ESA Earth Home Missions Data Products Resources Applications
    24-Jul-2014
EO Data Access
How to Apply
How to Access
Index
MIPAS Data Formats Products
Records
2 MDSR per MDS 1 forward sweep 1 reverse sweep
2 MDSRs per MDS 1 forward sweep 1 reverse sweep
LOS calibration GADS
Spectral Lines MDS
P T Retrieval MW ADS
VMR Retrieval Parameters GADS
P t Retrieval GADS
Framework Parameters GADS
Processing Parameters GADS
Inverse LOS VCM matrices MDS
General GADS
Occupation matrices for vmr#1 retrieval MDS
MDS2 -- 1 mdsr forward sweep 1 mdsr reverse
Occupation matrices for p T retrieval MDS
General GADS
Priority of p T retrieval occupation matices
P T occupation matrices ADS
MIP_NLE_2P SPH
Summary Quality ADS
Instrument and Processing Parameters ADS
Microwindows occupation matrices for p T and trace gas retrievals
Scan information MDS
Level 2 product SPH
MDS1 -- 1 mdsr forward sweep 1 mdsr reverse sweep
H2O Target Species MDS
P T and Height Correction Profiles MDS
Continuum Contribution and Radiance Offset MDS
Structure ADS
Summary Quality ADS
Residual Spectra mean values and standard deviation data ADS
PCD Information of Individual Scans ADS
Instrument and Processing Parameters ADS
Microwindows Occupation Matrices ADS
Scan Information MDS
1 MDSR per MDS
Scan Geolocation ADS
Mipas Level 1B SPH
Calibrated Spectra MDS
Structure ADS
Summary Quality ADS
Offset Calibration ADS
Scan Information ADS
Geolocation ADS (LADS)
Gain Calibration ADS #2
Gain Calibration ADS #1
Level 0 SPH
DSD#1 for MDS containing VMR retrieval microwindows data
DSD for MDS containing p T retrieval microwindows data
VMR #1 retrieval microwindows ADS
P T retrieval microwindows ADS
1 MDSR per MDS
VMR profiles MDS (same format as for MIP_IG2_AX)
Temperature profiles MDS (same format as for MIP_IG2_AX)
Pressure profile MDS (same format as for MIP_IG2_AX)
P T continuum profiles MDS (same format as for MIP_IG2_AX)
GADS General (same format as for MIP_IG2_AX)
Level 0 MDSR
Values of unknown parameters MDS
Computed spectra MDS
Jacobian matrices MDS
General data
Data depending on occupation matrix location ADS
Microwindow grouping data ADS
LUTs for p T retrieval microwindows MDS
GADS General
P T retrieval microwindows ADS
ILS Calibration GADS
Auxilliary Products
MIP_MW1_AX: Level 1B Microwindow dictionary
MIP_IG2_AX: Initial Guess Profile data
MIP_FM2_AX: Forward Calculation Results
MIP_CS2_AX: Cross Sections Lookup Table
MIP_CS1_AX: MIPAS ILS and Spectral calibration
MIP_CO1_AX: MIPAS offset validation
MIP_CL1_AX: Line of sight calibration
MIP_CG1_AX: MIPAS Gain calibration
MIP_SP2_AX: Spectroscopic data
MIP_PS2_AX: Level 2 Processing Parameters
MIP_PS1_AX: Level 1B Processing Parameters
MIP_PI2_AX: A Priori Pointing Information
MIP_OM2_AX: Microwindow Occupation Matrix
MIP_MW2_AX: Level 2 Microwindows data
MIP_CA1_AX: Instrument characterization data
Level 0 Products
MIP_RW__0P: MIPAS Raw Data and SPE Self Test Mode
MIP_NL__0P: MIPAS Nominal Level 0
MIP_LS__0P: MIPAS Line of Sight (LOS) Level 0
Level 1 Products
MIP_NL__1P: MIPAS Geolocated and Calibrated Spectra
Level 2 Products
MIP_NLE_2P: MIPAS Extracted Temperature , Pressure and Atmospheric Constituents Profiles
MIP_NL__2P: MIPAS Temperature , Pressure and Atmospheric Constituents Profiles
Glossary
References
Glossaries of technical terms
Level 2 processing
Pointing
Miscellaneous hardware and optical terms
Spectrometry and radiometry
Data Processing
Alphabetical index of technical terms
Frequently Asked Questions
The MIPAS Instrument
Inflight performance verification
Instrument characteristics and performances
Preflight characteristics and expected performances
Subsystem description
Payload description and position on the platform
MIPAS Products and Algorithms
Data handling cookbook
Characterisation and calibration
Calibration
Latency, throughput and data volume
Auxiliary products
Level 2
Instrument specific topics
Algorithms and products
Level 2 products and algorithms
Products
The retrieval modules
Computation of cross-sections
Level 1b products and algorithms
Products
Calculate ILS Retrieval function
Level 1a intermediary products and algorithms
Product evolution history
Definition and convention
MIPAS Products User Guide
Image gallery
Further reading
How to use MIPAS data?
Summary of applications and products
Peculiarities of MIPAS
Geophysical coverage
Principles of measurement
Scientific background
MIPAS Product Handbook
Services
Site Map
Frequently asked questions
Glossary
Credits
Terms of use
Contact us
Search


