Error sources on the land surface temperature retrieved from thermal infrared single channel remote sensing data

Transcription

1 Internationa Journa of Remote Sensing Vo. 27, No. 5, 10 March 2006, Error sources on the and surface temperature retrieved from therma infrared singe channe remote sensing data J. C. JIMÉNEZ-MUÑOZ and J. A. SOBRINO* Goba Change Unit, Department of Thermodynamics, Facuty of Physics, University of Vaencia, Dr. Moiner 50, 46100, Burjassot, Spain (Received 30 Apri 2004; in fina form 27 October 2004 ) In this paper, a theoretica study compementary to others given in the iterature about the errors committed on the and surface temperature retrieved from the radiative transfer equation in the therma infrared region by remote sensing techniques has been anaysed. For this purpose, the MODTRAN 3.5 code has been used in order to simuate different conditions and evauate the infuence of severa parameters on the and surface temperature accuracy: atmospheric correction, noise of the sensor, and surface emissivity, aerosos and other gaseous absorbers, anguar effects, waveength uncertainty, fu-width hafmaximum of the sensor and band-pass effects. The resuts show that the most important error source is due to atmospheric effects, which eads to an error on surface temperature between 0.2 K and 0.7 K, and and surface emissivity uncertainty, which eads to an error on surface temperature between 0.2 and 0.4 K. Hence, assuming typica uncertainties for remote sensing measurements, a tota error for and surface temperature between 0.3 K and 0.8 K has been found, so it is difficut to achieve an accuracy ower than these vaues uness more accurate in situ vaues for emissivity and atmospheric parameters are avaiabe. 1. Introduction The importance of and surface temperature for environmenta studies has been pointed out by different authors (Barton 1992, Lagouarde et a. 1995, Qin and Karniei 1999, Dash et a. 2002, Schmugge et a. 2002, etc.), so it is needed for water and energy budgets at the soi/atmosphere interface and evapotranspiration estimations among others. In order to retrieve the and surface temperature (hereinafter referred to as T s or LST) from remote sensing data in the therma infrared region, different techniques and agorithms have been appied, namey, singe-channe equations, two-channe or spit-window agorithms, dua-ange agorithms, etc. Most of them are based on approximations of the radiative transfer equation (RTE), which can be written in the therma infrared region as L at-sensor n ~ e B ðt s Þzð1{e ÞL atm; o t zl atm: where e is the surface emissivity, B(, T s ) is the radiance emitted by a backbody at temperature T s, L atmq is the downweing atmospheric radiance, in which the diffusive approximation for the downweing atmospheric fux F5p L atmq has been considered, t is the tota transmission of the atmosphere (transmissivity) and ð1þ *Corresponding author. Emai: sobrino@uv.es Internationa Journa of Remote Sensing ISSN print/issn onine # 2006 Tayor & Francis DOI: /

2 1000 J. C. Jiménez-Muñoz and J. A. Sobrino L atmq is the upweing atmospheric radiance. A these magnitudes aso depend on the observation ange. However, it is aso possibe to obtain T s directy from equation (1) by correcting the atmospheric effects and aso the emissivity effect. Then, T s can be cacuated by inversion of the Panck s aw. The main goa of this paper is to obtain a quantitative estimation of the error associated with the T s retrieved from the RTE (equation (1)) in order to contribute to other theoretica and experimenta studies given in the iterature, which provide errors committed on LST retrieved from other techniques (see, for exampe, Becker and Li 1990a, b, 1995, Li and Becker 1993, Prata 1993, 1994, Sobrino et a. 1993, 1994, 1996, Wan and Li 1997, etc.). 2. Sensitivity anaysis The purpose of this section is to carry out a sensitivity anaysis of the RTE and its infuence on the T s. From equation (1), it is cear to notice that T s depend on the atmospheric parameters (atmospheric transmissivity, t, upweing atmospheric radiance or path radiance, L atmq, and downweing atmospheric radiance, L atmq ), the at-sensor radiance (L at-sensor ) and the and surface emissivity (e ). The error on the T s due to the uncertainties of these parameters is anaysed in the foowing sections. For this purpose, the MODTRAN 3.5 (Abreu and Anderson 1996) code has been used in order to simuate different conditions and evauate the infuence of the parameters above-mentioned on the LST accuracy. In order to focus the study, the midatitude summer (MLS) atmosphere has been chosen from the MODTRAN standard atmospheres. In figure 1 the transmissivity spectrum for the MLS atmosphere is represented. The main atmospheric absorbers are aso indicated. The foowing we-known regions can be observed: in the range 8 9 mm an increasing tendency with strong absorptions is observed. The absorption for this range is due mainy to the atmospheric water vapour content and secondy to other atmospheric absorbers as N 2 O, CO + 2 and CH 4. In the range 9 10 mm a strong absorption is observed. This absorption is mainy due to the ozone and it is Figure 1. Transmissivity spectrum for the midatitude summer atmosphere.

