New Measurement Kalman ﬁlter physical retrieval of surface … · 2020. 7. 16. · Measurement Techniques Open Access Kalman ﬁlter physical retrieval of surface emissivity and

Atmos. Meas. Tech., 6, 3613–3634, 2013www.atmos-meas-tech.net/6/3613/2013/doi:10.5194/amt-6-3613-2013© Author(s) 2013. CC Attribution 3.0 License.

Atmospheric Measurement

TechniquesO

pen Access

Kalman filter physical retrieval of surface emissivity andtemperature from geostationary infrared radiances

G. Masiello1, C. Serio1, I. De Feis2, M. Amoroso1, S. Venafra1, I. F. Trigo 3, and P. Watts4

1Scuola di Ingegneria, Università della Basilicata, Potenza, Italy2Istituto per le Applicazioni del Calcolo “Mauro Picone” (CNR), Naples, Italy3Instituto Portugues do Mar e da Atmosfera IP, Land SAF, Lisbon, Portugal4EUMETSAT, Darmstadt, Germany

Correspondence to:C. Serio ([email protected])

Received: 7 July 2013 – Published in Atmos. Meas. Tech. Discuss.: 26 July 2013Revised: 16 October 2013 – Accepted: 22 November 2013 – Published: 20 December 2013

Abstract. The high temporal resolution of data acquisitionby geostationary satellites and their capability to resolve thediurnal cycle allows for the retrieval of a valuable source ofinformation about geophysical parameters. In this paper, weimplement a Kalman filter approach to apply temporal con-straints on the retrieval of surface emissivity and temperaturefrom radiance measurements made from geostationary plat-forms. Although we consider a case study in which we ap-ply a strictly temporal constraint alone, the methodology willbe presented in its general four-dimensional, i.e., space-time,setting. The case study we consider is the retrieval of emis-sivity and surface temperature from SEVIRI (Spinning En-hanced Visible and Infrared Imager) observations over a tar-get area encompassing the Iberian Peninsula and northwest-ern Africa. The retrievals are then compared with in situ dataand other similar satellite products. Our findings show thatthe Kalman filter strategy can simultaneously retrieve sur-face emissivity and temperature with an accuracy of± 0.005and±0.2 K, respectively.

1 Introduction

Infrared instrumentation on geostationary satellites is nowrapidly approaching the spectral quality and accuracy ofmodern sensors on board polar platforms. Currently at thecore of European Organisation for the Exploitation of Me-teorological Satellites (EUMETSAT) geostationary meteo-rological programme is the Meteosat (meteorological satel-lite) Second Generation (MSG). EUMETSAT is preparing

for Meteosat Third Generation (MTG), which will carry theFlexible Combined Imager (FCI) with a spatial resolution of1–2 km at the sub-satellite point and 16 channels (8 in thethermal band), and an infrared sounder (IRS) that will beable to provide unprecedented information on horizontally,vertically, and temporally (four-dimensional; 4-D) resolvedwater vapor and temperature structures of the atmosphere.The IRS will have a hyperspectral resolution of 0.625 cm−1

wave numbers, will take measurements in two bands, thelong-wave infrared (LWIR) (14.3–8.3 µm) and the mid-waveinfrared (MWIR) (6.25–4.6 µm), and with a spatial resolutionof 4 km and a repeat cycle of 60 min.

Because geostationary satellites are capable of resolvingthe diurnal cycle, and hence providing time-resolved se-quences or times series of observations, they are a sourceof information which can suitably constrain the derivationof geophysical parameters. In this paper, we implement aKalman filter (KF) approach for applying temporal con-straints on the retrieval of surface emissivity and tempera-ture from radiance measurements made from MSG SEVIRI(Spinning Enhanced Visible and Infrared Imager). The studyhas been performed also in view of future applications to theMTG mission. This mission should improve sounding den-sity, quality and accuracy of surface and atmospheric param-eters.

The Kalman filter (Kalman, 1960; Kalman and Bucy,1961) has received widespread attention in the context of nu-merical weather prediction (NWP) and in the broad researcharea of data assimilation (e.g.,Lorenc, 1986; Evensen, 1994;Talagrand, 1997; Nychka and Anderson, 2010). Specific

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3614 G. Masiello et al.: Kalman filter surface temperature and emissivity retrieval from geostationary platforms

applications to atmospheric chemistry with satellite data havebeen described by, e.g.,Khattatov et al.(1999); Lamarque etal. (1999); Levelt et al.(1998). For a detailed review and tuto-rial of the theoretical background of KF the reader is referredto, e.g.,Wikle and Berliner(2007); Nychka and Anderson(2010).

The present paper addresses the capability of the Kalmanfilter to convey temporal constraint in the retrieval of sur-face parameters though time series of geostationary satellitedata. The fact that time continuity of the observations bringsmuch information about atmospheric processes is normallyevidenced by the pronounced dynamical correlation, whichin many instances can be modeled with Markov chains orMarkov stochastic processes (e.g.,Serio, 1992; Cuomo et al.,1994; Serio, 1994, and references therein).

To exploit the temporal information, we focus on imple-menting the KF so that it is capable of incorporating dynam-ical correlation within the retrieval process without makinguse of a full dynamical NWP system. We aim at develop-ing a retrieval strategy for surface emissivity and tempera-ture which allows us insight into understanding how we canbetter exploit satellite data per se. In other words, the analy-sis is conducted within a context which envisages an almostentirely data-driven system. In this respect, we clarify that al-though we try to exploit tools such as the KF which are gen-erally used in a data assimilation context, we aim at address-ing a retrieval problem limited to surface parameters. We donot want to solve an assimilation problem according to thecommon method (e.g., seeNychka and Anderson, 2010) ofcombining a NWP model with observations.

In view of future MTG applications, the KF methodologyis presented in a general context which applies to both spa-tial and temporal constraints. However, it will be exempli-fied for a case study in which we consider a strictly temporalconstraint alone. As said, this is the problem of surface tem-perature (Ts) and emissivity (�) separation, that is, the simul-taneous retrieval of(Ts,�) from SEVIRI infrared channels.Toward this objective, a case study has been defined whichincludes a specific target area characterized by a large vari-ety of surface features.

The problem of retrieving surface emissivity and temper-ature from satellite data has long been studied. A recent re-view of the subject has been provided byLi et al. (2013).According to this review, our KF approach from a geosta-tionary platform is novel. A similarity could be found withthe scheme developed byLi et al. (2011). However, while inLi et al. (2011) the observations are accumulated for a pre-scribed time slot (normally six hours), we pursue a genuinedynamical strategy which exploits the sequential approach ofthe Kalman filter. This results in an algorithm which does notneed to increase the dimensionality of the data space becauseof time accumulation, while preserving the highest time res-olution prescribed by the repeat time of the geostationary in-strumentation (15 min for SEVIRI).

To check the quality and performance of our approach, KFretrieval results will be compared with in situ data, space-time collocated ECMWF (European Centre for Medium-Range Weather Forecasts) analysis and other similar satelliteproducts, such as those from the AVHRR (Advanced VeryHigh Resolution Radiometer).

The study is organized as follows. Section2 will presentthe data used in the analysis. This section will also providesome details about the forward model we have developed forSEVIRI. Section3 will deal with the retrieval methodology,whereas Sect.4 will exemplify the application of the method-ology to a SEVIRI case study. Finally, conclusions will bemade in Sect.5.

2 Data and forward modeling

In this paper, the KF methodology will be applied for the re-trieval of surface emissivities and surface temperature fromSEVIRI infrared channels in the atmospheric window overa target area, covering a geographic region with very differ-ent surface features: sea water, arid, vegetated and cultivatedland, and urban areas.

The SEVIRI imager on board Meteosat-9 allows for acomplete image scan (full Earth scan) once every 15 min pe-riod with a spatial resolution of 3 km for 12 channels (8 inthe thermal band), over the full disk covering Europe, Africaand part of South America.

SEVIRI infrared channels range from 3.9 µm to 12 µm.Their conventional definition in terms of channel number isgiven in Table1, whereas their spectral response is shown inFig. 1. The figure also provides a comparison with a typicalIASI (Infrared Atmospheric Sounder Interferometer) spec-trum at a spectral sampling of 0.25 cm−1.

IASI has been developed in France by the Centre Nationald’Etudes Spatiales (CNES) and is on board the Metop (Me-teorological Operational Satellite) platform, a series of threesatellites belonging to the EUMETSAT European Polar Sys-tem (EPS). The instrument has a spectral coverage extend-ing from 645 to 2760 cm−1, which, with a sampling interval1σ = 0.25 cm−1, gives 8461 data points or channels for eachsingle spectrum. Data samples are taken at intervals of 25 kmalong and across track, each sample having a minimum diam-eter of about 12 km. Further details on IASI and its missionobjectives can be found inHilton et al.(2012). Atmosphericparameters (temperature, water vapor and ozone profiles) de-rived from IASI spectral radiances will be used in this paperto assess the sensitivity of SEVIRI atmospheric window in-frared channels to the atmospheric state vector.

