**Fourier Transform Spectroscopy with Partially Scanned Interferograms as a Tool to Retrieve Atmospheric Gases Concentrations from High Spectral Resolution Satellite Observations – Methodological Aspects and Application to IASI**

Carmine Serio1, Guido Masiello1 and Giuseppe Grieco2 *1CNISM, Unitá di Ricerca di Potenza, Universitá della Basilicata, Potenza 2DIFA, Universitá della Basilicata, Potenza Italy* 

#### **1. Introduction**

246 Atmospheric Model Applications

Zhao,J., Khalizov, A., Zhang, R. *Hydrogen-Bonding Interaction in Molecular Complexes and Clusters of Aerosol Nucleation Precursors, J. Phys. Chem. A* 2009, 113, 680.

364.

*Thermochemical Kinetics, and Noncovalent Interactions J. Chem. Theory Comput.* 2006, *2*,

Fourier Transform Spectroscopy with partially scanned interferograms (FTS\*PSI) is a technique to obtain the difference between spectra of atmospheric radiance at two diverse spectral resolutions (Grieco et al., 2010; 2011; Kyle, 1977; Smith et al., 1979).

In the context of infrared remote sensing, the idea of using partially scanned interferograms for the retrieval of atmospheric parameters dates back to Kyle (1977) who argued that large portions of the spectrum (the Fourier transform of the interferogram and vice versa) could bring poor or no information for a given atmospheric parameter, whereas small ranges in the interferogram domain could concentrate much information about the parameter at hand. Kyle (1977) exemplified the technique for temperature, whereas a correlation interferometer was proposed for the observation of atmospheric trace gases by Goldstein et al. (1978). The direct inversion of small segments of interferometric radiances for the purpose of temperature retrieval was further analyzed and exemplified in Smith et al. (1979).

In some circumstances, according to Kyle (1977) the interferogram domain can provide a data space in which information about the atmospheric thermodynamical state can be encoded much more efficiently than in the spectral domain. Exploiting this idea, Grieco et al. (2010) has shown how to perform a dimensionality reduction of high spectral resolution infrared observations, which preserves the spectral coverage of the original spectrum . Once compared to the usual way of reducing a high amount of spectral data by simply considering a sparse (optimal) selection of the spectral channels, the methodology has shown a better performance mostly for the retrieval of water vapor.

Unlike the previous works by Grieco et al. (2010); Kyle (1977); Smith et al. (1979), which concentrated on the direct use of inteferogram radiances, in this study, we consider the point of transforming back to the spectral domain the partial scanned interferogram, in such a way to obtain a difference spectrum with improved signal-to-noise ratio. This difference spectrum, instead of the partial interferogram, is considered for the retrieval of atmospheric gases.

Observations. Methodological Aspects and Application to IASI 3

Atmospheric Gases Concentrations from High Spectral Resolution Satellite Observations...

<sup>249</sup> Fourier Transform Spectroscopy with Partially Scanned Interferograms as a Tool to Retrieve

3, 4, 5 and 6, respectively. Section 3, which covers the case of CO2 is mainly intended to exemplify the overall methodology. Results for CO2 concerning the quality and validation of the methodology have been recently presented in Grieco et al. (2011). These results will not be shown here again, but only summarized for the benefit of the reader in section 3.3. However, the material shown in the two methodological sections, 3.1, and 3.2 is largely complementary to that presented in Grieco et al. (2011) and addresses important aspects, which generalize the application of the methodology to the full earth disk. The results for CO, CH4 and N2O, shown in sections 4, 5 and 6, are presented here for the first time and have not been covered

For the purpose of simulations we use the Chevalier data set (Chevalier, 2001). This set is

Real IASI observations and related atmospheric state vectors come from the 2007 Joint Airborne IASI Validation Experiment (JAIVEx) campaign (JAIVEx, 2007) over the Gulf of Mexico. We have a series of 6 spectra for 29 April 2007, 16 spectra for 30 April 2007, and finally 3 spectra for 04 May 2007. The spectra were recorded for clear sky fields of view, selected based on high resolution satellite imagery from AVHRR (Advanced Very High Resolution Radiometer) on MetOp (Meteorological Operational Satellite) and in-flight observations. Furthermore, the data for 29 April 2007 correspond to a nadir IASI field view,

Dropsonde and ECMWF (European Centre for Medium range Weather Forecasts) model data, which have been used to have a best estimate of the atmospheric state during each MetOp overpass, have been prepared and made available to us by the JAIVEx team. The JAIVEx dataset contains dropsonde profiles of temperature, water vapor, ozone, and carbon monoxide which extend up to 400 hPa. Above 400 hPa only ECMWF model data are available. The atmospheric state vectors used in this study use JAIVex dropsonde data up to 400 hPa supplemented by collocated ECMWF forecasts of temperature, water vapor and ozone from

For the purpose of atmospheric gas retrieval we have also used IASI data for a monthly acquisition on July 2010 over the Mediterranean area. These data set has been used to produce monthly maps for the four gases at hand. The spectra were checked for clear sky using the IASI stand alone cloud detection scheme developed by Grieco et al. (2007); Masiello et al.

The principles of FTSPSI have been recently discussed in Grieco et al. (2011), here we limit to show the basic aspects. FTSPSI is a particular application of Fourier spectroscopy, which today counts widespread applications in many fields. Fourier spectroscopy fundamentals can be found in appropriate textbooks (see e.g. Bell (1972)). A summary of the basic definitions,

in previous publications by the authors. Conclusions are drawn in section 7.

mostly used to define pairs of IASI spectra and atmospheric state vectors.

whereas those for the other two days to a field of view of 22.50 degrees.

400 hPa to 0.1 hPa (corresponding to about 65 km).

which are relevant to our methodology is now presented.

(2002; 2003; 2004); Serio et al. (2000).

**2.2 Basic aspects of FTSPSI**

**2. Background**

**2.1 IASI data**

Depending on the interferogram range, the difference spectrum can isolate emission features of atmospheric gases from, e.g., the strong surface emission. In this respect the technique is mostly suited for nadir looking radiometers/spectrometers on board of satellites, since for this instrumentation the observed infrared radiance is made up by the atmospheric component plus that of surface emission. The issue of properly choosing the partial interferogram has to do with correlation interferometry. Because of the Wiener-Khinchin-Einstein theorem (e.g. see Bell (1972)), the interferogram is the *auto*-correlation function of the light spectrum, which means that periodic or almost periodic features present in the spectrum can yield sharp peaks (constructive interference). in the interferogram signal. As an example for the specific case of CO2, rotation transitions yield a spectrum characterized by a periodic pattern with a period of about 1.5 cm−1. This periodic pattern determines a strong signature (coherent interference) in the interferogram domain at about 1/1.5 cm=0.66 cm and overtones. Thus, for the case of CO2, molecular spectroscopy fundamentals tell us how to exactly choose the proper partial interferogram.

In this study, FTS\*PSI will be exemplified to show how we can handle and partly separate from the spectrum the surface emission in order to develop and implement suitable schemes for the retrieval of the columnar load of CO2, CO, CH4 and N2O. The methodology will be applied to the Infrared Atmospheric Sounding Interferometer (IASI), which is flying on board the Metop-A (Meteorological Operational Satellite) platform. IASI was developed in France by the Centre National d'Etudes Spatiales (CNES) and is the first of three satellites of the European Organization for the Exploitation of Meteorological Satellites (EUMETSAT) European Polar System (EPS). The instrument spectral bandwidth covers the range from 645 to 2760 cm−<sup>1</sup> (3.62 to 15.50 *μ*m), with a sampling interval Δ*σ* = 0.25 cm−1. Thus, each single spectrum yields 8461 data points or channels. The calibrated IASI interferogram extends from 0 to a maximum OPD of 2 cm.

As already outlined, the main objective of this study is to illustrate, demonstrate and exemplify the potential advantages of high infrared spectral resolution observations data analysis with partially scanned interferograms, through the exploitation of IASI data. Towards this objective, we have used a forward/inverse methodology, which we refer to as *ϕ*-IASI, whose mathematical aspects and validation has been largely documented (see e.g. Amato et al. (2002); Carissimo et al. (2005); Grieco et al. (2007); Masiello et al. (2002; 2003; 2004); Masiello & Serio (2004); Masiello et al. (2009; 2011); Serio et al. (2000)).

The remote sensing of atmospheric minor and trace gases from nadir looking instrumentation on board polar satellites is not a new subject. Among many others we quote here the experience with the Japanese IMG (Interferometric Monitoring of Greenhouse Gases) (Lubrano et al., 2004), the American AIRS (Atmospheric Infrared Radiometer Sounder) (Chahine et al, 2005; 2008; McMillan et al., 2005) and the European IASI (Boynard et al., 2009; Clarisse et al., 2008; 2009; Clerbaux et al., 2009; Crevoisier et al., 2009; Grieco et al., 2011; Ricaud et al., 2009). However, this work is focused mostly on the novel methodology of FTS\*PSI rather than particular applications, which nevertheless are here considered to exemplify the use of the procedure for the remote sensing of minor and trace gases.

