IASI Level 1 processing algorithms desciption

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IASI Level 0 and 1 processing algorithms description

Bernard TOURNIER Denis BLUMSTEIN François-Régis CAYLA

NOVELTIS CNES* SISCLE

Toulouse Toulouse Saint-Siscle

FRANCE FRANCE FRANCE

Abstract

IASI is an infrared atmospheric sounder. CNES is leading the IASI program in association with EUMETSAT and is supported by METEO-FRANCE for scientific aspects.

The instrument is composed of a Fourier transform interferometer and an associated infrared imager. The interferometer provides a spectral range from 645 cm-1 to 2760 cm-1 (3.6 µm to 15.5 µm) and a spectral resolution of 0.5 cm-1 (FWHM) after apodisation. The sounder pixel size is 12 km (at nadir). The infrared imager allows the co-registration of IASI sounding with AVHRR images. A moving mirror with 30 scan positions provides a swath of about 2000 km (120 spectra for each 8 seconds).

The interferograms are processed by an on-board digital processing subsystem which performs the inverse Fourier transform and the radiometric calibration in order to decrease the IASI transmission rate from 45 Mbits/s to 1.5 Mbits/s. This part of the processing is the level 0 processing. The level 1 processing is performed on ground and produces resampled, apodised and calibrated spectra (radiometric post calibration and spectral calibration). An AVHRR radiances classification inside IASI sounder pixels is also provided.

The aim of this presentation is to describe the physical and mathematical content of the IASI level 0 and 1 processing algorithms and their architecture. It also describe the initialisation parameters required for execution of these algorithms.

Introduction

IASI sounder is a Fourier Transform spectrometer. This means that the physical measurement is not directly a spectrum and that a lot of mathematical transformations are necessary to obtain this physical data.

The goal of this paper is, after a brief reminder of the Fourier transform spectroscopy principles, to present the set of these transformations.

The description of the IASI instrument itself has been limited in this paper to the strict necessary. For a more detailed presentation, the reader should refer to the companion paper by [Phulpin & al.], to [Cayla 2001] or [Chalon & al].

Fourier transform spectroscopy

Sounder part of the IASI instrument is a Michelson interferometer (see fig. 1). In such an instrument the input light is divided by a semi-reflective surface S in two beams of approximately equal intensity. These are reflected by the mirrors M1 and M2 and return to S where they are re-combined to enter the detection system D. M1 is fixed, while M2 can be moved towards and away from D. It is then possible to introduce an Optical Path Difference (OPD) x between the two beams.

[pic]

Fig. 1 – Michelson interferometer schematics

With monochromatic input light of wavelength λ and wavenumber ν (ν = 1/λ) , the recombination of the two coherent waves leads to interference. Indeed the amplitude A of the output wave depends on the phase difference φ between the two waves which in turn depends on the OPD x

[pic]

The light intensity I(x), which is proportional to the square of the amplitude can be written

[pic] (1)

If we now consider a continuous spectrum S(ν) instead of the monochromatic wave, it is easy to show that the interferogram I(x) will be given by

[pic] (2)

So the interferogram is the sum of a constant term and the real part of the Direct Fourier Transform of S(ν). The first term is of no use for spectroscopy but the second term can be used to retrieve the spectrum S of the incident light by simply taking the Inverse Fourier Transform of the recorded interferogram I. So the name of the technique.

[pic] (3)

In practice, before it can be used, the interferogram must be sampled at very stable and very regularly spaced positions of the moving mirror. Using a very stable clock would not be sufficient as it would induce very stringent specifications on the speed of the moving mirror mechanism. Instead the sampling is triggered by the zero-crossings of a reference laser interferogram whose frequency ([pic]cm-1) is locked to an acetylene absorption line. Thus the interferogram is only known for a finite number of discrete values xi limited by the maximum optical path difference xmax allowed by the moving mirror mechanism. The computation used to retrieve the spectrum is thus a Discrete Fourier Transform and as a consequence, the spectrum is known with a sampling Δν = 1/(2xmax) and only for wave numbers below νrpd.

The rest of the paper will go into the details of the physics of the measurement process : radiometric calibration and spectral properties of the instrument.

Instrument design and its consequences on the processing

Radiometric calibration

The instrument should be linear in energy. Therefore a non linearity correction step is performed on the interferograms before the raw spectra are computed. Indeed, it is known that non linearity effect can lead, with other effects, to significant radiometric calibration errors.

