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

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

U. Amatoa, G. Masiellob, C. Serioc,∗, M. Viggianob

a Istituto per le Applicazioni della Matematica CNR, Via Pietro Castellino 111, 80131 Napoli, Italyb Istituto di Metodologie Avanzate di Analisi Ambientale del CNR Tito Scalo, Pz, Italy

c 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 radianceand its Jacobians with respect to any set of geophysical parameters both for clear and cloudy sky, and presents the software whichimplements the procedure. The performance of the code has been evaluated with respect to accuracy and speediness through acomparison with a state-of-art line-by-line radiative transfer model. The new code is well suited for nadir viewing satellite andairplane 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 tech-niques; Infrared

1. Software availability

σ-IASI is a FORTRAN based radiative transfer codedesigned to match the spectral range of the InfraredAtmospheric Sounding Interferometer (IASI). Withminor modifications, it could also be used to performradiative transfer for other sounding instruments with thesame or comparable spectral quality as that of IASI. Thefunctionality of σ-IASI software package has been setup to be as independent as possible of the details of thecomputer environment in which it is run. The packagewas originally developed for the ALPHA UNIX plat-form and then extended to a wide range of computersystems, including Personal Computers with MicrosoftWindows operating system and HP UNIX workstations.

Minimal requirements: 1 GB hard disk free and 128MB RAM.

σ-IASI both in UNIX and Personal Computer(Microsoft Windows) implementation is available onrequest from: Professor Carmine Serio, Dipartimento diIngegneria e Fisica dell’Ambiente, Universita` della

∗ Corresponding author. Fax:+39-971427271.E-mail address: serio@unibas.it (C. Serio).

1364-8152/02/$ - see front matter 2002 Elsevier Science Ltd. All rights reserved.PII: S1364-8152 (02)00027-0

Basilicata, C.da Macchia Romana, 85100 Potenza, Italy.(e-mail: serio@unibas.it)

2. Introduction

The development of an accurate long-term obser-vational data set to study global change will be one ofthe major challenges of Earth science in the next decadeand beyond. Towards this objective, new observationswith better accuracy and resolution than now availableare required. While humidity and temperature have beenidentified as major variables in assessing climate changeand they play a fundamental role in Earth’s energy andwater cycle processes, current vertical sounding instru-ments on board of operational satellites are inadequate.An analysis of this inadequacy brings into evidence thetoo low spectral resolution of the infrared sounders(resolving power 50 to 100) whose main consequence isa degradation of the vertical resolution and of the accu-racy of the retrieved quantities. It is now recognized thatan improvement by one order of magnitude of the spec-tral resolution meets the specifications issued by theWorld Meteorological Organization (WMO) on accuracyneeded to improve weather forecasts (temperature with

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

an average error of 1 K; humidity with an average errorof 10%–20%; vertical resolution of 1 km, at least in thelower troposphere).

There are two incoming space missions which will flyhigh spectral resolution infrared sensors: the AmericanAdvanced Infrared Radiometer Sounder (AIRS)(Aumann and Strow, 2001) and the European InfraredAdvanced Sounding Interferometer (IASI) (Cayla,1995). AIRS is a grating spectrometer with a resolvingpower s/�s�1200 (s being the wavenumber), whereasIASI is a Fourier Transform Interferometer with a sam-pling rate �s=0.25 cm�1. Both spectrometers will coverthe spectral range 3.5–16 µm and are being developed tomeet the accuracy specifications issued by WMO. Totalamount of ozone and some information about its verticaldistribution, fractional cloud cover and cloud toptemperature/pressure are mission objectives, as well.

Because of the relatively high spectral resolution,modern sensors are posing new and challenging prob-lems in the area of remote sensing of geophysical para-meters from infrared radiances. New tools have to bedeveloped in order to exploit all the potential infor-mation from observations.

This paper describes and fully documents the physicalmodeling of radiative transfer in the atmosphere, in aform which is suitable for next generation infrared sen-sors aiming at meterorological missions, and presents itssoftware implementation.

Because of the much higher spectral quality of newmeteorological infrared sensors, the generation of syn-thetic spectral radiance needs line-by-line calculations.There are at present two line-by-line radiative transfermodels which are widely used by the community of newgeneration infrared sensors people: GENLN2 (Edwards,1992) and LBLRTM (Clough et al., 1992; Clough andIacono, 1995), the second one being much faster thanthe first one but still to slow and complex to be used inform of a subroutine in a nearly real-time ground-seg-ment processing chain. It should be stressed, however,that neither GENLN2 nor LBLRTM have been intendedto be used as general purpose subroutines. They wantrather to provide the basis to check accuracy of spec-troscopy in the form needed for atmospheric remotesensing and to develop suitable parameterized forwardmodels.

In this context, GENLN2 has been used by variousauthors as the father code from which to yield muchfaster software programmes. Exploiting an idea whichgoes back to Scott and Chedin (1981), Strow et al.(1998) used GENLN2 to generate compressed look-tables of monochromatic optical depth. The look-uptables formed then the basis of a fast line-by-line radiat-ive transfer code (DeSouza-Machado et al., 1998).

Our main aim is to implement a software easy to inte-grate in other applications which need nadir viewingradiative transfer. This includes nadir viewing satellite

and airborne sensors of the same spectral quality as thatof IASI and AIRS, that is sampling rate, �s, in the range0.1–2 cm�1. The code is expected to provide the neededforward model calculations for meteorological appli-cations in which the emphasis is put on the retrieval oftemperature and water vapour profiles.

To attain the above objective

1. LBLRTM has been used to yield suitable monochro-matic optical depth look-tables,

2. all the mathematical radiative transfer formulas havebeen rewritten in an analytical fashion in order to gainefficiency and save computing time.