The study is organized as follows. Section 2 is mainly devoted to the methodological aspects: in section 2.1 we present the IASI data, whereas the description of the fundamentals of the partially scanned interferogram approach is presented in sections 2.2, 2.2.1 and 2.3. Application to IASI for the retrieval of CO2, CO, CH4 and N2O is discussed in sections 3, 4, 5 and 6, respectively. Section 3, which covers the case of CO2 is mainly intended to exemplify the overall methodology. Results for CO2 concerning the quality and validation of the methodology have been recently presented in Grieco et al. (2011). These results will not be shown here again, but only summarized for the benefit of the reader in section 3.3. However, the material shown in the two methodological sections, 3.1, and 3.2 is largely complementary to that presented in Grieco et al. (2011) and addresses important aspects, which generalize the application of the methodology to the full earth disk. The results for CO, CH4 and N2O, shown in sections 4, 5 and 6, are presented here for the first time and have not been covered in previous publications by the authors. Conclusions are drawn in section 7.

#### **2. Background**

#### **2.1 IASI data**

2 Will-be-set-by-IN-TECH

Depending on the interferogram range, the difference spectrum can isolate emission features of atmospheric gases from, e.g., the strong surface emission. In this respect the technique is mostly suited for nadir looking radiometers/spectrometers on board of satellites, since for this instrumentation the observed infrared radiance is made up by the atmospheric component plus that of surface emission. The issue of properly choosing the partial interferogram has to do with correlation interferometry. Because of the Wiener-Khinchin-Einstein theorem (e.g. see Bell (1972)), the interferogram is the *auto*-correlation function of the light spectrum, which means that periodic or almost periodic features present in the spectrum can yield sharp peaks (constructive interference). in the interferogram signal. As an example for the specific case of CO2, rotation transitions yield a spectrum characterized by a periodic pattern with a period of about 1.5 cm−1. This periodic pattern determines a strong signature (coherent interference) in the interferogram domain at about 1/1.5 cm=0.66 cm and overtones. Thus, for the case of CO2, molecular spectroscopy fundamentals tell us how to exactly choose the proper partial

In this study, FTS\*PSI will be exemplified to show how we can handle and partly separate from the spectrum the surface emission in order to develop and implement suitable schemes for the retrieval of the columnar load of CO2, CO, CH4 and N2O. The methodology will be applied to the Infrared Atmospheric Sounding Interferometer (IASI), which is flying on board the Metop-A (Meteorological Operational Satellite) platform. IASI was developed in France by the Centre National d'Etudes Spatiales (CNES) and is the first of three satellites of the European Organization for the Exploitation of Meteorological Satellites (EUMETSAT) European Polar System (EPS). The instrument spectral bandwidth covers the range from 645 to 2760 cm−<sup>1</sup> (3.62 to 15.50 *μ*m), with a sampling interval Δ*σ* = 0.25 cm−1. Thus, each single spectrum yields 8461 data points or channels. The calibrated IASI interferogram extends from

As already outlined, the main objective of this study is to illustrate, demonstrate and exemplify the potential advantages of high infrared spectral resolution observations data analysis with partially scanned interferograms, through the exploitation of IASI data. Towards this objective, we have used a forward/inverse methodology, which we refer to as *ϕ*-IASI, whose mathematical aspects and validation has been largely documented (see e.g. Amato et al. (2002); Carissimo et al. (2005); Grieco et al. (2007); Masiello et al. (2002; 2003;

The remote sensing of atmospheric minor and trace gases from nadir looking instrumentation on board polar satellites is not a new subject. Among many others we quote here the experience with the Japanese IMG (Interferometric Monitoring of Greenhouse Gases) (Lubrano et al., 2004), the American AIRS (Atmospheric Infrared Radiometer Sounder) (Chahine et al, 2005; 2008; McMillan et al., 2005) and the European IASI (Boynard et al., 2009; Clarisse et al., 2008; 2009; Clerbaux et al., 2009; Crevoisier et al., 2009; Grieco et al., 2011; Ricaud et al., 2009). However, this work is focused mostly on the novel methodology of FTS\*PSI rather than particular applications, which nevertheless are here considered to exemplify the use of

The study is organized as follows. Section 2 is mainly devoted to the methodological aspects: in section 2.1 we present the IASI data, whereas the description of the fundamentals of the partially scanned interferogram approach is presented in sections 2.2, 2.2.1 and 2.3. Application to IASI for the retrieval of CO2, CO, CH4 and N2O is discussed in sections

2004); Masiello & Serio (2004); Masiello et al. (2009; 2011); Serio et al. (2000)).

the procedure for the remote sensing of minor and trace gases.

interferogram.

0 to a maximum OPD of 2 cm.

For the purpose of simulations we use the Chevalier data set (Chevalier, 2001). This set is mostly used to define pairs of IASI spectra and atmospheric state vectors.

Real IASI observations and related atmospheric state vectors come from the 2007 Joint Airborne IASI Validation Experiment (JAIVEx) campaign (JAIVEx, 2007) over the Gulf of Mexico. We have a series of 6 spectra for 29 April 2007, 16 spectra for 30 April 2007, and finally 3 spectra for 04 May 2007. The spectra were recorded for clear sky fields of view, selected based on high resolution satellite imagery from AVHRR (Advanced Very High Resolution Radiometer) on MetOp (Meteorological Operational Satellite) and in-flight observations. Furthermore, the data for 29 April 2007 correspond to a nadir IASI field view, whereas those for the other two days to a field of view of 22.50 degrees.

Dropsonde and ECMWF (European Centre for Medium range Weather Forecasts) model data, which have been used to have a best estimate of the atmospheric state during each MetOp overpass, have been prepared and made available to us by the JAIVEx team. The JAIVEx dataset contains dropsonde profiles of temperature, water vapor, ozone, and carbon monoxide which extend up to 400 hPa. Above 400 hPa only ECMWF model data are available. The atmospheric state vectors used in this study use JAIVex dropsonde data up to 400 hPa supplemented by collocated ECMWF forecasts of temperature, water vapor and ozone from 400 hPa to 0.1 hPa (corresponding to about 65 km).

For the purpose of atmospheric gas retrieval we have also used IASI data for a monthly acquisition on July 2010 over the Mediterranean area. These data set has been used to produce monthly maps for the four gases at hand. The spectra were checked for clear sky using the IASI stand alone cloud detection scheme developed by Grieco et al. (2007); Masiello et al. (2002; 2003; 2004); Serio et al. (2000).

#### **2.2 Basic aspects of FTSPSI**

The principles of FTSPSI have been recently discussed in Grieco et al. (2011), here we limit to show the basic aspects. FTSPSI is a particular application of Fourier spectroscopy, which today counts widespread applications in many fields. Fourier spectroscopy fundamentals can be found in appropriate textbooks (see e.g. Bell (1972)). A summary of the basic definitions, which are relevant to our methodology is now presented.

Observations. Methodological Aspects and Application to IASI 5

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<sup>251</sup> Fourier Transform Spectroscopy with Partially Scanned Interferograms as a Tool to Retrieve

As opposite to gas emission, surface emission varies smoothly with the wave number, *σ*. Thus, if we consider two spectral samplings, Δ*σ*1, Δ*σ*2, which are close each other but small enough so that the surface emission has been resolved at either samplings, then the difference

contains mostly the atmospheric emission alone. In Eq. 6, *ri* is the spectrum at sampling Δ*σi*,

In reality, the above differentiation cannot get completely rid of the surface radiation, because the corresponding emission, which reaches the Top of the Atmosphere (TOA) is modulated by the total transmittance function, which is itself a highly oscillatory function. However, in window regions where the transmittance approximates the unity, and where gases have only weak absorption bands, the difference *d*(*σ*) yields a signal, which is mostly the result of

It is important here to stress that to compute the difference spectrum, *d*(*σ*) we do not need to measure the spectrum twice. The operation can be done by simply Fourier transforming one single *partial* interferogram. In fact, with reference to Eq. 6 and provided that, Δ*σ<sup>i</sup>* > Δ*σ* (*i* = 1, 2), and with Δ*σ* defined by Eq. 5, that is the sampling rate corresponding to the maximum optical path difference, we have that *d*(*σ*) corresponds to the interferogram, *I*(*x*) computed

where, to fix the idea we have assumed Δ*σ*<sup>1</sup> < Δ*σ*2. Therefore, *d*(*σ*) can be computed by

*Ip*(*x*) = 0 for *x* < *x*<sup>1</sup> *Ip*(*x*) = *I*(*x*) for *x*<sup>1</sup> ≤ *x* ≤ *x*<sup>2</sup> *Ip*(*x*) = 0 for *x*<sup>2</sup> < *x* ≤ *xmax*

The left and right zero-padding ensures that the difference spectrum is defined on the same

For the purpose of Fourier Transform computations, the above concept of partial interferogram is better formalized by considering an appropriate data-sampling window with

*<sup>W</sup>*˜ (*x*) = � 1 for *<sup>x</sup>*<sup>1</sup> ≤| *<sup>x</sup>* |≤ *<sup>x</sup>*<sup>2</sup>

This data-sampling window removes from the spectrum all those broad features, which are represented by interferometric radiances below the left truncation point, *x*1. In fact we have

where *W* is the box-car window as defined in Eq. (4) and where the under scripts 1 and 2 identify the window function with cutting points, *x*<sup>1</sup> and *x*2, respectively. Because the Fourier transform is linear, within the spectral domain, the double-truncation data-sampling window

Fourier transforming the *partial* interferogram, *Ip*(*x*) defined by

a lower and upper truncation point (Grieco et al., 2011; Kyle, 1977),

⎧ ⎨ ⎩

*σ*-grid as that corresponding to the sampling rate Δ*σ*.

is equivalent to take the difference of Eq. 6.