As stated before, for the purpose of this study, SEVIRIMeteosat-9 high rate level 1.5 image data and IASI (level 1C)observations have been collected for the target area shown inFig. 2 for the full month of July 2010. The area is coveredwith 392 088 Meteosat-9 pixels and includes Spain, Portugal,

Atmos. Meas. Tech., 6, 3613–3634, 2013 www.atmos-meas-tech.net/6/3613/2013/

G. Masiello et al.: Kalman filter surface temperature and emissivity retrieval from geostationary platforms 3615

Table 1. Definition of SEVIRI infrared channels and radiometricnoise in noise equivalent difference temperature (NEDT) at a scenetemperature of 280 K.

Channel wave no.Number (cm−1) wavelength (µm) NEDT at 280 K (K)

1 2564.10 3.92 1612.90 6.2 0.123 1369.90 7.3 0.204 1149.40 8.7 0.135 1030.9 9.7 0.216 925.90 10.8 0.137 833.30 12.0 0.188 746.30 13.4 0.37


D2, 2105-2116, 1999.Menglin, J.: Interpolation of surface radiative temperature measured

from polar orbiting satellites to a diurnal cycle: 2. Cloudy-pixeltreatment. Journal of Geophysical Research, 105, D3, 4061-4076, 2000.

Mira, M., Valor, E., Boluda, R., Caselles, V., and Coll, C.: Influenceof soil water content on the thermal infrared emissivity of baresoils: Implication for land surface temperature determination, J.Geophys. Res., 112, F04003, 2007. doi:10.1029/2007JF000749

Nychka, D. W., and, Anderson, J.L.: Data Assimilation. Handbookof Spatial Statistics ed. A. Gelfand, P. Diggle, P. Guttorp and M.Fuentes. Chapman & Hall/CRC, New York, 2010.

Rodgers, C. D.: Inverse methods for atmospheric sounding. Theoryand Practice World Scientific, Singapore, 2000.

Seemann,S. W., Borbas, E. E., Knuteson, R. O., Stephenson, G. R.,and Huang, H. L.: Development of a Global Infrared Land Sur-face Emissivity Database for Application to Clear Sky SoundingRetrievals from Multispectral Satellite Radiance Measurements,Journal of Applied Meteorology and Climatology, 47, 108-123,2008. doi:10.1175/2007JAMC1590.1.

Serio, C.: Discriminating low-dimensional chaos from randomness:A parametric time series modelling approach. Il Nuovo Cimento,107B, 681-702, 1992.

Serio, C.: Autoregressive Representation of Time Series as a Toolto Diagnose the Presence of Chaos. Europhys. Lett., 27, 103,doi:10.1209/0295-5075/27/2/005, 1994.

Serio, C., Masiello, G., Amoroso, M., Venafra, S., Amato, U., andDe Feis, I.: Study on Space-Time Constrained Parameter Estima-tion from Geostationary Data. Final Progress Report, EUMET-SAT contract No. EUM/CO/11/4600000996/PDW, 2013.

Talagrand, O.: Assimilation of observations, an introduction. J. Me-teorol. Soc. Jpn, 75, 191-209, 1997.

Tarantola, A.: Inverse Problem Theory: Methods for Data Fittingand Model Parameter Estimation. Elsevier, New York, 1987.

Trigo, I. F., Monteiro, I.T., Olesen, F., and Kabsch, E.: An assess-ment of remotely sensed land surface temperature, J. Geophys.Res., 113, D17108, doi:10.1029/2008JD010035, 2008.

Trigo I. F., and Viterbo, P.: Clear sky window channel adiances:a comparison between observations and the ECMWF model. J.App. Met., 42, 1463-1479, 2003

Wikle, C.K., and Berliner, L.M.: A Bayesian Tutorialfor data assimilation. Physica D, 230 (1-2), 1-26,doi:10.1016/j.physd.2006.09.017, 2007.

Wikle, C.K., and Hooten M.B.: A general science-based frame-work for dynamical spatio-temporal models. Test, 19, 417-451,doi:10.1007/s11749-010-0209-z, 2010.

500 1000 1500 2000 2500 30000

0.05

0.1

0.15

wave number (cm−1)

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spe

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1

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iri s

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Seviri Ch1Seviri Ch2Seviri Ch3Seviri Ch4Seviri Ch5Seviri Ch6Seviri Ch7Seviri Ch8IASI

Fig. 1. SEVIRI channel spectral response over-imposed to a typicalIASI spectrum.

20° W 10° W 0° 10° E

20° N

30° N

40° N

50° N

Seville test area

Sahara desert test area

Mediterranean Seatest area

Fig. 2. Target area (rectangled in blue) used to check the retrievalalgorithms. The figure also show the position of smaller test areassituated East of Seville (Spain), in a flat dune area in the Saharadesert, and below Sardinia island in the Mediterranean sea.

Fig. 1.SEVIRI channel spectral response superimposed on a typicalIASI spectrum.

part of the northwestern Africa, and the western part of theMediterranean Basin.

To check the performance of the scheme, we have alsoselected three smaller areas (also shown in Fig.2 with redboxes) in Spain, the Sahara desert, and the MediterraneanBasin, which have a size of 0.5× 0.5 degrees and each cor-respond to one box of the ECMWF analysis grid mesh (e.g.,see Fig.3). For the Spanish location the area includes 187SEVIRI pixels, 219 for the Sahara desert and 178 for theMediterranean Basin. The land coverage for the small tar-get area close to Seville is a mosaic of cultivated areas, withgreen grass, foliage, bare soil and urban areas. For this typeof coverage we expect an emissivity at atmospheric windowwell above 0.90. The small Sahara desert area is just a desertsand homogeneous flat area, with no vegetation. In this casewe know that emissivity is dominated by quartz particles,which yield a characteristic fingerprint at 8.6 µm (reststrahlendoublet of quartz). This strong signature is in the middle ofthe SEVIRI channel at 8.7 µm, and, therefore, the retrievedemissivity at this channel has to show the quartz fingerprint.


D2, 2105-2116, 1999.Menglin, J.: Interpolation of surface radiative temperature measured

from polar orbiting satellites to a diurnal cycle: 2. Cloudy-pixeltreatment. Journal of Geophysical Research, 105, D3, 4061-4076, 2000.

Mira, M., Valor, E., Boluda, R., Caselles, V., and Coll, C.: Influenceof soil water content on the thermal infrared emissivity of baresoils: Implication for land surface temperature determination, J.Geophys. Res., 112, F04003, 2007. doi:10.1029/2007JF000749

Nychka, D. W., and, Anderson, J.L.: Data Assimilation. Handbookof Spatial Statistics ed. A. Gelfand, P. Diggle, P. Guttorp and M.Fuentes. Chapman & Hall/CRC, New York, 2010.

Rodgers, C. D.: Inverse methods for atmospheric sounding. Theoryand Practice World Scientific, Singapore, 2000.

Seemann,S. W., Borbas, E. E., Knuteson, R. O., Stephenson, G. R.,and Huang, H. L.: Development of a Global Infrared Land Sur-face Emissivity Database for Application to Clear Sky SoundingRetrievals from Multispectral Satellite Radiance Measurements,Journal of Applied Meteorology and Climatology, 47, 108-123,2008. doi:10.1175/2007JAMC1590.1.

Serio, C.: Discriminating low-dimensional chaos from randomness:A parametric time series modelling approach. Il Nuovo Cimento,107B, 681-702, 1992.

Serio, C.: Autoregressive Representation of Time Series as a Toolto Diagnose the Presence of Chaos. Europhys. Lett., 27, 103,doi:10.1209/0295-5075/27/2/005, 1994.

Serio, C., Masiello, G., Amoroso, M., Venafra, S., Amato, U., andDe Feis, I.: Study on Space-Time Constrained Parameter Estima-tion from Geostationary Data. Final Progress Report, EUMET-SAT contract No. EUM/CO/11/4600000996/PDW, 2013.

Talagrand, O.: Assimilation of observations, an introduction. J. Me-teorol. Soc. Jpn, 75, 191-209, 1997.

Tarantola, A.: Inverse Problem Theory: Methods for Data Fittingand Model Parameter Estimation. Elsevier, New York, 1987.

Trigo, I. F., Monteiro, I.T., Olesen, F., and Kabsch, E.: An assess-ment of remotely sensed land surface temperature, J. Geophys.Res., 113, D17108, doi:10.1029/2008JD010035, 2008.

Trigo I. F., and Viterbo, P.: Clear sky window channel adiances:a comparison between observations and the ECMWF model. J.App. Met., 42, 1463-1479, 2003

Wikle, C.K., and Berliner, L.M.: A Bayesian Tutorialfor data assimilation. Physica D, 230 (1-2), 1-26,doi:10.1016/j.physd.2006.09.017, 2007.

Wikle, C.K., and Hooten M.B.: A general science-based frame-work for dynamical spatio-temporal models. Test, 19, 417-451,doi:10.1007/s11749-010-0209-z, 2010.