Item (2) is of a particular importance to us. To date,a fully documented scheme for the computation of ana-lytical Jacobians with respect, to any geophysical para-meter is still lacking. The capability of analytical Jacobi-ans is of paramount importance to speed up calculationssince they are much faster than finite-difference Jacobi-ans, presently available with GENLN2 and LBLRTM.

Our analysis was born because of the need to developa suitable forward model for IASI which joined fastnessto accuracy and ability of radiance and derivative matr-ices or Jacobians calculations. However, the scheme maybe easily adapted to other sensors, similar to IASI, byadjusting for the spectral ranges and by inputting theproper Instrumental Line Shape function.

An early version of the scheme described in this papermay be found in Amato et al. (1998). Applications ofthe scheme to infrared spectral radiance for the inversionof geophysical parameters have been mostly describedand discussed in the framework of the IMG project(Interferometric Monitoring of Greenhouse gases)(Amato et al., 1999; Lubrano et al., 2000, 2002; Masielloet al., 2002). IMG (Kobayashi et al., 1999) is a Fouriertransform spectrometer which has successfully flown onboard on the Japanese polar platform Advanced EarthObserving Satellite (ADEOS) from August 1996 toJune 1997.

The paper is organized as follows. In Section 2, wedeal with the basics of radiative transfer. The fully ana-lytical scheme, suitable for software implementation, isdeveloped in Sections 3 and 4 for clear and cloudy sky,respectively. Section 5 is dedicated to the optical depthdata base description and organization. Application ofthe methodology to IASI is given in Section 6, whereσ-IASI is compared to LBLRTM in terms of accuracyand computational effort. Conclusions are drawn in Sec-tion 7.

3. The basics of radiative transfer in theatmosphere

Under the assumption of non-scattering atmosphere,supposing the gas in local thermodynamic equilibrium

653U. Amato et al. / Environmental Modelling & Software 17 (2002) 651–667

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

dR(s,s)ds

� �K(s,s)[R(s,s)�B(s,T(s))] (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 monochromaticabsorption coefficient [cm2/g] and r(s) the absorber den-sity [g/cm3]. The explicit expression of the Planck func-tion 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 heightcoordinate from surface to the top of the atmospheregives

R(s) � egB(Tg)t0 � �� �

0

B(T)∂t∂z

dz (3)

where Tg is the surface temperature and t0 indicates thetotal transmittance from 0 to �. Here, the dependenceof eg, t0, t, B, on the wavenumber s and the angle q aswell as the dependence of T on the altitude z have beenimplicitly assumed; moreover, it is also obvious that tdenotes the total transmittance function from the level zto � along the q direction.

In the same way, we can compute the atmosphericdownwelling radiance, that is the radiance emitted fromthe atmosphere and reaching the surface along the lineof sight specified from the zenith angle q,

R(s) � �0

� �

B(T)∂t∗

∂zdz (4)

where now t∗ is the downwelling transmittance from the

Fig. 1. Radiation traveling a slant path s forming an angle q with the vertical z. In a practical case q is the satellite viewing angle.

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

t(z;s,m)t∗(z;s,m) � t0(s,m) (5)

so that Eq. (4) can be written in terms of upwelling trans-mittances as

R(s) � �t0 �� �

0

B(T)∂∂z�1t�dz (6)

The downwelling radiance is emitted in all the direc-tions. This radiance is back reflected to the space andmay reach the satellite. In the following, we will assumespecular reflection, so that only the radiance reaching thesurface along the direction—q will be reflected in the q-direction. It is also important to note that the downwel-ling term is of second order with respect to the upwellingone. It gives an important contribution only in the win-dow regions of earth’s thermal emission and can be neg-lected in the opaque spectral intervals. An additionalterm should eventually arise from the contribution fromthe solar irradiance. Solar irradiance is transmittedthrough the atmosphere to the surface, and here it canbe partially reflected to the outer space. If we assumethe surface to be a Lambertian diffuser, the total radiat-ive transfer equation taking into account also both backreflected and solar contribution can be written as

R(s) � egB(Tg)t0 � �� �

0

B(T)∂t∂z

dz � (eg (7)

�1)t20 �� �

0

B(T)∂∂z�1t�dz �

1�egpt�msIs(s)

where Is(s) is the extra-terrestrial solar spectralirradiance 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-

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

plied by the transmittance along the upwelling q-direction). Note that the expression of solar contributionin Eq. (7) holds only for the infrared region. For visiblelight it is no more correct since it neglects scattering. Inaddition, in the thermal band the solar radiance adds asignificant contribution only for wavenumbers greaterthan �2000 cm�1.

In deriving Eq. (7) we assumed specular reflection atthe surface for earth’s radiation and Lambertian dif-fusion for solar radiation, which may seem inconsistentsince we have two different assumptions for the samesurface and the same wavenumber. However, the prob-lem here is that the Lambertian assumption transformsthe highly directional solar radiation field in an uni-formly diffuse one which is much more consistent withearth’s emission radiation. To this end, note that in theabove equation the solar term has units of spectralirradiance [W m�2 cm�1] and becomes a spectral radi-ance after dividing it by the solid angle p.

In this form the radiative transfer model is suitablefor applications in the broad field of remote sensing fromspace with infrared sensors. It should be also noticedthat the model we have presented is mostly intended forinfrared sensors which operate in the so-called nadirlooking mode. These sensors are aimed at profiling thelower atmosphere for temperature and absorbing gasconstituent profiles. For sensors operating in the limblooking mode the basic Eq. (1) has to be solved for ageometry of a limb path through a refractive atmosphere.These last sensors are intended for the gas profiling ofthe stratosphere.