 
 
 


2.4.4.1.3 The retrieval modules

2.4.4.1.3.1 General features of the adopted approach

In contrast with the data processing of already flown operational limb sounding instruments (e.g. on the Nimbus-7 and UARS Satellites), which have been mainly of radiometric type, MIPAS data analysis will be confronted with the exploitation of broad band and high resolution spectral measurements which contain information about several atmospheric constituents that are each observed in several spectral elements. The multiplicity of unknowns and the redundancy of the data to be handled lead to the adoption of a retrieval strategy based upon the following three choices.

2.4.4.1.3.1.1 Use of Microwindows

The redundancy of information coming from MIPAS measurements makes it possible to select a set of narrow (less than 3 cm-1 width) spectral intervals containing the best information on the target parameters, while the intervals containing little or no information can be ignored. The use of selected spectral intervals, called  'microwindows', allows the size of analyzed spectral elements to be limited and avoids the analysis of spectral regions which are characterized by uncertain spectroscopic data, interference by non-target species, Non-Local Thermal Equilibrium (NLTE) and line mixing effects. More generally, priority can be given to the analysis of spectral elements with most information on the target species and less affected by systematic errors, e.g. for VMR retrievals transitions with weak temperature dependence can be preferred in order to minimize mapping of temperature uncertainties on to the resulting VMR vertical profiles. 2.4.4.3.
By analyzing the sensitivity of the radiance to changes of target parameters, lists of appropriate microwindows have been selected for the retrieval of H2O, O3, HNO3, CH4, N2O and NO2 as well as for the joint retrieval of pressure and temperature Clarmann T. v., A. Dudhia, and CO Ref. [1.17 ] . A microwindow database 2.4.4.4. has been created and is currently being refined with respect to minimization of retrieval errors, following the approach illustrated in Clarmann T. v., and G. Echle Ref. [1.28 ] .

2.4.4.1.3.1.2 Sequential retrieval of the species

The unknowns of the retrieval problem are:

  • the observation geometries, that are identified by the 'tangent pressures', i.e. the values of pressure corresponding to the tangent points of the limb measurements (pressure is the independent variable and it is used as the altitude coordinate of all the profiles) ;
  • the profile of temperature;
  • the profile of  VMR of the five target species;
  • the atmospheric continuum, that includes all the emission sources that are frequency independent within a microwindow.
  • zero-level calibration correction, that accounts for additive microwindow dependent offsets that could remain uncorrected after Level 1b 2.4.3.2. processing.
These unknowns are retrieved using the following sequence of operations. First temperature and 'tangent pressures' are retrieved simultaneously (p,T retrieval), then the target species VMR profiles are retrieved individually in sequence. The reason of this approach is that a simultaneous retrieval of all the species would require a huge amount of computer memory, because the size of the matrices to be handled by the retrieval is proportional to the product between the unknown parameters and the number of observations.
The feasibility of the simultaneous retrieval of pressure and temperature has been investigated in Carlotti M., and M. Ridolfi Ref. [1.13 ] and Clarmann T. v., A. Linden,and CO Ref. [1.19 ] . Simultaneous p,T retrieval exploits the hydrostatic equilibrium assumption, that provides a relationship between temperature, pressure and geometrical altitude. The use of the hydrostatic equilibrium assumption is discussed in this  section eq. 2.37 .
The sequence of the target species retrieval has been determined according to the degree of their reciprocal spectral interference and is: H2O, O3, HNO3, CH4, N2O and NO2 .
In p, T retrieval the quantities to be fitted are the 'tangent pressures', the temperature profile sampled in correspondence of the 'tangent pressures' and the parameters of atmospheric continuum and zero-level calibration correction. In each VMR retrieval, the fitted quantities are the VMR altitude distribution of the considered gas sampled at the tangent pressures and the parameters of atmospheric continuumand instrumental zero-level calibration correction.

2.4.4.1.3.1.3 Global Fit analysis of the Limb-Scanning sequence

A global fit approach Carlotti M Ref. [1.15 ] is adopted for the retrieval of each vertical profile. This means that spectral data relating to a complete limb scan sequence are fitted simultaneously. Compared to the onion-peeling method McKee T. B., R. I. Whitman, J. J. Lambiotte jr Ref. [1.52 ] , Goldman A., R. S. Saunders Ref. [1.42 ] , the global fit provides a more comprehensive exploitation of the information and a rigorous determination of the correlations between atmospheric parameters at the different altitudes. Besides, it permits the full exploitation of hydrostatic equilibrium condition and is better compatible with the modeling of the finite field of view (FOV) of the instrument.

2.4.4.1.3.2 Mathematics of the retrieval

In the inversion algorithm the syntetic spectra simulated using a radiative transfer model (forward model) through an inhomogeneous atmosphere are fitted to the observed spectra. The simulations are fitted to the observations by varying the input parameters of the model (such as pressure, temperature, VMR, etc...) according to a non-linear Gauss-Newton procedure. Further details of the mathematics of the retrieval model can be found here 2.4.4.1.3.2.1. .

2.4.4.1.3.2.1 Details

The problem of retrieving the altitude distribution of a physical or chemical quantity from limb-scanning observations of the atmosphere falls within the general class of problems that requires the fitting of a theoretical model, describing the behavior of the observed system, to a set of available observations Ref. [1.65 ] , Ref. [1.41 ] , Ref. [1.53 ] , Ref. [1.61 ] .
The instrument observes the radiance image emitted by the atmosphere at different values of the spectral frequency image and of the limb-viewing angle θ. The theoretical model (or forward model image) simulates the observations through a set of parameters p and q: p represents the quantities that affect the radiance but are not retrieved, and q the quantities that are retrieved, i. e. the distribution profile of the atmospheric quantity under investigation.
The retrieval procedure consists of the search for the set of values of the parameters q that produces the "best" simulation of the observations.
It has to be noted that the real atmospheric profile is a continuous function, but, since there will always be a finite number of measurements (n), in order to constrain the problem, the unknown function is approximated with a discrete representation that we indicate with a vector x of dimension m, where m < n. Between the m discrete points, an interpolated value is then used in the forward model.
Assuming a normal (Gaussian) distribution for the measurement errors, the approach generally used for determining the parameters which produce the best simulation of the observation is the least-square fit. This procedure, deriving from the theory of the maximum likelihood estimation Ref. [1.61 ] , looks for the solution x that minimizes the χ 2 function, defined as the square summation of the differences between observations and simulations, weighted by the measurement noise.
Given n measurements Si and the corresponding simulations Fi ( p ,image) calculated by the forward model using the assumed profile image, and calling n the vector of the differences between observations and simulations and V n the variance-covariance matrix (VCM) associated to the vector n, the quantity to be minimized with respect to the unknown parameters x is:

image eq 2.6
If the observations are suitably chosen, matrix V n is not singular and its inverse exists. However, in case the spectrum is sampled on a grid finer than 1/(2 * MPD) where MPD is the Maximum Path Difference, the inverse of matrix V n does not exist. In this case the generalized inverse Ref. [1.46 ] of V n is used in equation eq. 2.6 .
In general, the observations do not depend linearly on the unknown parameters x. As a consequence equation eq. 2.6 is not a quadratic function of the unknowns, and an analytic expression for the values of the unknowns which minimizes equation eq. 2.6 cannot be determined.
However, sufficiently close to the minimum, we may assume that the χ 2 function is well approximated by a quadratic form, obtained expanding equation eq. 2.6 in the Taylor series about the initial guess profile:
image eq 2.7
where∇ and ∇ 2 indicate respectively the gradient and the Hessian matrix of the χ 2 function and image is a vector of dimension m, providing the correction to be applied to the assumed value of parameter imagein order to obtain its correct value x.
Writing the gradient and the Hessian matrix of the χ 2 function explicitly, and indicating with K the Jacobian matrix, i.e. a matrix of n rows and m columns, whose entry kij is the derivative of simulation i made with respect to parameter j, we obtain:
image eq 2.8
If the problem is linear or if near the minimum the residuals have a null average, the term with image/image can be neglected in equation eq. 2.8 and the value of y which minimizes the χ 2 function is the Gauss-Newton solution Ref. [1.41 ] :
image eq 2.9
Therefore, the solution matrix of the inverse problem, defined as the matrix that calculates the unknowns from the measured quantities, is equal to:
image eq 2.10
If the hypothesis of linearity is not satisfied, with this procedure the minimum of the χ 2 function is not reached but only a step is done toward the minimum. The vector image , with y computed using equation eq. 2.9 , represents only a better estimate of the parameters than image. In this case the whole procedure must be reiterated starting from the new estimate of the parameters (Newtonian iteration) and equation eq. 2.9 has to be written as:
image eq 2.11
where iter indicates the iteration index, imagethe result of the previous iteration,image image the Jacobian relative to the profile imageimage the residuals.
Convergence criteria are therefore needed in order to establish when a value close enough to the minimum of the χ 2 function has been reached. However, this procedure is successful only in the case of sufficiently weak non-linearities. If we start from a position of the χ 2 function that is far from the minimum, its second order expansion may be a poor approximation of the shape of the function, so that the calculated correction can be misleading, and increase rather than decrease the residuals. For this reason, a modification of the Gauss-Newton method, the Levenberg Ref. [1.47 ] - Marquardt Ref. [1.51 ] , Ref. [1.56 ] method, is used. The modification involves the introduction in equation eq. 2.11 of a factor λ which reduces the amplitude of the parameter correction vector.
image eq 2.12
The factor λ is initialized to a user-defined (less than 1) number, and during the retrieval iterations it is increased or decreased depending on whether the χ 2 function increases or decreases. At convergence the Levenberg - Marquardt method provides the same solution that is obtained with equation eq. 2.11 because they aim at the same minimum of the χ 2 function.
If V n is a correct estimate of the errors of the observations and the real minimum of the χ 2 function is found, the quantity defined by equation eq. 2.6 has an expectation value of (n - m) and a standard deviation equal to image. The value of the quantity image provides therefore a good estimate of the quality of the retrieval. Values of imagethat deviate too much from unity indicate the presence of incorrect assumptions in the retrieval.
The errors associated with the solution of the inversion procedure can be characterized by the variance-covariance matrix V x of x given by:
image eq 2.13
where D c and K c are respectively the solution matrix and the Jacobian matrix evaluated at convergence.
Matrix Vx maps the experimental random errors onto the uncertainty of the values of the retrieved parameters. Actually, the square root of the diagonal elements of Vx measures the root mean square (r.m.s.) error of the corresponding parameter. The off-diagonal element (V x )ij of matrix Vx , normalized to the square root of the product of the two diagonal elements (V x )ii and (V x )jj , provides the correlation coefficient between parameters i and j.
According to the theory of maximum likelihood, if an a-priori information on the unknown parameters is available, (x a and V -1 a being respectively the vector containing the a-priori values of the unknown and its VCM) the χ 2 expression to be minimized is Ref. [1.59 ] :
image eq 2.14
and equation eq. 2.11 becomes:
image eq 2.15
while equation eq. 2.13 becomes:
image eq 2.16
If we assume that the complementary information consists of a vector n1 with a VCM image connected by the Jacobian K1 to the unknowns y, equation eq. 2.15 can be written as:
image eq 2.17
This expression, which is implicitly used in equation eq. 2.9 for the combination of the information provided by independent microwindows, is used for the exploitation of non-radiometric information (external information). In section 5.1. the problems associated with the use of external information on the unknown parameters will be reviewed on the light of the choices implemented in the MIPAS Level 2 retrieval modules.
 
