The σ-IASI code for the calculation of infrared atmospheric radiance and its derivatives

Environmental Modelling & Software 17 (2002) 651–667 www.elsevier.com/locate/envsoft

The σ-IASI code for the calculation of infrared atmospheric radiance and its derivatives U. Amato a, G. Masiello b, C. Serio c,∗, M. Viggiano b a

c

Istituto per le Applicazioni della Matematica CNR, Via Pietro Castellino 111, 80131 Napoli, Italy b Istituto di Metodologie Avanzate di Analisi Ambientale del CNR Tito Scalo, Pz, Italy Istituto Nazionale per la Fisica della Materia, Unita` di Napoli, Gruppo Collegato di Potenza, C. da Macchia Romana, 85100 Potenza, Italy Received 4 September 2001; received in revised form 24 November 2001; accepted 12 March 2002

Abstract This paper describes a new fast line-by-line radiative transfer scheme which computes top of the atmosphere spectral radiance and its Jacobians with respect to any set of geophysical parameters both for clear and cloudy sky, and presents the software which implements the procedure. The performance of the code has been evaluated with respect to accuracy and speediness through a comparison with a state-of-art line-by-line radiative transfer model. The new code is well suited for nadir viewing satellite and airplane infrared sensors with a sampling rate in the range 0.1–2 cm⫺1.  2002 Elsevier Science Ltd. All rights reserved. Keywords: Observation and data reduction techniques; Computer modeling and simulation; Spectroscopy; Interferometry; Remote observing techniques; Infrared

1. Software availability σ-IASI is a FORTRAN based radiative transfer code designed to match the spectral range of the Infrared Atmospheric Sounding Interferometer (IASI). With minor modifications, it could also be used to perform radiative transfer for other sounding instruments with the same or comparable spectral quality as that of IASI. The functionality of σ-IASI software package has been set up to be as independent as possible of the details of the computer environment in which it is run. The package was originally developed for the ALPHA UNIX platform and then extended to a wide range of computer systems, including Personal Computers with Microsoft Windows operating system and HP UNIX workstations. Minimal requirements: 1 GB hard disk free and 128 MB RAM. σ-IASI both in UNIX and Personal Computer (Microsoft Windows) implementation is available on request from: Professor Carmine Serio, Dipartimento di Ingegneria e Fisica dell’Ambiente, Universita` della

∗

Corresponding author. Fax: +39-971427271. E-mail address: [email protected] (C. Serio).

Basilicata, C.da Macchia Romana, 85100 Potenza, Italy. (e-mail: [email protected])

2. Introduction The development of an accurate long-term observational data set to study global change will be one of the major challenges of Earth science in the next decade and beyond. Towards this objective, new observations with better accuracy and resolution than now available are required. While humidity and temperature have been identified as major variables in assessing climate change and they play a fundamental role in Earth’s energy and water cycle processes, current vertical sounding instruments on board of operational satellites are inadequate. An analysis of this inadequacy brings into evidence the too low spectral resolution of the infrared sounders (resolving power 50 to 100) whose main consequence is a degradation of the vertical resolution and of the accuracy of the retrieved quantities. It is now recognized that an improvement by one order of magnitude of the spectral resolution meets the specifications issued by the World Meteorological Organization (WMO) on accuracy needed to improve weather forecasts (temperature with

1364-8152/02/$ - see front matter  2002 Elsevier Science Ltd. All rights reserved. PII: S 1 3 6 4 - 8 1 5 2 ( 0 2 ) 0 0 0 2 7 - 0

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an average error of 1 K; humidity with an average error of 10%–20%; vertical resolution of 1 km, at least in the lower troposphere). There are two incoming space missions which will fly high spectral resolution infrared sensors: the American Advanced Infrared Radiometer Sounder (AIRS) (Aumann and Strow, 2001) and the European Infrared Advanced Sounding Interferometer (IASI) (Cayla, 1995). AIRS is a grating spectrometer with a resolving power s/⌬s⬇1200 (s being the wavenumber), whereas IASI is a Fourier Transform Interferometer with a sampling rate ⌬s=0.25 cm⫺1. Both spectrometers will cover the spectral range 3.5–16 µm and are being developed to meet the accuracy specifications issued by WMO. Total amount of ozone and some information about its vertical distribution, fractional cloud cover and cloud top temperature/pressure are mission objectives, as well. Because of the relatively high spectral resolution, modern sensors are posing new and challenging problems in the area of remote sensing of geophysical parameters from infrared radiances. New tools have to be developed in order to exploit all the potential information from observations. This paper describes and fully documents the physical modeling of radiative transfer in the atmosphere, in a form which is suitable for next generation infrared sensors aiming at meterorological missions, and presents its software implementation. Because of the much higher spectral quality of new meteorological infrared sensors, the generation of synthetic spectral radiance needs line-by-line calculations. There are at present two line-by-line radiative transfer models which are widely used by the community of new generation infrared sensors people: GENLN2 (Edwards, 1992) and LBLRTM (Clough et al., 1992; Clough and Iacono, 1995), the second one being much faster than the first one but still to slow and complex to be used in form of a subroutine in a nearly real-time ground-segment processing chain. It should be stressed, however, that neither GENLN2 nor LBLRTM have been intended to be used as general purpose subroutines. They want rather to provide the basis to check accuracy of spectroscopy in the form needed for atmospheric remote sensing and to develop suitable parameterized forward models. In this context, GENLN2 has been used by various authors as the father code from which to yield much faster software programmes. Exploiting an idea which goes back to Scott and Chedin (1981), Strow et al. (1998) used GENLN2 to generate compressed looktables of monochromatic optical depth. The look-up tables formed then the basis of a fast line-by-line radiative transfer code (DeSouza-Machado et al., 1998). Our main aim is to implement a software easy to integrate in other applications which need nadir viewing radiative transfer. This includes nadir viewing satellite

