2.4.4.1.3.3.1.3 Computation of crosssections
The computation of
crosssections in equation
is a very time consuming
part of the forward model,
due to the large number of
spectral lines to be
considered, the high
spectral resolution
required and the number of
p, T combinations for which
they have to be
computed. In the
code the crosssections can
either be calculated line by
line (LBL), using a
preselected spectroscopic
database and fast Voigt
profile computation, or be
read from lookup tables
(LUTs). The use of
LUTs is
advantageous only if they
can be stored in random
access memory (RAM), and
the feasibility of this
depends on the amount of
memory required.
Crosssection LUTs have to
be tabulated for all the
considered microwindows,
for each absorber
g, and for an
appropriate range of
p,T
values. For a
single microwindow
(typical width 0.5
cm^{1}), the number
of data points required
is of the order of:
1000 spectral
pts. x 100 pressures x
10 temperature pts. =
10^{6} grid points   eq 2.24 
A pictorial
representation of a typical
table is shown in figure2.25 . For a
retrieval, up to 100 such tables
have to be stored (tens of microwindows
with a few absorbers per
microwindow), requiring too much
memory for most practical
applications. In
order to overcome this
difficulty, the solution
suggested by Strow in
Strow L. L., H. E. Motteler
Ref. [1.62 ]
was adopted. It consists of
compressing this information
using `Singular Value
Decomposition'. Any matrix
K (m x
n) can be decomposed as
the product of three other
matrices:
  eq 2.25 
where
U (m x
n) and
V (n x
n) are orthonormal
matrices, and is a diagonal
matrix containing n
singular values. Assuming
that most of the information is
contained in the j
(<< n) largest
singular values, the n
dimension of the decomposition
matrices can be truncated to
give:
  eq 2.26 
where the
reduced matrices
U'
(m x j) and
W'
(j x n) are
much smaller matrices than
the original matrix
K.
In this application, the matrix
K contains the
logarithm of the crosssection
and is decomposed and stored as
matrices
U'(
,j) and
W'(j,x),
where x represents any
pair of (p,T)
conditions. Typically the number
j of singular
values is less than 10, giving
compression factors of the order
of 10  100. The
number of singular values, as
well as the number of pressure
and temperature increments, are
optimized so that the maximum
difference between the
resulting LUT and LBL
convolved limb radiance is below
the microwindow
radiance noise criteria
(equal to NESR/10). In
figure2.26 the
differences between LUT and LBL
limb radiances before and
after AILS
convolution are reported for a
representative microwindow.
The combined use of
irregular
grids and LUTs means that
the absorption
crosssections need only be
reconstructed at a subset of
fine grid points. Therefore the
m dimension is
effectively reduced by an
order of magnitude, which
usually makes
U' the
smaller of the two components of
the LUT.

Figure 2.25 Plot of the logarithm of the CO<sub>2</sub> crosssection (k) tabulated for the spectral interval 693.45693.725 cm<sup>1. Each of the 12 major cycles in p, T axis corresponds to a different temperature, and within each are 6 different ln) values varying from Lorentz to Doppler broadening. 