Model (7) applies to clear sky. In the assumption ofa single cloud layer, the extension to cloudy atmosphereis straightforward (Chahine, 1974):

R(s) � (1�a)·R0(s) � a·Rcld(s) (8)

here a is the fractional amount of clouds in the field ofview; 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 com-

puted to obtain the final cloudy sky radiance. To limitthe cloudy-sky radiance computational burden , the clearsky term, given by Eq. (7), is simplified by cancelingthe two last terms. This yields

R0(s) � egB(Tg)t0 � �� �

0

B(T)∂t∂z

dz (9)

The rationale to drop the solar term is that in a cloudyatmosphere, scattering cannot be neglected in the nearinfrared spectral region and demands for an ad hoc treat-ment which is not compatible with the need of savingcomputational time. Because of this limitation, σ-IASIis not suitable, in cloudy conditions, for the computationof synthetic radiance spectra at wave numbers above�2000 cm�1.

The fact that we neglect the downwelling term is inline with the simplified nature of the cloud model weconsider, that is infinitesimally thin cloud and singlelayer cloud. Keeping second order terms in the totaltransmittance function, such as the reflected downwel-ling contribution, would add only computational burdenlikely to be nullified by the simplified cloud model.

Nevertheless, the cloud model and the approximationswe adopt do combine in forming cloudy spectral radi-ance which has proved to be useful for real applications.The model we use for cloudy sky is, for example, thatused to assimilate data from the HIRS radiometer thatflies on existing NOAA weather satellites (e.g. Eyre,1991). Moreover, the model implemented in σ-IASI isthat used for the formulation of the CO2 slicing tech-nique to estimate cloud top pressure and temperature(e.g. Wylie and Menzel, 1989).

To sum up, for a cloudy atmosphere, the radiativetransfer model implemented in the present version of σ-IASI is given by Eq. (8), with the clear term given byEq. (9) and the overcast radiance by

Rcld(s) � eg(1�ec)B(Tg)t0 � (1�ec)�Lc

0

B(T)∂t∂z

dz (10)

� ecB(Tc)tc � �� �

Lc

B(T)∂t∂z

dz

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

4. Radiance and Jacobian calculation: clear sky

In this section we exploit the basic formulas derivedin the previous section to define a suitable analyticalscheme for the software implementation of the calcu-lation of spectral radiance and its derivative with respectto geophysical parameters. Aim of this and next sectionis to fully document equations and formulas which formthe body of the radiative transfer embedded in the soft-ware implementation of σ-IASI. To begin with, we con-sider first the case of clear sky.

4.1. Radiance computation

The inhomogeneous nature of the atmosphere along aradiation path is most readily treated by sub-dividing itinto a set of layers; in this way the integration over z inEq. (7) becomes a summation over the constituent lay-ers:

R(s)�egB(Tg)t0 � �Lj � 1

B(Tj)(tj�tj�1) � (eg (11)

655U. Amato et al. / Environmental Modelling & Software 17 (2002) 651–667

Fig. 2. Layering of the atmosphere and definition of the layers, levels and related transmittances.

�1)t20 �Lj � 1

B(Tj)(t�1j �t�1

j�1) �(1�eg)pt�msIs(s)

where L is the number of layers in which the atmospherehas been divided, tj is the total transmittance from thetop of the jth layer to � and tj�1 is the total transmittancefrom the bottom of the j-th layer to �. Note that tL=1.It should be stressed that B is computed at the averagetemperature Tj of the jth layer of the atmosphere. Thecode σ-IASI borrows the procedure to form the averagelayer temperature and average layer of other geophysicalparameters from LBLRTM. The average operationinvolves basically the hydrostatic equation along withthe perfect gas law to derive a physical interpolation ofthe various parameters within the layer. For temperaturethe layer average corresponds to the ratio between theaverage layer pressure and air density.

The layer boundaries should be chosen in such a waythat the gas within the layer can be considered homo-geneous. In our approximation of a plane-parallel atmos-phere the layers take the form of horizontal slabs asshown in Fig. 2. For the work here shown the atmos-phere has been divided into 43 pressure layers extendingfrom the surface pressure to 0.005 mbar (see Table 1).

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

Layer Pressure (hPa) Layer Pressure (hPa) Layer Pressure (hPa)

1 1013.25–1005.43 16 436.95–396.81 31 45.29–35.512 1005.43–985.88 17 396.81–358.28 32 35.51–27.263 985.88–957.44 18 358.28–321.50 33 27.26–20.404 957.44–922.46 19 321.50–286.60 34 20.40–14.815 922.46–882.80 20 286.60–253.71 35 14.81–10.376 882.80–839.95 21 253.71–222.94 36 10.37–6.957 839.95–795.09 22 222.94–194.36 37 6.95–4.418 795.09–749.12 23 194.36–167.95 38 4.41–2.619 749.12–702.73 24 167.95–143.84 39 2.61–1.4210 702.73–656.43 25 143.84–122.04 40 1.42–0.6911 656.43–610.60 26 122.04–102.05 41 0.69–0.2912 610.60–565.54 27 102.05–85.18 42 0.29–0.1013 565.54–521.46 28 85.18–69.97 43 0.10–0.00514 521.45–478.54 29 69.97–56.7315 478.54–436.95 30 56.73–45.29

The layering of the atmosphere we have adopted hasbeen first developed and used by Matricardi and Saund-ers (1999) who showed that the error introduced by lim-iting the number of layers to 43 is negligible. This wasdone by computing the difference between brightnesstemperature spectra obtained by dividing the atmosphereinto 43 and 98 layers, respectively. A similar exercisehave been repeated by us and we have found that differ-ences reach at most 0.1 K.

