Multiple limb sounder measurements of the same atmospheric region taken from different directions can be combined in a 3-D tomographic retrieval. Mathematically, this is a computationally expensive inverse modelling problem. It typically requires an introduction of some general knowledge of the atmosphere (regularisation) due to its underdetermined nature.

This paper introduces a consistent, physically motivated (no ad-hoc parameters) variant of the Tikhonov regularisation scheme based on spatial derivatives of the first-order and Laplacian. As shown by a case study with synthetic data, this scheme, combined with irregular grid retrieval methods employing Delaunay triangulation, improves both upon the quality and the computational cost of 3-D tomography. It also eliminates grid dependence and the need to tune parameters for each use case. The few physical parameters required can be derived from in situ measurements and model data. Tests show that a 82 % reduction in the number of grid points and 50 % reduction in total computation time, compared to previous methods, could be achieved without compromising results. An efficient Monte Carlo technique was also adopted for accuracy estimation of the new retrievals.

Dynamics and mixing processes in the upper troposphere and lower
stratosphere (UTLS) are of great interest. They control the exchange between
these layers

Infrared limb sounding is an important tool for measuring the temperature and
volume mixing ratios of trace gases in UTLS with high resolution, especially
in the vertical direction, which is critical for resolving typical structures
there (e.g.

A tomographic data retrieval uses multiple measurements of the same air mass,
taken from different directions, to obtain high-resolution 3-D temperature and
trace gas concentration data. There is, as it often happens with remote
sensing retrievals, a large number of the atmospheric states that would agree
with the given observations within their expected precision. A regularisation
algorithm is employed to pick the solution that is in best agreement with our
prior knowledge of the atmospheric state and general understanding of the
physics involved. The mathematical framework of the tomographic retrieval is
outlined in Sect.

The classic Tikhonov regularisation scheme

Data retrievals from limb sounders are typically performed on a rectilinear
grid. In this paper, we define a rectilinear grid by taking a set of
longitudes, a set of latitudes and a set of altitudes and placing a grid
point at each of the possible combinations of these coordinates. Such grids
can be a limiting factor for efficiency of numerical calculations. Due to the
exponential nature of atmosphere density distribution with altitude, most of
the radiance along any line of sight of a limb imager comes from the vicinity
of the lowest altitude point on a given line of sight (called the “tangent
point”). The resolution of the retrieved data depends on the density of
tangent points in the area. For airborne observations this density is highly
inhomogeneous. The densely measured regions are limited in size, located
below flight altitude only and rarely rectangular. The large, poorly resolved
areas with little or no tangent points still need to be included in the grid
as long as lines of sight of any measurements pass through them. A
rectilinear grid retrieval tends to either underresolve the well-measured
area, or waste memory and computation time for regions with few measurements.
To avoid this problem, a tomographic retrieval on an irregular grid with
Delaunay triangulation was developed (Sect.

Most retrieval error estimation techniques would be unreasonably
computationally expensive in the case of a 3-D tomographic retrieval. Monte
Carlo methods allow a relatively quick error estimation. In order to apply
Monte Carlo for a 3-D tomographic retrieval on an irregular grid and using our
newly developed regularisation, dedicated algorithms have to be developed.
These are presented in Sect.

The new methods described in Sects.

The main focus of this paper is the algorithm for retrieving atmospheric quantities, such as temperature or trace gas mixing ratios, from limb sounder measurements in 3-D. In this section, we outline the general inverse modelling approach to this problem and identify some aspects we aim to improve upon.

Let an atmospheric state vector

Finding the precision matrix

We chose a slightly different, more physically and statistically motivated approach to regularisation, which is introduced in the following section.

This section describes the theoretical background for the regularisation
algorithm used for our retrieval, i.e. the motivation for

If the atmospheric state

Let us denote the departure of the atmospheric quantity

It now remains to find an appropriate covariance kernel

It can be shown

This general expression can now be discretised for use in the retrieval. We
can represent the scalar field

Other terms of the integral in Eq. (

We aim to perform a retrieval on an arbitrary, finite grid. This section explains how vertically stretched Delaunay triangulation with linear interpolation is employed for that purpose.

To maintain generality and compatibility with arbitrary grids, we partition
the retrieval volume into Euclidean simplexes (tetrahedrons in our 3-D case)
with vertices at grid points. Many such partitions exist, but it is
beneficial to ensure that the number of very elongated tetrahedrons, which
increase interpolation and volume integration errors, are kept to a minimum.
The standard technique used to achieve this is a Delaunay triangulation
(

In our implementation, we retrieve several quantities on the same grid, and

In order to keep the computational costs low, we use a simple linear
interpolation scheme: in each Delaunay cell we express an atmospheric
quantity

In order to evaluate the cost function based on 3-D exponential covariance
(Eqs.