Figure 2.26 44 km tangent height radiance from a microwindow (frequency range 693.45  693.725 cm1) selected for p,T retrieval. The upper plot shows the highresolution radiance obtained with LBL calculation of crosssection (solid line) and with the use of LUTs (dotted line), as well as the convolved radiance (dashed line for LBL calculation and (+) for use of LUTs). The lower plot shows the difference between the two methods, before AILS convolution (solid line, left axis) and after AILS convolution (dashed line, right axis). 
2.4.4.1.3.3.1.4 Computation of the
radiative transfer integral
The radiative transfer
integral (equation)
is computed as a
summation over discrete
layers as follows:
  eq 2.27 
where τ
_{l,t}
is the single layer
optical depth:
  eq 2.28 
l = 1 and l =
N (N is equal to twice the
number of layers) are
respectively the index of the
farthest and the nearest
integration step along the line
of sight with respect to the
observer, t is the
index of the limb view, B_{l,t}
is the Planck function
computed for the equivalent
temperature of the main gas of
the retrieval in layer
l and limb view
t.
and are respectively the
absorption cross section and the
column of the gas
g. The
second term of equation
eq.
2.28 is
the contribution to the
single layer optical depth of
all emission sources
characterized by a constant
amplitude in a microwindow,
the socalled 'atmospheric
continuum': is an altitude and
microwindow dependent absorption
crosssection, that is fitted by
the retrieval program,is the air column.
Since the atmosphere
is assumed horizontally
homogeneous and with a spherical
symmetry, the computation of equation
eq.
2.27 is
further accelerated taking into
account that the two
contributions of the same layer
from opposite sides of the
tangent point are characterized
by different transmissions,
but by the same emission.
Therefore equation
eq.
2.27
becomes:
  eq 2.29 
This expression is
computed at the irregular
frequency grid and at the
grid of the simulated
tangent altitudes, i.e.
the grid containing the
measured limb
views 5.3.5. and the additional
ones needed for the FOV
convolution (see section 2.4.4.1.3.3.1.6. ).
The zerolevel
calibration correction
(caused by
selfemission of the
instrument, scattering of
light into the instrument or
third order nonlinearity of
the detectors) has to be
finally applied to the
spectrum. This is performed
adding to the atmospheric
spectrum equation
eq.
2.29 a
microwindow
dependent (but tangent
altitude independent) offset
which is then fitted
by the retrieval program.
2.4.4.1.3.3.1.5 AILS convolution
The
Instrument Line Shape (ILS)
function is, by definition,
the response of the
spectrometer to a
monochromatic radiance. In
the case of a perfect
Fourier transform
spectrometer, observing a
stable source of evenly
distributed radiance, the
ILS is equal to the
convolution of the
sinc
eq.
5.3
function, associated
with the finite spectral
resolution of the
instrument, with a term due
to the finite angular
aperture. If the angular
aperture is circular, this
term is equal to a
rectangular function shifted
in wavenumber and with a
width that varies
linearly with the wavelength
Bell R. J.
Ref. [1.2 ]
. In a real instrument
alignment errors and
irregular angular aperture
lead to a more elaborate
ILS. Furthermore, the uneven
distribution of the source
radiance could introduce
the major complication of a
frequency and tangent
altitude dependent ILS
distortion. The atmospheric
radiance at the limb is
indeed characterized by an
exponential energy
distribution through the
input diaphragm.
Fortunately, it has been
demonstrated
Delbouille L. and G. Roland
Ref. [1.24 ]
that even the strongest
exponential energy
distribution across the
field of view does not
significantly affect the
ILS. The main effect of the
nonuniform energy
distribution across the
input aperture is
instead that a significant
difference can exist between
the effective tangent
altitude and the geometrical
one, and this effect is
taken into account by FOV
convolution (section 2.4.4.1.3.3.1.6. ).
The ILS function, which is
an input of the forward
model, is therefore a
function independent of the
tangent altitude and will
take into account
alignment and aperture
effects, but not the
instrument responsivity and
phase error corrections,
which are corrected for in
the Level
1B 2.4.3.2. processing.
Measured MIPAS spectra are
apodized before entering the
retrieval process. This
choice is dictated by the
use of selected spectral
intervals ('microwindows')
for the retrieval of the VMR
profiles. The advantages of
using
'microwindows'
have been already discussed
in section 2.4.4.1.3.1.1. .
The measured spectrum is
equal to the convolution
of the atmospheric spectrum
with the ILS, and
since the ILS is
characterized by '
sidelobes' decreasing
linearly in amplitude, the
atmospheric signal at a
given frequency affects the
measured signal in a
broad spectral interval.
Therefore, the simulation of
a narrow microwindow
of the measured spectrum
requires the calculation of
a broad spectral interval.
Apodization reduces the
amplitude of the side lobes
of the ILS and accordingly
the size of the spectral
interval in which the
atmospheric spectrum must be
calculated. It causes a loss
of spectral resolution, but
if a reversible apodization
function is used, no
information is lost. When
the spectrum is
convolved with the
apodization function,
correlation between
different spectral points is
introduced which appear as
offdiagonal terms in the VCM of the
observations. The
'NortonBeer
strong'
Norton R. H., R.Beer
Ref. [1.54 ]
function is used as
apodizing function. The
effects of the ILS and
the apodization are
simultaneously
considered by convolving the
atmospheric radiance with
the apodized instrument line
shape (AILS), obtained
convolving the ILS with the
apodization function.
The atmospheric radiance
computed at the irregular
grid points should first be
interpolated on the regular
fine grid (spacing 0.0005
cm^{1}) and then
convolved with the AILS
function in order to obtain
the spectrum on the
coarse grid (spacing 0.025
cm^{1}). Actually,
these two operations are
simultaneously performed
skipping the step of
interpolating the spectrum
on the regular fine grid,
with a saving in both
computation time and memory requirements.
2.4.4.1.3.3.1.6 Instrument field of
view convolution
The main effect of the
exponential variation of the
atmospheric radiance as a
function of the limb angle
through the input aperture
is that a nonnegligible
difference exists between
the signal observed along
the central line of the FOV and
the integrated signal.
The spread of the FOV in the
altitude domain is measured
experimentally and tabulated
in a FOV pattern
distribution FOV(z) whose
shape is constructed
with a linear interpolation
from a tabulated function.
The FOV pattern is assumed
to be constant as a function
of the scan angle.
The effect of field of view
is taken into account
performing, for each
spectral frequency, the
convolution between the
tangent altitude
dependent spectrum and the
FOV pattern. This
convolution requires the
forward model calculation
for a number of lines of
sight that span the
vertical range of the FOV
around the tangent altitude.
In order to reduce the
number of computations, in
the Level 2 algorithm, the
variation of the spectrum as
a function of tangent
altitude is determined by
interpolating a polynomial
through the spectra
calculated at contiguous
tangent altitudes in the
range of the FOV
pattern. Where the
radiance profile varies
rapidly with tangent
altitude, additional spectra
may be simulated at
intermediate tangent levels
to maintain the
numerical accuracy of the
convolution. In figure2.27 the
values of the spectrum at a
significant frequency
calculated with an
analytical convolution at
altitudes between 9.5
and 12.5 km are plotted as a
function of the
corresponding values of the
spectrum calculated with a
reference numerical
convolution. The results of
the analytical convolution
for both quartic and
parabolic interpolation are
shown for a critical case
(in the atmospheric
model used for these
simulations, the tropopause
is located at 11 km). The
deviation of the curve from
a straight line
indicates the presence of a
potential error in the
computation of the
analytical derivative of the
spectrum with respect to the
tangent pressure (section 2.4.4.1.3.3.2.
). The
simultaneous computation of
the whole sequence of limb
scanning spectra,
which is required by the
global fit approach, allows
a simple and efficient
computation of the FOV
convolution and avoids the
reiterated computation of
spectra with adjacent
tangent altitudes.