In Eq. (11) we pose

S+ � �Lj � 1

B(Tj)(tj�tj�1) and S� � �Lj � 1

B(Tj)(t�1j (12)

�t�1j�1)

so that the radiance expression becomes

R(s)�egB(Tg)t0 � S+ � (eg�1)t20S� (13)

�(1�eg)pt�msIs(s)

4.2. Surface emissivity Jacobian

From Eq. (13) it easily follows

∂R(s)∂eg

� B(Tg)t0 � t20S��mst�Is

p(14)

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

4.3. Surface temperature Jacobian

From Eq. (13) it easily follows

∂R(s)∂Tg

� egt0∂B∂Tg

(15)

where the derivative of the Planck function can be com-puted according to

∂B∂T

�c1c2s4exp(c2s /T)

T2[exp(c2s /T)�1]2 (16)

4.4. Atmospheric temperature Jacobian

For temperature Jacobian we have to consider eachlayer separately. In Eq. (11) the mean temperature Tj

appears in three different pieces, t0 the Planck functionB and the total transmittances t1,…, tj�1.For the termsrelated to the Planck function we have the following ana-lytical contribution

∂B∂Tj

(tj�tj�1) � (eg�1)t20∂B∂Tj

(t�1j �t�1

j�1) (17)

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

t0 � �L

j � 1hj (18)

with vl the spectral optical depth of the lth layer of theatmosphere (contribution from all gases). Then,

∂t0∂Tj

�∂hj

∂Tj

�L

k � 1k � j

hk � �t0∂vj

∂Tj

(19)

The last contribution to the derivative with respect tothe temperature comes from the terms of the type(tk�tk�1) of S+ and (t�1

k �t�1k�1) of S�. Note that if the

derivative is taken with respect to the temperature of thejth layer, then only transmittances with index kj�1 areinvolved. Analogously to Eq. (19), we have

∂tk∂Tj

� �tk∂vj

∂Tj

, for k j

∂tk∂Tj

� 0, for k�j

(20)

Now, by using this expression, it easily follows∂R∂Tj

� D0j � D+

j � D�j

D0j � ��egt0B(Tg) � 2(eg�1)t20S� �

1�egpmst�(1 �

ms

m)Is�∂vj

∂Tj

D+j � [ �j

k � 1

tk�1(B(Tk)�B(Tk�1))]∂vj

∂Tj

�∂B∂Tj

(tj�tj�1)

D�j � (eg�1)t20��[ �j

k � 1

t�1k�1(B(Tk)�B(Tk�1))]

∂vj

∂Tj

�∂B∂Tj

(t�1j �t�1

j�1)(21)

In the above formula, to gain symmetry we havedefined the dummy variable B(T0)=0 (T0 is a dummyvariable and has nothing to do with Tg). In addition, werecall that m=cos(q) is the cosine of the viewing zenithangle, and ms=cos(qs) is the cosine of the solar zenithangle; t� is the two path transmittance along the pathsun–surface–satellite.

The terms in Eq. (21) containing the Planck functionderivative are completely analytical and can be com-puted directly without any further approximation. Inprinciple, the terms that contain the derivative ∂vj /∂Tj

might not be computed explicitly because fully line-by-line radiative transfer models deal with the dependenceof the optical depth on temperature through look uptables. However, it will be shown in Section 5 that themonochromatic optical depth may be validly para-meterized through a low order polynomial with respectto temperature which allows us to derive a simple ana-lytical forms of its temperature derivative.

4.5. Gas concentration Jacobian (line contribution)

The spectral radiance R depends on gas concentrationthrough the transmittance function which can be writ-ten as

tj � �NGAS

h � 1t(h)

j , (22)

where NGAS indicates the number of absorbing gas con-stituents. The following obvious relation holds, too,

t(h)j � �

L

l � j � 1h(h)

l (23)

where h(h)j is the transmittance of the jth layer for the

hth gas. Furthermore, we have

h(h)j � exp(�v(h)

j ) with v(h)j � k(h)

j q(h)j Lj (24)

where h(h)j is the monochromatic optical depth, k(h)

j is theabsorbing coefficient of the hth gas and jth layer; q(h)

j isthe gas concentration of the hth gas and jth layer;Lj=Hj/m is the optical path length, with Hj being thewidth of the layer j, and m=cos(q).

Keeping in mind the basic relation (13), we have thatthe Jacobian with respect to any given gas has two con-tributions: the first one from the transmittance t0 and thesecond one from the terms S� and S+. The applicationof relations of Eq. (24) gives

∂t0∂q(h)

j

�∂hj

∂q(h)j

�L

k � 1k � j

hk � �∂v(h)

j

∂q(h)j

t0 (25)

for the first term. For the terms S� and S+ we need thederivative ∂tk /∂q(h)

j . Using the basic relations (22) to(24), we have

657U. Amato et al. / Environmental Modelling & Software 17 (2002) 651–667

∂tk∂q(h)

j

� �tk∂v(h)

j

∂q(h)j

, for k j

∂tk∂q(h)

j

� 0, for k�j

(26)

from which the derivative of the radiance with respectto the hth absorbing gas constituent easily follows:

∂R(s)∂q(h)

j

� G0j � G+

j � G�j ,

G0j � ��egt0B(Tg) � 2(eg�1)t20S� �

1�egpmst�(1 �

ms

m)Is�∂v(h)

j

∂q(h)j

G+j � ��egt0B(Tg) � 2(eg�1)t20S� �

1�egpmst�(1 �

ms

m)Is�∂v(h)

j

∂q(h)j

G�j � �(eg�1)t20[ �j

k � 1

t�1k�1(B(Tk)�B(Tk�1))]