500 1000 1500 2000 2500 30000

0.05

0.1

0.15


Iasi

spe

ctru

m (

W m

−2

sr−

1 (c

m−

1 )−

1

500 1000 1500 2000 2500 30000

0.2

0.4

0.6

0.8

1

Sev

iri s

pect

ral r

espo

nse

Seviri Ch1Seviri Ch2Seviri Ch3Seviri Ch4Seviri Ch5Seviri Ch6Seviri Ch7Seviri Ch8IASI

Fig. 1. SEVIRI channel spectral response over-imposed to a typicalIASI spectrum.

20° W 10° W 0° 10° E

20° N

30° N

40° N

50° N

Seville test area

Sahara desert test area

Mediterranean Seatest area

Fig. 2. Target area (rectangled in blue) used to check the retrievalalgorithms. The figure also show the position of smaller test areassituated East of Seville (Spain), in a flat dune area in the Saharadesert, and below Sardinia island in the Mediterranean sea.

Fig. 2. Target area (blue rectangle) used to check the retrieval algo-rithms. The figure also show the position of smaller test areas situ-ated east of Seville (Spain), in a flat dune area in the Sahara desert,and below island of Sardinia in the Mediterranean sea.

For the whole target area shown in Fig.2, we have alsoacquired ancillary information for the characterization of thethermodynamical atmospheric state. This information is pro-vided by ECMWF analysis products for the surface temper-ature,Ts and the atmospheric profiles of temperature, wa-ter vapor and ozone(T ,Q,O) at the canonical hours 00:00,06:00, 12:00 and 18:00 UTC. ECMWF model data are pro-vided on 0.5× 0.5 degree grid. In each ECMWF grid boxthere are on average≈ 200 SEVIRI pixels, for which we as-sume that the atmospheric state vector is the time collocatedECMWF analysis (e.g., see Fig.3).

Within the inverse scheme, an important issue concernsa priori information to constrain the retrieval of emissiv-ity. To this end, we have used the University of Wiscon-sin Baseline Fit Global Infrared Land Surface Emissiv-ity Database (UW/BFEMIS database, e.g.,http://cimss.ssec.wisc.edu/iremis/) (Seemann et al., 2008; Borbas and Ruston,2010). The UW/BFEMIS database is available for years 2003to 2012, globally, with 0.05 degree spatial resolution. Detailsof how to transform UW/BFEMIS database emissivity to SE-VIRI channel emissivity can be found inSerio et al.(2013);Masiello et al.(2013).

For the purpose of comparison, we have used also NOAA(National Ocean and Atmosphere Administration) Opti-mum Interpolation 1/4 degree daily Sea-Surface Tempera-ture (OISST) analyses for the month of July 2010. The anal-ysis, which is a product of the processing of AMSR (Ad-vanced Microwave Scanning Radiometer) and AVHRR, willbe compared to that obtained by SEVIRI for sea surface. The

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http://cimss.ssec.wisc.edu/iremis/http://cimss.ssec.wisc.edu/iremis/



−5.6 −5.4 −5.2 −5 −4.8 −4.6 −4.4 −4.2 −4 −3.833.8

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Fig. 3. Example of overlapping between the SEVIRI fine mesh andthat coarse corresponding to the ECMWF analysis.

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Fig. 4. Simulation of SEVIRI atmospheric window channels 7, 6and 5 respectively (12 µm, 10.8 µm and 8.7 µm) response (panel b)to the forcing of the daily temperature cycle shown in panel a). Panelc) shows the derivative ratio D (see text for details) correspondingto the three SEVIRI atmospheric window channels. In panel b) r.u.stands for radiance units; 1 r.u.=1 W m−2 (cm−1)−1 sr−1.

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Fig. 5. Retrieval exercise using simulations with a variance of thestochastic term for Ts equal to 1 K2 and f = 10. Upper panel: skintemperature retrieval; lower panel: emissivity retrieval at the threeSEVIRI atmospheric window channels.


analysis will be referred to as AMSR+AVHRR OISST in theremainder of this paper. The AMSR+AVHRR OISST analy-sis has been downloaded from the websiteftp://eclipse.ncdc.noaa.gov/pub/OI-daily-v2/NetCDF/2010/AVHRR-AMSR/.

Finally, we have also used data collected at the Evoraground site (38.55◦ N, 8.01◦ W) located in southern Portu-gal and maintained by the EUMETSAT Satellite Applica-tions Facility on Land Surface Analysis (LSA SAF) team.The area surrounding the site is dominated byQuercuswood-land plains and is fairly homogeneous at the SEVIRI spatialscales (Dash et al., 2004). The ground station is equippedwith a suite of radiometers (9.6–11.5 µm range) providingtemperatures of tree canopies and of ground in the sun andin the shade. These are combined to provide a compositeground temperature representative of SEVIRI pixels, consid-ering that the fractional area coverage of canopies is 0.32(Trigo et al., 2008). It is worth mentioning that the groundand the treetop canopy present contrasting temperatures par-ticularly during daytime, when differences can easily reach15 K. As a consequence, the composite ground temperaturesare fairly sensitive to the fraction of trees being considered.For this purpose, the area surrounding the Evora station wascarefully characterized using very high resolution IKONOSsatellite images (Kabsch et al., 2008; Trigo et al., 2008). Inaddition, our Kalman Filter retrievals for the pixels closer toEvora are also compared with the operational land-surfacetemperature product provided by the LSA SAF (Freitas etal., 2010).

2.1 Forward models:σ -IASI and σ -SEVIRI

Forward calculations for the SEVIRI channels 2–8 (see Ta-ble 1) are obtained by theσ -SEVIRI code that we have de-veloped specifically for this study.

We do not consider the SEVIRI channel at 3.9 µm sinceduring daytime it is contaminated by reflected solar radiationand affected by non-local thermodynamic equilibrium (non-LTE) effects. Furthermore, the CO2 line mixing at 4.3 µmCO2 band head is poorly modeled in state-of-the-art radiativetransfer and can add potentially large bias.

Regarding channels 2 to 8, the forward model,σ -SEVIRIhas been derived fromσ -IASI (Amato et al., 2002) which is amonochromatic radiative transfer designed for the fast com-putation of spectral radiance and its derivatives (Jacobian)with respect to a given set of geophysical parameters.

The form of the radiative transfer equation, whichσ -IASIand henceσ -SEVIRI consider in its numerical scheme, hasbeen recently reviewed and presented inMasiello and Serio(2013), to which the interested reader is referred. The modelalso takes into account the radiance term, which is the ra-diation reflected from the surface back to the satellite. BothLambertian and specular reflections can be modeled.

To accomplish the radiative transfer calculationσ -IASIuses a lookup table for the optical depth; this table was de-veloped from one of the most popular line-by-line forwardmodels, Line-By-Line Radiative Transfer Model (LBLRTM)(Clough et al., 2005).

The modelσ -SEVIRI is itself based on a lookup table,which is obtained by a proper down-sampling of the lookuptable forσ -IASI. For this reason we need to give some detailsaboutσ -IASI in order to describe howσ -SEVIRI works.

Theσ -IASI model (Amato et al., 2002) parameterizes themonochromatic optical depth with a second-order polyno-mial. At a given pressure-layer and wave numberσ (in cm−1

units), the optical depth for the a genericith molecule is com-puted according to

χσ,i = qi

2∑j=0

cσ,j,iTj , (1)

whereT is the temperature,qi the molecule concentrationandcσ,j,i with j = 0,1,2, are fitted coefficients, which areactually stored in the optical depth lookup table.

For water vapor, unlike other gases, in order to take intoaccount effects depending on the gas concentration, such asself-broadening, a bi-dimensional lookup table created byMasiello and Serio(2003) is used. Thus, for water vapor,identified withi = 1, the optical depth is calculated accord-ing to

χσ,1 = q1

(2∑

j=0

cσ,j,1Tj+ cσ,3,1q1

). (2)

The subscriptσ indicates the monochromatic quantities. Inthe case of hyperspectral instruments, such as IASI, themonochromatic optical depths are computed and parameter-ized at the spectral sampling interval of 10−4 cm−1.

This spectral sampling is much too fine for a band instru-ment such as SEVIRI. In the case of SEVIRI, the spectral


ftp://eclipse.ncdc.noaa.gov/pub/OI-daily-v2/NetCDF/2010/AVHRR-AMSR/ftp://eclipse.ncdc.noaa.gov/pub/OI-daily-v2/NetCDF/2010/AVHRR-AMSR/


sampling can be averaged and sampled at a rate of 10−1 cm−1

without sacrificing accuracy. Also in this case the opticaldepth can be parameterized with a low-order polynomial, andits coefficients are obtained as explained below.

For each species,i, we can define an equivalent opticaldepth which can be parameterized with respect to tempera-ture in the same way we do for monochromatic quantities(Eqs.1 and 2). Considering the larger channel bandwidthsof the SEVIRI measurements, averaging is applied over thespectral wave-number band of each channel. This averagingis identified by the angular brackets〈·〉. The equivalent opti-cal depth is

χ〈σ 〉,i = qi

2∑j=0

c〈σ 〉,j,iTj , (3)

where the equivalent coefficientsc〈σ 〉,j,i with j = 0,1,2, areobtained by fitting the layer transmittance averaged over thecoarse sampling of 10−1 cm−1,

qi

2∑j=0

c〈σ 〉,j,iTj

= − log[〈

exp(−χσ,i

)〉]. (4)

Because of this down-samplingσ -SEVIRI, which is basedon the coarse-mesh lookup table, runs≈ 1000 times fasterthan σ -IASI. As the parent code,σ -IASI, σ -SEVIRI cancompute the analytical Jacobian derivative for a large set ofsurface and atmospheric parameters:�, Ts and(T ,Q,O).