We begin by establishing some requirements that our derivative-estimation
algorithm will have to meet. Let us write

We deal with this problem by explicitly ensuring that

Since Eq. (

A solution satisfying the above criterion and able to deal with grid points
with neighbours at irregular positions is based on polynomial fitting. For
six grid points around a grid point

The only remaining issue is selecting a suitable set of grid points

Let

Let

For each Delaunay grid neighbour

If

Repeat step 2 for

If neither condition of step 2 is satisfied for at least 1 dimension, repeat steps 1–3 with not only direct Delaunay grid neighbours of

Computing Eq. (

Let a Delaunay cell have vertices

We can now see that Eq. (

Estimating the precision of remote-sensing data products generated by means
of inverse modelling is essential for the users of the final data and also
valuable for evaluation and optimisation of the inverse modelling techniques.
Detailed quantitative descriptions of data accuracy can be derived in theory
(see validation in Sect.

In order to implement this technique, one needs a way to generate a random
vector

A widely used technique for obtaining the matrix

Cholesky decomposition does not preserve the sparse structure of

The situation is very different in the higher dimensions. We will use some
results of graph theory to show that Cholesky decomposition is not practical
in those cases. The

For these higher dimensions, we need to use sparse matrix iterative
techniques to reduce computational cost and memory storage requirements, such
as Krylov subspace methods. In general, it is rather difficult to find a
simple iteration scheme that would compute the square root of a matrix and
converge reasonably fast. Here, we will follow an algorithm proposed by

We chose a classical approach – a Runge–Kutta method, to solve the linear
ODE system numerically. Using constant step size proved to be inefficient, so
adaptive step size control was introduced (in particular, a fifth-order
Runge–Kutta–Fehlberg method by

Although this root-finding algorithm does not provide a way to directly
multiply the matrix root with itself, there is an inexpensive way to verify
that the matrix root was computed correctly. Let

The Gimballed Limb Observer of Air Radiance in the Atmosphere (GLORIA) is an
aircraft-based infrared limb imager. It is a Michelson interferometer with a
2-D detector array, spectral coverage from 770 to 1400

The implementations of the algorithms described in the Sects.

Panel

A hexagonal flight pattern intended for
tomography was realised on 25 January 2016, as part of flight 10 of the
POLSTRACC measurement campaign. The flight path of the HALO research aircraft
contained a regular hexagon with a diameter (distance between the opposite
vertices) of around 500

The test case used in this paper is based on the actual aircraft path,
measurement locations, spectral lines used for retrieval and meteorological
situation during this flight. Synthetic measurement data were used instead of
real GLORIA observations: the forward model of JURASSIC2 was employed to
simulate the observed radiances in the atmospheric state given by the ECWMF
temperature and WACCM model data for trace gases. Simulated instrument noise
was subsequently added to those radiances. This set-up allows us to use the model
data as a reference (the “true” atmospheric state) for evaluation of
retrieval quality. The same set of simulated measurements was used for all
test retrievals described in this paper. These measurements were obtained by
running the forward model on a very dense grid (about twice as dense in each
dimension as those of the densest test retrievals). This was done to ensure
that the discretisation errors in the simulated measurements would be minimal
and would not favour any retrieval (as could happen if they were
generated on the same grid). An evaluation of forward model errors on
different grids is presented in Appendix

The retrieval derives temperature and volume mixing ratios of

Panels

Retrieval parameters.

Regularisation weights

The standard deviation

3-D tomographic retrievals are most useful for those trace gases that have
high vertical gradients and complex spatial structure. The spatial
correlation lengths of concentrations of such gases are mostly determined by
stirring, mixing and other dynamical processes and are therefore similar.
Furthermore,

An analysis of spatial variability of some airborne in situ measurements was
performed by

The spatial structure of retrieved temperature differs from that of the trace
gases. It is determined not only by mixing and stirring, but also radiative
processes and gravity waves. Radiative transfer tends to erase some of the
fine vertical structure determined by isentropic transport; hence one should
use a greater vertical correlation length than in the case of trace gases.
Gravity waves have a variety of length scales, but due to the finite resolution
of the instrument not all of them can be retrieved. Following the gravity
wave observational filter study for the GLORIA instrument in

Grid densities in different regions for retrieval

The test data described in the previous section were processed with several different regularisation set-ups and retrieval grids. Here we present the results for four different runs.

Firstly, as a reference, the latest version of JURRASIC2 without the
implementation of any new algorithms described in this paper was used
(retrieval A). It uses a rectangular grid that covers altitudes from 1
to 64

Panels show the difference between retrieved temperature and true temperature of the simulated atmosphere. Rows A–D show the results of the respective retrievals. In the first column, black lines represent the contours of the a priori temperature value. In columns 2 and 3, the black line is the flight path, the thick grey line is the position of the cut shown in the first column and the thin grey line is the topography (Icelandic coastline visible).