∂v(h)j

∂q(h)j

(27)

In the approximation in which the optical depth scaleswith the the absorber amount, the derivative in Eq. (27)is completely analytical, since, because of Eq. (24) wehave

∂v(h)j

∂q(h)j

�v(h)

j

q(h)j

(28)

In case this is a too much crude approximation, whichis the case of H2O continuum to be treated next section,

the derivative∂vj

∂qj

has to be approximated through finite

difference or a suitable interpolation scheme. In thepresent σ-IASI version, except that for water vapor con-tinuum, the approximation of optical depth scaling withthe absorber amount is used. It will be shown in Section6 that this approximation introduces unimportant errorsat 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 radi-ance with respect to gas concentration ultimatelydepends on the derivative of the gas layer optical depthwith respect to its concentration. The assumption thatthe optical depth scales with the absorber amount is nomore valid for the continuum absorption of water vaporwhich depends on the gas concentration itself.

For this case, to derive a more appropriate relation for

the derivative∂vj

∂qj

to be inserted in Eq. (27), it has to be

noted that H2O continuum absorption has two main con-tributions: self and foreign contribution. The foreigncontribution is dominant for in-band absorption whereasthe self continuum is dominant for out-of-band absorp-tion (Clough et al., 1989). Furthermore the foreignabsorption coefficient may be considered largely inde-

pendent of the gas concentration, whereas the selfabsorption coefficient is proportional to the H2O concen-tration. For this reason we may consider that for in-bandcalculation the water vapor continuum optical depth isproportional to the gas concentration, and to the squareof the gas concentration for out-of-band calculation,that is

∂vwj

∂qwj

� v(w)

j

q(w)j

, s�[1200,2000]cm�1 (in � band)

2v(w)j

q(w)j

, otherwise (out of � band),

(29)

where vwj and qw

j are the continuum-absorption watervapor optical depth at wavelength s and the H2O con-centration, respectively. Although approximated, theabove derivative is analytical and has to be preferred toa finite difference derivative.

To summarize, the rate of change of the radiance withwater vapor continuum is still expressed by Eq. (27) but

with∂vj

∂qj

computed according to the formula above.

Finally, the water vapor line and continuum layeroptical depth derivatives have to be added before inser-tion in Eq. (27) to form the derivative of radiance withrespect to water vapor concentration.

5. Radiance and Jacobian calculation: cloudyatmosphere

In this section we extend the forward model to acloudy atmosphere. As for the previous section, the mainaim is to give a detailed account of the relevant equa-tions and their form suitable for a direct analytical com-putation.

According to the basic equation for the radiance, R(s),in a cloudy atmosphere see Eq. (8) we have to considera numerical computation of the overcast-sky radiance,Rcld, and derivatives with respect to the four new para-meters α, hc, Tc and εc as well. Because of the additiveform [see Eq. (8)] of the cloudy-sky radiance, we needto concern here only with the overcast-sky term, sincethe clear-sky term has been already dealt with in theprevious section. To avoid long and uneasy to handlemathematical equations, we limit here to write down allthe calculations which apply to the overcast-sky term,Rcld(s). It is intended that this term has to be composedto the clear-sky one, according to Eq. (8), to form thecloudy-sky radiance.

5.1. Radiance computation

As we have pointed out in Section 3.1, atmosphericlayering allows a straight-forward integration over z in

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

Eq. (10), so that the integration sign can be replaced bya summation over the constituent layers. Thus: for Rcld

the following expression yields

Rcld(s)�ecB(Tc)tc � �Lj � 1

B(Tj)(tj�tj�1) � eg(1 (30)

�ec)B(Tg)t0�ec �Lc

j � 1

B(Tj)(tj�tj�1)

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

5.2. Surface emissivity Jacobian

From Eq. (30) it easily follows

∂Rcld

∂eg� (1�ec)t0B(Tg) (31)

5.3. Cloud emissivity Jacobian

There is no contribution to this coefficient from clear-sky radiance, so that the contribution to this term comesentirely from Eq. (30)

∂R∂ec

� tcB(Tc)��Lc

i � 1

B(Ti)(ti�ti�1)�egt0B(Tg) (32)

5.4. Surface temperature Jacobian

The cloud contribution to this term is given by

∂Rcld

∂Tg

� eg(1�ec)t0∂B∂Tg

(33)

where again the derivative of the Planck function can becomputed according to Eq. (16).

5.5. Cloud top temperature Jacobian

Eq. (30) gives the complete expression for the cloudtop temperature Jacobian as follows

∂R∂Tc

� ectc∂B∂Tc

(34)

5.6. Atmospheric temperature Jacobian

The derivatives with respect to the temperature profileare much more complicated. Due to definition of thetotal transmittance and Eq. (30), the calculation forRcld yields

∂Rcld

∂Tk

� (1�ec)�∂B∂Tk

(tk�tk�1) � ��k

i � 1

(B(Ti) (35)

�B(Ti�1))ti�1�∂vk

∂Tk�t0(1�ec)egB(Tg)

∂vk

∂Tk

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

∂Rcld

∂Tk

� �tcecB(Tc)∂vk

∂Tk

�t0(1�ec)egB(Tg)∂vk

∂Tk

� ��k

i � 1

(B(Ti)�B(Ti�1))ti�1�∂vk

∂Tk

�∂B∂Tk

(tk�tk�1) (36)

�ec[�Lc

i � 1

B(Ti)(ti�ti�1)]∂vk

∂Tk

otherwise.