3 The retrieval framework

Before showing the retrieval problem for the pair of surfaceparameters(Ts,�), we briefly review the concept of the op-timal estimation in the general context of data assimilation(Lorenc, 1986; Talagrand, 1997; Wikle and Berliner, 2007;Nychka and Anderson, 2010; Rodgers, 2000), which allowsus to describe the retrieval methodology in its general spa-tiotemporal framework and also to put in evidence its com-monalities with the KF methodology.

For the benefit of the reader, we will try to stay as closeas possible to the notation used inRodgers(2000), thereforethe symbol and subscriptε will be used to denote the obser-vational covariance matrix and hence the noise term affectingthe spectral radiance. For emissivity we will use the symbol�, which should not be confused withε.

3.1 Static a priori background

To simplify the exposition let us assume that the times areindexed by integers,t = 1,2, . . ., although handling unevenlyspaced times does not add any fundamental difficulty.

The derivation of the thermodynamical state of the atmo-sphere, at a given timet , given a set of independent obser-vations of the spectral radiance,Rt (σ ), is well established

when each timet is considered independent from past andfuture measures. LetRt be the radiance vector

Rt = (Rt (σ1), . . .,Rt (σm))T , (5)

with m the number of spectral radiances, and where the su-perscriptT stands for transpose. Under the assumption ofmultivariate normality the retrieval problem can be seen asone of variational analysis in which a suitable estimation ofthe state vector is obtained by minimizing the form (see e.g.,Courtier, 1997; Talagrand, 1997; Tarantola, 1987; Carissimoet al., 2005):

minv

1

2(Rt − F(v))

T S−1ε (Rt − F(v))

+1

2(v − va))

T S−1a (v − va)) , (6)

whereF is the forward model function;v is the atmosphericstate vector, of sizen; va is the atmospheric background statevector, of sizen; Sε is the observational covariance matrix, ofsizem × m; andSa is the background covariance matrix, ofsizen×n. Equation (6) is commonly linearized and a GaussNewton iterative method is used to solve the quadratic form

minx

1

2(yt − Kx)T S−1ε (yt − Kx)

+1

2(x − xa)

T S−1a (x − xa) , (7)

whereK is ∂F (v)∂v

|v=vo ; yt = Rt −Rot ; andx is v −vo;xa =va − vo. It should be stressed that, formally, the state vector,v can be thought of as a 3-D geophysical field, and not nec-essarily of a vector in one dimension (altitude coordinate).

The formal solution of Eq. (7) is well established (e.g.,Tarantola, 1987; Rodgers, 2000).

x̂ = xa +(KT S−1ε K + S

−1a

)−1KT S−1ε (yt − Kxa)

Ŝ =(KT S−1ε K + S

−1a

)−1 (8)In the context ofdata assimilation, xa is normally the fore-cast at timet , andSa is the forecast error covariance matrix.The estimation,̂x, is referred to as the analysis.

3.2 The Kalman filter

The Kalman filter was first developed byKalman(1960) andKalman and Bucy(1961) in an engineering context and asa linear filter. Its derivation from the Bayes formalism hasbeen shown by many authors (e.g., see the review byWikleand Berliner, 2007).

With our notation, the formal filter can be summarizedwith the two equations below which are often referred to astheobservation equation(or data model) and thestate equa-tion (or dynamic model or system model), respectively.{

Rt = F(vt ) + εtvt+1 = Mvt + ηt+1

(9)



Here M is a linear operator and the noise model term,ηthas covariance,Sη. The remaining parameters appearingin Eq. (9) have the same meaning as those introduced inSect.3.1. KF is intrinsically linear, therefore the observationequation has to be linearized in order to write down the op-timal estimation for the state vector. With the same notationwe have used until now, we have the linear KF form{

yt = K txt + εtvt+1 = Mvt + ηt+1

, (10)

where we use the notationK t for the Jacobian to stress thatit depends on time,t .

It should be noted that we assume that both the noise termsεt andηt are independent of the state vector.

3.2.1 The KF update step or analysis

Under the same assumption of multivariate normal statisticsas that used in Sect.3.1, we have that the optimal KF esti-mate,x̂t at timet is given by (e.g.,Wikle and Berliner, 2007)

x̂t = xa +(KTt S

−1ε K t + S

−1a

)−1KTt S

−1ε (yt − K txa)

Ŝt =(KTt S

−1ε K t + S

−1a

)−1 . (11)We see that the optimal KF estimate forx̂t is formally equiv-alent to that obtained by the variational or optimal estimationapproach in Sect.3.1. We recall, once again, that in the con-text of data assimilation, xa is normally the forecast at timet , andSa is the error forecast covariance matrix. The estima-tion, x̂t , is referred to as theanalysisat time t , which hascovariance matrix given bŷSt .

One important aspect of the formal solution is that theanalysis update depends only on the data at timet and noton that at previous times. This property is referred to as theMarkov property. In fact, the formal solution for the anal-ysis does not depend on the dynamical system directly. Wecan see that the expression in Eq. (11) does not contain theoperatorM .

The above property is also referred to as the regularizationproperty of KF. New data comes in att and the KF updatedstate estimate is the minimizer of the quadratic form or costfunction,S:

S = minx

1

2(yt − K txt )T S−1ε (yt − K txt )

+1

2(xt − xa)

T S−1a (xt − xa) . (12)

However, an important distinction regarding data assimi-lation is thatSa is potentially generated from the process andnot from an external spatial model. In factSa is iterated withthe process, as will become clear in examining the forecaststep for the linear KF. It is important here to stress that theminimization of the form (12) needs an iterative approach

because of the nonlinearity of the forward model and a crite-rion to stop iterations. We use the usualχ2 criterion. In fact,under linearity, the value of twice the quadraticS (Eq. 12)at the minimum is distributed as aχ2 variable withm de-grees of freedom (Tarantola, 1987). A χ2 threshold,χ2th, atthree sigma confidence intervals, can be then obtained ac-cording toχ2th = m + 3

√2m. Therefore, the iterative proce-

dure is stopped when

χ2 = 2× S ≤ χ2th. (13)

3.2.2 The KF forecast step

In our notation,x̂ = v̂−vo andx̂a = v̂a −vo, so that the for-mal KF estimate for the state vector is

v̂t = va +(KTt S

−1ε K t + S

−1a

)−1KTt S

−1ε (yt − K txa) . (14)

For the forecast step the KF assumes that the process evolvesin a linear way, according to the operatorM ; therefore, wecan obtain an estimate of the forecast at timet + 1, standingat timet , through the linear transform

v̂f

t+1 = M v̂t , (15)

where the superscriptf stands for forecast. The forecast hasuncertainty given by

Ŝf

t = MŜtMT

+ Sη (16)

whereSη is the covariance matrix of the noise termηt (seeEq.10).

As soon as new data comes in at timet + 1, the forecastbecomes the background,

va = v̂f

t+1, Sa = Ŝf

t , (17)

and we are ready to obtain the new analysis,v̂t+1.An important concept to draw from this sequential updat-

ing is that spatial information about the distribution ofvt canbe generated from the dynamics of the process. In fact, ana-lyzing the forecast covariance matrix (Eq.16), it is seen thatit is based on the previous forecast covariance matrix and alsoinherits the dynamical relationship from the previous time.Thus, in the situation of assimilation for a space-time pro-cess, the spatial covariance for inference is built up sequen-tially based on past updates with observations and propagat-ing the posterior forward in time as a forecast distribution.We stress that this spatial information is the difference or er-ror between the conditional mean and the true field and is notthe covariance of the process itself.

However, the goodness of this spatial information mostlyrelies on the quality of the physics we model with the op-eratorM . Typically, the forecast step is completed by a de-terministic, physically based model. In this case, the spatialinformation has value. However, in a case in which we want



the problem driven from the data, the model can be very sim-plistic and inherently inadequate to describe the real-worldprocess. In this case, spatial information has to be providedexternally through a proper definition ofSa .

3.3 A formulation of the emissivity/temperatureretrieval with KF

As stated at the beginning of Sect.3, the general KF formal-ism has been described and presented in a full 4-D setting.In this section we will deal with an application to the(Ts,�)problem where we apply a strictly temporal only method.

To begin with, we introduce a transform for the emissiv-ity, which allows us to constrain the retrieval to the physicalemissivity range of 0–1. Letting� be the emissivity at any ofthe channels, we consider thelogit transform

e = log�

1− �, (18)

which has inverse

� =exp(e)

1+ exp(e). (19)

The transform maps 0–1 into the interval[−∞,+∞] andvice versa. Therefore if we work with the variablee, retrievalpositiveness for� is ensured.