and airborne sensors of the same spectral quality as that of IASI and AIRS, that is sampling rate, ⌬s, in the range 0.1–2 cm⫺1. The code is expected to provide the needed forward model calculations for meteorological applications in which the emphasis is put on the retrieval of temperature and water vapour profiles. To attain the above objective 1. LBLRTM has been used to yield suitable monochromatic optical depth look-tables, 2. all the mathematical radiative transfer formulas have been rewritten in an analytical fashion in order to gain efficiency and save computing time. Item (2) is of a particular importance to us. To date, a fully documented scheme for the computation of analytical Jacobians with respect, to any geophysical parameter is still lacking. The capability of analytical Jacobians is of paramount importance to speed up calculations since they are much faster than finite-difference Jacobians, presently available with GENLN2 and LBLRTM. Our analysis was born because of the need to develop a suitable forward model for IASI which joined fastness to accuracy and ability of radiance and derivative matrices or Jacobians calculations. However, the scheme may be easily adapted to other sensors, similar to IASI, by adjusting for the spectral ranges and by inputting the proper Instrumental Line Shape function. An early version of the scheme described in this paper may be found in Amato et al. (1998). Applications of the scheme to infrared spectral radiance for the inversion of geophysical parameters have been mostly described and discussed in the framework of the IMG project (Interferometric Monitoring of Greenhouse gases) (Amato et al., 1999; Lubrano et al., 2000, 2002; Masiello et al., 2002). IMG (Kobayashi et al., 1999) is a Fourier transform spectrometer which has successfully flown on board on the Japanese polar platform Advanced Earth Observing Satellite (ADEOS) from August 1996 to June 1997. The paper is organized as follows. In Section 2, we deal with the basics of radiative transfer. The fully analytical scheme, suitable for software implementation, is developed in Sections 3 and 4 for clear and cloudy sky, respectively. Section 5 is dedicated to the optical depth data base description and organization. Application of the methodology to IASI is given in Section 6, where σ-IASI is compared to LBLRTM in terms of accuracy and computational effort. Conclusions are drawn in Section 7. 3. The basics of radiative transfer in the atmosphere Under the assumption of non-scattering atmosphere, supposing the gas in local thermodynamic equilibrium

U. Amato et al. / Environmental Modelling & Software 17 (2002) 651–667

653

and invoking Kirchhoff’s law, the equation which governs the change of intensity R(s) [W m⫺2 sr cm⫺1] of wavenumber s [cm⫺1] in the q-direction along a path ds [cm] can be written as (Lenoble, 1993; Liou, 1992)

level z to the surface. The relation between t∗ and the upwelling t in Eq. (3) is

dR(s,s) ⫽ ⫺K(s,s)[R(s,s)⫺B(s,T(s))] ds

so that Eq. (4) can be written in terms of upwelling transmittances as

(1)

where s is the coordinate along the slant path (see Fig. 1), B(s, T(s)) is the Planck function at temperature T(s), and K(s,s)=k(s,s)r(s), with k(s,s) the monochromatic absorption coefficient [cm2/g] and r(s) the absorber density [g/cm3]. The explicit expression of the Planck function B(T), in the wave number domain is B(s,T) ⫽

c1s3 exp(c2s / T)⫺1

(2)

with c1=1.1911×10⫺8 W m⫺2 sr⫺1(cm⫺1)⫺4 and c2= 1.4388 K(cm⫺1)⫺1. Integrating Eq. (1) in the height coordinate from surface to the top of the atmosphere gives

冕

⫹⬁

R(s) ⫽ egB(Tg)t0 ⫹

∂t B(T) dz ∂z

(3)

0

where Tg is the surface temperature and t0 indicates the total transmittance from 0 to ⬁. Here, the dependence of eg, t0, t, B, on the wavenumber s and the angle q as well as the dependence of T on the altitude z have been implicitly assumed; moreover, it is also obvious that t denotes the total transmittance function from the level z to ⬁ along the q direction. In the same way, we can compute the atmospheric downwelling radiance, that is the radiance emitted from the atmosphere and reaching the surface along the line of sight specified from the zenith angle q,

冕 0

R(s) ⫽

⫹⬁

∂t∗ B(T) dz ∂z

(4)

where now t∗ is the downwelling transmittance from the

Fig. 1.

t(z;s,m)t∗(z;s,m) ⫽ t0(s,m)

冕

⫹⬁

R(s) ⫽ ⫺t0

冉冊

∂ 1 B(T) dz ∂z t

0

(5)

(6)

The downwelling radiance is emitted in all the directions. This radiance is back reflected to the space and may reach the satellite. In the following, we will assume specular reflection, so that only the radiance reaching the surface along the direction—q will be reflected in the qdirection. It is also important to note that the downwelling term is of second order with respect to the upwelling one. It gives an important contribution only in the window regions of earth’s thermal emission and can be neglected in the opaque spectral intervals. An additional term should eventually arise from the contribution from the solar irradiance. Solar irradiance is transmitted through the atmosphere to the surface, and here it can be partially reflected to the outer space. If we assume the surface to be a Lambertian diffuser, the total radiative transfer equation taking into account also both back reflected and solar contribution can be written as

冕

⫹⬁

R(s) ⫽ egB(Tg)t0 ⫹

∂t B(T) dz ⫹ (eg ∂z

(7)

0

冕

⫹⬁

⫺1)t20

0

冉冊

∂ 1 1⫺eg B(T) dz ⫹ t⬘msIs(s) ∂z t p

where Is(s) is the extra-terrestrial solar spectral irradiance impinging on a normal surface [W m⫺2 cm⫺1], ms=cos(qs) is the cosine of the solar zenith angle, and t⬘ ⫽ t0(m)t0 (ms) is the two path transmittance (total transmittance along the qs solar direction multi-

Radiation traveling a slant path s forming an angle q with the vertical z. In a practical case q is the satellite viewing angle.

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U. Amato et al. / Environmental Modelling & Software 17 (2002) 651–667

plied by the transmittance along the upwelling qdirection). Note that the expression of solar contribution in Eq. (7) holds only for the infrared region. For visible light it is no more correct since it neglects scattering. In addition, in the thermal band the solar radiance adds a significant contribution only for wavenumbers greater than ⬇2000 cm⫺1. In deriving Eq. (7) we assumed specular reflection at the surface for earth’s radiation and Lambertian diffusion for solar radiation, which may seem inconsistent since we have two different assumptions for the same surface and the same wavenumber. However, the problem here is that the Lambertian assumption transforms the highly directional solar radiation field in an uniformly diffuse one which is much more consistent with earth’s emission radiation. To this end, note that in the above equation the solar term has units of spectral irradiance [W m⫺2 cm⫺1] and becomes a spectral radiance after dividing it by the solid angle p. In this form the radiative transfer model is suitable for applications in the broad field of remote sensing from space with infrared sensors. It should be also noticed that the model we have presented is mostly intended for infrared sensors which operate in the so-called nadir looking mode. These sensors are aimed at profiling the lower atmosphere for temperature and absorbing gas constituent profiles. For sensors operating in the limb looking mode the basic Eq. (1) has to be solved for a geometry of a limb path through a refractive atmosphere. These last sensors are intended for the gas profiling of the stratosphere. Model (7) applies to clear sky. In the assumption of a single cloud layer, the extension to cloudy atmosphere is straightforward (Chahine, 1974):