 

2.4.4.1.3.3 Main components of the retrieval

As already stated in the previous section, the objective of the retrieval program is the determination of the atmospheric parameters that better fit the simulations to the observations.
Starting from some first-guess values of the unknown parameters 2.4.4.5. and using information on observation geometry and instrumental characteristics, the forward model computes the simulated spectra, which are compared with the measured spectra provided by MIPAS Level 1b processor 2.4.3.2. . The difference between simulated and measured spectra provides the vector of the residuals used to evaluate the cost function ( χ; 2 ) to be minimized (see mathematical details here) 2.4.4.1.3.2.1. . A new profile is generated by modifying the first guess with the corrections provided by the Gauss-Newton formula. The inversion requires the knowledge of the Jacobian matrix. The improved profile can be used as new guess for generating simulated spectra which are again compared with the measured ones. The iterative procedure stops when the convergence criteria are fulfilled.
The main components of the retrieval algorithm are therefore:


2.4.4.1.3.3.1 Forward model

The purpose of the forward model is to simulate the spectra measured by the instrument in the case of known atmospheric composition. The signal measured by the spectrometer is equal to the atmospheric radiance which reaches the spectrometer modified by the instrumental effects, mainly due to the finite spectral resolution and the finite Field of View (FOV) of the instrument. The atmospheric radiance that reaches the instrument when pointing to the limb at tangent altitude zt is calculated by means of the radiative transfer equation:

image eq 2.18
where x is the position along the line of sight between the observation point x0 and the point xi at the farthest extent of the line of sight, B(image, x) is the source function, c(image,x) is the absorption cross-section, η(x) is the number density of absorbing molecules. The exponential term represents the atmospheric transmittance between x and x0 . In the case of local thermodynamic equilibrium B(image, x) is equal to the Planck function.
The measured signal S(image,zt) is simulated convolving the atmospheric limb radiance image with the Apodized Instrument Line Shape (AILS, defined in section 2.4.4.1.3.3.1.5. ) and with the MIPAS FOV function (FOV(zt ), discussed in section 2.4.4.1.3.3.1.6. ):
image eq 2.19
The computation of equation eq. 2.18 and equation eq. 2.19 requires many operations that must be repeated for several variables, each with numerous values. The search for a sequence of operations that avoids repetition of the same calculations and minimizes the number of memorized quantities is the first objective of the optimization process.
The following sequence of operations has been chosen:
1. definition of the frequency grid in which the atmospheric radiance is calculated (section 2.4.4.1.3.3.1.1. );
2. definition of the tangent altitude grid of the simulations and of an atmospheric layering (common to all the simulations) used for discretising their radiative transfer integral; determination of the 'paths' (i.e. combination of layer and limb view) that require a customized calculation of the values of pressure and temperature and definition of the values of pressure and temperature characteristics of each of these 'paths' (section 2.4.4.1.3.3.1.2. );
3. computation of absorption cross-sections, relating to all the selected paths and all the gases (section 2.4.4.1.3.3.1.3. );
4. computation of radiative transfer integral (section 2.4.4.1.3.3.1.4. );
5. AILS convolution (section 2.4.4.1.3.3.1.5. );
6. FOV convolution (section 2.4.4.1.3.3.1.6. )
The implementation of each of these operations is described below.

