a global prior covariance includes greater variation than
a latitude specific atmosphere may experience. Therefore, the fine grid DFS
represents an upper limit to what may be achieved in an operational
retrieval. Figure

A typical mid-latitude summer atmosphere was chosen from the ensemble to help
visualise the differences of selecting coarse grids from the methods in
Sect.

These off-diagonal responses in the fine grid AKMs for

Figure

Cumulative diagonals of the AKM for a temperature retrieval on the 101 level fine grid as well as 18 level coarse
grid AKMs (Sects.

Vertical selection comparisons were made using the same number of vertical
levels specific to each atmosphere to isolate differences solely due to the
method. The selection method that retains the greatest DFS on average for the
80 atmospheres is the iterative method from Sect.

Figure

Histograms of 15 equally spaced bins showing the percent loss in DFS by using a coarse vertical grid with equal pressure spacing vs. the coarse grid from the iterative method. The dotted vertical lines show the median values of the 80-atmosphere distribution.

The loss of DFS by using a vertical grid selected from the cumulative
diagonal of

Finally, when constructing a retrieval algorithm with IASI or another
infrared sounding instrument, the developer may consider whether to use
a constant vertical grid or one that optimally adapts to the observed
atmosphere based on the prior state and converging profile.
Figure

Histograms of 15 equally spaced bins showing the percent loss in DFS by creating a coarse vertical grid from the
cumulative trace of

Notice that the median DFS losses are less than 3

To summarise Figs.

Summaries of the ensemble results from
Figs.

Histograms of 15 equally spaced bins showing the loss in DFS by using a constant globally optimised vertical grid
vs. an atmosphere specific grid. The dotted vertical lines show median values of the 80-atmosphere distribution, where the temperature and

When retrieving atmospheric profiles of temperature and trace gases from infrared spectral radiances, it is important to consider where in the vertical profile the estimates are made. A new iterative method for selecting a vertical grid was proposed and shown to outperform previously used selection methods by accounting for correlations and sensitivities between different vertical levels. Other compared methods of establishing a vertical grid coarser than the radiative transfer grid were using levels equally spaced in pressure and selecting levels by interpolating along the cumulative diagonal of the fine grid AKM.

The 80-atmosphere ensemble created to parametrise RTTOV was used to
systematically compare the different vertical grid selection methods for
temperature,

Comparing to the cumulative diagonal of

Finally, much effort is spent making retrieval schemes run faster. Naturally,
one would prefer to design a coarse vertical grid just once and apply that to
all scenarios, rather than optimise a grid for each retrieved atmosphere.
This depends upon the variability of vertical information content. The median
loss of DFS for

Portions of this work were funded by the United States Air Force. The views expressed in this article are those of the author and do not reflect the official policy or position of the United States Air Force, Department of Defense or the US Government. Edited by: A. Butz