Then, two reference views are used for radiometric calibration. An internal black body whose temperature (TBB) is precisely monitored provides the hot reference (SBB) and external view to the space provides the cold reference (SCS).

The radiometric calibration is performed in two steps. A first calibration is done on-board in the complex plane according to equation (4)1

[pic] (4)

Then a post-calibration step is performed on-ground to compensate for physical effects not taken into account in the first step (compensation of spectral calibration, scanning mirror incidence correction, non-blackness effect, etc.). The justification for separating these two steps is given in the section on data processing.

Equation (4) is discussed in [Revercomb et al.] as a mean to solve a problem with the radiometric calibration of the HIS sounder. This problem is induced by the fact that the phase of the cold raw spectrum2 can be different from the phase of the hot raw spectrum and this lead to very big calibration errors when using the modulus of these as a measure of the radiance.

The phase of the raw spectra are not necessary null because the interferogram is not an even function of the OPD x. There are many causes for this (like e.g. electronic delays in the acquisition chains or chromatism of the optics) but the main reason is that the interferogram sample used as the centre point for performing the inverse FFT does not necessary correspond to the position of Zero Path Difference (ZPD). Indeed it is an easy matter to see that if the ZPD is at a distance ε of the NZPD3 sample, then the phase of the spectrum will differ from zero. It will be equal to

[pic] (5)

So, the use of equation (4) solves the initial problem, but induces the requirement that the interferogram sample chosen as NZPD must correspond to the same OPD in the three interferograms used in radiometric calibration. As there is no hardware detection of the ZPD position in the IASI instrument, the selection of the NZPD is done by software.

The principle of this selection is to notice that, if the NZPD is correctly chosen, then the representative point of any raw spectrum S(ν), in the complex plane, (either atmospheric or calibration view) must stand on or very near to the straight line joining the calibration raw spectra SCS(ν) and SBB(ν), which is called the calibration line. On the opposite, any error on the NZPD selection will move the point S(ν) far from this line, except for some particular values of the wave number ν. The method used is then to choose, for each view, the NZPD value that minimises the sum of the distances between S(ν) and the corresponding calibration lines. The sum is over a small number of ν values taken nominally in the B3 band.

Field Of View effects and spectral properties

The figure 2 below is a simplified view of the IASI interferometer which will allow to introduce the vocabulary useful for describing the data processing. The first thing that the reader can notice is the use of cube corner mirrors. This optical configuration was chosen because these devices are less sensitive to orientation errors than flat mirrors.

The second important thing is that, while we have presented for simplification the Fourier transform spectroscopy principles with a collimated light on axis of the instrument, IASI will in fact be used over a relatively large field of view of about 3.3 degrees (see the four off axis pixels which are defined by a cold field stop in the focal plane). This will have important effects on the physics of the measurements.

[pic]

fig.2 – IASI Interferometer simplified spectral model3

Indeed, for a monochromatic plane wave ν coming in the instrument with an off axis angle θ, it can be shown that the OPD in formula (1) must be replaced by [pic], where x is the OPD measured by the reference laser.

This is the cause of a continuous change of the interferometric state over the field of view that induces the so-called fringes of equal inclination of the Michelson interferometer. These fringes can be observed as concentric circles in the focal plane (see fig.2). It can be shown that the centre of these circles is the direction defined by the two cube corner apexes C1 and C24. Ideally the trajectory of C2 is a straight line passing through C1, then the centre of the fringes is fixed. This direction is called the interferometric axis. When the former conditions are not perfectly met, the centre of the fringes will stay fixed only when the OPD is far from zero. This allows to define the interferometric axis in more realistic conditions (errors coming from optical alignment or moving mirror guiding mechanism imperfections).