*d*(*σ*) = *r*1(*σ*) − *r*2(*σ*) (6)

*x*<sup>1</sup> = (2Δ*σ*2)<sup>−</sup>1; *x*<sup>2</sup> = (2Δ*σ*1)−<sup>1</sup> (7)

0 otherwise (9)

*<sup>W</sup>*˜ (*x*) = *<sup>W</sup>*2(*x*) <sup>−</sup> *<sup>W</sup>*1(*x*) (10)

(8)

spectrum

with *i* = 1, 2.

atmospheric emission alone.

over the range [*x*1, *x*2], with

In Fourier spectroscopy the spectrum, *r*(*σ*) (with *σ* the wavenumber) and the interferogram, *I*(*x*) (with *x* the optical path difference) constitute a Fourier pair defined by the couple of equations (see e.g. Bell (1972)),

$$\sigma(\sigma) = \int\_{-\infty}^{+\infty} I(\mathbf{x}) \exp(-2\pi i \sigma \mathbf{x}) d\mathbf{x} \tag{1}$$

$$I(\mathbf{x}) = \int\_{-\infty}^{+\infty} r(\sigma) \exp(2\pi i \sigma \mathbf{x}) d\sigma \tag{2}$$

with *i* the imaginary unit. The spectrum and the interferogram are in practice band-limited functions, therefore taking into account that the interferogram is sampled up to a given maximum optical path difference, *xmax*, we modify Eq. (1) by introducing the data-sampling window, *W*(*x*)

$$\sigma(\sigma) = \int\_{-\infty}^{+\infty} W(\mathbf{x}) I(\mathbf{x}) \exp(-2\pi i \sigma \mathbf{x}) d\mathbf{x} \tag{3}$$

with

$$\mathcal{W}(\mathbf{x}) = \begin{cases} 1 \text{ for } |\mathbf{x}| \le \mathbf{x}\_{\text{max}} \\ 0 & \text{otherwise} \end{cases} \tag{4}$$

where |·| means absolute value, *xmax* is the maximum optical path difference.

The maximum optical path difference, *xmax* also determines the sampling rate, Δ*σ* within the spectral domain. According to the Nyquist rule, the relation is

$$
\Delta \sigma = \frac{1}{2\chi\_{\text{max}}}, \quad \Delta \mathbf{x} = \frac{1}{2(\sigma\_2 - \sigma\_1)} \tag{5}
$$

where *<sup>σ</sup>*<sup>2</sup> <sup>−</sup> *<sup>σ</sup>*<sup>1</sup> is the spectral band-width. For IASI we have *<sup>σ</sup>*<sup>1</sup> <sup>=</sup> 645 cm−1, *<sup>σ</sup>*<sup>2</sup> <sup>=</sup> 2760 cm−1, hence *xmax* = 2 cm and Δ*σ* = 0.25 cm−1.

The fact that IASI is apodized (see e.g. Amato et al. (1998)) is of no concern here. For IASI we just consider the interferogram obtained from the calibrated, apodized spectrum. One could also consider to obtain the interferogram corresponding to each IASI band one at a time. This is appropriate, e.g., when dealing with a given gas whose absorption bands are confined within a single IASI band.

According to the Shannon-Whittaker sampling theorem (e.g. see (Robinson & Silvia, 1981)), in case we want to re-sample the spectrum at a sampling rate lower than the original, we just have to introduce in Eq. (3) a data-sampling window with its new cutting point at *x<sup>τ</sup>* < *xmax*. The new sampling rate, Δ*στ* appropriate to *x<sup>τ</sup>* involves again the Nyquist rule, we have Δ*στ* = (2*x<sup>τ</sup>* )−1.

#### **2.2.1 The couple partial interferogram, difference spectrum**

For a nadir viewing instrument such as IASI, one of the most prominent feature within the observations is the surface emission. This emission allows us to retrieve skin temperature and has information about surface emissivity. However, in case we are interested in retrieving atmospheric parameters, such as minor gases, we would like to have information only about atmospheric emission, since the surface emission interferes with the signal we want to exploit for the retrieval process.

4 Will-be-set-by-IN-TECH

In Fourier spectroscopy the spectrum, *r*(*σ*) (with *σ* the wavenumber) and the interferogram, *I*(*x*) (with *x* the optical path difference) constitute a Fourier pair defined by the couple of

with *i* the imaginary unit. The spectrum and the interferogram are in practice band-limited functions, therefore taking into account that the interferogram is sampled up to a given maximum optical path difference, *xmax*, we modify Eq. (1) by introducing the data-sampling

1 for <sup>|</sup> *<sup>x</sup>* |≤ *xmax*

, <sup>Δ</sup>*<sup>x</sup>* <sup>=</sup> <sup>1</sup>

The maximum optical path difference, *xmax* also determines the sampling rate, Δ*σ* within the

where *<sup>σ</sup>*<sup>2</sup> <sup>−</sup> *<sup>σ</sup>*<sup>1</sup> is the spectral band-width. For IASI we have *<sup>σ</sup>*<sup>1</sup> <sup>=</sup> 645 cm−1, *<sup>σ</sup>*<sup>2</sup> <sup>=</sup> 2760 cm−1,

The fact that IASI is apodized (see e.g. Amato et al. (1998)) is of no concern here. For IASI we just consider the interferogram obtained from the calibrated, apodized spectrum. One could also consider to obtain the interferogram corresponding to each IASI band one at a time. This is appropriate, e.g., when dealing with a given gas whose absorption bands are

According to the Shannon-Whittaker sampling theorem (e.g. see (Robinson & Silvia, 1981)), in case we want to re-sample the spectrum at a sampling rate lower than the original, we just have to introduce in Eq. (3) a data-sampling window with its new cutting point at *x<sup>τ</sup>* < *xmax*. The new sampling rate, Δ*στ* appropriate to *x<sup>τ</sup>* involves again the Nyquist rule, we have Δ*στ* =

For a nadir viewing instrument such as IASI, one of the most prominent feature within the observations is the surface emission. This emission allows us to retrieve skin temperature and has information about surface emissivity. However, in case we are interested in retrieving atmospheric parameters, such as minor gases, we would like to have information only about atmospheric emission, since the surface emission interferes with the signal we want to exploit

*I*(*x*) exp(−2*πiσx*)*dx* (1)

*r*(*σ*) exp(2*πiσx*)*dσ* (2)

*W*(*x*)*I*(*x*) exp(−2*πiσx*)*dx* (3)

0 otherwise (4)

<sup>2</sup>(*σ*<sup>2</sup> <sup>−</sup> *<sup>σ</sup>*1) (5)

*r*(*σ*) =

*I*(*x*) =

 +∞ −∞

*W*(*x*) =

<sup>Δ</sup>*<sup>σ</sup>* <sup>=</sup> <sup>1</sup>

where |·| means absolute value, *xmax* is the maximum optical path difference.

2*xmax*

*r*(*σ*) =

spectral domain. According to the Nyquist rule, the relation is

**2.2.1 The couple partial interferogram, difference spectrum**

hence *xmax* = 2 cm and Δ*σ* = 0.25 cm−1.

confined within a single IASI band.

 +∞ −∞

> +∞ −∞

equations (see e.g. Bell (1972)),

window, *W*(*x*)

with

(2*x<sup>τ</sup>* )−1.

for the retrieval process.

As opposite to gas emission, surface emission varies smoothly with the wave number, *σ*. Thus, if we consider two spectral samplings, Δ*σ*1, Δ*σ*2, which are close each other but small enough so that the surface emission has been resolved at either samplings, then the difference spectrum

$$d(\sigma) = r\_1(\sigma) - r\_2(\sigma) \tag{6}$$

contains mostly the atmospheric emission alone. In Eq. 6, *ri* is the spectrum at sampling Δ*σi*, with *i* = 1, 2.

In reality, the above differentiation cannot get completely rid of the surface radiation, because the corresponding emission, which reaches the Top of the Atmosphere (TOA) is modulated by the total transmittance function, which is itself a highly oscillatory function. However, in window regions where the transmittance approximates the unity, and where gases have only weak absorption bands, the difference *d*(*σ*) yields a signal, which is mostly the result of atmospheric emission alone.

It is important here to stress that to compute the difference spectrum, *d*(*σ*) we do not need to measure the spectrum twice. The operation can be done by simply Fourier transforming one single *partial* interferogram. In fact, with reference to Eq. 6 and provided that, Δ*σ<sup>i</sup>* > Δ*σ* (*i* = 1, 2), and with Δ*σ* defined by Eq. 5, that is the sampling rate corresponding to the maximum optical path difference, we have that *d*(*σ*) corresponds to the interferogram, *I*(*x*) computed over the range [*x*1, *x*2], with

$$\mathbf{x}\_1 = \left(2\Delta\sigma\_2\right)^{-1}; \quad \mathbf{x}\_2 = \left(2\Delta\sigma\_1\right)^{-1} \tag{7}$$

where, to fix the idea we have assumed Δ*σ*<sup>1</sup> < Δ*σ*2. Therefore, *d*(*σ*) can be computed by Fourier transforming the *partial* interferogram, *Ip*(*x*) defined by

$$\begin{cases} I\_p(\mathbf{x}) = 0 & \text{for} \quad \mathbf{x} < \mathbf{x}\_1 \\ I\_p(\mathbf{x}) = I(\mathbf{x}) & \text{for} \quad \mathbf{x}\_1 \le \mathbf{x} \le \mathbf{x}\_2 \\ I\_p(\mathbf{x}) = \mathbf{0} & \text{for} \quad \mathbf{x}\_2 < \mathbf{x} \le \mathbf{x}\_{\max} \end{cases} \tag{8}$$

The left and right zero-padding ensures that the difference spectrum is defined on the same *σ*-grid as that corresponding to the sampling rate Δ*σ*.