5.7. Gas concentration Jacobian (line contribution)

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

∂Rcld

∂q(h)k

� (ec�1)�egB(Tg)t0��k

i � 1

(B(Ti) (37)

�B(Ti�1))ti�1�∂v(h)k

∂q(h)k

if Lc�k and

∂Rcld

∂q(h)k

� ��egB(Tg)t0 � ��k

i � 1

(B(Ti�1))ti�1�� egecB(Tg)t0�ecB(Tc)tc�ec �Lc

i � 1

B(Ti)(ti (38)

�ti�1)∂v(h)k

∂q(h)k

if Lck. For the line contribution we have, as for theclear sky case and by assuming optical depth scalingwith absorber amount,

∂v(h)k

∂q(h)k

�v(h)

k

q(h)k

(39)

5.8. Water vapor Jacobian (continuum contribution)

The computation of the derivative∂vk

∂qk

to be inserted

in Eqs. (37) and (38) follows the scheme described inSection 3.6. Again, note that the continuum and line con-tributions for water vapor absorption have to be added

659U. Amato et al. / Environmental Modelling & Software 17 (2002) 651–667

to have the total rate of change of radiance with watervapor concentration.

5.9. Derivative with respect to the top cloud height

The derivative with respect to the cloud top height isgiven by

∂R∂Lc

� a�ecB(Tc)∂tc∂Lc

�B(T∗c )

∂tc∂Lc

� (1 (40)

�ec)∂tc∂Lc

�Lc

j � 1

B(Tj)(tj�tj�1) � (1�ec)B(Tc)∂tc∂Lc

�which, by noting that

∂tc∂Lc

� tck(Lc)r(Lc) (41)

becomes

∂R∂Lc

� a�[ecB(Tc)�B(T∗c ) � (1�ec)B(Tc)]tck(Lc)r(Lc)

� (1�ec)�Lc

j � 1

B(Tj)(tj�tj�1) (42)

where T∗c , k(Lc) and r(Lc) are the air temperature, mono-

chromatic absorbing coefficient and density just abovethe clouds, respectively. For practical purposes they canbe approximated by the corresponding average-layervalues of the layer just above the cloud.

5.10. Derivative with respect to the fractional amountof clouds

Finally, keeping in mind Eq. (8), the obviousexpression for this coefficient is found to be

∂R∂a � �R0 � Rcld (43)

with R0 being the clear-sky radiance given by

R0(s)�egB(Tg)t0 � �Lj � 1

B(Tj)(tj�tj�1) (44)

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 thecomputations described in the previous sections are

1. temperature and gas constituent profiles, which werefers to as geophysical parameters,

2. monochromatic layer optical depth.

The layer optical depth depends itself on item (1)above and its generation is one of the principal sourceof the enormous computational time needed for radiativetransfer calculations. The basic idea to speed up the com-putation is to compute only once the layer optical depthand to store it in a suitable look-up table.

For a given atmospheric layer, the basic formula forthe monochromatic optical depth is given by Eq. (24)which shows that it is directly proportional to absorberamount so that it may be easily scaled for gas variables.In addition, the monochromatic optical depths vary quitesmoothly with temperature which makes it possible todevelop a fully parameterization on the basis of the geo-physical parameters.

In practice, we have generated a large look-up tableof layer monochromatic optical depths that can be interp-olated in temperature by low-order polynomials, andsimply scaled for the absorber amount. The polynomialcoefficients are then stored rather than the optical depthitself: which saves Hard Disk memory and speed up theinput process.

More in detail, the coefficients stored in the look-uptables are the results of three main steps: (1) generation,(2) two-sided thresholding of the monochromatic opticaldepth for the gas species and (3) its parameterizationwith respect to temperature.

6.1. Generation

Using version 4.3 of LBLRTM (Clough et al., 1992;Clough and Iacono, 1995), monochromatic line contri-bution optical depth has been computed for a referenceatmospheric state (US Standard Atmosphere 1962, And-erson et al., 1986) plus eight evenly spaced temperatures(±10 K, …, ±40 K).

For each individual species, the monochromatic layeroptical 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 agiven layer, we set �s equal to 0.25 times the width ofthe narrowest line, which gives for the highest-altitudelayer �s=0.000169 cm�1 Because of the LBLRTMstrategy, the sampling rate depends on the layer. To limitthe size of the look-up table to a size below 1 GB, theoptical depth below the layer L=28 is stored as generatedby LBLRTM, whereas for L�28 it is averaged in binsof width �s=10�3 cm�1 before storing it in the look-uptable. We have checked that in doing that the error inthe convolved brightness temperature spectra is below0.08 K.

Individual look-tables have been generated for the fol-lowing gas species: water vapor, carbon dioxide, ozone,nitrous oxide, carbon monoxide, methane, oxygen andnitrogen, a choice which enables σ-IASI to performJacobian calculations for these gases. For the potentialusers of σ-IASI, this means that any single gas of the

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

above set of gases, a group of them, or all of them simul-taneously may be dealt with as retrievable geophysicalparameters in inversion schemes. The other atmosphericgases (see Table 2) form the mixed-gas optical depthand their composition may not be altered by users. Thereference concentration values for the mixed species arethose compiled in the AFGL library (Anderson et al.,1986). It should be stressed here that the number of gasspecies that could have a measurable effect on nadirviewing satellite instruments, of spectral quality compa-rable to that of IASI, is limited to 3–6, according to thespectral range. Our data base includes, in form of vari-able absorbing constituents whose concentration may bespecified by the user, the main species which dominateupwelling radiance once convolved to a spectral resol-ution of 0.1–2 cm�1. The remaining species are not neg-lected, but they are taken into account as a unit whichcannot be altered by the user.