The substitution of x by [pic] in equation (1) shows that the interferogram of a monochromatic wave ν at an angle θ of the interferometric axis is exactly the same as the one corresponding to a wave [pic] coming from the direction of the interferometric axis. Therefore when the field of view will be uniformly illuminated by a monochromatic wave, the raw spectrum obtained by (3) will be the sum of many cardinal sine at slightly different ν positions. Two effects are then observed :

• the main one is that the spectral line is observed at a position [pic] where θ is averaged over the whole pixel,

• the second one is that the instrument response to such an input is not the theoretical cardinal sine (see fig. 3 below)

[pic]

fig.3 – IASI ISRF depends on the wave number

To cope with the first effect, the wave number associated to the k spectral sample is given by the spectral calibration equation (given here at the first order of precision)

[pic], for k = 0,1,...,N/2 (6)

where N is the size of the FFT and θ is the off axis angle. This equation is valid directly for pixel of small size. For a larger pixel the angle θ must be averaged over the whole pixel. In order to be able to apply this spectral calibration, the value of the θp angle must be known for each of the IASI pixels. The first step is to estimate the position of the interferometric axis by comparing the known spectral position of some characteristic features in atmospheric spectra and their measured spectral position by the four pixels. Roughly speaking this will place the interferometric axis on circles centred on the pixels centre so the problem is solved in principle (see section on level 1a processing, 3rd bullet, for a more precise description of the process).

Many observations can be drawn from the figure 3. The IASI instrument response function (ISRF) is not symmetric5 about its gravity centre and depends on the wave number ν. This last effect is due to integration of an increasing number of fringes inside the pixel which causes more self-apodisation when the wave number increases. This self-apodisation effect is named after the apodisation technique which consists in multiplying interferograms by a function decreasing slowly to 0 when the OPD increases before taking the inverse Fourier transform. This has the beneficial effect of reducing the size of the sidelobes in the ISRF and the detrimental effect of decreasing the spectral resolution.

It has to be noticed that the exact form of the ISRF is not the same for the 4 pixels and that it will also depends on a lot of instrument characteristics which are mainly the position of the interferometric axis but also OPD distorsion due to cube corner trajectory and pixel response function in the field of view.

Therefore, in order to make the user task easier, higher level spectra 1C are made independent of instrument characteristics. This is done by the equation (7) below5.

[pic] (7)

where A(x), the self-apodisation function is a complex function which is the ratio between complex (simulated) interferograms obtained for a monochromatic light of wave number ν. The numerator is related to the actual instrument and the denominator to the theoretical interferogram given by modulated part of (1).

[pic] (8)

The corresponding ISRF described by the figure 4 below.

[pic]

fig.4 – IASI ISRF 1C (independent of varying instrument characteristics)

To conclude this review of the relations between field of view effects and spectral properties of the instrument, it remains to say that spatial non uniformity of the scene radiance will also have spectral effects, which have to be taken into account at user level.

IASI data processing

The IASI measurement data processing up to level 1 is in charge of the transformation of interferograms measurements into fully calibrated spectra independent of the instantaneous state of the instrument.

The transmission rate allocated to IASI measurements is 1.5 Megabits/s and the instrument data production rate is 45 Megabits/s. In order to reach the allocated data transmission rate it was necessary to implement on board the instrument a substantial part of the IASI data processing. The trade off between on board complexity and data rate led to compute a roughly calibrated and band merged spectrum and to encode it before transmission.

Consequently, the Nearly Real Time (NRT) IASI data processing is split in two processing chains : the level 0 processing located on board the instrument and the level 1 processing located on ground.

Moreover, an offline processing chain located at the Technical Expertise Centre (TEC) is in charge of

• the monitoring of the instrument performance,

• the computation of the level 0 and 1 initialisation parameters in relation to preceding point,

• the computation of the long term varying IASI products,

• the monitoring of NRT processing (Level 0 and 1)

Level 0 processing

The main goal of the IASI level 0 processing is to reduce the transmission data rate; the choice done is to produce calibrated spectra in terms of radiometry and to merge the three spectral bands. The radiometric calibration is done in the complex plane after inverse Fourier transform of the interferograms. The raw spectra involved in the radiometric calibration law are cold space, hot black body and atmospherics spectra. The complex arguments of these spectra have to be consistent as explained in the previous paragraph, the consistency is achieved thanks to the NZPD algorithm which select the interferograms pivot samples passed to the Fourier transform algorithm which insures coherency of the complex arguments for all the spectra involved in the calibration law. As the Fourier transform is applied on board, the non linearity correction have to be applied before.

The architecture of the Level 0 processing chain is given by fig. 5 below.

[pic]

fig.5 – General architecture of Level 0 processing

This chain is divided into three processing sub chains.

• The interferograms pre-processing is dedicated to the non linearity correction, the spikes detection, the computation of the Number of the interferograms sample associated to the Zero Path Difference (NZPD) and applies finally the Fourier Transform to the interferogram.

