We review the main factors driving the calculation of the tangent height of spaceborne limb
measurements: the ray-tracing method, the refractive index model and the assumed atmosphere. We
find that commonly used ray tracing and refraction models are very accurate, at least in the
mid-infrared. The factor with largest effect in the tangent height calculation is the assumed
atmosphere. Using a climatological model in place of the real atmosphere may cause tangent height
errors up to

Inversion algorithms for atmospheric limb measurements from space usually
retrieve profiles on pressure coordinates due to the significant uncertainty
of the line of sight. Accurate knowledge of the line of sight is however
needed to reconstruct the altitude grid of the retrieved profiles, as it may
be necessary for comparison to correlative measurements (such as those
obtained from lidar) that are intrinsically represented on an absolute
altitude grid. Spectral measurements also contain information on the
instrument viewing direction; however, the spectral resolution and the
signal-to-noise ratio are often insufficient to determine accurate estimates
of the line of sight. The accurate calculation of the line of sight from the
engineering estimates of the instrument pointing angles and instrument
position relies, however, on the accuracy of both the ray-tracing algorithm
and the model used for atmospheric refraction. In this work, we compare the
accuracy of a few ray-tracing and atmospheric refraction models using the
tangent height error as a quantifier. The tests presented are based on
measurements of the Michelson Interferometer for Passive Atmospheric Sounding
(MIPAS,

The propagation path of electromagnetic rays through an inhomogeneous medium
can be deduced from the eikonal equation

After some algebraic manipulation, using Eq. (

This is a vectorial second-order differential equation that permits one to derive the full ray path
across an inhomogeneous medium, if

direct numerical solution of Eq. (

tangential displacement method

iterative Snell's law

In Thayer's implementation of Snell's law the atmosphere is assumed
horizontally homogeneous. This implementation is one of the fastest
ray-tracing methods, however, if the horizontal variability of the atmosphere
is taken into account, this method is not adequate. For horizontally varying
atmosphere, the level lines of

The tangential displacement method (henceforth referred to as TD) is an
iterative approach for the solution of the eikonal equation, using an
approximation to avoid the calculation of the second derivatives of

For the direct numerical solution of the eikonal equation we implemented
a multi-step predictor–corrector method (henceforth referred to as EIK) using
the two-step Adams–Bashforth formula for the predictor and the BDF2 formula
for the corrector

All the implemented methods can be applied to a three-dimensional
ray tracing. Our implementation is however planned for inclusion in the ESA
retrieval model for the Michelson Interferometer for Passive Atmospheric Sounding
(MIPAS) routine data processing

In this work we considered three refractive index models:

Barrel–Sears formula

simplified Edlén formula

Ciddor formula

The Barrel–Sears empirical formula has been used for a long time for atmospheric infrared
applications. We include this formula in our tests mainly for historical reasons. The version
implemented in our code is

The simplified Edlén formula is the model currently implemented in the ESA retrieval code for
routine MIPAS data inversion. The formula implemented in our code is

Ciddor's formula models the refractive index as a function of wavelength, pressure, temperature,
water vapor and carbon dioxide content. The formula was originally tested with experimental data
extending only up to 1.7

Whatever refraction model is chosen, the refractivity depends on pressure, temperature and,
possibly, water vapor and carbon dioxide. Thus, the ray tracing depends on the assumed atmosphere. To
evaluate the impact of the selected atmosphere we considered the following models:

the US Standard Atmosphere, 1976

the IG2 atmosphere

the atmosphere retrieved in a previous processing version of MIPAS data

atmospheric refractivity profiles determined from co-located Radio Occultation (RO) measurements

The US Standard Atmosphere (US76), together with the simplified Edlén formula for refraction, is
the model currently adopted by ESA in MIPAS level 1b data processing

The IG2 climatological database is a collection of atmospheric profiles used
as initial guess (IG) or assumed profiles in MIPAS routine level 2 retrievals

The last two atmospheres rely on experimental data. The tests with RO
refractivity measurements considered in this work are limited to the MIPAS
orbit 43442 acquired on 21 June 2010. In this orbit, there are 16 MIPAS limb
scans for which a co-located RO measurement exists within 300

The calculated ray path depends on the ray-tracing method, the refractive
index model and the assumed atmosphere. To evaluate the impact of each of
these factors, we use the error on the calculated tangent height of the limb
measurements as a quantifier. The MIPAS level 1b data files provide the
geolocation of the tangent points of the limb measurements. These are
determined using the position and attitude of the satellite via a ray-tracing
algorithm that uses the US76 atmosphere and the simplified Edlén formula
for refractivity. Since we have no access to the algorithm details, however,
we are not able to reproduce exactly the tangent height values reported in
the level 1b data files. To maintain full consistency with the level 1b data,
instead of starting the ray tracing from the satellite position and attitude,
we use the following approach. We start from the level 1b tangent point
geolocation and use the same assumptions of the level 1b processor to
back-calculate the latitude and the slope of the ray path at the intersection
with the atmospheric boundary, fixed at 120

All the tests reported below were carried out using the MIPAS measurements acquired on 12 orbits (3 for each season) in the year 2010. We find that the estimated errors are correlated with latitude and season; however, their maximal amplitude changes very little in our sample of orbits. For this reason, here we only present the results of orbit 43442, for which correlative RO refractivity measurements are also available for some scans.

In order to assess the accuracy and computational efficiency of the
ray-tracing methods described in Sect.

Efficiency of the tested ray-tracing methods, for step sizes of 0.1, 0.25, 0.5 and
1.0

In Fig.

With the fixed US76 atmosphere, we then tested the impact of the refractive index
model used for the incoming path. In the case of a horizontally homogeneous
atmosphere such as the US76, the impact of changing the refraction model
can be estimated by the change in tangent height calculated from the
constant value of the product

Differences between recalculated and original tangent heights. Assumed atmospheres: IG2

Finally, with the same strategy we studied the impact of the assumed
atmosphere on the ray tracing. As expected, we found that this is the
assumption with the largest impact on the calculation of the height of the
tangent points. In Fig.

For the 16 limb scans of orbit 43442 for which a co-located RO measurement is
available, we repeated the test of Fig.

Note that the error in the tangent height shown in Fig.

In order to assess the influence of the newly calculated tangent heights on
the level 2 products, we retrieved orbit 43442 starting from the recalculated
tangent heights using the ESA operational algorithm

Average difference between temperature retrieved with recalculated (

The small size of the differences in the retrieved profiles is due to the
ability of the ESA retrieval code to adjust the pressure at the tangent
points and recalculate tangent height increments using the hydrostatic
equilibrium. Like the ESA operational code, several MIPAS inversion
algorithms

We analyzed the main factors driving the calculation of the tangent heights
of spaceborne limb measurements. We found that the factor with largest effect
in the tangent height calculation is the assumed atmosphere. Using
a climatological model in place of the real atmosphere may cause tangent
height errors up to

This study was supported by the ESA-ESRIN contract 21719/08/I-OL.Edited by: M. Nicolls