The look-up tables which form our layer optical depthdata base have been generated by making full use of theLBLRTM capabilities and borrows from this code theCO2 line mixing scheme; the reference value for the CO2

mixing ratio is 350 ppmv and may be changed in inputby the user. The required spectroscopic parameters havebeen extracted from the database HITRAN 96 (Rothmanet al., 1998).

6.2. Thresholding

Let dl,ss represent the layer optical depth, for the chemi-

cal species s at wavenumber s. We have introduced adouble threshold according to the following rules forall species:

�dl,ss h1⇒dl,s

s � 0;

dl,ss � h2⇒dl,s

s � h2.(45)

After extensive experiments, we have assumedh1=10�4 and h2=10. This corresponds to a maximumerror on transmittances of 10�4 and 5×10�5, respect-ively.

Table 2Chemical species included in the σ-IASI data-base

Species Symbolic Species Symbolic Species Symbolicnumber name number name number name

1 H2O 11 NH3 21 HOCl2 CO2 12 HNO3 22 N2

3 O3 13 OH 23 HCN4 N2O 14 HF 24 CH3Cl5 CO 15 HCl 25 H2O2

6 CH4 16 HBR 26 C2H2

7 O2 17 HI 27 C2H6

8 NO 18 ClO 28 PH3

9 SO 19 OCS 29 COF2

10 NO2 20 H2CO 30 SF6

6.3. Parameterization

Monochromatic optical depth for each species hasbeen parameterized as a function of temperature and fit-ted by a low-order polynomial: second order has beenproved to be more than adequate to best-fit optical depth.The results of the thresholding and parameterizationphase for all atmospheric layers show that 88.5% ofoptical depths are below threshold h1, 0.3% of them areabove threshold h2 (both do not need any representation)and the remaining 11.3% can be represented by a secondorder polynomial (e.g. Fig. 3). Because of parabolicparameterization, thresholding and layer dependent sam-pling rate strategy, the final size of the look-up table is726 MB against the initial 27.7 GB.

6.4. Further remarks

The main hypthosesis embedded in our look-up tableis that the optical depth is directly proportional to theabsorber amonut and can, therefore, easily scaled forvariable H2O, O3, CO2, CH4, CO, N2O and so on. Thisstrategy still allows for the correct computation of gasline absorption effects which may be parameterized interms of the air pressure. However, line absorptioneffects which depend on the gas concentration and there-fore on the partial pressure of individual gases are neg-lected. This is the case of the effects of variable self-broadening of gases on the optical depths. Since the typi-cal concentration of minor and trace species is of orderof ppmv or below, the effect of self-broadening ofatmospheric gases is not really a concern. For monochro-matic radiance, self-broadening of H2O may beimportant because H2O may reach concentrations in vol-ume up to 2–4% in moist atmospheres. However, forspectral radiance sensed through sensors such as AIRSand IASI, the effect of self-broadening is small (e.g.Strow et al., 1998). We have checked that at the IASIspectral sampling rate of 0.25 cm�1 the larger errorintroduced in the spectral radiance is of 0.4%. If the userapplication warrants second order self-broadeningeffects to be taken into account, then s-IASI is not themost appropriate code for it.

Cross sections of chlorofluorocarbons are not para-meterized in terms of look-up tables, since absorptioncan be quickly computed separately. Our σ-IASIaccounts for the presence of CFC species, namelyCC13F (CFC-11) and CC12F2 (CFC-12).

Finally, water vapor continuum absorption is not para-meterized in form of a look-up table, it is computed sep-arately. We use the CKD standard (Clough et al., 1989),version 2.4.

661U. Amato et al. / Environmental Modelling & Software 17 (2002) 651–667

Fig. 3. Parameterizing optical depth with temperature. The figure shows an example of parabolic fit to layer optical depth. The case shown refersto 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 previoussections applies to monochromatic spectral radiance. Tospecialize the computations to a given instrument onemore operation is needed which consists in convolvingthe monochromatic radiance and Jacobians with theInstrumental Line Shape of the sensor. The convolutionoperation is embedded in the σ-IASI software in a propermodule which may be changed by the user.

In the remaining of this section we describe the testwe have performed to check the accuracy and fastnessof our methodology once applied to yield IASI syn-thetic radiance.

IASI has a spectral coverage from 640 cm�1 to 2760cm�1. The full spectral range is divided in three bandswhich are here defined:

� Band 1: 640 to 1210 cm�1;� Band 2: 1210 to 2000 cm�1;� Band 3: 2000 to 2760 cm�1.

The instrument is still under industrial developementand its radiometric accuracy has not yet, definitely char-cacterized. At present, various scenarios for the radio-metric noise are considered which give the reasonablerange of radiometric performanmce of the sensor. Thesescenarios are shown in Fig. 4. The lower scenario is usu-ally referred to as Cannes specifications. It is expectedthat IASI radiometric noise will be in between theCannes specifications and the reasonable scenario (seeFig. 4) with the best possible commitment being the mostlikely final outcome.

The accuracy of the forward model in the IASI-modeoperation has been tested by comparing σ-IASI radianceoutput to IASI spectral radiance obtained by LBLRTM.

Table 3List of AFGL climatological models of atmosphere

AFGL n Atmosphere

1 Tropical2 Mid-Latitude Summer3 Mid-Latitude Winter4 High-Latitude Summer5 High-Latitude Winter6 US Standard

Six test models of atmosphere (see Table 3) have beenconsidered which encompass polar to tropical air masses(Anderson et al., 1986).