➢ The non linearity correction is done by the application of look up tables. It corrects sequentially (inverse order of defect apparition) the non linearities of the detection and acquisition chains.

➢ The spikes detection prevents the use of corrupted interferogram in the calibration processing. If the filtered interferogram by the application of a band pass filter is higher than a given threshold, a flag is raised. A 13 taps filter is used inside the central fringe and a 5 taps filter is apply outside.

➢ The NZPD computation is dedicated to the determination of the pivot sample of the Fourier transform. This algorithm computes, after reduced inverse Fourier transform of the interferogram and for selected pivot sample around a first approximation of the NZPD, the distance between the complex spectra samples and the calibration line. The NZPD is the sample which gives the minimum of the distance. This algorithm run whatever the target is, in case of calibration target the calibration line (reduced complex spectra) is updated after filtering.

➢ The Fourier transform algorithm applied to the interferogram centred at the NZPD sample position gives us the complex spectrum corresponding to the measured interferogram.

• The radiometric calibration coefficients computation and filtering :

➢ The cold space spectrum and the hot black body spectrum are used to compute the instantaneous radiometric calibration coefficients in the complex plane following the classical linear formula expressed in equation (4).

➢ If the distance between the previous filtered coefficients and the instantaneous ones is inside a given threshold, the filtered coefficients are updated following the recursive formula : with LN line number and N length of the filter

• [pic]

• the same for B

• The atmospheric spectra computation dedicated to the application of the calibration coefficients, the band merging and the coding of the spectra.

➢ The spectral calibration coefficients are applied to the atmospheric spectra in the complex plane. Following the formulation in equation (4), this operation removes most of the imaginary part of the spectrum. The residual imaginary part due to the instrument noise and calibration instabilities is monitored and allow the flagging of wrong measurements.

➢ The three spectral bands are merged by linear combination of the overlapping areas and the weights of the combination are the normalised inverse of the spectral radiometric noise.

➢ The merged spectra are coded by application of a spectral scaling law, offset removing and applying a bit mask. This operation allow us to transmit an average of 8.2 bits per spectral sample without loss of useful information.

Level 1 processing

In a retrieval process, the direct model errors are of the same importance as the instrument errors. A special effort was made on the instrument and processing design in order to minimise the instrumental error component of the direct model (i.e. the ISRF).

As explained before, the measured spectra are strongly dependent on the spectral variability of the ISRF, the variability of the ISRF between pixels and between scan lines and the induced variability of the spectral sampling. The aim of the level 1 processing is to provide the best estimation of the interferometer geometry at the measurement time. This estimation model will be used at each IASI product levels. The estimation of the instrument model is done thanks to the estimation of the parameters describing the interferometer geometry during the measurements.

Some long terms parameters of the estimation model are computed by the TEC processing chain and are input parameters of the level 1 estimation process. The only short term parameter of the estimation is the interferometer axis position in the focal plane. This last parameter which is the most sensitive is susceptible to be rapidly variable, so it is estimated for each scan position, averaged over a full scan line and filtered.

Once the estimated model is known, the corresponding spectral calibration and apodisation functions are computed and applied in order to remove all the spectral variability of the measurements.

The estimation model is used at level 1a to give the correct spectral positions of the spectra samples, these position are varying form one pixel to another. It is used also at level 1b to give spectra sampled on a common spectral basis. Then it is used at level 1c in order to give spectra convolved by a unique ISRF in order to minimise the computation load for the simulation of IASI spectra in the level 2 processing.

The other goals are to complete the radiometric calibration by taking into account the neglected components in the level0 processing , to process the integrated imager radiometric calibration, to compute the co-registration vector between IASI sounder and AVHRR measurements and finally to analyse the AVHRR radiances in the Field Of View of the IASI Integrated Imager (IIS), and to describe the positions, the coverage and the characterics of the resultant clusters inside the IASI sounder FOV.

The architecture of the Level 1 processing chain is given by fig. 6 below.

[pic]

fig.6 – General architecture of Level 1 processing

The level 1 processing chain is divided into three processing sub chain.