Figure 2.27 FOV convolved limb radiance values, at a significant frequency, calculated analytically at altitudes between 9.5 and 12.5 km versus corresponding reference values obtained by means of numerical convolution. The analytical convolution performed using a quadratic interpolation (made drawing a parabola through 3 spectra with tangent altitudes in the range of FOV pattern) is compared with the one obtained drawing a polynomial of the fourth degree (quartic interpolation) through 5 spectra with tangent altitudes in the FOV pattern range. The reference FOV convolved spectral values are computed simulating spectra at 100 m distant tangent altitudes. The deviation of the curves from a straight line indicates the presence of an error in the interpolated spectrum and hence of a potential error in the computation of the analytical derivatives. 
2.4.4.1.3.3.2 Jacobian calculation
Another important part of the
retrieval code is the fast
determination of the derivatives
of the radiance with respect to
the retrieval parameters. In
the following we will first
present the five different types
of derivatives which have to be
computed and then explain the
procedure implemented for their
calculation. As already stated
in section 2.4.4.1.3.1. , the
retrieval parameters are: (1)
volume mixing ratios of
atmospheric trace gases at
tangent pressures, (2) atmospheric
continuum values at tangent
pressures, (3) the tangent
pressures themselves, (4)
temperature at tangent
pressures, and (5) the zerolevel
calibration correction. As
described in section 2.4.4.1.3.4.4. , a
retrieval grid is chosen which
coincides with the tangent
points. As a consequence, the
number of parameters in (1), (3)
and (4) is equal to the number
of tangent points. By contrast,
the atmospheric continuum
(2) and the zerolevel
calibration correction (5) are
assumed to be microwindow
dependent, and the atmospheric
continuum also tangent altitude
dependent. Thus, the number of
atmospheric continuum parameters
is equal to the number of
microwindows times the
number of tangent altitudes at
which the individual
microwindows are used, while the
number of zerolevel calibration
correction parameters is
equal to the number of
microwindows. The
numerical computation of the
derivatives requires, for each
retrieved parameter, an extra
forward model calculation with
an increment applied to that
parameter. Whenever possible,
derivatives are computed
analytically in the sense that
analytical formulas of the
derivatives are implemented in
the program (this is the case of
the retrieval parameters (1),
(2), (3) and (5)). Where the
calculation of sufficiently
precise analytical derivatives
requires computations as time
consuming as the calculation of
spectra (parameters (4)), an
optimized numerical procedure is
implemented.
Derivatives with respect to
parameters (1) and (2) are
handled in a similar way. The
aim is the calculation of the
derivatives of the spectrum
L provided by equation
eq.
2.27 with
respect to the atmospheric
retrieval parameters x_{n}
on each tangent level
n. The expression for
these derivatives reads:
  eq 2.30 
. If
we neglect the 2^{nd} order
dependence of equivalent
temperatures on the parameters x_{n}
, the first term of (4.12)
eq.
2.30 is equal to
0. The second term requires the
calculation of the derivative of the
radiance with respect to the optical
depth and the derivative of the
optical depth with respect to the
retrieval parameter. The derivative
of the radiance with respect to the
optical depth is equal to:
  eq 2.31 
In this
expression, the first term is the
derivative of the emission of layer
j attenuated by all layers
between j and the
observer, while the second term is
the derivative of the attenuation
for the radiation of each layer up
to layer j1 and
represents the emission spectrum
measured by the observer due to the
first (j1) layers.
The derivative of the optical depth
with respect to each parameter of
the atmospheric continuum reads
simply:
  eq 2.32 
whereis the continuum cross
section of the jth
layer and the continuum
parameter, i.e the continuum cross
section value at the nth
tangent level relative to a given
microwindow. Due to the limited
influence of the single change of a
continuum parameter on the whole
continuum profile, the only
terms which are different
from 0 are the ones corresponding to
layers between tangent levels
(n+1) and (n1).
The derivative of the
optical depth with respect to the
VMR parameters is equal to:
  eq 2.33 
Neglecting the
dependence of equivalent temperature
and pressure on the VMR parameters
it follows that . During the
calculation of the CurtisGodson
equivalent quantities, only the
derivatives of the partial columns
of each layer with respect to the VMR at the tangent
level have to be determined.
The analytical derivative of the
spectrum with respect to the tangent
pressure (3) can be very
complicated, since changes in the
column, in the lineshapes and
in the temperature have to be
considered when tangent pressure is
perturbed. The problem was overcome
by exploiting the fact that the
effect of FOV convolution is
to shift the 'effective'
tangent pressure of the
spectrum. Therefore, the derivative
of the spectrum with respect to
tangent pressure is calculated
performing an analytical
derivative of the expression that
provides the convolution between the
spectrum interpolated as a function
of tangent pressure and the FOV pattern. The
accuracy of the derivative
calculated in this way is strictly
connected with the accuracy of
the interpolated spectrum in the
range where the FOV pattern is
defined (figure2.27 ).
The temperature derivatives are
determined in a 'fast
numerical' way. In contrast
with slow numerical derivatives, in
the 'fast numerical'
calculation the derivatives are
computed in parallel with the
spectra, avoiding unnecessary
repeated calculations. The
implemented fast numerical
derivatives with respect to tangent
temperature make use of the limited
influence of the change of one
temperature parameter on the overall
temperature profile: only the
CurtisGodson equivalent
temperatures of the neighboring
layers above and below the
tangent altitude (relating to the
considered parameter) are affected.
Hence, in addition to the
calculation of crosssections
and radiative transfer corresponding
to the unperturbed original
temperature profile, needed for the
forward calculation, only
crosssections relating to the
layers affected by the current
temperature parameter change and
radiative transfer as modified by
the changed crosssections are
computed. The derivatives
with respect to the zerolevel
calibration correction (5) are equal
to 1, since the zerolevel
calibration correction is simply
an additive term of the simulated
spectrum. With the
adopted optimizations the additional
computing effort required for the
Jacobian calculation is only twice
the effort needed for one
forward model run. This is a very
interesting result, considering that
the full numerical calculation of
the derivatives would require as
many forward model reruns as many
are the fitting parameters (≈
100).
2.4.4.1.3.3.3 Convergence criteria
The convergence criteria adopted
in our code are a compromise
between the required accuracy of
the parameters and the computing
speed of the algorithm. The
following three convergence
conditions are used to achieve this:
 Condition on linearity: at
the current iteration
'iter'
the relative difference
between the actual and the
expected value of chisquare
computed in the linear
approximation () must be less
than a fixed threshold t
_{1}:

  eq 2.34 
where is computed
using the expression:
  eq 2.35 
 Condition on attained
accuracy: the relative
correction that has to be
applied to the parameters
for the subsequent iteration
is below a fixed
threshold t_{2}
i.e.:

  eq 2.36 
. Different
thresholds are used for the
different types of
parameters depending on
their required accuracy.
Furthermore, whenever an
absolute accuracy
requirement is present for a
parameter, the absolute
variation of the parameter
is checked instead of the
relative variation
considered in equation
eq.
2.36 .
The nontarget
parameters of the retrieval,
such as continuum and
instrumental offset
parameters are not included
in this check. Parameters
equal to zero are
excluded from this check as well.
 Condition on computing time:
the maximum number of
iterations must be less than
a given threshold.
The convergence is reached
if either condition equation
eq.
2.34 or
condition equation
eq.
2.36 is
satisfied. If only the condition on
computing time is satisfied, the
retrieval is considered
unsuccessful.
2.4.4.1.3.4 Choices and assumptions in
the forward and retrieval algorithms
In this section we report de
description of specific issues
regarding both the strategy adopted
in the version algorithm and the
simplifications / shourtcomings
implemented in the forward model
2.4.4.1.3.4.1 Use of complementary
information in the inversion model
MIPAS limb scanning retrievals
have redundancy of measurements,
such that stable vertical
profiles of atmospheric state
parameters can be retrieved
without constraining the
retrieval with apriori
knowledge. On the other hand,
when some complementary
information on the unknown
parameters is available, the
quality of retrieved parameters
can be improved by including
this information in the
retrieval process.
The external information can be
either of 'general'
type (as in the case of
climatological data or model
forecasts) which, as such,
may apply to several MIPAS
observations made at different
locations and at different time,
or of 'specific' type,
which relates to individual
MIPAS scans. The
exploitation of
'general' information
is not desirable because the
same external information can be
used for the retrievals of
several sequences so that the
results of these retrievals are
affected by a common bias and
can no longer be considered
independent measurements.
The correlation between
subsequent profiles would add
extra complexity in the use of
the data in geographical maps
and averages. On
the other hand, complementary
information relating to the
individual MIPAS sequences (e.g.
line of sight data, or data
relating to the same air
mass actually sounded by the
considered MIPAS scan) can be
profitably included in the
retrieval, as stated in section 2.4.4.1.3.2.1. , using
equation
instead of equation. In
fact, engineering data defining
the instrument Line Of Sight
(LOS) are
updated at each scan and
therefore constitute an
effective and
'specific' source of
information which can be
routinely used in p,T retrievals
without introducing a bias in
the retrieved profiles.
The LOS
engineering information consists
of both a vector
z
containing the tangent altitudes
of the current scan and of a VCM
V
_{z} related to the
vector
z. A relationship
between the engineering
tangent heights and the unknowns
of the p,T retrieval is provided
by hydrostatic equilibrium
constraint. This law,
generally fulfilled in normal
atmospheric conditions,
especially in the stratosphere,
provides a relationship between
pressure, temperature and
altitude. Assuming known
pressure and temperature
distributions and a reference
altitude , the
altitude relating to
pressure is calculated as:
  eq 2.37 
where M
is the average molecular weight of
the atmosphere, R the
universal gas constant, g_{j}
the acceleration of
gravity relating to the
jth atmospheric layer (its
dependence on altitude and latitude
is derived from the WGS84 model
Department of Defense
Ref. [1.25 ]
) and is the average
temperature of the jth
atmospheric layer. The
engineering measurements of tangent
altitudes are linked with the
unknowns of the inversion problem
(p, T ) by equation
eq.
2.37 . In
particular, equation
eq.
2.37 can be
locally linearized providing:
  eq 2.38 
Matrix K
_{z}
is the Jacobian connecting the
engineering tangent altitudes with
the unknowns p,T and is
obtained deriving equation with
respect to p and
T. In this case the
solution of the retrieval problem is
found by simultaneously inverting equation and equation
eq.
2.38 . The
solution formula is in this case
provided by equation and is
equal to:
  eq 2.39 
In our approach
climatological data only contribute
to the definition of the first guess
profile, and profiles which are used
within the forward calculation
but not retrieved (i.e.
contaminants). The first guess is
obtained by combining climatological
data and the previously
retrieved profile with optimal
estimation equation. In
this case, the optimal estimation
helps in reducing the number of
retrieval iterations, but does not
directly contribute to the retrieval
error.
2.4.4.1.3.4.2 Profile regularization
In some cases the retrieved
profiles vary as a function of
altitude with an oscillation
that is greater than it is
physically reasonable to
expect. This oscillation is
intrinsic to the retrieval
problem, because the solution is
represented in a base of
functions different from the
base of the observations
identified by the Jacobian: if
the base of the solution
contains some components that
are nearly orthogonal to the
base of the measurements, these
components are sensitive to
small variations of noise with a
consequent instability of
the solution. The
techniques intended to reduce
these instabilities are called
'regularisation'
techniques. For instance,
TikhonovPhillips [26]
Ref. [1.63 ]
regularisation consists in the
minimization of the function:
,   eq 2.40 
where
x is the unknown,
L is an appropriate
operator which determines the type
of constraint, m is
the regularisation parameter which
determines the relative weight of
the two conditions and x
_{0} is the apriori
estimate of the solution, which is
usually set equal to zero.
The solution minimizing the function
equation
eq.