For the purpose of Fourier Transform computations, the above concept of partial interferogram is better formalized by considering an appropriate data-sampling window with a lower and upper truncation point (Grieco et al., 2011; Kyle, 1977),

$$\tilde{\mathcal{W}}(\mathbf{x}) = \begin{cases} 1 \text{ for } \mathbf{x}\_1 \le |\mathbf{x}| \le \mathbf{x}\_2\\ 0 & \text{otherwise} \end{cases} \tag{9}$$

This data-sampling window removes from the spectrum all those broad features, which are represented by interferometric radiances below the left truncation point, *x*1. In fact we have

$$
\tilde{\mathcal{W}}(\mathbf{x}) = \mathcal{W}\_2(\mathbf{x}) - \mathcal{W}\_1(\mathbf{x}) \tag{10}
$$

where *W* is the box-car window as defined in Eq. (4) and where the under scripts 1 and 2 identify the window function with cutting points, *x*<sup>1</sup> and *x*2, respectively. Because the Fourier transform is linear, within the spectral domain, the double-truncation data-sampling window is equivalent to take the difference of Eq. 6.

Observations. Methodological Aspects and Application to IASI 7

Atmospheric Gases Concentrations from High Spectral Resolution Satellite Observations...

<sup>253</sup> Fourier Transform Spectroscopy with Partially Scanned Interferograms as a Tool to Retrieve

<sup>1000</sup> <sup>1500</sup> <sup>2000</sup> <sup>2500</sup> <sup>0</sup>

Fig. 2. IASI level 1C radiometric noise in terms of Noise Equivalent Difference Temperature

because IASI data are Gaussian apodized (Amato et al., 1998). However, in section 3 we will see that, at least for IASI, Eq. 11 provides a good reference to have the order of magnitude of

However, whatever the kind of noise (correlated or uncorrelated) in the original spectrum space may be, to obtain *d*(*σ*) we use zero padding in order not to modify the IASI sampling

When properly defined, the difference spectrum, *d*(*σ*) is expected to be less sensitive to the atmospheric state, but the given parameter whose signal we want to amplify, than the original

To quantitatively analyze this aspect we introduce an obvious measure of the sensitivity of

For each given channel or wavenumber, *σ*, the noise-normalized sensitivity to a generic

where *ε*Δ(*σ*) is the radiometric noise affecting the difference spectrum, *d*(*σ*). The above sensitivity expression says how large is the noise-normalized variation of the signal at *σ* to a unitary perturbation of **X**. Note that in general, **X** can a be a scalar (e.g. surface temperature and emissivity, in which case the sensitivity is a scalar itself, function of the wave number, *σ*), or a vector (e.g., temperature profile, water vapour profile), in which case the sensitivity is a

For purpose of comparison, the above sensitivity can be also defined for the case of the original

*ε*(*σ*)

*∂r*(*σ*) *∂***X**

**<sup>S</sup>***X*(*σ*) = <sup>1</sup>

*ε*Δ(*σ*)

*∂d*(*σ*) *∂***X**

(12)

(13)

wave number (cm−1)

0.5

the noise reduction in the transformed difference-spectrum space.

**2.3 Sensitivity of** *d*(*σ*)**-channels to the atmospheric state vector**

parameter, **X**, is defined and computed according to

vector itself, function of the wave number.

of 0.25 cm−1. This means that the noise in the *d*-spectrum space is correlated.

*d*(*σ*) to a given parameter. This measure makes use of the Jacobian derivative.

**<sup>S</sup>***X*,Δ(*σ*) = <sup>1</sup>

1

1.5

NEDT at 280 K (K)

(NEDT) at as scene temperature of 280 K.

spectrum, *r*(*σ*).

IASI spectrum, *r*(*σ*)

2

2.5

3

Fig. 1. Examples of couples (interferogram, spectrum) for different optical path differences. The couple a) b) refers to *x* = 0.1 cm, the couple c) d), to *x* = 0.2 cm. Couple e) f) exemplifies that the partial interferogram in the range [0.1, 0.2] cm corresponds to the difference of the spectra d)-b).

An example of partial interferogram is provided in Fig. 1, where it is also exemplified that the difference spectrum enhances the atmospheric emission, while the broad surface emission is almost zeroed.

Another important characteristic of FTSPSI is that the difference spectrum can be observed with enhanced signal-to-noise ratio with respect to the whole spectrum, *r*(*σ*), that is to the spectrum corresponding to the maximum optical path difference. According to a well-known results of Fourier spectroscopy (Pichett & Strauss, 1972), the spectral noise variance is proportional to the interferogram bandwidth Δ*x* = *x*<sup>2</sup> − *x*1. In case we consider the full interferogram, Δ*x* is exactly the maximum optical path difference, which for IASI is 2 cm. As a consequence, the variance of *d*(*σ*) is simply computed in case we know the variance of *r*(*σ*), that is the radiometric noise affecting the spectral radiances.

The radiometric noise for *r*(*σ*) we have used in this paper is the IASI L1C radiometric noise, which is shown in Fig. 2. Let *ε*(*σ*) be the radiometric noise affecting the spectrum, *r*(*σ*). For the difference spectrum *d*(*σ*), the noise is

$$
\varepsilon\_{\Delta}(\sigma) = \varepsilon(\sigma) \sqrt{\frac{\mathbf{x}\_2 - \mathbf{x}\_1}{\mathbf{x}\_{\max}}} \tag{11}
$$

where the underscript Δ refers to the difference spectrum.

Strictly speaking, Eq. 11 applies to the case of uncorrelated noise, that is in case the noise affecting the original spectrum, *r*(*σ*) is truly random. However, this is not the case for IASI, 6 Will-be-set-by-IN-TECH

−0.05 0 0.05

−0.05 0 0.05 0.1

0.02

−0.02

Fig. 1. Examples of couples (interferogram, spectrum) for different optical path differences. The couple a) b) refers to *x* = 0.1 cm, the couple c) d), to *x* = 0.2 cm. Couple e) f) exemplifies that the partial interferogram in the range [0.1, 0.2] cm corresponds to the difference of the

An example of partial interferogram is provided in Fig. 1, where it is also exemplified that the difference spectrum enhances the atmospheric emission, while the broad surface emission is

Another important characteristic of FTSPSI is that the difference spectrum can be observed with enhanced signal-to-noise ratio with respect to the whole spectrum, *r*(*σ*), that is to the spectrum corresponding to the maximum optical path difference. According to a well-known results of Fourier spectroscopy (Pichett & Strauss, 1972), the spectral noise variance is proportional to the interferogram bandwidth Δ*x* = *x*<sup>2</sup> − *x*1. In case we consider the full interferogram, Δ*x* is exactly the maximum optical path difference, which for IASI is 2 cm. As a consequence, the variance of *d*(*σ*) is simply computed in case we know the variance of *r*(*σ*),

The radiometric noise for *r*(*σ*) we have used in this paper is the IASI L1C radiometric noise, which is shown in Fig. 2. Let *ε*(*σ*) be the radiometric noise affecting the spectrum, *r*(*σ*). For

Strictly speaking, Eq. 11 applies to the case of uncorrelated noise, that is in case the noise affecting the original spectrum, *r*(*σ*) is truly random. However, this is not the case for IASI,

 *<sup>x</sup>*<sup>2</sup> <sup>−</sup> *<sup>x</sup>*<sup>1</sup> *xmax*

*ε*Δ(*σ*) = *ε*(*σ*)

0

r(σ) (W m−2 (cm−1)−1 sr−1)

1000 1500 2000 2500

d)

f)

(11)

1000 1500 2000 2500

1000 1500 2000 2500

wave number, σ (cm−1)

0.1 b)

a)

c)

e)

0 0.005 0.01

0 0.02 0.04 0.06

0 0.02 0.04 0.06 0.08

optical path difference, x (cm)

that is the radiometric noise affecting the spectral radiances.

where the underscript Δ refers to the difference spectrum.

the difference spectrum *d*(*σ*), the noise is

−2

spectra d)-b).

almost zeroed.

0

2

I(x) (W m−2 sr−1)

Fig. 2. IASI level 1C radiometric noise in terms of Noise Equivalent Difference Temperature (NEDT) at as scene temperature of 280 K.

because IASI data are Gaussian apodized (Amato et al., 1998). However, in section 3 we will see that, at least for IASI, Eq. 11 provides a good reference to have the order of magnitude of the noise reduction in the transformed difference-spectrum space.

However, whatever the kind of noise (correlated or uncorrelated) in the original spectrum space may be, to obtain *d*(*σ*) we use zero padding in order not to modify the IASI sampling of 0.25 cm−1. This means that the noise in the *d*-spectrum space is correlated.

#### **2.3 Sensitivity of** *d*(*σ*)**-channels to the atmospheric state vector**

When properly defined, the difference spectrum, *d*(*σ*) is expected to be less sensitive to the atmospheric state, but the given parameter whose signal we want to amplify, than the original spectrum, *r*(*σ*).

To quantitatively analyze this aspect we introduce an obvious measure of the sensitivity of *d*(*σ*) to a given parameter. This measure makes use of the Jacobian derivative.