The fact that we neglect the downwelling term is in line with the simplified nature of the cloud model we consider, that is infinitesimally thin cloud and single layer cloud. Keeping second order terms in the total transmittance function, such as the reflected downwelling contribution, would add only computational burden likely to be nullified by the simplified cloud model. Nevertheless, the cloud model and the approximations we adopt do combine in forming cloudy spectral radiance which has proved to be useful for real applications. The model we use for cloudy sky is, for example, that used to assimilate data from the HIRS radiometer that flies on existing NOAA weather satellites (e.g. Eyre, 1991). Moreover, the model implemented in σ-IASI is that used for the formulation of the CO2 slicing technique to estimate cloud top pressure and temperature (e.g. Wylie and Menzel, 1989). To sum up, for a cloudy atmosphere, the radiative transfer model implemented in the present version of σIASI is given by Eq. (8), with the clear term given by Eq. (9) and the overcast radiance by

R(s) ⫽ (1⫺a)·R0(s) ⫹ a·Rcld(s)

4. Radiance and Jacobian calculation: clear sky

(8)

here a is the fractional amount of clouds in the field of view; R0 is the clear-sky radiance, given by Eq. (7); Rcld is the overcast-sky radiance to be defined in a while. According to Eq. (8), both R0 and Rcld have to be computed to obtain the final cloudy sky radiance. To limit the cloudy-sky radiance computational burden , the clear sky term, given by Eq. (7), is simplified by canceling the two last terms. This yields

冕

⫹⬁

R0(s) ⫽ egB(Tg)t0 ⫹

∂t B(T) dz ∂z

(9)

冕

Lc

∂t Rcld(s) ⫽ eg(1⫺ec)B(Tg)t0 ⫹ (1⫺ec) B(T) dz ∂z

(10)

0

冕

⫹⬁

⫹ ecB(Tc)tc ⫹

∂t B(T) dz ∂z

Lc

where Lc and Tc are the cloud top height and temperature, tc is the total transmittance from Lc to the space, and ec is the cloud top emissivity.

In this section we exploit the basic formulas derived in the previous section to define a suitable analytical scheme for the software implementation of the calculation of spectral radiance and its derivative with respect to geophysical parameters. Aim of this and next section is to fully document equations and formulas which form the body of the radiative transfer embedded in the software implementation of σ-IASI. To begin with, we consider first the case of clear sky. 4.1. Radiance computation

0

The rationale to drop the solar term is that in a cloudy atmosphere, scattering cannot be neglected in the near infrared spectral region and demands for an ad hoc treatment which is not compatible with the need of saving computational time. Because of this limitation, σ-IASI is not suitable, in cloudy conditions, for the computation of synthetic radiance spectra at wave numbers above ⬇2000 cm⫺1.

The inhomogeneous nature of the atmosphere along a radiation path is most readily treated by sub-dividing it into a set of layers; in this way the integration over z in Eq. (7) becomes a summation over the constituent layers:

冘 L

R(s)⬇egB(Tg)t0 ⫹

j⫽1

B(Tj)(tj⫺tj⫺1) ⫹ (eg

(11)

U. Amato et al. / Environmental Modelling & Software 17 (2002) 651–667

Fig. 2.

冘

Layering of the atmosphere and definition of the layers, levels and related transmittances.

L

⫺1)t

2 0

B(Tj)(t ⫺t ⫺1 j

655

(1⫺eg) )⫹ t⬘msIs(s) p

⫺1 j⫺1

j⫽1

where L is the number of layers in which the atmosphere has been divided, tj is the total transmittance from the top of the jth layer to ⬁ and tj⫺1 is the total transmittance from the bottom of the j-th layer to ⬁. Note that tL=1. It should be stressed that B is computed at the average temperature Tj of the jth layer of the atmosphere. The code σ-IASI borrows the procedure to form the average layer temperature and average layer of other geophysical parameters from LBLRTM. The average operation involves basically the hydrostatic equation along with the perfect gas law to derive a physical interpolation of the various parameters within the layer. For temperature the layer average corresponds to the ratio between the average layer pressure and air density. The layer boundaries should be chosen in such a way that the gas within the layer can be considered homogeneous. In our approximation of a plane-parallel atmosphere the layers take the form of horizontal slabs as shown in Fig. 2. For the work here shown the atmosphere has been divided into 43 pressure layers extending from the surface pressure to 0.005 mbar (see Table 1).

The layering of the atmosphere we have adopted has been first developed and used by Matricardi and Saunders (1999) who showed that the error introduced by limiting the number of layers to 43 is negligible. This was done by computing the difference between brightness temperature spectra obtained by dividing the atmosphere into 43 and 98 layers, respectively. A similar exercise have been repeated by us and we have found that differences reach at most 0.1 K. In Eq. (11) we pose

冘 L

S+ ⫽

冘 L

B(Tj)(tj⫺tj⫺1) and S⫺ ⫽

j⫽1

B(Tj)(t⫺1 j

⫺t⫺1 j⫺1) so that the radiance expression becomes R(s)⬇egB(Tg)t0 ⫹ S+ ⫹ (eg⫺1)t20S⫺ ⫹

(12)

j⫽1

(13)

(1⫺eg) t⬘msIs(s) p

4.2. Surface emissivity Jacobian From Eq. (13) it easily follows mst⬘Is ∂R(s) ⫽ B(Tg)t0 ⫹ t20S⫺⫺ ∂eg p

Table 1 Pressure layering of the atmosphere used in σ-IASI Layer

Pressure (hPa)

Layer

Pressure (hPa)

Layer

Pressure (hPa)

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

1013.25–1005.43 1005.43–985.88 985.88–957.44 957.44–922.46 922.46–882.80 882.80–839.95 839.95–795.09 795.09–749.12 749.12–702.73 702.73–656.43 656.43–610.60 610.60–565.54 565.54–521.46 521.45–478.54 478.54–436.95