Level 1a :

• Estimate the Instrument geometry occurring during the measurements, deduce the spectral calibration functions and the apodisation functions : the instrument model estimation starts with the computation of spectral shift between a modelled calibrated spectrum and every measured spectra for a full scan line (4 pixels by 30 scan positions), a filtering of these spectral shifts in done pixels by pixels, the interferometer axis position for a scan line into the focal plane is obtained by the minimisation of the sum over the pixels of the distances between the computed spectral shifts and the modelled spectral shifts for a grid of the possible values of the interferometer axis position in the focal plane. The position of the minimum of these distances gives the interferometer axis. Finally this axis position is filtered by a polynomial fitting of the axis determination during the filtering period. This filtered position is used in a bilinear interpolation in the grid to compute the interpolated spectral functions which are : Spectral Calibration Functions (SCF), Apodisation Functions (AF) and Instrument Spectral Response Functions (ISRF).

• Refine the radiometric calibration.

➢ Spectral calibration of the Planck function applied on board : apply the SCF on the wave number basis of the Planck function used for the computation of the radiometric coefficient calibration applied on board.

➢ Apply the hot black body emissivity function : the Planck function used on board have to be corrected again for neglecting on board the hot black body emissivity function.

➢ Correct for the scan mirror reflectivity angular law : the absolute radiometric is completed by correcting each spectrum for the spectral and incidence variation of the scan mirror reflectivity. This algorithm corrects also for the polarisation effect.

• Compute the co-registration vector between IASI Integrated Imager (IIS) and AVHRR : knowing the IASI and AVHRR scanning laws, the image from IIS is projected and interpolated onto the AVHRR raster, a correlation coefficient between this image and a linear combination of AVHRR channel 4 and 5 is computed for various values of the possible offset between the two images finally the position of the maximum of the correlation is determined. This position gives the co-registration vector between the IIS and the AVHRR. This vector has to be calculated for all scan positions.

• Analyse the AVHRR radiances in the IASI FOV. The AVHRR radiances covering the 50 by 50 km square are decomposed into 6 clusters (at most), their barycentre position and coverage fraction are computed for each IASI FOV : a dedicated algorithm based on the Freeman technique analyses the 5 channels AVHRR radiances covering the IIS FOV and determines the clusters. Then, knowing the co-registration vector between IASI and IIS and the co-registration vector between IIS and AVHRR, the IASI sounder IPSF are projected onto the classified AVHRR image and the coverage fraction and barycentre position of the clusters present in the IASI FOVS are determined.

• Geo-locate the IASI FOV measurements is done while preserving accuracy of the co-registration with AVHRR images.

Level 1b :

• Apply the estimated spectral calibration functions : the calibrated wave numbers of the spectrum samples are computed.

• Resample the spectra : the spectra defined on the calibrated wave number basis are resampled on a constant wave number basis (0.25 cm-1). This process involves oversampling with a large factor (e.g. 10) of the level 1a spectra.

Level 1c :

• Apply the estimated apodisation functions : the estimated apodisation functions as defined in Equation (7) are defined in the interferogram space for a subset of the IASI wave numbers (142 functions instead of 8461), so the convolution is done in the interferogram space by multiplying the direct Fourier transform of the level 1b spectrum with the 142 apodisation functions and then by inverse Fourier transform of the products. The apodisation is only exact for the 142 subset of spectrum samples, the others samples are obtained by linear interpolation as defined in the following equation :

[pic] (9)

where

➢ [pic] are the subset of IASI wave numbers for which the apodisation functions are computed,

and [pic]

➢ and [pic] the spectrum sample at the position ( of the direct Fourier transform of the level 1b spectrum multiplied by apodisation functions defined at (j.

TEC processing

The architecture of the TEC processing chain is given by fig. 7 below.

[pic]

fig.7 – Architecture of TEC processing (off line)

Initialisation of on board processing

• Compute and update the non-linearity correction tables.

➢ The non linearity of the analog to digital converter is characterised once on ground for various temperature. The on board tables are updated if the temperature of this device varies.

➢ The non linearity of the detectors and the analog acquisition chain is determined on each interferogram down loaded in the verification data. The on board tables are updated if necessary.

• Compute and update the Reduced Spectra for the NZPD detection : the reduced spectra necessary for the initialisation of the on board NZPD algorithm are computed for each interferogram (from calibration targets) down loaded in the verification data. The on board initial reduced spectra are updated if necessary.

• Compute and update the spectra coding table : the spectra coding table is computed and updated if the minimum and maximum of energy expected for spectra samples vary significantly.