For each given channel or wavenumber, *σ*, the noise-normalized sensitivity to a generic parameter, **X**, is defined and computed according to

$$\mathbf{S}\_{X,\Delta}(\sigma) = \left(\frac{1}{\varepsilon\_{\Delta}(\sigma)} \frac{\partial d(\sigma)}{\partial \mathbf{X}}\right) \tag{12}$$

where *ε*Δ(*σ*) is the radiometric noise affecting the difference spectrum, *d*(*σ*). The above sensitivity expression says how large is the noise-normalized variation of the signal at *σ* to a unitary perturbation of **X**. Note that in general, **X** can a be a scalar (e.g. surface temperature and emissivity, in which case the sensitivity is a scalar itself, function of the wave number, *σ*), or a vector (e.g., temperature profile, water vapour profile), in which case the sensitivity is a vector itself, function of the wave number.

For purpose of comparison, the above sensitivity can be also defined for the case of the original IASI spectrum, *r*(*σ*)

$$\mathbf{S}\_{X}(\sigma) = \left(\frac{1}{\varepsilon(\sigma)} \frac{\partial r(\sigma)}{\partial \mathbf{X}}\right) \tag{13}$$

Observations. Methodological Aspects and Application to IASI 9

Atmospheric Gases Concentrations from High Spectral Resolution Satellite Observations...

<sup>255</sup> Fourier Transform Spectroscopy with Partially Scanned Interferograms as a Tool to Retrieve

Pressure (hPa)

Pressure (hPa)

Fig. 4. The two reference atmosphere models used to simulate IASI synthetic radiances. The models belong to two opposite weather conditions: tropical and High Latitude Winter. As a rule, for the reference state, the CO2 profile is assumed constant with altitude (as shown in (c)). For checking a possible dependence of the methodology on the shape of the CO2 profile, the analysis has also considered a case of CO2 profile non uniform with altitude (shown in

Figure 5(a) exemplifies, for the case of the channel at *σ*<sup>1</sup> = 783.75, that the relation between *Dj*

The solid lines shown in Fig. 5 are the linear best fit to the data points. It is important to stress that the regression coefficients *aj* and *bj* depend on the Field of View angle. Unless otherwise stated, the results shown in this section apply to the nadir angle. In addition, it is also important to stress that even for noisy-free radiances the estimation of the column-integrated CO2 from Eq. 15 has a residual uncertainty. This uncertainty is of the order od 1-1.5 ppmv and can be thought of as a sort of error inherited from the linearization of the dependence of *qCO*<sup>2</sup> on *dj*. The exact amount of this uncertainty can depend on the wave number. Actually, the search of the *linear* channels have been done by computing the linear regression standard error at each single channel and choosing only those with standard error less than 1.5 ppmv. The typical accuracy or estimation error of *qCO*<sup>2</sup> estimated by Eq. 15 for the four channels at

Table 1 (see the accuracy for the noiseless case) also shows that a linear relation holds for the

remaining three channels, *σ*<sup>1</sup> = 809.25 cm−1, *σ*<sup>3</sup> = 976.75 cm−1, *σ*<sup>4</sup> = 2105 cm−1.

0 2 4 6 8 10 12 14

(b) H2O profile

375 380 385 390 395 400

(d) CO2 non-uniform profile

*qCO*<sup>2</sup> = *ajDj* + *bj* (15)

CO2 mixing ratio (ppmv)

O mixing ratio (g/kg)

Tropical High Latitude Winter

Tropical

H2

200 220 240 260 280 300

Tropical High Latitude Winter

Temperature (K)

Tropical High Latitude Winter

(a) Temperature profile

386 387 388 389 390 391 392

(c) CO2 uniform profile

mixing ratio (ppmv)

(d)); this case was used in combination with the tropical model alone.

CO2

and *qCO*<sup>2</sup> is perfectly linear, that is

hand is shown in Tab. 1.

Pressure (hPa)

Pressure (hPa)

### **3. Application to CO**<sup>2</sup>

As said in section 1, for the case of CO2 we have that a small range around the optical path difference at 0.66 cm is mostly dominated by CO2 emission. The resulting CO2 signature is clearly visible in Fig. 3, which shows a partial interferogram in the range 0.65 to 0.68 cm. Transforming back to the spectral domain the partial interferogram, we obtain the difference

Fig. 3. a) Example of partial interferogram in the range [*x* = 0.65, *x* = 0.68] cm for a case of CO2 load equal to 0 and 385 ppmv, respectively; b) the difference spectrum *d*(*σ*). The full IASI spectral coverage 645 to 2760 cm−<sup>1</sup> has been considered.

spectrum shown in 3(b), where the CO2-lines periodic pattern is clearly amplified.

Figure 3(b) suggests that *d*(*σ*) is mostly a function of the CO2 amount alone. This can be checked in simulation through generation and analysis of difference-spectra corresponding to diverse amount of CO2.

#### **3.1 The case of noise free radiances**

Radiative transfer calculations were performed based on the atmospheric states summarized in Fig. 4. With reference to the tropical model of atmosphere shown in Fig. 4 (panels (a), (b) and (c)), difference spectra were generated for sixteen different columnar amounts, *qCO*<sup>2</sup> (*i*), (*i* = 1 . . . 16) of CO2 ranging from 300 to 450 ppmv with a step of 10 ppmv. To perform the radiative transfer calculations, we had to assume a model of dependence on altitude for the CO2 mixing ratio. As shown in Fig. 4, this was assumed to be constant with altitude. A search of the channels that are best linearly correlated with the CO2 columnar amount shows that there are many and a few of this are largely insensitive to surface emissivity and temperature. Four of these good channels have been identified and they correspond to the wave numbers *σ*<sup>1</sup> = 783.75 cm−1, *σ*<sup>2</sup> = 809.25 cm−1, *σ*<sup>3</sup> = 976.75 cm−1, *σ*<sup>4</sup> = 2105 cm−1, respectively.

Let *dj*(*i*) be the difference spectrum ordinate corresponding to the wave number *σ<sup>j</sup>* (with *j* = 1 . . . 4) and the CO2 columnar amount, *qCO*<sup>2</sup> (*i*), (with *i* = 1 . . . 16), respectively. For each wave number, *σj*, we consider the standardized quantity,

$$D\_{\dot{j}}(i) = \frac{d\_{\dot{j}}(i) - \mu\_{\dot{j}}}{s\_{\dot{j}}} \tag{14}$$

with *μ<sup>j</sup>* and *sj* the mean and standard deviation of the sample, {*dj*(*i*)}*i*=1...16, respectively.

8 Will-be-set-by-IN-TECH

As said in section 1, for the case of CO2 we have that a small range around the optical path difference at 0.66 cm is mostly dominated by CO2 emission. The resulting CO2 signature is clearly visible in Fig. 3, which shows a partial interferogram in the range 0.65 to 0.68 cm. Transforming back to the spectral domain the partial interferogram, we obtain the difference

0.65 0.655 0.66 0.665 0.67 0.675 0.68 −4

a)

with CO2 zero CO2

b)

with CO2 zero CO2

Optical path difference, x (cm)

<sup>800</sup> <sup>1000</sup> <sup>1200</sup> <sup>1400</sup> <sup>1600</sup> <sup>1800</sup> <sup>2000</sup> <sup>2200</sup> <sup>2400</sup> <sup>2600</sup> −0.02

Fig. 3. a) Example of partial interferogram in the range [*x* = 0.65, *x* = 0.68] cm for a case of CO2 load equal to 0 and 385 ppmv, respectively; b) the difference spectrum *d*(*σ*). The full

Figure 3(b) suggests that *d*(*σ*) is mostly a function of the CO2 amount alone. This can be checked in simulation through generation and analysis of difference-spectra corresponding to

Radiative transfer calculations were performed based on the atmospheric states summarized in Fig. 4. With reference to the tropical model of atmosphere shown in Fig. 4 (panels (a), (b) and (c)), difference spectra were generated for sixteen different columnar amounts, *qCO*<sup>2</sup> (*i*), (*i* = 1 . . . 16) of CO2 ranging from 300 to 450 ppmv with a step of 10 ppmv. To perform the radiative transfer calculations, we had to assume a model of dependence on altitude for the CO2 mixing ratio. As shown in Fig. 4, this was assumed to be constant with altitude. A search of the channels that are best linearly correlated with the CO2 columnar amount shows that there are many and a few of this are largely insensitive to surface emissivity and temperature. Four of these good channels have been identified and they correspond to the wave numbers *σ*<sup>1</sup> = 783.75 cm−1, *σ*<sup>2</sup> = 809.25 cm−1, *σ*<sup>3</sup> = 976.75 cm−1, *σ*<sup>4</sup> = 2105 cm−1, respectively.

Let *dj*(*i*) be the difference spectrum ordinate corresponding to the wave number *σ<sup>j</sup>* (with *j* = 1 . . . 4) and the CO2 columnar amount, *qCO*<sup>2</sup> (*i*), (with *i* = 1 . . . 16), respectively. For each wave

> *dj*(*i*) − *μ<sup>j</sup> sj*

(14)

*Dj*(*i*) =

with *μ<sup>j</sup>* and *sj* the mean and standard deviation of the sample, {*dj*(*i*)}*i*=1...16, respectively.

spectrum shown in 3(b), where the CO2-lines periodic pattern is clearly amplified.

wave number (cm−1)

−0.01 0 0.01 0.02

IASI spectral coverage 645 to 2760 cm−<sup>1</sup> has been considered.

d(σ), W m−2 (cm−1)−1 sr−1

I(x), W m−2 sr−1

**3. Application to CO**<sup>2</sup>

diverse amount of CO2.