16 17 18 19 20 21 22 23 24 25 26 27 28 29 30

436.95–396.81 396.81–358.28 358.28–321.50 321.50–286.60 286.60–253.71 253.71–222.94 222.94–194.36 194.36–167.95 167.95–143.84 143.84–122.04 122.04–102.05 102.05–85.18 85.18–69.97 69.97–56.73 56.73–45.29

31 32 33 34 35 36 37 38 39 40 41 42 43

45.29–35.51 35.51–27.26 27.26–20.40 20.40–14.81 14.81–10.37 10.37–6.95 6.95–4.41 4.41–2.61 2.61–1.42 1.42–0.69 0.69–0.29 0.29–0.10 0.10–0.005

(14)

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U. Amato et al. / Environmental Modelling & Software 17 (2002) 651–667

For temperature Jacobian we have to consider each layer separately. In Eq. (11) the mean temperature Tj appears in three different pieces, t0 the Planck function B and the total transmittances t1,…, tj⫺1.For the terms related to the Planck function we have the following analytical contribution

In the above formula, to gain symmetry we have defined the dummy variable B(T0)=0 (T0 is a dummy variable and has nothing to do with Tg). In addition, we recall that m=cos(q) is the cosine of the viewing zenith angle, and ms=cos(qs) is the cosine of the solar zenith angle; t⬘ is the two path transmittance along the path sun–surface–satellite. The terms in Eq. (21) containing the Planck function derivative are completely analytical and can be computed directly without any further approximation. In principle, the terms that contain the derivative ∂vj /∂Tj might not be computed explicitly because fully line-byline radiative transfer models deal with the dependence of the optical depth on temperature through look up tables. However, it will be shown in Section 5 that the monochromatic optical depth may be validly parameterized through a low order polynomial with respect to temperature which allows us to derive a simple analytical forms of its temperature derivative.

∂B ∂B (tj⫺tj⫺1) ⫹ (eg⫺1)t20 (t⫺1 ⫺t⫺1 j⫺1) ∂Tj ∂Tj j

4.5. Gas concentration Jacobian (line contribution)

4.3. Surface temperature Jacobian From Eq. (13) it easily follows ∂B ∂R(s) ⫽ egt0 ∂Tg ∂Tg

(15)

where the derivative of the Planck function can be computed according to c1c2s4exp(c2s / T) ∂B ⫽ 2 ∂T T [exp(c2s / T)⫺1]2

(16)

4.4. Atmospheric temperature Jacobian

(17)

The total transmittance t0 is expressed in terms of the transmittance (contribution from all gas constituents) of the lth layer hl=exp(⫺vl) by means of the relation L

t0 ⫽ ⌸ hj

(18)

j⫽1

with vl the spectral optical depth of the lth layer of the atmosphere (contribution from all gases). Then, ∂vj ∂t0 ∂hj L ⫽ ⌸ hk ⫽ ⫺t0 ∂Tj ∂Tj k ⫽ 1 ∂Tj

(19)

k⫽j

The last contribution to the derivative with respect to the temperature comes from the terms of the type ⫺1 ⫺ (tk⫺tk⫺1) of S+ and (t⫺1 k ⫺tk⫺1) of S . Note that if the derivative is taken with respect to the temperature of the jth layer, then only transmittances with index kⱕj⫺1 are involved. Analogously to Eq. (19), we have ∂vj ∂tk ⫽ ⫺tk , for k ⬍ j ∂Tj ∂Tj

(20)

∂tk ⫽ 0, for kⱖj ∂Tj Now, by using this expression, it easily follows ∂R ⫽ D0j ⫹ D+j ⫹ D⫺ j ∂Tj

冋

D0j ⫽ ⫺ egt0B(Tg) ⫹ 2(eg⫺1)t20S⫺ ⫹

冘

册

1⫺eg ms ∂vj m t⬘(1 ⫹ )Is p s m ∂Tj

∂vj ∂B D ⫽[ tk⫺1(B(Tk)⫺B(Tk⫺1))] ⫹ (tj⫺tj⫺1) ∂T ∂T j j k⫽1

再 冘 j

D ⫽ (eg⫺1)t ⫺[ ⫺ j

2 0

k⫽1

t

(22)

h⫽1

where NGAS indicates the number of absorbing gas constituents. The following obvious relation holds, too, tj(h) ⫽

L

⌸ hl(h)

l⫽j⫹1

(23)

where hj(h) is the transmittance of the jth layer for the hth gas. Furthermore, we have hj(h) ⫽ exp(⫺vj(h)) with vj(h) ⫽ kj(h)qj(h)Lj

(24)

where hj(h) is the monochromatic optical depth, kj(h) is the absorbing coefficient of the hth gas and jth layer; qj(h) is the gas concentration of the hth gas and jth layer; Lj=Hj/m is the optical path length, with Hj being the width of the layer j, and m=cos(q). Keeping in mind the basic relation (13), we have that the Jacobian with respect to any given gas has two contributions: the first one from the transmittance t0 and the second one from the terms S⫺ and S+. The application of relations of Eq. (24) gives (25)

k⫽j

冎

∂vj ∂B ⫺1 ⫺1 (B(Tk)⫺B(Tk⫺1))] ⫹ (t ⫺tj⫺1) ∂Tj ∂Tj j

⫺1 k⫺1

NGAS

tj ⫽ ⌸ tj(h),

∂hj L ∂vj(h) ∂t0 ⫽ ⌸ h ⫽ ⫺ t k ∂qj(h) ∂qj(h)k ⫽ 1 ∂qj(h) 0

j

+ j

The spectral radiance R depends on gas concentration through the transmittance function which can be written as

(21)

for the first term. For the terms S⫺ and S+ we need the derivative ∂tk / ∂qj(h). Using the basic relations (22) to (24), we have

U. Amato et al. / Environmental Modelling & Software 17 (2002) 651–667

∂tk ∂vj(h) ⫽ ⫺tk (h), for k ⬍ j (h) ∂qj ∂qj

(26)

∂tk ⫽ 0, for kⱖj ∂qj(h)

from which the derivative of the radiance with respect to the hth absorbing gas constituent easily follows: ∂R(s) ⫽ G0j ⫹ G+j ⫹ G⫺ j , ∂q(h) j