• Compute and update the spectral band limits : the spectral band limits are computed and updated if necessary depending on the radiometric noise spectrum.

Initialisation of ground processing

• Compute and update the long term offset of the fixed corner cube : a dedicated algorithm determine the offset of the fixed corner cube for each interferogram (from calibration targets) down loaded in the verification data. This offset is passed to the ISRF model which is in charge of the spectral functions data bank computation.

• Compute and update the spectral functions data bank : the ISRF model compute and update the spectra function data bank which is partially passed to the level 1 processing and to the users. The model compute the Self Apodisation Functions (SAF), the Instrument Spectral Response Functions (ISRF), the Spectral Calibration Functions (SCF) and the Apodisation Functions (AF) for a grid of possible interferometer axis positions in the focal plane. The input data of the model are : the Instrument Point Spread Functions (IPSF) of each 4 pixels, the fixed corner cube offset, the moving corner cube displacement law.

• Compute and update the scan mirror angular reflectivity laws : the on ground characterised scan mirror angular reflectivity laws are adjusted by the means of a dedicated external calibration sequence looking at a second cold space view.

• Compute and update the IASI sounder and IASI integrated imager co-registration vector : the co-registration vector between IASI sounder and Integrated Imager, which is supposed to be slowly variable, is computed for various non homogeneous scenes along the METOP orbit in order to derive an orbital co-registration model. This model is passed to the level 1 processing chain to complete the co-registration between IASI sounder and AVHRR imager.

• Compute and update the radiometric instrument noise spectrum : the radiometric noise spectrum is computed for each external calibration sequence dedicated to this purpose by looking at the hot black body target during the 30 scan number sub cycles.

• Compute and update the IASI covariance matrix : the IASI covariance matrix is computed and passed to the users. The update is done for every significant change in the radiometric noise figure.

References

Cayla F.R. 2001, L’interféromètre IASI : un nouveau sondeur satellitaire à haute résolution, La Météorologie, 8e série, No 32, 23–39.

Chalon G., Cayla F.R., Diebel D. 2001, IASI : An advance sounder for operational meteorology, IAF 2001 conference proceedings.

Genest J. and Tremblay P. 1999, Instrument Line Shape of Fourier transform spectrometers : analytic solutions for non uniformly illuminated off-axis detectors, Applied Optics, Vol. 38, No. 25.

Griffiths P.R. and De Haseth J.A. 1986, Fourier transform infrared spectrometry, John Wiley & Sons, New York, 656 p.

Phulpin T., Cayla F.R., Chalon G., Diebel D. and Schluessel P. 2002, IASI on board METOP : Project status and scientific preparation, ITSC-XII conference proceedings.

Prunet P., Thépaut J.-N. et Cassé V., 1998, The information content of clear sky IASI radiances and their potential for numerical weather prediction. Quart. J. Roy. Meteor. Soc., 124, 211-241.

Revercomb H.E., Buijs H., Howell H.B., LaPorte D.D., Smith. W.L. and Sromovsky L.A 1988, Radiometric calibration of IR Fourier transform spectrometers : solution to a problem with the High-Resolution Interferometer Sounder, Applied Optics, Vol. 27, No. 15.

* Centre National d’Etudes Spatiales

1 what is not apparent in the equation (4) is that the calibration coefficients A and B are in fact updated every 8 seconds through a first order recursive filter with time constant 80 seconds.

2 From now on we will use the term raw spectrum for the Inverse Fourier Transform of an interferogram.

3 Number of sample of the Zero Path Difference

3 This is only a simplified model which must be refined for actual computations. In particular, a compensating blade is introduced in the optical configuration, which is a variant of the Connes one, in order to compensate for the finite thickness of the beamsplitter. Another point apparent on the figure 2 is that the pupil of the instrument is imaged on both the cube corners and the scan mirror through an afocal telescope. This pupil is defined physically by a cold aperture stop.

4 in fact C2 is the image through the beamsplitter (i.e. in the optical space of C1) of the moving cube corner apex

5 This effect is due to the shape of the IASI pixel. See Genest & al. for a nice demonstration of this fact.

5 for the ease of exposition the dependency with nð of the function A is ignored in this section. See the section on Level 1 processing for treatment of this matter.

this fact.

5 for the ease of exposition the dependency with ν of the function A is ignored in this section. See the section on Level 1 processing for treatment of this matter.

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