**3.1 The case of noise free radiances**

number, *σj*, we consider the standardized quantity,

Fig. 4. The two reference atmosphere models used to simulate IASI synthetic radiances. The models belong to two opposite weather conditions: tropical and High Latitude Winter. As a rule, for the reference state, the CO2 profile is assumed constant with altitude (as shown in (c)). For checking a possible dependence of the methodology on the shape of the CO2 profile, the analysis has also considered a case of CO2 profile non uniform with altitude (shown in (d)); this case was used in combination with the tropical model alone.

Figure 5(a) exemplifies, for the case of the channel at *σ*<sup>1</sup> = 783.75, that the relation between *Dj* and *qCO*<sup>2</sup> is perfectly linear, that is

$$q\_{CO\_2} = a\_{\dot{\jmath}} D\_{\dot{\jmath}} + b\_{\dot{\jmath}} \tag{15}$$

The solid lines shown in Fig. 5 are the linear best fit to the data points. It is important to stress that the regression coefficients *aj* and *bj* depend on the Field of View angle. Unless otherwise stated, the results shown in this section apply to the nadir angle. In addition, it is also important to stress that even for noisy-free radiances the estimation of the column-integrated CO2 from Eq. 15 has a residual uncertainty. This uncertainty is of the order od 1-1.5 ppmv and can be thought of as a sort of error inherited from the linearization of the dependence of *qCO*<sup>2</sup> on *dj*. The exact amount of this uncertainty can depend on the wave number. Actually, the search of the *linear* channels have been done by computing the linear regression standard error at each single channel and choosing only those with standard error less than 1.5 ppmv. The typical accuracy or estimation error of *qCO*<sup>2</sup> estimated by Eq. 15 for the four channels at hand is shown in Tab. 1.

Table 1 (see the accuracy for the noiseless case) also shows that a linear relation holds for the remaining three channels, *σ*<sup>1</sup> = 809.25 cm−1, *σ*<sup>3</sup> = 976.75 cm−1, *σ*<sup>4</sup> = 2105 cm−1.

Observations. Methodological Aspects and Application to IASI 11

<sup>257</sup> Fourier Transform Spectroscopy with Partially Scanned Interferograms as a Tool to Retrieve

atmosphere (see Fig. 4 from (a) to (c)), we obtain the result shown in Fig. 5(c), which says that

In addition, not only the linear shape does not change with the state vector and the assumed shape of the CO2 profile, it is exactly the same (as it is shown in Fig. 5(d)). That is the regression coefficients remain constant under a varying state vector, even in the CO2 profile

Once again, we stress that although Fig. 5 focuses on the channel at *σ*<sup>1</sup> = 783.75 cm−1, the same results hold for all the four channels, *σj*, *j* = 1, . . . , 4. The linearity for these channels is likely to be a results of the fact that they correspond to wing regions of strong absorption bands, or to moderate absorption bands. In general, channels inside strong absorptions bands

It is important to stress that while the linear dependence of *qCO*<sup>2</sup> on *Dj* is universal, this is not true for the case in which we consider the difference spectrum, *dj*, since the standardization

Since in practice we observe *dj*, in case we have one single IASI spectrum, we are faced with the problem of computing *μ<sup>j</sup>* and *sj* to perform the standardization, which yield, *Dj* (see Eq. 14). The only way to go in this case is to compute a series of synthetic IASI spectra with different load of CO2. The synthetic spectra have to be computed on the basis of the possibly best estimate of the atmospheric state vector corresponding to the given IASI observation. Thus, for *weather* applications in which we need a space-time resolved CO2 estimation, we do

For *climate* analysis, we need to average over many weather states. Thus, the fact that *qCO*<sup>2</sup> has an universal linear dependence on *Dj* means that, by considering averages, we get rid of the dependence on the state vector. In practice, if we compare *dj*-values properly averaged on climate time scales, these values are directly proportional to *qCO*<sup>2</sup> -values averaged over the same climate time scales. As an example if we take all the IASI clear sky sea-surface tropical spectra for the year, say *Y*<sup>1</sup> and compute the mean value of *dj* for whatever *j* (call this average < *dj*,*Y*<sup>1</sup> >), and redo this operation for a second year *Y*2, the difference < *dj*,*Y*<sup>2</sup> > − < *dj*,*Y*<sup>1</sup> > is directly proportional to the variation of the CO2 over the time span *Y*<sup>2</sup> − *Y*1. To accurately estimate CO2 changes on a climate time span is an easy task from satellite

With radiances affected by noise we can again just use Eq. 15 to estimates the CO2 columnar amount. However, now the variability of the estimate is expected to increase according to the

In case of noisy radiances, taking into account Eq. 11, the accuracy can be computed through

*<sup>j</sup>* /*s*<sup>2</sup>

*<sup>j</sup>*)var *dj* 

(16)

the usual rule of variance propagation. With reference to the basic Eq. 15, we have

var(*qCO*<sup>2</sup> ) = (*a*<sup>2</sup>

the dependence of *qCO*<sup>2</sup> on *Dj* is linear whatever the atmospheric state vector may be.

Atmospheric Gases Concentrations from High Spectral Resolution Satellite Observations...

shape, which means that Eq. 15 has a general or universal validity.

show a behavior, which is largely non linear.

parameters, *μ<sup>j</sup>* and *sj* do depend on the state vector.

need information on the atmospheric state vector.

observations, provided we use FTS\*PSI.

**3.2 Noisy radiances**

noise affecting IASI radiances.

where var(·) stand for variance.

and (c) are intercompared.

Fig. 5. Exemplifying the dependence of *qCO*<sup>2</sup> on *Dj* for the channel at *<sup>σ</sup>*<sup>1</sup> = 783.75 cm−<sup>1</sup> as a function of the atmosphere model (tropical and High Latitude Winter, HLW) and shape of the CO2 profile.


Table 1. Accuracy of the column-integrated CO2 amount estimated through the linear fit of Eq. 15 for the case of noise-free and noisy radiances.

More important here is the fact that the functional relation between *qCO*<sup>2</sup> and *Dj* does not depend on the assumed shape for the CO2 profile. To check this dependence we have re-done the calculations, but now with a CO2 mixing ratio profile, which is not constant with altitude. This altitude-varying profile (shown in Fig. 4(d)) represents a realistic situation. In fact, it is the result of the ECMWF analysis Engelen et al. (2009) corresponding to the date (29 April 2007) and location of the JAIVEx experiment. Figure 5(b) shows that the functional relation remains perfectly linear.

Moreover, the shape of the functional relation is completely independent of the state vector, as well. In fact, if we redo the calculations, but now with the High Latitude Winter model of atmosphere (see Fig. 4 from (a) to (c)), we obtain the result shown in Fig. 5(c), which says that the dependence of *qCO*<sup>2</sup> on *Dj* is linear whatever the atmospheric state vector may be.

In addition, not only the linear shape does not change with the state vector and the assumed shape of the CO2 profile, it is exactly the same (as it is shown in Fig. 5(d)). That is the regression coefficients remain constant under a varying state vector, even in the CO2 profile shape, which means that Eq. 15 has a general or universal validity.

Once again, we stress that although Fig. 5 focuses on the channel at *σ*<sup>1</sup> = 783.75 cm−1, the same results hold for all the four channels, *σj*, *j* = 1, . . . , 4. The linearity for these channels is likely to be a results of the fact that they correspond to wing regions of strong absorption bands, or to moderate absorption bands. In general, channels inside strong absorptions bands show a behavior, which is largely non linear.

It is important to stress that while the linear dependence of *qCO*<sup>2</sup> on *Dj* is universal, this is not true for the case in which we consider the difference spectrum, *dj*, since the standardization parameters, *μ<sup>j</sup>* and *sj* do depend on the state vector.

Since in practice we observe *dj*, in case we have one single IASI spectrum, we are faced with the problem of computing *μ<sup>j</sup>* and *sj* to perform the standardization, which yield, *Dj* (see Eq. 14). The only way to go in this case is to compute a series of synthetic IASI spectra with different load of CO2. The synthetic spectra have to be computed on the basis of the possibly best estimate of the atmospheric state vector corresponding to the given IASI observation. Thus, for *weather* applications in which we need a space-time resolved CO2 estimation, we do need information on the atmospheric state vector.

For *climate* analysis, we need to average over many weather states. Thus, the fact that *qCO*<sup>2</sup> has an universal linear dependence on *Dj* means that, by considering averages, we get rid of the dependence on the state vector. In practice, if we compare *dj*-values properly averaged on climate time scales, these values are directly proportional to *qCO*<sup>2</sup> -values averaged over the same climate time scales. As an example if we take all the IASI clear sky sea-surface tropical spectra for the year, say *Y*<sup>1</sup> and compute the mean value of *dj* for whatever *j* (call this average < *dj*,*Y*<sup>1</sup> >), and redo this operation for a second year *Y*2, the difference < *dj*,*Y*<sup>2</sup> > − < *dj*,*Y*<sup>1</sup> > is directly proportional to the variation of the CO2 over the time span *Y*<sup>2</sup> − *Y*1. To accurately estimate CO2 changes on a climate time span is an easy task from satellite observations, provided we use FTS\*PSI.