冋 冋

册 册

G0j ⫽ ⫺ egt0B(Tg) ⫹ 2(eg⫺1)t20S⫺ ⫹

1⫺eg ms ∂v m t⬘(1 ⫹ )Is p s m ∂q

G+j ⫽ ⫺ egt0B(Tg) ⫹ 2(eg⫺1)t20S⫺ ⫹

1⫺eg ms ∂v(h) j mst⬘(1 ⫹ )Is (h) p m ∂qj

冘 j

Gj⫺ ⫽ ⫺(eg⫺1)t20[

k⫽1

t⫺1 k⫺1(B(Tk)⫺B(Tk⫺1))]

∂v(h) j ∂q(h) j

(27) In the approximation in which the optical depth scales with the the absorber amount, the derivative in Eq. (27) is completely analytical, since, because of Eq. (24) we have ∂vj(h) vj(h) ⫽ ∂qj(h) qj(h)

pendent of the gas concentration, whereas the self absorption coefficient is proportional to the H2O concentration. For this reason we may consider that for in-band calculation the water vapor continuum optical depth is proportional to the gas concentration, and to the square of the gas concentration for out-of-band calculation, that is

冦

vj(w) , s苸[1200,2000]cm⫺1 (in ⫺ band) qj(w)

(h) j (h) j

(28)

In case this is a too much crude approximation, which is the case of H2O continuum to be treated next section, ∂vj has to be approximated through finite the derivative ∂qj difference or a suitable interpolation scheme. In the present σ-IASI version, except that for water vapor continuum, the approximation of optical depth scaling with the absorber amount is used. It will be shown in Section 6 that this approximation introduces unimportant errors at the level of the convolved IASI spectral radiance. 4.6. Water vapor Jacobian (continuum contribution) It is seen by Eq. (27) that the rate of change of radiance with respect to gas concentration ultimately depends on the derivative of the gas layer optical depth with respect to its concentration. The assumption that the optical depth scales with the absorber amount is no more valid for the continuum absorption of water vapor which depends on the gas concentration itself. For this case, to derive a more appropriate relation for ∂vj to be inserted in Eq. (27), it has to be the derivative ∂qj noted that H2O continuum absorption has two main contributions: self and foreign contribution. The foreign contribution is dominant for in-band absorption whereas the self continuum is dominant for out-of-band absorption (Clough et al., 1989). Furthermore the foreign absorption coefficient may be considered largely inde-

657

∂vwj ⫽ ∂qwj 2vj(w) , otherwise (out of ⫺ band), qj(w)

(29)

where vwj and qwj are the continuum-absorption water vapor optical depth at wavelength s and the H2O concentration, respectively. Although approximated, the above derivative is analytical and has to be preferred to a finite difference derivative. To summarize, the rate of change of the radiance with water vapor continuum is still expressed by Eq. (27) but ∂vj computed according to the formula above. with ∂qj Finally, the water vapor line and continuum layer optical depth derivatives have to be added before insertion in Eq. (27) to form the derivative of radiance with respect to water vapor concentration.

5. Radiance and Jacobian calculation: cloudy atmosphere In this section we extend the forward model to a cloudy atmosphere. As for the previous section, the main aim is to give a detailed account of the relevant equations and their form suitable for a direct analytical computation. According to the basic equation for the radiance, R(s), in a cloudy atmosphere see Eq. (8) we have to consider a numerical computation of the overcast-sky radiance, Rcld, and derivatives with respect to the four new parameters α, hc, Tc and εc as well. Because of the additive form [see Eq. (8)] of the cloudy-sky radiance, we need to concern here only with the overcast-sky term, since the clear-sky term has been already dealt with in the previous section. To avoid long and uneasy to handle mathematical equations, we limit here to write down all the calculations which apply to the overcast-sky term, Rcld(s). It is intended that this term has to be composed to the clear-sky one, according to Eq. (8), to form the cloudy-sky radiance. 5.1. Radiance computation As we have pointed out in Section 3.1, atmospheric layering allows a straight-forward integration over z in

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Eq. (10), so that the integration sign can be replaced by a summation over the constituent layers. Thus: for Rcld the following expression yields

冘

再

L

Rcld(s)⬇ecB(Tc)tc ⫹

B(Tj)(tj⫺tj⫺1) ⫹ eg(1

冘

B(Tj)(tj⫺tj⫺1)

j⫽1

where Lc is the level where the cloud is, and again L is the number of layers in which atmosphere is divided, tj is the total transmittance from the top of the jth layer to ⬁ and tj⫺1 is the total transmittance from the bottom of the jth layer to ⬁.

(31)

冋冘

册

k

⫹

(B(Ti)⫺B(Ti⫺1))ti⫺1

i⫽1

冘 Lc

B(Ti)(ti⫺ti⫺1)]

∂B ∂vk ⫹ (t ⫺t ) ∂Tk ∂Tk k k⫺1

(36)

∂vk ∂Tk

5.7. Gas concentration Jacobian (line contribution)

冋

(32)

k

册

再

∂Rcld ⫽ ⫺egB(Tg)t0 ⫹ ∂qk(h)

The cloud contribution to this term is given by

(37)

∂vk(h) ∂qk(h)

if Lcⱖk and

5.4. Surface temperature Jacobian

冘

∂Rcld ⫽ (ec⫺1) egB(Tg)t0⫺ (B(Ti) ∂qk(h) i⫽1 ⫺B(Ti⫺1))ti⫺1

Lc

冋冘

册

k

(B(Ti⫺1))ti⫺1

i⫽1

冘 Lc

(33)

where again the derivative of the Planck function can be computed according to Eq. (16). 5.5. Cloud top temperature Jacobian Eq. (30) gives the complete expression for the cloud top temperature Jacobian as follows ∂B ∂R ⫽ ectc ∂Tc ∂Tc

∂vk ∂vk ⫺t0(1⫺ec)egB(Tg) ∂Tk ∂Tk

The basic relation (30), along with Eqs. (22), (23) and (24)

There is no contribution to this coefficient from clearsky radiance, so that the contribution to this term comes entirely from Eq. (30)

∂B ∂Rcld ⫽ eg(1⫺ec)t0 ∂Tg ∂Tg

(35)

otherwise.