In order to work with the parametere we have to properlytransform the Jacobian. It easily follows from Eq. (18) that

∂R

∂e=

∂R

∂��(1− �), (20)

whereR is the radiance at a generic channel.If we linearize the forward model, at timet , with respect

to e andTs, we obtain

yt = Atδet + BtδTst , (21)

with δe = e − eo of dimensionm× 1, δTst = Tst −Tsto. Thematrix At is the emissivity Jacobian, a diagonal matrix ofsize m × m, and Bt is the surface temperature Jacobian, avector of dimensionm × 1. We have that the size of the ob-servation vector,yt is m × 1, the dimension of the JacobianKt = (At ,Bt ) is m × (m + 1), and that the state vector,

xt =

δe1tδe2t. . .

δeMtδTst

, (22)has dimension(m + 1) × 1. As regards the state or modelequation for emissivity, an evolution equation is straightfor-ward if we consider the high repeat rate of SEVIRI observa-tions (15 min). This leads us to assume that the evolution of� has a low variability on a time scale of few hours. This isparticularly true for emissivity, but much less for temperature

over land, which is strongly influenced from the daily cycle.For sea surface the assumption of a low time variability ontime scales of several hours is good both forTs and�.

With this in mind, let v = (e1, . . .,em,Ts)T be theemissivity–temperature vector, a suitable dynamical equationis then a simple persistence

vt+1 = Mvt + ηt+1, (23)

where, according to our notation (see Sect.3.2), ηt is a noiseterm with covariance,Sη, andM is the identity propagationoperator.

We know that the persistence model of Eq. (23) is notphysically correct since it cannot reproduce the strong dailycyclic behavior ofTs expected in clear sky for land surface(Gottsche and Olesen, 2009; Menglin and Dickinson, 1999;Menglin, 2000). It could be a fair model for sea surface,where thermal inertia of water strongly damps the effect ofthe solar cycle; however, it cannot represent a good modelfor land surface.

Nevertheless, it has to be stressed that within the con-text of the Kalman filter methodology we can accommo-date our knowledge about the adequacy of the model (Wikleand Berliner, 2007). In practice, provided that the parame-ters are strongly constrained by the data, the precise form ofthe evolutionary equation is not important for the estimationproblem as long as the error covariance appropriately reflectsthe uncertainty of the current state estimate. To this end, animportant role is played by the stochastic noise covariance,Sη. By properly tuning the stochastic noise covariance, wecan have a retrieval which is either dominated by the data(Sη → +∞, model inadequate), or the state model (Sη → 0,model adequate).

SEVIRI atmospheric window channels are strongly dom-inated byTs. This is exemplified in Fig.4, which shows asimulation of the daily evolution ofTs for a desert site andthe corresponding radiance signal at channel 7 (12 µm). Thesimulation has been obtained using the daily cycle model de-veloped byGottsche and Olesen(2009) on the basis of insitu observations made at a station in the Namib desert. Themodel fits the data with an accuracy of≈ 1–2 K, thereforetheTs evolution shown in Fig.4 reflects a realistic situation.

The corresponding radiance has been obtained throughσ -SEVIRI. The state vector needed for the computation of theradiance has been obtained from the ECMWF analysis for adesert site.

Another way to assess the strong dependence of the SE-VIRI atmospheric window channels on temperature is tocompute the indexD between two consecutive times,t andt + 1, defined according to

D =Bt (σj ) + B(t+1)σj

2×

Ts,t+1 − Ts,t

Rt+1(σj ) − Rt (σj ), (24)

where σj denotes the wave number of a generic SEVIRIchannel. This index is the ratio of the derivative Jacobian of




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(°C

)Temperature forcing

a)

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Rad

ianc

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.u.)

Radiance response

b)

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1.05


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ensi

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ss)

c)

12 μm10.8 μm8.7 μm

8.7 μm

10.8 μm

12 μm


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Ts

(K)

Temperature


0 5 10 15 20 25 300.94

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0.98

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Em

issi

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Channel at 12 μm

0 5 10 15 20 25 300.92

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0.98


Em

issi

vity

Channel at 10.8 μm

0 5 10 15 20 25 300.7

0.75

0.8


Em

issi

vity

Channel at 8.7 μm





Fig. 4. Simulation of the response of the SEVIRI atmospheric win-dow channels 7, 6 and 5 (12 µm, 10.8 µm and 8.7 µm, respectively)(b) to the forcing of the daily temperature cycle shown in(a). Panel(c) shows the derivative ratioD (see text for details) correspond-ing to the three SEVIRI atmospheric window channels. In(b) r.u.stands for radiance units; 1 r.u. = 1 W m−2 (cm−1)−1 sr−1.

the surface temperature to the increment of the radiance dueto the variation of the surface temperature within the timeinterval (t, t + 1). Because of the meaning of the Jacobian,this index has to be close to 1 in case the channel stronglydepends onTs. Note that the second factor in Eq. (24) isthe finite-difference-based calculation of the inverse of theJacobian itself. For the case shown in Fig.4a, b, Fig.4cshows the indexD for the three SEVIRI atmospheric chan-nels. From this figure it is immediately seen that the radiancetime-behavior is completely dominated by the time-evolutionof Ts. This is a helpful situation because, at least for temper-ature, we can design a Kalman filter which is strongly drivenby the data.

To this end, we first clarify how we build upSη andSa onthe basis of the related matrices for emissivity and surfacetemperature.

We do not consider correlation between emissivity andsurface temperature; therefore,

Sη =(

Sηe, 00, SηTs

), (25)

and

Sa =(

Se, 00, STs

), (26)

whereSηe is the covariance matrix of the emissivity stochas-tic term; SηTs is the variance (scalar) of the surface-temperature stochastic term;Se is the initial background co-variance matrix of the emissivity vector; andSTs is the initialbackground variance (scalar) of the surface-temperature pa-rameter.

At this point we have defined all the components of our(Ts,�) problem which are needed to run the Kalman filter.The flow of operations is here summarized for the benefit ofthe reader. First, obtain the analysis update through Eq. (14);second, compute the forecast with Eq. (15); third, find theforecast covariance matrix through Eq. (16); and, finally, de-fine the forecast to be the new background (Eq.17) and returnto Eq. (14) for a new cycle.

Further details of how we build up the above ingredi-ents are given below. To begin with,Se is derived from theUW/BFEMIS database (see Sect.2). Its definition and cal-culation is space-time localized. For a given month and SE-VIRI pixel location, UW/BFEMIS yields ten different sam-ples of the emissivity vector from ten different years. TheUW/BFEMIS emissivity-vector samples undergo thelogittransform (see Eq.18) and are used to compute the covari-ance matrixSe. It could be argued thatSe built up on tensamples implies an unrealistically large statistical uncertaintyfor the covariances. This is true and reflects the present stageof our knowledge about surface emissivity. Because of thisuncertainty we will be forced to apply somewhat ad hoc fur-ther manipulations of this covariance matrix to get realisticretrievals.

An example ofSe for the set of seven SEVIRI channels(2 to 8 in Table1), for the month of July and for a SEVIRIpixel corresponding to a site in the Sahara desert is shown inTable2. Se shown in Table2 makes reference to the emissiv-ity vector ordered from longest to shortest wave number. Theelement (5,5) corresponds to the channel at 8.7 µm, whichis in the middle of the quartz reststrahlen band and hence ischaracterized by the strongest variability.

The covariance matrixSηe is derived fromSe with a scal-ing procedure. This is justified because of the need to scaledownSηe in order to correctly take into account the expectedvariation of emissivity on a time scale comparable to the SE-VIRI repeat time of 15 min. However, this is a rather ad hocinflation/deflation procedure, which is performed on the ba-sis of trial and error until we yield realistic retrievals.

The covariance matrixSe is scaled according to the fol-lowing procedure. LetSe(i,j); i,j = 1, . . .,m = 7 be the el-ements ofSe. The correlation matrixCe is defined accordingto

Ce(i,j) =Se(i,j)

√Se(i, i)Se(j,j)

; i,j = 1, . . .,m = 7, (27)

and the matrixSe is scaled according to

S(s)e (i,j) =

√Se(i, i)

f 2(i)×

Se(j,j)

f 2(j)Ce(i,j);

i,j = 1, . . .,m = 7, (28)

whereS(s)e is the matrix scaled by the vector of scaling fac-tors,f . The scaling operation above preserves the correlationstructure and in practice we consider a constant scaling fac-tor, f 2, that does not change along the diagonal.



Table 2. Example of the matrixSe for a SEVIRI pixel corresponding to a desert site (30.66◦ N, 5.56◦ E). The covariance matrix has beencomputed for the SEVIRI channels 2 to 8 in Table1 and makes reference to the emissivity vector ordered from longest to shortest wave. Theelement (5,5) corresponds to the channel at 8.7 µm, which is in the middle of the quartz reststrahlen band and hence is characterized by thestrongest variability.