Then, to evaluate the performance of second-order regularisation, retrieval B
was performed. The grid and interpolation methods were identical to retrieval
A, but the regularisation was replaced with the second-order scheme from
Eqs. (

We compare the quality of different retrievals by inspecting the differences
between the retrieved temperature and temperature used to generate the
synthetic measurements (“true” temperature). These differences are shown on
horizontal and vertical slices of the observed atmospheric volume in Fig.

Retrieved temperature at different altitudes in the vertical cut
shown in Figs.

A comparison of Fig.

The third retrieval (C) was performed on the same set of grid points as B and
with the same input parameters, but the grid was treated as irregular; i.e. a
Delaunay triangulation was found. The algorithms described in Sect.

Finally, retrieval D was performed with the same input parameters and methods
as C, but using only the subset (22 %) of the original grid points to reduce
computational cost. These points were chosen so that grid density would be
highest in the volumes best resolved by GLORIA and sparser where little
measurement data are available. In particular, a set of axially symmetric
regions around the centre of original retrieval (i.e. symmetric with respect to
vertical axis at 66

Panels B and D show the Monte Carlo estimate of the total temperature error the retrievals B and D would have if the measurements were real. Panel Ds shows an error estimate for the synthetic retrieval D (this assumes that unretrieved quantities, e.g. pressure, are known exactly). In columns 2 and 3, the black line is the flight path, the thick grey line is the position of the cut shown in the first column and the thin grey line is the topography.

JURASSIC2 finds an atmospheric state minimising Eq. (

All calculations were performed on the Jülich Research on Exascale Cluster
Architectures (JURECA) supercomputer operated by the Jülich Supercomputing
Centre

Table

The results in Table

Computational cost of retrievals (calculation times given in minutes).

Retrieval D gives a further 45 % reduction of the computation time by removing
82 % of points in the grid. A speed-up of such an order was expected. Only a
minority of grid points contribute to a forward model (and hence Jacobian)
calculations, and JURASSIC2 code is heavily optimised for sparse matrix
operations. Most of the grid points contributing to the forward model are in the
areas well resolved by the instrument and thus cannot be removed from the
grid without compromising data products. Therefore, the actual savings in
computation time mostly come from removing points above a flight level and
choosing a more appropriate non-rectangular shape for the densest part of the
grid when measurement geometry requires it. Some of the points above the flight
path must be included since infrared radiation from there is measured by the
instrument, but very little information about atmosphere far above the flight
level can be retrieved, so very low grid densities are sufficient (Table

We follow

These results confirm that temperature can only be reliably retrieved within
the flight hexagon or very close to the actual path. Also, the error
estimates are higher in the central area of hexagon, close to flight altitude,
in agreement with test results (compare column 3 of Figs.

Monte Carlo retrieval results provided here were verified by estimating
the temperature error at one grid point using a more direct approach.

This approach allows for error estimation for a small number of grid points
in a reasonable amount of time. We chose one point at the centre of the hexagon
(66

A new regularisation algorithm for 3-D
tomographic limb sounder retrievals employing second spatial derivatives
(based on Eqs.

A Delaunay-triangulation-based approach for retrievals on irregular grids was developed. By better adapting the grid to the retrieval geometry and thinning it out in the regions where a lack of measurement data limits the resolution, the computation time of a tomographic retrieval was reduced by a total of 50 % without significant deterioration of the results. Irregular grids can be further employed for efficient tomographic retrievals in non-standard measurement geometries. At the time of writing, the methods developed in this paper were already in use to process GLORIA limb sounder data.

Monte Carlo diagnostics were newly implemented for a 3-D tomographic retrieval, allowing for quick and reliable evaluation of measurement quality, identification of reliably resolved volumes and selection of optimal 3-D tomography set-ups. Error estimates for retrieved atmospheric quantities can now be calculated at each grid point, while the previous approaches to diagnostics only allowed us to do that for a few selected points within a similar time frame.

The output of the test retrievals A–D discussed in the paper are available in the Supplement.

Equation (

Precision matrix

Forward model tests.

Difference in forward model results compared to original synthetic measurement set without noise. A–D refer to the forward model results of respective retrievals.

The exponential covariance relation given in Eq. (

As pointed out in Sect.

The supplement related to this article is available online at:

The authors declare that they have no conflict of interest.

The POLSTRACC campaign was supported by the German Research Foundation (Deutsche Forschungsgemeinschaft, DFG Priority Program SPP 1294). The authors gratefully acknowledge the computing time granted through JARA-HPC on the supercomputer JURECA at Forschungszentrum Jülich. The European Centre for Medium-Range Weather Forecasts (ECMWF) is acknowledged for meteorological data support. The authors especially thank the GLORIA team and the POLSTRACC flight planning and coordination team for their great work during the POLSTRACC campaign on which the test cases in this paper are based. The article processing charges for this open-access publication were covered by a Research Centre of the Helmholtz Association. Edited by: Markus Rapp Reviewed by: two anonymous referees