5.3. Cloud emissivity Jacobian

冘

册 冎

i⫽1

From Eq. (30) it easily follows

∂R ⫽ tcB(Tc)⫺ B(Ti)(ti⫺ti⫺1)⫺egt0B(Tg) ∂ec i⫽1

(B(Ti)

i⫽1

∂vk ∂vk ∂Rcld ⫽ ⫺tcecB(Tc) ⫺t0(1⫺ec)egB(Tg) ∂Tk ∂Tk ∂Tk

⫺ec[

5.2. Surface emissivity Jacobian

∂Rcld ⫽ (1⫺ec)t0B(Tg) ∂eg

k

if the cloud top height is greater (or equal) than the kth level, and

Lc

⫺ec)B(Tg)t0⫺ec

⫺B(Ti⫺1))ti⫺1

(30)

j⫽1

冋冘

∂Rcld ∂B ⫽ (1⫺ec) (t ⫺t ) ⫹ ∂Tk ∂Tk k k⫺1

(34)

5.6. Atmospheric temperature Jacobian The derivatives with respect to the temperature profile are much more complicated. Due to definition of the total transmittance and Eq. (30), the calculation for Rcld yields

⫹ egecB(Tg)t0⫺ecB(Tc)tc⫺ec ⫺ti⫺1)

冎

B(Ti)(ti

(38)

i⫽1

∂vk(h) ∂qk(h)

if Lc⬍k. For the line contribution we have, as for the clear sky case and by assuming optical depth scaling with absorber amount, ∂vk(h) vk(h) ⫽ ∂qk(h) qk(h)

(39)

5.8. Water vapor Jacobian (continuum contribution) ∂vk to be inserted ∂qk in Eqs. (37) and (38) follows the scheme described in Section 3.6. Again, note that the continuum and line contributions for water vapor absorption have to be added The computation of the derivative

U. Amato et al. / Environmental Modelling & Software 17 (2002) 651–667

to have the total rate of change of radiance with water vapor concentration. 5.9. Derivative with respect to the top cloud height The derivative with respect to the cloud top height is given by

冋

∂tc ∂tc ∂R ⫽ a ecB(Tc) ⫺B(T∗c ) ⫹ (1 ∂Lc ∂Lc ∂Lc

冘 Lc

∂tc ∂tc ⫺ec) B(Tj)(tj⫺tj⫺1) ⫹ (1⫺ec)B(Tc) ∂Lcj ⫽ 1 ∂Lc

(40)

册

which, by noting that ∂tc ⫽ tck(Lc)r(Lc) ∂Lc

(41)

becomes

再

∂R ⫽ a [ecB(Tc)⫺B(Tc∗) ⫹ (1⫺ec)B(Tc)]tck(Lc)r(Lc) ∂Lc

冘 Lc

⫹ (1⫺ec)

B(Tj)(tj⫺tj⫺1)

j⫽1

冎

(42)

where Tc∗, k(Lc) and r(Lc) are the air temperature, monochromatic absorbing coefficient and density just above the clouds, respectively. For practical purposes they can be approximated by the corresponding average-layer values of the layer just above the cloud. 5.10. Derivative with respect to the fractional amount of clouds Finally, keeping in mind Eq. (8), the obvious expression for this coefficient is found to be ∂R ⫽ ⫺R0 ⫹ Rcld ∂a

(43)

with R0 being the clear-sky radiance given by

冘 L

R0(s)⬇egB(Tg)t0 ⫹

B(Tj)(tj⫺tj⫺1)

(44)

j⫽1

and Rcld the cloudy atmosphere radiance of Eq. (30).

6. The layer optical depth data-base The basic input ingredients needed to carry out all the computations described in the previous sections are 1. temperature and gas constituent profiles, which we refers to as geophysical parameters, 2. monochromatic layer optical depth.

659

The layer optical depth depends itself on item (1) above and its generation is one of the principal source of the enormous computational time needed for radiative transfer calculations. The basic idea to speed up the computation is to compute only once the layer optical depth and to store it in a suitable look-up table. For a given atmospheric layer, the basic formula for the monochromatic optical depth is given by Eq. (24) which shows that it is directly proportional to absorber amount so that it may be easily scaled for gas variables. In addition, the monochromatic optical depths vary quite smoothly with temperature which makes it possible to develop a fully parameterization on the basis of the geophysical parameters. In practice, we have generated a large look-up table of layer monochromatic optical depths that can be interpolated in temperature by low-order polynomials, and simply scaled for the absorber amount. The polynomial coefficients are then stored rather than the optical depth itself: which saves Hard Disk memory and speed up the input process. More in detail, the coefficients stored in the look-up tables are the results of three main steps: (1) generation, (2) two-sided thresholding of the monochromatic optical depth for the gas species and (3) its parameterization with respect to temperature. 6.1. Generation Using version 4.3 of LBLRTM (Clough et al., 1992; Clough and Iacono, 1995), monochromatic line contribution optical depth has been computed for a reference atmospheric state (US Standard Atmosphere 1962, Anderson et al., 1986) plus eight evenly spaced temperatures (±10 K, …, ±40 K). For each individual species, the monochromatic layer optical depth is generated at high spectral sampling rate, ⌬s. Following the strategy which is used in LBLRTM (Clough et al., 1992; Clough and Iacono, 1995), for a given layer, we set ⌬s equal to 0.25 times the width of the narrowest line, which gives for the highest-altitude layer ⌬s=0.000169 cm⫺1 Because of the LBLRTM strategy, the sampling rate depends on the layer. To limit the size of the look-up table to a size below 1 GB, the optical depth below the layer L=28 is stored as generated by LBLRTM, whereas for Lⱖ28 it is averaged in bins of width ⌬s=10⫺3 cm⫺1 before storing it in the look-up table. We have checked that in doing that the error in the convolved brightness temperature spectra is below 0.08 K. Individual look-tables have been generated for the following gas species: water vapor, carbon dioxide, ozone, nitrous oxide, carbon monoxide, methane, oxygen and nitrogen, a choice which enables σ-IASI to perform Jacobian calculations for these gases. For the potential users of σ-IASI, this means that any single gas of the

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above set of gases, a group of them, or all of them simultaneously may be dealt with as retrievable geophysical parameters in inversion schemes. The other atmospheric gases (see Table 2) form the mixed-gas optical depth and their composition may not be altered by users. The reference concentration values for the mixed species are those compiled in the AFGL library (Anderson et al., 1986). It should be stressed here that the number of gas species that could have a measurable effect on nadir viewing satellite instruments, of spectral quality comparable to that of IASI, is limited to 3–6, according to the spectral range. Our data base includes, in form of variable absorbing constituents whose concentration may be specified by the user, the main species which dominate upwelling radiance once convolved to a spectral resolution of 0.1–2 cm⫺1. The remaining species are not neglected, but they are taken into account as a unit which cannot be altered by the user. The look-up tables which form our layer optical depth data base have been generated by making full use of the LBLRTM capabilities and borrows from this code the CO2 line mixing scheme; the reference value for the CO2 mixing ratio is 350 ppmv and may be changed in input by the user. The required spectroscopic parameters have been extracted from the database HITRAN 96 (Rothman et al., 1998). 6.2. Thresholding Let dl,s s represent the layer optical depth, for the chemical species s at wavenumber s. We have introduced a double threshold according to the following rules for all species:

再

l,s dl,s s ⬍ h1⇒ds ⫽ 0;

(45)

l,s dl,s s ⬎ h2⇒ds ⫽ h2.