#### **3.2 Noisy radiances**

10 Will-be-set-by-IN-TECH

qCO2 (ppmv)

Fig. 5. Exemplifying the dependence of *qCO*<sup>2</sup> on *Dj* for the channel at *<sup>σ</sup>*<sup>1</sup> = 783.75 cm−<sup>1</sup> as a function of the atmosphere model (tropical and High Latitude Winter, HLW) and shape of

Channel Accuracy Accuracy

(cm−1) (ppmv) (ppmv) 783.75 1.4 17.5 809.25 1.5 10.8 976.75 1.2 28.7 2105 1.1 26.2 Table 1. Accuracy of the column-integrated CO2 amount estimated through the linear fit of

More important here is the fact that the functional relation between *qCO*<sup>2</sup> and *Dj* does not depend on the assumed shape for the CO2 profile. To check this dependence we have re-done the calculations, but now with a CO2 mixing ratio profile, which is not constant with altitude. This altitude-varying profile (shown in Fig. 4(d)) represents a realistic situation. In fact, it is the result of the ECMWF analysis Engelen et al. (2009) corresponding to the date (29 April 2007) and location of the JAIVEx experiment. Figure 5(b) shows that the functional relation

Moreover, the shape of the functional relation is completely independent of the state vector, as well. In fact, if we redo the calculations, but now with the High Latitude Winter model of

(Noise-free case) (Noisy case)

qCO2 (ppmv)

−2 −1.5 −1 −0.5 <sup>0</sup> 0.5 <sup>1</sup> 1.5 <sup>2</sup> <sup>280</sup>

(b) Tropical, CO2 non unifrom

−2 −1.5 −1 −0.5 <sup>0</sup> 0.5 <sup>1</sup> 1.5 <sup>2</sup> <sup>280</sup>

(d) The three linear best fits in (a), (b)

and (c) are intercompared.

D1

Tropical, Uniform Tropical, non−Uniform HLW, Uniform

D1

Data Points Linear Fit

Data points Linear Fit

Data Points Linear Fit

−2 −1.5 −1 −0.5 <sup>0</sup> 0.5 <sup>1</sup> 1.5 <sup>2</sup> <sup>280</sup>

(a) Tropical, CO2 unifrom

−2 −1.5 −1 −0.5 <sup>0</sup> 0.5 <sup>1</sup> 1.5 <sup>2</sup> <sup>280</sup>

(c) HLW, CO2 unifrom

Eq. 15 for the case of noise-free and noisy radiances.

D1

D1

qCO2 (ppmv)

the CO2 profile.

remains perfectly linear.

qCO2 (ppmv)

With radiances affected by noise we can again just use Eq. 15 to estimates the CO2 columnar amount. However, now the variability of the estimate is expected to increase according to the noise affecting IASI radiances.

In case of noisy radiances, taking into account Eq. 11, the accuracy can be computed through the usual rule of variance propagation. With reference to the basic Eq. 15, we have

$$\text{var}\left(q\_{\text{CO}\_2}\right) = \left(a\_{\text{j}}^2 / s\_{\text{j}}^2\right) \text{var}\left(d\_{\text{j}}\right) \tag{16}$$

where var(·) stand for variance.

Observations. Methodological Aspects and Application to IASI 13

<sup>259</sup> Fourier Transform Spectroscopy with Partially Scanned Interferograms as a Tool to Retrieve

Pressure (hPa)

0 2000 4000 6000 8000

(ppmv−1)

Fig. 7. For the four channels listed in the legend, panel a) shows the sensitivity of *r*(*σ*) and

have that the couple (*d*1, *d*2) is slightly correlated with a correlation of 0.34, whereas the other

For a given IASI observation, let us suppose that we have computed through Eq. 15, the CO2 amount, say *qj*, corresponding to the four channels at *σj*, *j* = 1, . . . , 4. Let us define

diagonal matrix, whose diagonal elements are the regression coefficients (*aj*/*sj*)2. Then the

**C**−<sup>1</sup> *q* **1***t***C**−<sup>1</sup> *<sup>q</sup>* **1**

> **1***t* **C**−<sup>1</sup> *<sup>q</sup>* **1** −<sup>1</sup>

One could say, why not to use more channels to get the accuracy close to the inherent limit of 1.5 ppmv? Well, the problem is that the more channels we use, the more the correlation among channels themselves increase. With high correlated channels, the optimal estimation does not improve so much in comparison to the channel with the best accuracy. Furthermore, another important point, which brings us to the next section, is the sensitivity of the channels to the state vector. Since we have only access to a (possibly) best estimate of the atmospheric state corresponding to the given IASI observation, we need to make sure to select channels,

In this respect, the four channels, we are considering so long, show a very high sensitivity to CO2. This is seen from Fig. 7, which compares **S**(*σ*) to **S**Δ(*σ*) for the case of a tropical model of the atmosphere. We see that the sensitivity to CO2 improves of a factor 4-5 when passing from *r*(*σ*) to *d*(*σ*). Conversely, the sensitivity to the temperature profile (not shown for the sake of brevity) largely decreases when transforming from *r*(*σ*) to *d*(*σ*). The sensitivity analysis

*<sup>q</sup>*¯*CO*<sup>2</sup> <sup>=</sup> **<sup>1</sup>***<sup>t</sup>*

var(*q*¯*CO*<sup>2</sup> ) =

0 2000 4000 6000 8000

, where the super-script *t* denotes transpose operation. Let **A***<sup>d</sup>* be the 4 × 4

. Combining this way the four estimates, we have that the accuracy

*<sup>d</sup>* and the optimal Least

(18)

**q** (17)

(ppmv−1)

b)

785.25 cm−1 809.25 cm−1 976.25 cm−1 2105 cm−1

Sensitivity, SX,<sup>Δ</sup>

a)

785.25 cm−1 809.25 cm−1 976.75 cm−1 2105 cm−1

Atmospheric Gases Concentrations from High Spectral Resolution Satellite Observations...

Sensitivity, SX

covariance matrix, **C***<sup>q</sup>* of the vector **q** is the product **C***<sup>q</sup>* = **A***d***C***d***A***<sup>t</sup>*

Squares estimation of, *q*¯*CO*<sup>2</sup> from the four estimates is given by

Pressure (hPa)

panel b) that of *d*(*σ*) to the CO2 mixing ratio profile.

combinations are truly un-correlated.

with the variance, var(*q*¯*CO*<sup>2</sup> ) given by

improves to the value of ≈ ±7 ppmv.

which are loosely sensitive to the state vector, but CO2.

**q** = (*q*1,..., *q*4)*<sup>t</sup>*

where **1** = (1, 1, 1, 1)*<sup>t</sup>*

Figure 6 shows the standard deviation, *ε*Δ(*σ*) of the noise affecting the difference spectrum, *d*(*σ*) under the assumptions of IASI correlated noise (Fig. 6(a)) and IASI uncorrelated noise (Fig. 6(b)).

These computations have been done in simulation. First, we generated samples of random numbers with zero mean and standard deviation according to the IASI radiometric noise figures in Fig. 2. Second, these samples were passed through the same series of calculations as those that transform *r*(*σ*) to *d*(*σ*). In case we consider the effect of correlation, random numbers were generated according to the full IASI level 1C covariance matrix. The noise shown in Fig. 2 corresponds to the square root of the diagonal of the IASI covariance matrix.

Figure 6 also provides a comparison with the standard deviation obtained from a direct use of Eq. 11, with the appropriate band factor equal to 2/(0.68 <sup>−</sup> 0.65) = 8.165. It is seen that the comparison between simulation and Eq. 11 is perfect for the IASI uncorrelated noise. For the real case of IASI correlated noise, Eq. 11 slightly underestimates the standard deviation of the difference spectrum, *d*(*σ*). Here and in the following the noise affecting *d*(*σ*) has been computed by considering correlated IASI noise at level 1C. Using the standard deviation

Fig. 6. Standard deviation of the noise affecting the difference spectrum in case of a partial interferogram extending from 0.65 to 0.68 cm. The computations have been made with IASI correlated noise (a) and IASI uncorrelated noise (b). A comparison is also provided with the standard deviation calculated with Eq. 11.

expected for the difference spectrum (Fig. 6(a)) together with Eq. 16, we can easily compute the accuracy (that is the square root of var(*qCO*<sup>2</sup> )) of the linear regression at each of the four channels at hand. The accuracy is shown in Tab. 1, which allows us to compare the results with the noise-free case.

Form Tab. 1 it is seen that the regression error increases of a factor 5 to 25, depending on the channel. However, we can optimally average the four different estimates, in order to improve the accuracy. To do so, it is important to realize that the *dj*-channels may be correlated, so that we have to consider their proper covariance matrix in order to compute the optimal average.

Let **C***<sup>d</sup>* be the covariance matrix of the four channels. The size of **C***<sup>d</sup>* is 4 × 4 and it can be computed in simulation or directly by transforming the IASI covariance matrix from the radiance space to the difference-radiance space. Whatever we do, this matrix can be computed in advance and stored for later applications, therefore it can be thought of as being a known parameter. Based on simulations, which made use of the IASI level 1C covariance matrix, we

Fig. 7. For the four channels listed in the legend, panel a) shows the sensitivity of *r*(*σ*) and panel b) that of *d*(*σ*) to the CO2 mixing ratio profile.

have that the couple (*d*1, *d*2) is slightly correlated with a correlation of 0.34, whereas the other combinations are truly un-correlated.