2.4.4.1.3.3.1.1 Dependence on spectral frequency

Limb radiance spectra contain spectral features varying from the sharp, isolated, Doppler-broadened lines at high altitudes to wide, overlapping, Lorentz-broadened lines at low altitudes. In order to resolve the sharp features at high altitude, a 'fine grid' of spectral resolution of the order of 0.0005 cm-1 is required. Furthermore the overlapping wings at low altitudes require the grid to be ubiquitous. The choice of a small spacing implies a large number of spectral points and an equally large number of calculations.
However, even if a fine grid of 0.0005 cm-1 is needed, not all the points of this grid are equally important for the reconstruction of the spectral distribution and the full radiative transfer calculation needs only be performed for a subset of fine grid points, the remaining spectrum being recovered by interpolation. This subset is denoted the 'irregular grid' and depends on the microwindow boundaries and on the Instrument Line Shape.
In principle it is possible to determine an optimal irregular grid for each tangent altitude, taking advantage of the particular line broadening associated with each observation geometry. However, while this might lead to a reduction in total number of fine grid points for which radiative transfer calculations are required, this prevents the savings obtained with calculations common to all tangent altitudes. As a consequence, an irregular grid is determined which is valid for all tangent altitudes, and therefore common to all observation geometries.
The number of points that are selected for the irregular grid depends upon the interpolation law that is used for the reconstruction of the spectral distribution. Several interpolation functions were investigated, but since the same interpolation is also required for the calculation of Jacobian, which, unlike the radiances, can have negative values, logarithmic and inverse interpolations had to be discarded and finally a simple linear interpolation was chosen.
The procedure for the generation of the 'irregular grid' is to start with a set of complete fine grid limb radiance spectra for different tangent altitudes. Then, for each point, a `cost' function is determined, representing the maximum radiance error (at any altitude, after AILS convolution) that would arise if the point were to be replaced by an interpolated value. The point with the smallest interpolation cost is then eliminated and the process repeated for the remaining grid. The iteration continues until no further points can be removed without exceeding a maximum error criterion, chosen to be 10 % of the noise-equivalent signal radiance. In figure2.23 an example of the high resolution radiance of a representative microwindow computed for the irregular grid points, as well as its convolved radiance and the difference between the original and interpolated radiance, is shown.
Typically it is found that only 5-10 % of the complete fine grid is sufficient for a satisfactory reconstruction of the spectral distribution.


image
Figure 2.23 44 km tangent height spectral radiance from a microwindow (frequency range 693.45 - 693.725 cm<sup>-1</sup>) selected for p, T retrieval. The upper plot shows the high-resolution radiance (solid line), the irregular grid points (+), and the convolved radiance (dashed line). The lower plot shows the difference between the original and interpolated radiances, on both the high-resolution grid (solid line, left axis) and the convolved radiances (dashed line, right axis).