After extensive experiments, we have assumed h1=10⫺4 and h2=10. This corresponds to a maximum error on transmittances of 10⫺4 and 5×10⫺5, respectively. Table 2 Chemical species included in the σ-IASI data-base Species number

Symbolic name

Species number

Symbolic name

Species number

Symbolic name

1 2 3 4 5 6 7 8 9 10

H2O CO2 O3 N2O CO CH4 O2 NO SO NO2

11 12 13 14 15 16 17 18 19 20

NH3 HNO3 OH HF HCl HBR HI ClO OCS H2CO

21 22 23 24 25 26 27 28 29 30

HOCl N2 HCN CH3Cl H2O2 C2H2 C2H6 PH3 COF2 SF6

6.3. Parameterization

Monochromatic optical depth for each species has been parameterized as a function of temperature and fitted by a low-order polynomial: second order has been proved to be more than adequate to best-fit optical depth. The results of the thresholding and parameterization phase for all atmospheric layers show that 88.5% of optical depths are below threshold h1, 0.3% of them are above threshold h2 (both do not need any representation) and the remaining 11.3% can be represented by a second order polynomial (e.g. Fig. 3). Because of parabolic parameterization, thresholding and layer dependent sampling rate strategy, the final size of the look-up table is 726 MB against the initial 27.7 GB.

6.4. Further remarks

The main hypthosesis embedded in our look-up table is that the optical depth is directly proportional to the absorber amonut and can, therefore, easily scaled for variable H2O, O3, CO2, CH4, CO, N2O and so on. This strategy still allows for the correct computation of gas line absorption effects which may be parameterized in terms of the air pressure. However, line absorption effects which depend on the gas concentration and therefore on the partial pressure of individual gases are neglected. This is the case of the effects of variable selfbroadening of gases on the optical depths. Since the typical concentration of minor and trace species is of order of ppmv or below, the effect of self-broadening of atmospheric gases is not really a concern. For monochromatic radiance, self-broadening of H2O may be important because H2O may reach concentrations in volume up to 2–4% in moist atmospheres. However, for spectral radiance sensed through sensors such as AIRS and IASI, the effect of self-broadening is small (e.g. Strow et al., 1998). We have checked that at the IASI spectral sampling rate of 0.25 cm⫺1 the larger error introduced in the spectral radiance is of 0.4%. If the user application warrants second order self-broadening effects to be taken into account, then s-IASI is not the most appropriate code for it. Cross sections of chlorofluorocarbons are not parameterized in terms of look-up tables, since absorption can be quickly computed separately. Our σ-IASI accounts for the presence of CFC species, namely CC13F (CFC-11) and CC12F2 (CFC-12). Finally, water vapor continuum absorption is not parameterized in form of a look-up table, it is computed separately. We use the CKD standard (Clough et al., 1989), version 2.4.

U. Amato et al. / Environmental Modelling & Software 17 (2002) 651–667

661

Fig. 3. Parameterizing optical depth with temperature. The figure shows an example of parabolic fit to layer optical depth. The case shown refers to water vapor optical depth at wave number 1174.5361 cm⫺1, the temperature scale is expressed as deviation from the reference temperature.

7. Application to IASI The methodology we have described in the previous sections applies to monochromatic spectral radiance. To specialize the computations to a given instrument one more operation is needed which consists in convolving the monochromatic radiance and Jacobians with the Instrumental Line Shape of the sensor. The convolution operation is embedded in the σ-IASI software in a proper module which may be changed by the user. In the remaining of this section we describe the test we have performed to check the accuracy and fastness of our methodology once applied to yield IASI synthetic radiance. IASI has a spectral coverage from 640 cm⫺1 to 2760 cm⫺1. The full spectral range is divided in three bands which are here defined: 앫 Band 1: 640 to 1210 cm⫺1; 앫 Band 2: 1210 to 2000 cm⫺1; 앫 Band 3: 2000 to 2760 cm⫺1. The instrument is still under industrial developement and its radiometric accuracy has not yet, definitely charcacterized. At present, various scenarios for the radiometric noise are considered which give the reasonable range of radiometric performanmce of the sensor. These scenarios are shown in Fig. 4. The lower scenario is usually referred to as Cannes specifications. It is expected that IASI radiometric noise will be in between the Cannes specifications and the reasonable scenario (see Fig. 4) with the best possible commitment being the most likely final outcome. The accuracy of the forward model in the IASI-mode operation has been tested by comparing σ-IASI radiance output to IASI spectral radiance obtained by LBLRTM.

Table 3 List of AFGL climatological models of atmosphere AFGL n

Atmosphere

1 2 3 4 5 6

Tropical Mid-Latitude Summer Mid-Latitude Winter High-Latitude Summer High-Latitude Winter US Standard

Six test models of atmosphere (see Table 3) have been considered which encompass polar to tropical air masses (Anderson et al., 1986). The differences between LBLRTM and σ-IASI calculations are shown in Figs. 5 to 10 and compared to the most demanding IASI radiometric noise scenario (Cannes scenario). It is possible to see that the agreement is very good. Some slight discrepancy is, however, evident for the warm and wet atmospheres, such as that tropical, at the beginning of the IASI band 2 which coincides with the first part of the H2O vibrational band at 6.7 µm. This error is only partly due to the fact that we neglect self-broadening effects of water vapor. The relatively large discrepancy is due to the concomitant effect of CH4 and H2O. However, the discrepancy is in any case confined within the IASI error bars. We have computed the root mean square difference in the range 1210–1350 cm⫺1 in the case of the tropical air mass. The value of 0.11 K has been found that may compared the IASI radiometric noise (NEDT) of ⬇0.34–0.4 K in that range. For the benefit of the reader more familiar to brightness temperature spectra and NEDT radiometric errors, Table 4 summarizes the comparison between σ-IASI and

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U. Amato et al. / Environmental Modelling & Software 17 (2002) 651–667

Fig. 4. IASI radiometric noise scenarios in terms of NEDT at a scene temperature of 280 K (upper panel) and in terms of NEDN (lower panel).