For a given IASI observation, let us suppose that we have computed through Eq. 15, the CO2 amount, say *qj*, corresponding to the four channels at *σj*, *j* = 1, . . . , 4. Let us define **q** = (*q*1,..., *q*4)*<sup>t</sup>* , where the super-script *t* denotes transpose operation. Let **A***<sup>d</sup>* be the 4 × 4 diagonal matrix, whose diagonal elements are the regression coefficients (*aj*/*sj*)2. Then the covariance matrix, **C***<sup>q</sup>* of the vector **q** is the product **C***<sup>q</sup>* = **A***d***C***d***A***<sup>t</sup> <sup>d</sup>* and the optimal Least Squares estimation of, *q*¯*CO*<sup>2</sup> from the four estimates is given by

$$\bar{\eta}\_{\rm CO\_2} = \frac{\mathbf{1}^t \mathbf{C}\_q^{-1}}{\mathbf{1}^t \mathbf{C}\_q^{-1} \mathbf{1}} \mathbf{q} \tag{17}$$

with the variance, var(*q*¯*CO*<sup>2</sup> ) given by

12 Will-be-set-by-IN-TECH

Figure 6 shows the standard deviation, *ε*Δ(*σ*) of the noise affecting the difference spectrum, *d*(*σ*) under the assumptions of IASI correlated noise (Fig. 6(a)) and IASI uncorrelated noise

These computations have been done in simulation. First, we generated samples of random numbers with zero mean and standard deviation according to the IASI radiometric noise figures in Fig. 2. Second, these samples were passed through the same series of calculations as those that transform *r*(*σ*) to *d*(*σ*). In case we consider the effect of correlation, random numbers were generated according to the full IASI level 1C covariance matrix. The noise shown in Fig. 2 corresponds to the square root of the diagonal of the IASI covariance matrix. Figure 6 also provides a comparison with the standard deviation obtained from a direct use of Eq. 11, with the appropriate band factor equal to 2/(0.68 <sup>−</sup> 0.65) = 8.165. It is seen that the comparison between simulation and Eq. 11 is perfect for the IASI uncorrelated noise. For the real case of IASI correlated noise, Eq. 11 slightly underestimates the standard deviation of the difference spectrum, *d*(*σ*). Here and in the following the noise affecting *d*(*σ*) has been computed by considering correlated IASI noise at level 1C. Using the standard deviation

<sup>1000</sup> <sup>1500</sup> <sup>2000</sup> <sup>2500</sup> <sup>0</sup>

IASI level 1C noise

wave number (cm−1)

(b) truly random

Noise Diff. Spectrum simulated with no correlation IASI level 1C noise scaled with the band factor

Fig. 6. Standard deviation of the noise affecting the difference spectrum in case of a partial interferogram extending from 0.65 to 0.68 cm. The computations have been made with IASI correlated noise (a) and IASI uncorrelated noise (b). A comparison is also provided with the

expected for the difference spectrum (Fig. 6(a)) together with Eq. 16, we can easily compute the accuracy (that is the square root of var(*qCO*<sup>2</sup> )) of the linear regression at each of the four channels at hand. The accuracy is shown in Tab. 1, which allows us to compare the results

Form Tab. 1 it is seen that the regression error increases of a factor 5 to 25, depending on the channel. However, we can optimally average the four different estimates, in order to improve the accuracy. To do so, it is important to realize that the *dj*-channels may be correlated, so that we have to consider their proper covariance matrix in order to compute the optimal average. Let **C***<sup>d</sup>* be the covariance matrix of the four channels. The size of **C***<sup>d</sup>* is 4 × 4 and it can be computed in simulation or directly by transforming the IASI covariance matrix from the radiance space to the difference-radiance space. Whatever we do, this matrix can be computed in advance and stored for later applications, therefore it can be thought of as being a known parameter. Based on simulations, which made use of the IASI level 1C covariance matrix, we

Radiometric noise, W m2 (cm−1)−1 sr−1

<sup>1000</sup> <sup>1500</sup> <sup>2000</sup> <sup>2500</sup> <sup>0</sup>

standard deviation calculated with Eq. 11.

IASI level 1C noise

wave number (cm−1)

(a) with correlation

Noise Diff. Spectrum simulated with correlation IASI level 1C noise scaled with the band factor

with the noise-free case.

Radiometric noise, W m2 (cm−1)−1 sr−1

(Fig. 6(b)).

$$\text{var}(\bar{q}\_{CO\_2}) = \left(\mathbf{1}^t \mathbf{C}\_q^{-1} \mathbf{1}\right)^{-1} \tag{18}$$

where **1** = (1, 1, 1, 1)*<sup>t</sup>* . Combining this way the four estimates, we have that the accuracy improves to the value of ≈ ±7 ppmv.

One could say, why not to use more channels to get the accuracy close to the inherent limit of 1.5 ppmv? Well, the problem is that the more channels we use, the more the correlation among channels themselves increase. With high correlated channels, the optimal estimation does not improve so much in comparison to the channel with the best accuracy. Furthermore, another important point, which brings us to the next section, is the sensitivity of the channels to the state vector. Since we have only access to a (possibly) best estimate of the atmospheric state corresponding to the given IASI observation, we need to make sure to select channels, which are loosely sensitive to the state vector, but CO2.

In this respect, the four channels, we are considering so long, show a very high sensitivity to CO2. This is seen from Fig. 7, which compares **S**(*σ*) to **S**Δ(*σ*) for the case of a tropical model of the atmosphere. We see that the sensitivity to CO2 improves of a factor 4-5 when passing from *r*(*σ*) to *d*(*σ*). Conversely, the sensitivity to the temperature profile (not shown for the sake of brevity) largely decreases when transforming from *r*(*σ*) to *d*(*σ*). The sensitivity analysis

Observations. Methodological Aspects and Application to IASI 15

Atmospheric Gases Concentrations from High Spectral Resolution Satellite Observations...

<sup>261</sup> Fourier Transform Spectroscopy with Partially Scanned Interferograms as a Tool to Retrieve

Fig. 8. (a)- CO2 amount estimated from IASI (this work) and ECMWF analysis. (b)- IASI CO2

However, it is also important to stress that with a polar satellite such as METOP/1, the time-space data coverage is not uniform over the Mediterranean area, so that the spatial gradient has to be considered with some care and its assessment needs a suitable transport model. However, our finding is in agreement with similar maps derived by the Atmospheric

As for the case of CO2, CO has well defined and known rotation transitions, which yield an absorption band, centered in between the atmospheric window at 4.67 *μ*m (2142 cm−1). Because CO is a linear molecule, its strongest absorption features are regularly spaced and yield a periodic pattern, whose period is <sup>≈</sup> 4 cm−<sup>1</sup> (see Fig. 9). For this reason, the

Pressure (hPa)

Infrared Radiometer Sounder (AIRS) for the month of July (Chahine et al, 2008).

zero CO with CO

Fig. 9. The figure shows (a) two synthetic IASI spectra in the spectral region of CO

(c) shows the CO reference profile used for radiative transfer calculations.

absorption; one of the spectra has been calculated with zero load of CO. (b) The difference between the two evidences the sinusoidal appearance of the CO absorption features. Panel

interferogram has to show a characteristic CO-beating at *x* = 1/4cm = 0.25 cm. We consider a partial interferogram extending from 0.21 to 0.31 cm for a width Δ*x* = 0.1 cm. With the choice Δ*x* = 0.1, the factor of noise reduction of Eq. 11 is approximately 4.45. In addition, since the CO band at 4.67 *μ*m is completely covered by IASI band 3 (this extends from 2000 to 2760 cm−1), the interferogram has been built up for band 3 alone. Finally, the reference CO profile

<sup>2000</sup> <sup>2050</sup> <sup>2100</sup> <sup>2150</sup> <sup>2200</sup> <sup>2250</sup> −5

(b) <sup>2000</sup> <sup>2050</sup> <sup>2100</sup> <sup>2150</sup> <sup>2200</sup> <sup>2250</sup> <sup>0</sup>

wave number (cm−1)

(a)

wave number (cm−1)

Average CO2

concentration for July 2010 (ppmv)

(b) Mediterranean case study

0 50 100 150 200

(c)

CO mixing ratio (ppbv)

<sup>0</sup> <sup>5</sup> <sup>10</sup> <sup>15</sup> <sup>20</sup> <sup>25</sup> <sup>350</sup>

Number of IASI Sounding

(a) JAIVEx case study

for July 2010 over the Mediterranean area.

IASI CO2 IASI error bar ECMWF CO2

**4. Application to CO**

2 4 <sup>6</sup> x 10−3

r(σ), W m−2 (cm−1)−1 sr−1

Δ r(σ), W m−2 (cm−1)−1 sr−1

qCO2 (ppmv)

shows that the *dj*-channels are sensitive to a large part of the troposphere, from to 900 to 100 hPa, which for the case of a tropical model of atmosphere encompasses all the troposphere above the Planetary Boundary Layer. Conversely, the *dj* channels are almost insensitive to what happen in the boundary layer either for CO2 or e.g. temperature, which is good because the lower troposphere is where we expect to have a larger uncertainty to the atmospheric state. In other words, also in case we implement the technique with a state vector which largely differs from the *truth* in the lower troposphere, we can still have valuable estimates for CO2 provided the state vector is sufficiently accurate for the rest of the troposphere.