2.4.4.1.3.3.1.2 Ray-tracing and definition of the atmospheric layering

The radiative transfer integral, given by equation, is a path integral along the line of sight in the atmosphere. The line of sight is determined by the viewing direction of the instrument and, due to refraction, is not a straight line, but bends towards the earth. The refractive index of air is a function of both pressure and temperature (the dependence on frequency is negligible in the spectral region of MIPAS measurements, as well as the dependence on water vapor, since MIPAS does not penetrate in the lower troposphere) and can be determined with the Edlen Edlen B. Ref. [1.32 ] model. Since the Earth is assumed locally spherical, the local radius of curvature being determined by the simplified WGS84 model Department of Defense Ref. [1.25 ] which has been adopted for the Earth shape, the atmospheric layering is spherical too. In these conditions, the optical path x is linked to the altitude r by the following expression:

image eq 2.20
with r altitude referred to the earth center, n(r) refractive index and rt tangent altitude.
Equation eq. 2.20 has a singularity at the tangent point (r = rt ), however the singularity can be removed by changing the integration variable from r toimage.
It results that:
image eq 2.21
and the limit of this expression for r --> rt can be computed analytically considering the dependence of the refractive index, the pressure and the temperature on the altitude.
The path integral equation is computed as a summation over a set of discrete layers. A common set of layers is defined for all the spectra of the sequence. The boundaries of these layers are defined in correspondence of the grid of the simulated tangent altitudes. We recall that spectra are simulated at the tangent altitudes of the limb scan sequence and at some additional tangent altitudes that are used for the FOV convolution (as discussed in section 2.4.4.1.3.3.1.6. ). Extra levels are added to the simulation grid as long as either the variation of temperature or the Voigt half-width of a reference line across each layer are greater than a maximum threshold supplied by the user.
For each layer that results from this process, appropriate 'equivalent' pressure and temperature, namely the Curtis-Godson Houghton J. T. Ref. [1.45 ] quantities, have to be determined. These are calculated weighting the pressure and temperature along the ray-path with the number density of each absorbing gas. This technique allows a coarse discretisation of the atmosphere to be implemented. The equivalent value of parameter G (pressure or temperature respectively) relative to the gas g, the layer l and the limb view t, is calculated as:
image eq 2.22
where z is the altitude, imageand image are the heights of the boundaries of the layer, image is the VMR of the g-th gas, x t is the line of sight characterized by the tangent altitude zt image is the air number density, andimage is the slant column relative to the considered gas, layer and limb view:
image eq 2.23
It has to be noted that in principle, Curtis-Godson pressures and temperatures have to be computed for each gas, each layer and each limb view. However, when the approximations of flat layers and straight line of sight are valid, both the numerator and the denominator of equation eq. 2.22 are proportional to the secant of the angle θ between the line of sight and the vertical direction and the same values of pe and Te are obtained independently of the limb angle. We have verified that the use of secant law approximation causes very small errors at all altitudes, except at the tangent layer, as it is shown in figure2.24 .
In this approximation, since a layering common to all spectra of the scan is defined, it is sufficient to calculate the values of pe and Te for all the layers of the lowest limb view, and only for the lowest layer of the other limb-views, all the other layers being characterized by the same values of pe and Te as the lowest limb view.
This optimization is crucial, not only because less equivalent pressures and temperatures have to be calculated, but mainly because absorption cross-sections have to be calculated and stored for a smaller number of p, T combinations.
image
Figure 2.24 Differences between Curtis-Godson pressures and temperatures of the layers of the limb view with tangent altitude 30 km and the corresponding quantities of the same layers in the case of the lowest limb view (tangent altitude 6 km). On the left axis the percent differences on equivalent pressures (squares) are reported, on the right axis the absolute differences on equivalent temperature (triangle) are shown. The layers are counted from the tangent layer at 30 km and are 2 km thick.

Keywords: ESA European Space Agency - Agence spatiale europeenne, observation de la terre, earth observation, satellite remote sensing, teledetection, geophysique, altimetrie, radar, chimique atmospherique, geophysics, altimetry, radar, atmospheric chemistry