ColumnRow 1 2 3 4 5 6 7

1 0.0273 0.0265 0.0136 0.0070 0.0109−0.0025 0.00532 0.0265 0.0262 0.0137 0.0068 0.0100−0.0025 0.00573 0.0136 0.0137 0.0075 0.0037 0.0056−0.0017 0.00324 0.0070 0.0068 0.0037 0.0023 0.0028−0.0008 0.00125 0.0109 0.0100 0.0056 0.0028 0.0067−0.0018 0.00186 −0.0025 −0.0025 −0.0017 −0.0008 −0.0018 0.0008 −0.00077 0.0053 0.0057 0.0032 0.0012 0.0018−0.0007 0.0017

We assumeSηe = S(s)e . As already mentioned, the appro-

priate value off has to be tuned in simulation. After ex-tensive simulations (Serio et al., 2013), we have found thatf = 10 is appropriate for this case study.

As far asTs is concerned, based on the evidence of Fig.4,we want to stay closer to the data than to the model. We havethat a variance of 1 K2 for the initial background and stochas-tic termsSTs andSηTs respectively, provides a balanced re-trieval. In other words, at least for land surface,SηTs does notneed to be downscaled with respect toSTs.

This can be seen in Fig.5, where we show the results of aretrieval exercise obtained in simulation for the case of desertsite (seeSerio et al., 2013, for full details). The case shownuses a persistence model for the state equation of both emis-sivity and skin temperature. For emissivity this is correct,since the simulation assumes a constant emissivity at eachSEVIRI channel. Conversely it is not correct for the surfacetemperature, whose true value follows the daily cycle shownin Fig. 5.

The example shown in Fig.5 and the error analysis inFig. 6 allows us to illustrate the property of the KF to ac-commodate the knowledge of the adequacy of thedetermin-istic model. As said before this is obtained by properly tun-ing the stochastic term. In the example shown in Fig.5, thestochastic variance forTs has been set to 1 K2. Because ofthis choice, we correctly follow the data and retrieve the truevalue of the surface temperature within the accuracy deter-mined by the a posteriori covariance matrix, that is≈ 0.2◦C.The same conclusion holds for emissivity, which is retrievedwithin an accuracy of≈ 0.005. For the stochastic variance,we can also prescribe a value just equal to zero, which willresult in a retrieval highly dominated by the model. The re-sults are presented in Fig.7. We see that after some iterations,the retrieval just follows the (inadequate) persistence model.The initialization point for skin temperature in both exercisesis the true temperature minus 4◦C. Note that we need to spec-ify only the initialization point att = 0, after that KF yieldsthe retrieval on the basis of the data points and model alone.


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.u.)

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b)

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Fig. 5. Retrieval exercise using simulations with a variance of thestochastic term forTs equal to 1 K2 andf = 10. Upper panel: skintemperature retrieval; lower panel: emissivity retrieval at the threeSEVIRI atmospheric window channels.



The results shown in Figs.5 to 7 justify the use of a sim-plistic persistence model forTs because the data – that is,observations – strongly constrain the phenomenon under in-vestigation. We can get to the same conclusion by observingthat, even by prescribing a stochastic variance forTs equal to1 K2, we obtain a precision for the final estimate of≈ 0.2◦C.This finding implies that the a priori information forTs haslittle to no impact on the final estimate, which is thereforelargely dominated by the data.

The retrieval exercise shown in Figs.5 to 7 allows us toaddress another important issue: how the time constraint im-posed with the persistence model improves the retrieval incomparison to a scheme where this constraint is not im-posed at all. If we try to solve the(Ts,�) retrieval problemwithin the usual context of least squares estimation with astatic background, we get a retrieval covariance matrix witha strong anticorrelation betweenTs and�. This anticorrela-tion is due to the relationship of these two parameters withinthe radiative transfer equation and makes their effective sep-aration unpractical. To exemplify this effect, we have run thesame retrieval exercise shown in Fig.5, but now withM = 0.In this way, the retrieval does not evolve through the stateequation and the surface parameters are estimated on the ba-sis of a static background. The error analysis for this exerciseis shown in Fig.8. The anticorrelation effect is soon evident.We find thatTs is biased significantly low and� significantlyhigh. Comparing Fig.6 with Fig. 8, we can clearly iden-tify the impact of temporal information propagation from theKF versus the case without this propagation. The comparisonconfirms the merit of the KF application for this problem.

It is also noteworthy that for land we have empiricalstate models, which can reproduce the surface temperaturedaily cycle with high accuracy (Gottsche and Olesen, 2009;Menglin and Dickinson, 1999; Menglin, 2000). Thereforethe question may be posed as to whether or not we can im-prove the results by using a more adequate model forTs. Thisexercise has been performed inSerio et al.(2013), where thetemperature daily cycle was modeled with a second-order au-toregressive process. However, no improvement was foundwith respect to a simple persistence model. The fact is thatthe daily cycle is reproduced in its very fine details by thedata, as it is possible to see, e.g., from Fig. 4. Therefore, forthe particular case of retrieving(Ts,�), there is no essentialneed to include the daily cycle information though an exter-nal model. However, it has also to be stressed that this maynot be the case, e.g., for atmospheric parameters where thesatellite infrared observations are less adequate and need tobe assimilated in a system with an accurate dynamical model.

Finally, we stress that for sea surfaces a simple persistencemodel is accurate also for the skin temperature, and there-fore SηTs needs to be downscaled with respect toSTs. WeuseSTs = 1 K

2 and obtainSηTs again by scaling with a factorf = 10, that isSηTs = 0.01 K

2.For the sea-surface emissivity covariance we use Ma-

suda’s model (Masuda et al., 1988). For any single SEVIRI


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Fig. 6. Error analysis for the retrieval exercise shown in Fig. 5. Theprecision of the retrieval (square root of the diagonal of the covari-ance matrix, Ŝt) is shown by the ±1σ tolerance interval.

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)


Fig. 7. Retrieval exercise similar to that shown in Fig.5, but now thevariance of stochastic term for Ts is equal to 0 K2 and f = 10.

Fig. 6.Error analysis for the retrieval exercise shown in Fig.5. Theprecision of the retrieval (square root of the diagonal of the covari-ance matrix,̂St ) is shown by the±1σ tolerance interval.

pixel field of view angle, we generate the emissivity vectorfor wind speed in the range 0–15 m s−1 and with a step of1.5 m s−1. In this way we have 11 emissivity vectors, whichare used to define background vector and covariance. Again,the resulting covariance is downscaled by a factorf = 10.

The validity of the persistence model for sea surface hasbeen checked directly on the basis of real observations, be-cause for sea surface the ECMWF analysis is credited withan accuracy within±1 K. Figure9 exhibits the results for thesea target area shown in Fig.2 and for 31 July 2010. We seethat a stochastic variance term below 0.25 K2 tends to have abetter agreement with the ECMWF model, which leads us toconclude that for sea surface a persistence model is effectivenot only for emissivity, but also forTs.

In passing, we also note from Fig.9 that the skin tem-perature reaches a maximum around 15:00 UTC. The maxi-mum around 15:00 UTC is in agreement withGentemann etal. (2003), who showed that during the daytime, solar heat-ing may lead to the formation of a near-surface diurnal warmlayer, particularly in regions with low wind speeds. Analysis




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(K

)

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Channel at 12 μm

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0

0.02


Δ ε

Channel at 10.8 μm

0 5 10 15 20 25 30 35−0.04−0.02

00.020.04


Δ ε

Channel at 8.7 μm




Fig. 6. Error analysis for the retrieval exercise shown in Fig. 5. Theprecision of the retrieval (square root of the diagonal of the covari-ance matrix, Ŝt) is shown by the ±1σ tolerance interval.

0 5 10 15 20 25 300

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(°C

)

Temperature


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−1

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5


Δ T

(°C

)


Fig. 7. Retrieval exercise similar to that shown in Fig.5, but now thevariance of stochastic term for Ts is equal to 0 K2 and f = 10.

Fig. 7. Retrieval exercise similar to that shown in Fig. 5, but nowthe variance of stochastic term forTs is equal to 0 K2 andf = 10.

of TMI (the Tropical Rainfall Measuring Mission (TRMM)Microwave Imager) and AVHRR skin temperature have re-vealed that the onset of warming begins as early as 08:00and peaks near 15:00 with a magnitude of 2.8◦C during fa-vorable conditions.

3.3.1 Sensitivity to the atmospheric state vector

For the problem of(Ts,�) retrieval, we consider SEVIRI at-mospheric window channels alone, namely channels 4, 6, and7 (see Table1). However, in practice atmospheric windowchannels can have a contribution from the atmospheric pa-rameters(T ,Q,O) which give the major emission contribu-tion between 8 and 12 µm.

In the present scheme, the retrieved state vector includes(Ts,�) alone, whereas the principal atmospheric parametersare obtained from the ECMWF analysis and are not fur-ther iterated within the retrieval scheme. Thus, the question


0 5 10 15 20 25 30 35−5

−4

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−1

0

1

2

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4

5

Time (Hour of the day)

Δ T

(K

)

Temperature


0 5 10 15 20 25 30 35−0.04−0.02

00.020.04


Δ ε

Channel at 12 μm


0 5 10 15 20 25 30 35−0.04−0.02

00.020.04


Δ ε

Channel at 10.8 μm


0 5 10 15 20 25 30 35−0.04−0.02

00.020.04


Δ ε

Channel at 8.7 μm


Fig. 8. Error analysis for the retrieval exercise similar to that shownin Fig.5, but now with the propagation operator M = 0. The vari-ance of the stochastic term for Ts is equal to 1 K2 and f = 10.