2.40 , in a
Newtonian iteration, is given by:
.   eq 2.41 
As indicated by equation
eq.
2.40 , the
TikhonovPhillips
regularization minimizes a
modified and, depending
on the choice of the apriori
estimate x
_{0}, a bias may be
introduced in the retrieved
profile. The bias can be
avoided by choosing the apriori
estimate equal to the profile
estimate of the current
iteration: . In this case the
solution equation
eq.
2.41 is
equal to:
  eq 2.42 
LevenbergMarquardt solution is a
particular case of equation
eq.
2.42 , obtained
with
L=I.
Like in the case of the
LevenbergMarquardt method, also in
the case of equation
eq.
2.42 the
regularization operates only on the
correction that is determined in
the current iteration and stepwise
the same exact solution is
eventually reached. For this
reason the choice of is considered in
the literature
Rodgers C. D.
Ref. [1.59 ]
not to be a real
regularization. We find
however that the choice of serves our
objective of not changing the
function to be minimized (in order
to avoid the associated bias) while
successfully damping in the
retrieval the undesirable
components. The operator
L provides a filter
to the oscillatory components of the
correction y.
These components are added to the
retrieved profile to the extent that
they are well measured. In this way
the convergence path, followed
by the Newtonian iterations, is
modified even if still aiming at the
same convergence point. When the
convergence criteria are satisfied a
different solution is found in which
the oscillatory components, if
poorly determined by the
measurements, have not yet been
amplified as much as otherwise
possible. This is obtained with a
possible increase of the number of
iterations but with an equally good
value of the function. The apparent
contradiction of an unbiased
solution different from the
'exact' one and satisfying
to the same convergence criteria
can be explained considering that
the ' damping' introduced
by the operator L
acts only on the poorly
determined components, which by
definition cause a small difference
in the observations and therefore do
not change significantly
the function. An
optional regularisation of the type
equation
eq.
2.42 is
implemented in MIPAS Level 2
processor with L
^{T}
L equal to a
stripdiagonal matrix in which
elements different from 0 are only
the diagonal elements and the first
offdiagonal ones. Due to the
constant (userdefined) value of
µ, this regularisation proves
to be more effective than the
LevenbergMarquardt method in
damping oscillations in both
iteration and altitude domain.
The value of µ is determined on
the basis of operational needs.
2.4.4.1.3.4.3 Levels versus layers
The retrieval allows only the
determination of a discrete
representation of the vertical
profile. Several options can be
considered for the
discretisation of the unknown
profile, the most common being a
representation at discrete
levels with linearly
interpolated values at
intermediate altitudes
Carlotti M. and B. Carli
Ref. [1.14 ]
and a representation at
discrete layers with
constant value within the
thickness of the layer.
Since the spectrum contains
information mainly on the
average VMR in each
layer and not on the
specific values of VMR at
adjacent levels and since a
nonunique relation exists
between VMR at
adjacent levels and the average
mixing ratio in the layer
between the levels, more stable
results can be obtained with
the layer representation rather
than with the level
representation
Clarmann T. v., H.
Fischer, and H. Oelhaf
Ref. [1.22 ]
. However, the
level representation is more
easily used in plots and is
considered to result into a more
user friendly product.
Considering the small
mathematical difference that in
any case exists between the two
representations, the
representation at levels has
been chosen.
2.4.4.1.3.4.4 Retrieval vertical grid
The vertical resolution and the
accuracy with which the
retrieved profile is determined
are generally anticorrelated
(see
Ref. [1.14 ]
) and
are strongly dependent on the
grid where the retrieved points
are represented ('retrieval
grid'). In the case of
onion peeling method, the
unknown values can only be
retrieved at the measurement
grid, i.e. the grid of the
measured tangent altitudes. With
global fit, the
'retrieval grid' can
be different from the
'measurement grid'.
Since in Level 3 data processing
global maps on pressure
surfaces are produced, the
possibility offered by the
global fit of using fixed
pressure levels which will in
general be different from the
tangent altitude levels could
directly meet user needs.
The performances of the
retrieval code when the two
different retrieval grids are
used have been compared for
different measurement and
retrieval scenarios with the
method described in
Carlotti M. and B. Carli
Ref. [1.14 ]
. This analytical
procedure allows the estimation
of the random error that is
associated with each value of
the retrieved profile when a
given vertical resolution is
adopted, or, conversely, of the
vertical resolution that can be
reached when the random errors
must be contained within a
given limit. When the
retrieval is performed on the
measurement grid and then the
profile is interpolated to a
fixed grid, the vertical
resolution of the resulting
profile is degraded and its
accuracy improved
Carlotti M. and B. Carli
Ref. [1.14 ]
,
Carli B., M. Ridolfi and CO
Ref. [1.11 ]
. It has been found that
there is little difference
in vertical resolution and
accuracy between retrieving on
the measured grid and retrieving
on a fixed grid shifted with
respect to the measured
one. On the other
hand, if the retrieval is
performed on the nominal 3 km
grid and the measurements have
been performed on a
'stretched' grid (e.g.
3.5 km spacing), undesirable
oscillations are present in the
retrieved profile. These can be
avoided only if the
retrieval grid is made coarser
than any possible measurement
grid or if some profile
regularisation is imposed.
The final choice was to retrieve
vertical profiles at an altitude
grid defined by the tangent
altitude levels, since this
provides the best
information that can be derived
from the measurements and the
most stable results.
2.4.4.1.3.4.5 Vertical resolution