Fig. 5. Radiance difference (LBLRTM minus σ-IASI) for the tropical model of atmosphere and comparison with the ±1σ-interval (dashed line) of the radiometric noise (Cannes specification, see Fig. 4) of IASI.

U. Amato et al. / Environmental Modelling & Software 17 (2002) 651–667

Fig. 6.

As Fig. 5, but for for the mid-latitude summer model of atmosphere.

Fig. 7. As Fig. 5, but for the mid-latitude winter model of atmosphere.

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U. Amato et al. / Environmental Modelling & Software 17 (2002) 651–667

Fig. 8.

As Fig. 5, but for the high-latitude summer model of atmosphere and comparison.

Fig. 9.

As Fig. 5, but for the high latitude winter model of atmosphere.

U. Amato et al. / Environmental Modelling & Software 17 (2002) 651–667

Fig. 10.

As Fig. 5, but for the US standard model of atmosphere.

Table 4 Root mean square difference between σ-IASI and LBLRTM brightness temperature calculations (units in Kelvin degrees) AFGL n Air mass type 1 2 3 4 5 6

Tropical Mid-Latitude Summer Mid-Latitude Winter High-Latitude Summer High-Latitude Winter US Standard

Band 1

Band 2

Band 3

0.063 0.041 0.039 0.036 0.032 0.028

0.068 0.044 0.027 0.029 0.054 0.032

0.056 0.038 0.020 0.037 0.027 0.019

LBLRTM in terms of brightness temperature difference. The table shows the root mean square difference, ed, for each IASI band and type of air mass: ed ⫽

冪

冘 N

1 (T (j)⫺T2(j))2 Nj ⫽ 1 1

665

(46)

where T1 and T2 are the two brightness temperature spectra computed according to σ-IASI and LBLRTM, respectively; N is the number of data points for the given IASI band. It is possible to see that these values are one to two orders of magnitude below the expected IASI radiometric noise shown in Fig. 4. For the six models of atmosphere we have also recorded the running time for the two packages. The two software packages have been run on ALPHA workst-

ation with 512 MB Ram and 400 MHz CPU clock. We have found that σ-IASI runs about five times faster than LBLRTM for radiance computations. To complete a run for the IASI band 1, which is the most important band for temperature sounding, σ-IASI takes 20 s against the 100–110 s of LBLRTM. The efficiency increase up to a factor 50 when Jacobians are considered since only finite-difference computations are presently available within LBLRTM. The package σ-IASI takes about 200 s to compute one Jacobian for one IASI band which may be compared to the about 10,000 s taken by LBLRTM. Finally, the Jacobian capability of σ-IASI is exemplified in Fig. 11 which shows examples of temperature, water vapor, ozone and methane Jacobians. Reliability and accuracy of these products have been proved mostly by extensive applications to real observations recorded from the Japanese Fourier transform spectrometer, IMG. The experience we have up to now is that σ-IASI is able to yield consistent inversions for temperature and water vapor which compare very well, for example, to ECMWF analysis (European Centre for Medium range Weather Forecasts). This experience has been documented mostly in Amato et al. (1999), Lubrano et al. (2000, 2002), and Masiello et al. (2002).

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U. Amato et al. / Environmental Modelling & Software 17 (2002) 651–667

Fig. 11.

Example of σ-IASI Jacobians for temperature and various gases.

8. Conclusions We have presented a radiative transfer methodology which is suitable for next generation satellite nadir viewing infrared sensors with a sampling rate in between 0.1– 2 cm⫺1. The methodology we have developed is fully analytical and may be implemented to provide fast and accurate computations of spectral radiance and its derivative with respect to any set of geophysical parameters. The capability of σ-IASI to yield on line calculations of Jacobians makes the code an attractive tool for sensitivity analysis or error analysis which are typically required when designing a new instrument. Moreover, σ-IASI is the perfect subroutine for inversion software algorithms using the Gauss-Newton minimization scheme or Statistical Regularization (e.g. Tarantola, 1987) which heavily depends on the availability of the Jacobian matrix. The scheme originated because of the IASI quest for a forward model which joined fastness to accuracy and

ability of radiance and Jacobians calculations. However, σ-IASI may be easily adapted to other nadir viewing sensors by adjusting the spectral ranges and by inputting the proper Instrumental Line Shape function. The performance of the scheme has been evaluated by running it in the IASI-mode. We have found that σ-IASI compares to LBLRTM in accuracy, while being five times faster as far as spectral radiance is concerned. However, a paramount computational efficiency is gained for the calculation of Jacobians which are fully analytical in our scheme. Compared to a finite difference scheme implemented through LBLRTM, σ-IASI performs 50 times faster. The software implementation of σ-IASI combines in a single package two basic modules: 앫 a look-table module for the generation of monochromatic optical depths 앫 a module for the compuation of Top of Atmosphere spectral radiance and Jacobians.

U. Amato et al. / Environmental Modelling & Software 17 (2002) 651–667

It is here important to stress that the look-up table does not yield accurate optical depths under all conditions. For nadir viewing satellite sensors such as IASI, AIRS and meteorological infrared sensors which will fly in the foreseeable future (e.g. GIFTS (Smith et al., 2001)) the errors introduced by the look-up table scheme are unimportant and produce convolved spectral radiance accurate enough for applications. In this context, the class of nadir viewing sensors which may benefit from σ-IASI are those with a sampling rate within 0.1– 2 cm⫺1. Below 0.1 cm⫺1 σ-IASI could be not accuarte enough, whereas above 2 cm⫺1 it would be too much complex. Finally, in its mathematical formulation and implementation the module for the computation of Top of Atmosphere radiance and Jacobians is independent of the look-up table scheme, therefore LBLRTM or GENLN2 users could benefit by adopting it in their own codes.

Acknowledgements Work supported by Italian Space Agency and EUMETSAT (contract EUM/CO/99/688/DD).

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