0 5 10 15 20 2523

23.5

24

24.5

25

25.5

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27

Times(hour of the day)

Ski

n T

empe

ratu

re (°

C)

Skin Temp. stochastic variance term: 1 K2

Skin Temp. stochastic variance term: 0.25 K2

Skin Temp. stochastic variance term: 0.0 1 K2

ECMWF analysis

Fig. 9. Kalman filter retrieval analysis for skin temperature as afunction of the stochastic variance term for Ts. The retrieval hasbeen spatially averaged over the grid box of size 0.5× 0.5 degreesshown in Fig. 1.

Fig. 8.Error analysis for the retrieval exercise similar to that shownin Fig. 5, but now with the propagation operatorM = 0. The vari-ance of the stochastic term forTs is equal to 1 K2 andf = 10.

arises concerning the potential bias on the retrieved param-eters(Ts,�) resulting from the uncertainty of those not re-trieved, that is(T ,Q,O).

We stress that in our scheme the (non-retrieved) atmo-spheric state vector(T ,Q,O) is obtained from the space-time collocated ECMWF analysis, which, especially for aridregions such as that analyzed in this paper, could be signifi-cantly in error in daytime (Masiello and Serio, 2013).

The assessment of the bias on the retrieved pair(Ts,�)which arises from a non-perfect knowledge of the atmo-spheric state vector, can be performed through a linear per-turbation analysis by dealing with a generic atmospheric pa-rameter, sayX, as a interfering factor.

Within the context of optimal estimation, (e.g.,Rodgers,2000), which (as shown in Sect.3.2.1) applies to any iter-ation step of the Kalman filter methodology, the sensitivityof the retrieved vector,̂v, to a difference1X = X − Xo ofthe given atmospheric parameter,X, with respect to the ref-erence stateXo assumed in the forward model calculationscan be computed according to (Carissimo et al., 2005)




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−2

−1

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1

2

3

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5


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(K

)

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0 5 10 15 20 25 30 35−0.04−0.02

00.020.04


Δ ε

Channel at 12 μm


0 5 10 15 20 25 30 35−0.04−0.02

00.020.04


Δ ε

Channel at 10.8 μm


0 5 10 15 20 25 30 35−0.04−0.02

00.020.04


Δ ε

Channel at 8.7 μm


Fig. 8. Error analysis for the retrieval exercise similar to that shownin Fig.5, but now with the propagation operator M = 0. The vari-ance of the stochastic term for Ts is equal to 1 K2 and f = 10.

0 5 10 15 20 2523

23.5

24

24.5

25

25.5

26

26.5

27

Times(hour of the day)

Ski

n T

empe

ratu

re (°

C)

Skin Temp. stochastic variance term: 1 K2

Skin Temp. stochastic variance term: 0.25 K2

Skin Temp. stochastic variance term: 0.0 1 K2

ECMWF analysis

Fig. 9. Kalman filter retrieval analysis for skin temperature as afunction of the stochastic variance term for Ts. The retrieval hasbeen spatially averaged over the grid box of size 0.5× 0.5 degreesshown in Fig. 1.

Fig. 9. Kalman filter retrieval analysis for skin temperature as afunction of the stochastic variance term forTs. The retrieval hasbeen spatially averaged over the grid box of size 0.5× 0.5 degreesshown in Fig.1.

1v̂ =(KT S−1ε K + S

−1a

)−1KT S−1ε KX1X, (29)

whereK is the Jacobian matrix of the retrieved vector andKX is the Jacobian matrix of the interfering factor computedat the reference stateXo.

Equation (29) can be used to check the impact of possi-ble biases in the ECMWF analysis on the retrieval for sur-face emissivity and temperature. To obtain realistic situa-tions we have used a couple of day–night IASI spectra (seeFig. 10) recorded on 10 July 2010 over the Sahara desertat two close locations which are included in the target areashown in Fig.2. These two IASI spectra have been invertedfor (Ts,�,T ,Q,O) using the so-calledϕ-IASI package (Am-ato et al., 1995; Masiello and Serio, 2004; Carissimo et al.,2005; Grieco et al., 2007; Masiello et al., 2009; Masiello andSerio, 2013). The IASI retrieved atmospheric state vector iscompared to the ECMWF reference state vector in Fig.10.We see that large differences arise in daytime, mostly con-cerning the lower troposphere. For nighttime we have a goodagreement for the surface temperature (303.9 K of ECMWFand 303 K of IASI), whereas for daytime we have a disagree-ment which is as large as 12 K (310.1 of ECMWF and 321.7of IASI).

We can take the difference,XIASI −XECMWF, as a realisticdeparture of the ECMWF analysis from thetrueatmosphericstate vector and compute, through Eq. (29), the resulting biasover the retrieved surface emissivity and temperature. In do-ing so, we have usedSa defined according to

Sa =(

Se, 00, 1.K2

), (30)

with Se obtained from the UW/BFEMIS database. It shouldbe stressed thatSa defined in Eq. (30) gives the less favorablesituation. In fact, as iterations evolve, the matrixSa evolvesas well according to Eq. (16) and its norm tends to decrease.In this situation, it can be shown (Carissimo et al., 2005) thatthe interfering effect also tends to decrease. Thus the calcu-lations we are going to show should be viewed as an upperboundary to the impact of interfering atmospheric factors.

With this in mind, Table3 shows the impact over the re-trieval of the interfering factors,(T ,Q,O). It can be seenthat even in this least favorable case, the impact is modestand much lower than the precision of the retrieval. As ex-pected, the impact is larger during daytime, although mostlyaffecting the second decimal digit for skin temperature andbelow the fourth decimal digit for emissivity.

Based on this result, we have implemented the(Ts,�)-version of the Kalman filter methodology by consideringthe simultaneous use of channels 4, 6, and 7. The retrievedstate vectors are constructed from surface temperature andemissivity alone. We do consider atmospheric parametersin the state vector. These come from space-time collocatedECMWF analysis; however, they are not retrieved.

It is worth mentioning that the conclusion reached in thissection applies to our retrieval scheme and does not have ageneral validity. The impact of possible interfering factorsdepends on the regularization determined by the matrixSa ,and tends to attain its largest value in the limitS−1a → 0(Carissimo et al., 2005), that is, for the case of unconstrainedleast squares.

4 Results

4.1 Assessing the performance of theemissivity/temperature retrieval

We begin with the description of results obtained by process-ing SEVIRI data at the three small test areas shown in Fig.2.With this first series of results, we want to address issues suchas the precision of the method and its convergence properties.Results obtained from the analysis of the full target area willbe shown later in this section.

The Kalman filter has been applied to SEVIRI atmo-spheric channels alone. These are the channels at 12, 10.8and 8.7 µm. The corresponding channel emissivity and thesurface temperature have been simultaneously retrieved witha time resolution of 15 min. This time step also coincideswith the sequential updating rate of the filter.

For the sake of clearness, Table4 summarizes the manysettings of the filter. Note that in computing background vec-tor and the related covariance matrix from the UW/BFEMISdatabase, we have not used the data for July 2010, which areused for comparison with our results.

Figure 11 allows us to exemplify the precision of themethodology. The retrieval has been obtained for one single




1000 1500 2000 2500 30000

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0.16


Rad

ianc

e S

pect

rum

(r.

u,)

a)

(32.90° N, 7.50° E); UTC: 10−Jul−2010 09:35:37 (30.90° N, 5.80° E); UTC: 10−Jul−2010 20:52:43

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ssur

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Daytime IASIDaytime ECMWFNighttime IASINighttime ECMWF

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x 10−5

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O3 mixing ratio (ppv)

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ssur

e (h

Pa)

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H2O mixing ratio (g/Kg)

Pre

ssur

e (h

Pa)

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Daytime ECMWFNighttime ECMWFDaytime IASINighttime IASI

Fig. 10. a) Day-Night pair of IASI observations over a Sahara desert site (1 r.u.=1 W m−2 sr−1 (cm−1)−1); b) temperature retrieval andcomparison with the space-time collocated ECMWF analysis; c) H2O retrieval and comparison with the space-time collocated ECMWFanalysis; d) O3 retrieval and comparison with the space-time collocated ECMWF analysis.

Fig. 10. (a)Day–night pair of IASI observations over a Sahara desert site (1 r.u. = 1 W m−2 sr−1 (cm−1)−1); (b) temperature retrieval andcomparison with the space-time collocated ECMWF analysis;(c) H2O retrieval and comparison with the space-time collocated ECMWFanalysis;(d) O3 retrieval and comparison with the space-time collocated ECMWF analysis.

day and one single SEVIRI pixel from the Sahara desert testarea, and therefore corresponds to the highest space-time res-olution of the methodology. It is possible to see that evenfor a time resolution of 15 min, temperature is obtained witha precision of≈ ±0.2 K and better, whereas emissivity isobtained with a precision better than≈ ±0.005. The emis-sivity retrieval shown in Fig.11 corresponds to the channelat 8.7 µm. This channel is in the middle of the quartz rest-strahlen band and has the higher contrast in the atmosphericwindow. From Fig.11, we see that the emissivity tends to fol-low the daily cycle, with larger values obtained during night-time/early morning.