The vertical resolution of the
retrievals depends on both
experimental choices
(instantaneous FOV and
scanning altitude step), and on
the profile representation
within the retrieval (spacing of
the retrieved points).
Improvements in the vertical
resolution are always obtained
at the expenses of either
retrieval accuracy or, through
the measurement time,
horizontal resolution. A
preliminary model of MIPAS FOV
has been used, which assumes the
FOV shape to be a trapezium with
the greater base equal to 4 km
and the smaller base equal to 3
km. Assuming this FOV shape,
a 3 km spacing has been found to
be a good resolution/accuracy
compromise for both the
measurements and the retrieval grid.
2.4.4.1.3.4.6 Assumptions
In order to limit the complexity
of the code and meet the
computing time requirements,
some simplifications have been
adopted in the forward
model. In particular, some
effects have up to now been
neglected in both the
spectroscopic and the
atmospheric model.
Neglected effects in the
spectroscopic model are:
 line mixing
Edwards D. P. and L.
L. Strow
Ref. [1.31 ]
,
Rosenkranz P. W
Ref. [1.60 ]
occurring when
collisions between a
radiating molecule and the
broadening gas molecules
cause the transfer of
population between
rotationalvibrational
states. Line mixing affects
especially the Qbranches
where transitions between
rovibrational energy levels
closer than K_{B}T
(K_{B} is the
Boltzmann constant, T is the
temperature) are packed
together. The most apparent
effect of linemixing is a
reduction of the
crosssection in the wings
of the branch. The impact of
linemixing effects, mainly
significant for
CO_{2} lines, is
reduced by using an
appropriate selection of microwindows.
 pressure shift
Rosenkranz P. W
Ref. [1.60 ]
, that is significant
only at high pressures,
is not foreseen to affect
MIPAS spectra, because MIPAS
penetrates to the tangent
altitude of 8 km as a minimum.
Both these effects could
be taken into account without an
increase of the computing time if
they are modeled by the program that
generates the LUTs.
Concerning the atmospheric
model, the following assumptions
have been made:
 Assumption on local
thermodynamic equilibrium (LTE).
Level 2 algorithm assumes the
atmosphere in local
thermodynamic equilibrium: this
means that the temperature of
the Boltzmann distribution
is equal to the kinetic
temperature and that the source
function in equation
eq.
2.18 is
equal to the Planck function at
the local kinetic
temperature. This LTE model is
expected to be valid at the
lower altitudes where
kinetic collisions are frequent.
In the stratosphere and
mesosphere excitation mechanisms
such as photochemical processes
and solar pumping,
combined with the lower
collision relaxation rates make
possible that many of the
vibrational levels of
atmospheric constituents
responsible for infrared
emissions have excitation
temperatures which differ from
the local kinetic temperature.
It has been found
LopezPuertas M. and CO
Ref. [1.48 ]
that many CO _{2}
bands are strongly affected by
non LTE. However,
since the handling of NonLTE
would severely increase the
retrieval computing time,
it was decided to select only
microwindows whose emission is
in thermodynamic equilibrium to
avoid NonLTE calculations
in the forward model.
 Assumption of horizontally
homogeneous atmosphere
Limb sounding attains good
sensitivity due to the long path
lengths obtainable, but this
necessarily requires measurements
which `average' the
atmosphere over long horizontal
distances. With limbscanning,
there is the associated problem that
the profile of acquired tangent
points is sheared horizontally,
partly by the variation in elevation
angle and partly by the satellite
motion. A third problem is the
assumption that the retrieved value
at one altitude can be used to model
the contribution of the atmosphere
at that level along the ray paths
for lower tangent heights,
whereas in reality these paths all
intersect the altitude surface at
different locations. Each of these
effects has a horizontal length
scale of the order of several
hundred kilometers, and ignoring
these effects is the equivalent of
assuming that the atmosphere is
horizontally homogeneous over this
distance. Studies
Carli B., M. Ridolfi and CO
Ref. [1.11 ]
have shown that the retrieval
accuracy is particularly
sensitive to horizontal temperature
gradients. For example, ignoring a
temperature gradient of 3 K / 100 km
(a typical maximum, e.g.
associated with crossing the polar
vortex) can lead to composition
retrieval errors of tens of %,
although these errors are
localized and usually associated
with regions in which the
atmospheric composition is also
changing rapidly. Several
approaches
Carlotti M., B.M.Dinelli and CO
Ref. [1.12 ]
,
Carlotti M., B.M.Dinelli,
P.Raspollini, M.Ridolfi
Ref. [1.71 ]
have been considered in order
to allow for horizontal
inhomogeneity, but none of them
appear to be suitable for NRT
operational processing. We must,
therefore, be aware of the
assumption of horizontally
homogeneous atmosphere when
observing air masses with steep
gradients.
2.4.4.1.3.5 Algorithm validation
To validate the approximations
implemented in the forward model
internal to the Level 2 processor
(that is called 'Optimized
Forward Model' = OFM),
comparisons were made with a
specially developed linebyline
code based on GENLN2
Edwards D. P
Ref. [1.30 ]
. This code was compared with
several existing codes and was
elected as our reference forward
model (RFM)
Edwards D. P
Ref. [1.29 ]
. The main results of the RFM /
OFM intercomparisons are:
 Raytracing: for N_{2}O,
10 km tangentheight path
(representing the most
'difficult' case
involving both large VMR gradients
and refraction effects) RFMOFM
calculations differ by less than
0.7% in the slant column
calculations, less than 0.004%
in the CurtisGodson pressure
calculation and less than 0.002
K in CurtisGodson temperature calculation.
 Crosssection calculations: RFM and OFM full
spectral calculations agree to
better than 1 % near major
absorption features.
 Limb spectral
calculations: RFMOFM limb
radiance calculations agree to
within NESR/4 (the
values of the NESR in the
different spectral ranges are
reported on the figure below:

Figure 2.28 
The factor 4 is
intended to account for the
possible building up of
systematic effects and for the
achievement of a NESR better
than the requirements.
The retrieval code has been
validated by performing retrievals from
spectra generated by its own forward
model and by the RFM. Tests are in
progress (January 2001) using with
spectra obtained with the balloon
instrument MIPASB2. The
results obtained so far indicate that
both forward model error, i.e. error due
to imperfect modeling of the atmosphere,
and convergence error, i.e. error
due to the fact that the inversion
procedure does not find the real minimum
of the χ^{2} function,
are much smaller than the measurement
error due to radiometric noise.
2.4.4.1.3.6 Performances
2.4.4.1.3.6.1 Accuracy performance of
Level 2 retrieval algorithm
The main error sources that
affect the accuracy of the
retrieved profiles are:
 noise error, due to the
mapping of radiometric noise
in the retrieved profiles;
 temperature error,
which maps into VMR
retrieved profiles;
 systematic error,
due to incorrect input parameters.
The amplitude of noise
error has been evaluated (through equation
eq.
2.13 ) with
test retrievals that use
observations simulated starting from
assumed atmospheric profiles
(reference profiles) and
perturbed with random noise of
amplitude consistent with MIPAS
noise specification. The
effect of temperature error on
VMR retrievals is determined
using tabulated propagation
matrices which estimate the
effect for different
measuring conditions. Current
results indicate that
temperature error can be a
significant component of the
error budget and
consideration is being given to
methods to improve the accuracy
of temperature retrieval. (details on p,T
error propagation 2.4.4.3. )
Errors of the third type include
systematic errors, such as
spectroscopic errors or errors
due to imperfect knowledge of
the VMR profiles of
nontarget species. These errors
are taken into account in the
definition of the optimum size
of each microwindow
and for the
selection of the optimal set
of microwindows 2.4.4.4. that
should be used for the
retrieval. The quantifiers that
are calculated for these
operations can also be used for
the determination of the total
error budget.
The relevance of systematic
errors in the total error budget
depends on whether optimistic or
conservative error estimates are
used. The current estimate
of the ultimate retrieval
accuracy is summarized in the
following plot that reports the
total error as a function of the
altitude for each of the
retrieved constituents and for temperature.