The differences between LBLRTM and σ-IASI calcu-lations are shown in Figs. 5 to 10 and compared to themost demanding IASI radiometric noise scenario(Cannes scenario). It is possible to see that the agreementis very good. Some slight discrepancy is, however, evi-dent for the warm and wet atmospheres, such as thattropical, at the beginning of the IASI band 2 whichcoincides with the first part of the H2O vibrational bandat 6.7 µm. This error is only partly due to the fact thatwe neglect self-broadening effects of water vapor. Therelatively large discrepancy is due to the concomitanteffect of CH4 and H2O. However, the discrepancy is inany case confined within the IASI error bars. We havecomputed the root mean square difference in the range1210–1350 cm�1 in the case of the tropical air mass.The value of 0.11 K has been found that may comparedthe IASI radiometric noise (NEDT) of �0.34–0.4 K inthat range.

For the benefit of the reader more familiar to bright-ness temperature spectra and NEDT radiometric errors,Table 4 summarizes the comparison between σ-IASI and

662 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.

663U. 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.

664 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.

665U. Amato et al. / Environmental Modelling & Software 17 (2002) 651–667

Fig. 10. As Fig. 5, but for the US standard model of atmosphere.

Table 4Root mean square difference between σ-IASI and LBLRTM brightnesstemperature calculations (units in Kelvin degrees)

AFGL n Air mass type Band 1 Band 2 Band 3

1 Tropical 0.063 0.068 0.0562 Mid-Latitude Summer 0.041 0.044 0.0383 Mid-Latitude Winter 0.039 0.027 0.0204 High-Latitude Summer 0.036 0.029 0.0375 High-Latitude Winter 0.032 0.054 0.0276 US Standard 0.028 0.032 0.019

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

ed � �1N�N

j � 1

(T1(j)�T2(j))2 (46)

where T1 and T2 are the two brightness temperature spec-tra computed according to σ-IASI and LBLRTM,respectively; N is the number of data points for the givenIASI band. It is possible to see that these values are oneto two orders of magnitude below the expected IASIradiometric noise shown in Fig. 4.

For the six models of atmosphere we have alsorecorded the running time for the two packages. The twosoftware packages have been run on ALPHA workst-

ation with 512 MB Ram and 400 MHz CPU clock. Wehave found that σ-IASI runs about five times faster thanLBLRTM for radiance computations. To complete a runfor the IASI band 1, which is the most important bandfor temperature sounding, σ-IASI takes 20 s against the100–110 s of LBLRTM. The efficiency increase up to afactor 50 when Jacobians are considered since onlyfinite-difference computations are presently availablewithin LBLRTM. The package σ-IASI takes about 200s to compute one Jacobian for one IASI band which maybe compared to the about 10,000 s taken by LBLRTM.

Finally, the Jacobian capability of σ-IASI is exem-plified in Fig. 11 which shows examples of temperature,water vapor, ozone and methane Jacobians. Reliabilityand accuracy of these products have been proved mostlyby extensive applications to real observations recordedfrom the Japanese Fourier transform spectrometer, IMG.The experience we have up to now is that σ-IASI is ableto yield consistent inversions for temperature and watervapor which compare very well, for example, toECMWF analysis (European Centre for Medium rangeWeather Forecasts). This experience has been docu-mented mostly in Amato et al. (1999), Lubrano et al.(2000, 2002), and Masiello et al. (2002).

666 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 methodologywhich is suitable for next generation satellite nadir view-ing infrared sensors with a sampling rate in between 0.1–2 cm�1. The methodology we have developed is fullyanalytical and may be implemented to provide fast andaccurate computations of spectral radiance and itsderivative with respect to any set of geophysical para-meters.

The capability of σ-IASI to yield on line calculationsof Jacobians makes the code an attractive tool for sensi-tivity analysis or error analysis which are typicallyrequired when designing a new instrument. Moreover,σ-IASI is the perfect subroutine for inversion softwarealgorithms using the Gauss-Newton minimizationscheme or Statistical Regularization (e.g. Tarantola,1987) which heavily depends on the availability of theJacobian matrix.

The scheme originated because of the IASI quest fora forward model which joined fastness to accuracy and

ability of radiance and Jacobians calculations. However,σ-IASI may be easily adapted to other nadir viewingsensors by adjusting the spectral ranges and by inputtingthe proper Instrumental Line Shape function.

The performance of the scheme has been evaluated byrunning it in the IASI-mode. We have found that σ-IASIcompares to LBLRTM in accuracy, while being fivetimes faster as far as spectral radiance is concerned.However, a paramount computational efficiency isgained for the calculation of Jacobians which are fullyanalytical in our scheme. Compared to a finite differencescheme implemented through LBLRTM, σ-IASI per-forms 50 times faster.

The software implementation of σ-IASI combines ina single package two basic modules:

� a look-table module for the generation of monochro-matic optical depths

� a module for the compuation of Top of Atmospherespectral radiance and Jacobians.

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

It is here important to stress that the look-up tabledoes not yield accurate optical depths under all con-ditions. For nadir viewing satellite sensors such as IASI,AIRS and meteorological infrared sensors which will flyin the foreseeable future (e.g. GIFTS (Smith et al.,2001)) the errors introduced by the look-up table schemeare unimportant and produce convolved spectral radi-ance accurate enough for applications. In this context,the class of nadir viewing sensors which may benefitfrom σ-IASI are those with a sampling rate within 0.1–2 cm�1. Below 0.1 cm�1 σ-IASI could be not accuarteenough, whereas above 2 cm�1 it would be too muchcomplex.

Finally, in its mathematical formulation andimplementation the module for the computation of Topof Atmosphere radiance and Jacobians is independent ofthe look-up table scheme, therefore LBLRTM orGENLN2 users could benefit by adopting it in theirown codes.

Acknowledgements

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

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