This is better evidenced from the analysis of the long se-quence of clear-sky days shown in Figs.12 and 13. Theanalysis refers to a single SEVIRI pixel in the Sahara desert(30.66◦ N, 5.56◦ E) and has been performed with a time res-olution of 15 min. According toLi et al. (2012), we have thatthe amplitude of the daily cycle is quite evident for the SE-VIRI channel at 8.7 µm where peak-to-peak variations canreach≈ 0.03. Smaller variations are found at 10.8 µm (below0.01). At 12 µm the variation is much less pronounced and insome cases it seems to have a reverse sign with respect to thepattern at 8.7 µm, an effect which has been reported also byLi et al. (2012).

These daily variations of emissivity over desert sand, evenin the dry (non-rainy) season, have been first reported byLi etal. (2012). It is very likely that these variations are the result

Table 3. Potential bias affecting the retrieval of surface emissivityand temperature due to atmospheric parameters. The bias is dimen-sionless for emissivity and in K for the surface temperature.

Interfering atmospheric parameter

Retrieved parameter Temp. profile H2O profile Ozone profile

Day

Emissivity at 12 µm 0.0000 0.0000 0.0000Emissivity at 10.8 µm 0.0000 0.0000 0.0000Emissivity at 8.7 µm 0.0000 0.0001 0.0000Surface temperature 0.0013 0.0021 0.0004

Night

Emissivity at 12 µm 0.0000 0.0000 0.0000Emissivity at 10.8 µm 0.0000 0.0000 0.0000Emissivity at 8.7 µm 0.0000 −0.0001 0.0000Surface temperature 0.0014 −0.0014 0.0003

of day–night sand evotranspiration, which occurs for directadsorption of water vapor at the surface (Agam and Berliner,2004, 2006; Mira et al., 2007). Clear-sky daily variation ofemissivity is more pronounced for desert sand because of thestrong contrast of quartz absorption band at 8.6 µm.

However, it should be noted that the a posteriori covari-ance matrix of the analysis,Ŝt (see Eq.11) shows a relativelystrong anticorrelation of the retrievedTs and�. This anticor-relation could introduce a systematic drift in the retrieved



Table 4.Summary of the settings for the KF scheme.

Element Setting/Reference

Emissivity model equation PersistenceSurface temp. model equation PersistenceEmissivity true values UnknownEmissivity initial background vector (at time=0) Average from UW/BFEMIS database, over the years 2003–2012, but not 2010Emissivity initial backgroundSe (at time=0) from UW/BFEMIS database, 2003–2012 years, but not 2010Emissivity stochastic covariance,Sηe as line above scaled down withf = 10Surface temp. true values UnknownSurface temp. initial value (at time = 0) ECMWF analysis at 00:00 hSurface temp. initial background,STs 1 K

2

Surface temp. stochastic varianceSηTs As STs for land,STs/f2, with f = 10 for sea surface

Observational covariance matrix,Sε Diagonal, from SEVIRI radiometric noiseAtmospheric profiles Assumed known, space-time collocated ECMWF analysisConvergence criterion Cost function (χ2 = 2S ≤ χ2th)


0 5 10 15 20 250.73

0.74

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0.79

0.8


Em

issi

vity

at 8

.7 μ

m

0 5 10 15 20 2520

25

30

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55

60


Ts

(°C

)

Fig. 11. Top: emissivity retrieval (channel at 8.7 µm) for one dayand one single SEVIRI pixel over the Sahara desert; bottom panel:same as top, but for temperature. Error bars are the square root ofthe corresponding diagonal elements of the covariance matrix, Ŝt .

0 50 100 150 200 25020

40

60

80

Time (hour of the day, zero at midnight of July the first)

Ts

(°C

)

0 50 100 150 200 2500.1

0.15

0.2


Rad

ianc

e (r

.u.)

Calc.Obs.

RetrievalECMWF T

s

ECMWF Ts analysis

Fig. 12. Retrieved surface temperature (bottom panel) for a site inthe Sahara desert. The retrieval has been obtained with the Kalmanfilter for ten consecutive days. In the legend, ECMWF Ts analy-sis refers to the surface temperature analysis at the canonical hourswithin a day, whereas ECMWF Ts is the ECMWF surface tempera-ture linearly extrapolated to the SEVIRI time steps. The upper panelin the figure also shows the quality of the reconstructed radiance(channel at 12 µm). 1 r.u.=1 W m−2 sr−1 (cm−1)−1Fig. 11. Upper panel: emissivity retrieval (channel at 8.7 µm) for

one day and one single SEVIRI pixel over the Sahara desert; bottompanel: same as top, but for temperature. Error bars are the squareroot of the corresponding diagonal elements of the covariance ma-trix, Ŝt .

parameters, and therefore could potentially be a spuriouscause of the diurnal variation seen in emissivity. Never-theless, based on our present retrieval exercises with realand simulated observations, it seems that a persistence statemodel for emissivity is capable to attenuate cyclic artefactsin the retrieval (as an example, see the simulation providedin Figs.5 and6). Moreover, for non-arid lands we have ob-served situations in which the day–night emissivity variation,despite the anticorrelation ofTs and�, is in phase with thedaily temperature cycle (e.g., see Sect.4.2), that is, the re-verse of the situation we have observed for conditions overdesert sand. Furthermore, IASI observations (Masiello et al.,2013) confirm that the daily variation of emissivity is a gen-uine feature in the data. Finally,Hulley et al. (2010) hasshown that emissivity retrieval from satellite observations issensitive to changes in soil moisture.

Figures11 to 13 are meant to exemplify that our physicalscheme is sensitive to these day–night emissivity variations.An in-depth assessment of this effect is ongoing. As alreadystressed in section1, the present study mostly focuses on thenovel aspect of the methodology and a comparison of its re-sults with in situ data and other similar satellite products.

Bearing this in mind, we go back to Fig.12, from which itis possible to see that a slight cloudiness affects the observa-tions at the beginning of the second day. We do not skip theseobservations when performing the retrieval, therefore Fig.12shows that slight cloudiness does not bring the Kalman filterto an unstable state. In other words, the stability of the filteris not influenced by slight cloudiness, although this informa-tion is forward-propagated through the forecast.

However, overcast conditions that persist a long time (e.g.,from t1 to t2, with t2 − t1 � 15 min) can (e.g., because ofrain) drastically change the emissivity at the endpointst1and t2. Furthermore, cloudy radiances could be undetected,in which case serious overcast conditions could negativelyinfluence the retrieval products. To avoid this effect, cloudyradiances (if detected) are just skipped within the KF scheme




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0.74

0.75

0.76

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0.8


Em

issi

vity

at 8

.7 μ

m

0 5 10 15 20 2520

25

30

35

40

45

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55

60


Ts

(°C

)

Fig. 11. Top: emissivity retrieval (channel at 8.7 µm) for one dayand one single SEVIRI pixel over the Sahara desert; bottom panel:same as top, but for temperature. Error bars are the square root ofthe corresponding diagonal elements of the covariance matrix, Ŝt .

0 50 100 150 200 25020

40

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80


Ts

(°C

)

0 50 100 150 200 2500.1

0.15

0.2


Rad

ianc

e (r

.u.)

Calc.Obs.

RetrievalECMWF T

s

ECMWF Ts analysis

Fig. 12. Retrieved surface temperature (bottom panel) for a site inthe Sahara desert. The retrieval has been obtained with the Kalmanfilter for ten consecutive days. In the legend, ECMWF Ts analy-sis refers to the surface temperature analysis at the canonical hourswithin a day, whereas ECMWF Ts is the ECMWF surface tempera-ture linearly extrapolated to the SEVIRI time steps. The upper panelin the figure also shows the quality of the reconstructed radiance(channel at 12 µm). 1 r.u.=1 W m−2 sr−1 (cm−1)−1

Fig. 12.Retrieved surface temperature (bottom panel) for a site inthe Sahara desert. The retrieval has been obtained with the Kalmanfilter for ten consecutive days. In the legend, ECMWFTs analy-sis refers to the surface temperature analysis at the canonical hourswithin a day, whereas ECMWFTs is the ECMWF surface tempera-ture linearly extrapolated to the SEVIRI time steps. The upper panelin the figure also shows the quality of the reconstructed radiance(channel at 12 µm). 1 r.u. = 1 W m−2 sr−1 (cm−1)−1.

and, furthermore, only retrieval which satisfies the cost func-tion condition of Eq. (13) are propagated through the filter.To this end, it should be stressed that KF does not need todeal with equally spaced times.

This is exemplified in Fig.14, which shows the retrievalfor surface temperature corresponding to whole month ofJuly for the Seville test area. The analysis has been per-formed only for clear-sky soundings (according to the oper-ational SEVIRI cloud mask) and has been spatially averagedover the 187 available SEVIRI pixels. Cloudy radian

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