Figure 2.29 
2.4.4.1.3.6.2 Runtime performance
Note: here
is a section illustrating
only the runtime performance
of the ORM. What about the
L2 processor ????
The runtime
performances of the ORM have been
tested using different
computers. Tests have been
performed on simulated
observations using two different
sets of microwindows,
a preliminary standard set and a
set which optimizes the
tradeoff between accuracy and
runtime performance. In
these tests we used initial
guess profiles of the retrieval
that are sufficiently close to
the reference profiles (the ones
used to simulate the
observations), so that
convergence is reached in only
one iteration. The results of
these tests are shown in the
table below.
Considering that the measurement
time per scan is 75 seconds and
that more than one computer can
be used for the operational
analysis data, we can
conclude that the runtime
requirements are fully satisfied
also for retrievals that need
more than one iteration.

Table 2.2 Table: Runtime (sec.) for p,T and 5 target species retrieval (1 iteration)

Computer description

Standard
set of MWs

Optimized
set of MWs

SUN
SPARC station 20
120
MHz CPU, 128 Mb RAM

550 (*)

348 (*)

PENTIUM PC
200
MHz CPU, 256 Mb RAM

352

210

Ultra
Sparc station 5

181

Not Available

IBM
RS6000 Model 397

149

Not Available

Digital
DECSERVER Mod. 4100
600
MHz CPU, 1 Gb RAM

74

51

(*) This
runtime is strongly
affected by the use of swap space