3 centred more or ess at 9.5 mm. In the range mm a sighty descending tendency is observed, but high atmospheric transmissivity vaues are obtained. The absorption peaks are weaker than the ones observed in the 8 9 mm, so this region is the most used in therma infrared remote sensing and it is caed atmospheric window. In the region mm the transmissivity vaues are ower than the ones obtained in mm, and the main absorption is due to the atmospheric water vapour content. Finay, in the range mm, a strong absorption due to the CO 2 is observed. In this region the atmosphere is practicay opaque to the therma radiance. It shoud be noted that the resuts shown in the next sections have been spectray obtained for the atmospheric window region, between 10 and 12 mm (in steps of 1 cm 21 ), and then a mean vaue with the standard deviation and the r.m.s. deviation has been cacuated. This option has been chosen instead of using a particuar fiter function in order to obtain genera resuts and not particuarized for any therma sensor. Anyway, a mean vaue is equivaent to a square fiter function from 10 to 12 mm. Uness otherwise stated, the MODTRAN 3.5 code has been executed in therma radiance mode for a MLS atmosphere, in cear-sky conditions, with a view ange of nadir, considering an emissivity vaue of 0.98 and a surface temperature of 300 K. 2.1 Atmospheric correction In order to obtain accurate and surface temperature vaues, atmospheric effects must be removed. The parameters invoved in the atmospheric correction are the atmospheric transmissivity (t ), the upweing atmospheric radiance or path radiance (L atmq ) and the downweing atmospheric radiance (L atmq ). These parameters are correated and they depend mainy on the atmospheric water vapour content (w). Hence, when the atmospheric water vapour increases, the atmospheric transmissivity decreases and the upweing and downweing radiances increase. So in order to evauate the error committed on T s due to atmospheric effects, e w (T s ), the foowing equation has been considered: e w ðt s Þ~ LT s Lw Error sources on and surface temperature 1001 ew ð Þ~ LB ðt s Þ LT s Lw LB ðt s Þ ew ð Þ:d B w dt B ew ð Þ where e(w) is the error on water vapour. Assuming that the atmospheric parameters depend on the water vapour content, the term D B w can be obtained according to the foowing equation: d B w : LB " # ðt s Þ Lw ~ LB ðt s ÞLt Lt Lw z LB ðt s ÞLL atm: Lw z LB ðt s ÞLL atm; Lw LL atm: LL atm; whereas the term d T B is obtained by deriving the Panck s function and is given by: d T B : LT s LB ðt s Þ ~ c 2 c 1 c 1 B ðt s ÞzB 2 ð T n 2 c {1 1 sþ B ðt s Þ z1 ð4þ with c ? and c In the previous equation radiance is given in watts m 22 mm 21 sr 21, temperature in K and waveength in mm. In order to appy the equation (3), the expression for B (T s ) as a function of the atmospheric parameters is needed, which can be easiy obtained from equation (1), and aso the dependence of the atmospheric parameters with the atmospheric water vapour ð2þ ð3þ

4 1002 J. C. Jiménez-Muñoz and J. A. Sobrino content. For this purpose, a simuation using MODTRAN 3.5 and a set of 60 radiosoundings extracted from the TIROS Operationa Vertica Sounder (TOVS) Initia Guess Retrieva (TIGR) database (Scott and Chedin 1981) has been carried out. The simuation procedure performed for the mm square fiter shows the foowing resuts: StT 10{12mm ~{0:1514 wz1:028ðr~0:996, s~0:02þ ð5aþ SL atm: T 10{12mm ~0:7194 w 1:358 ðr~0:997, s~0:1þ ð5bþ SL atm; T 10{12mm ~1:161 w 1:228 ðr~0:996, s~0:1þ ð5cþ where r is the correation coefficient and s the standard error of estimate. Hence, from equation (3) it is easy to obtain the fina expression for d B w : d B w : LB ðt s Þ Lw ~ {0:1514 L atm: {L at-sensor e t 2 { 0:9769w0:358 z 1{ 1 ð6þ 1:4257w 0:228 e t e Finay, from equations (2), (4) and (6) it is possibe to obtain the vaues shown in figure 2, in which the ratio between the error on T s, e(t s ), and the error on water vapour, e(w), versus the water vapour is graphed. It shoud be noted that greater errors are committed for ow and high atmospheric water vapour contents, with a minimum error ocated at 3 g cm 22. Hence, assuming a typica water vapour uncertainty of 0.5 g cm 22 (Sobrino et a. 2002), an error on T s of 2.6 K, 1.1 K, 0.3 K and 1 K is obtained when the tota atmospheric water vapour content is 1 g cm 22, Figure 2. Ratio between the error on T s and the error on atmospheric water vapour versus the atmospheric water vapour.

5 Error sources on and surface temperature gcm 22,3gcm 22 and 4 g cm 22, respectivey. Taking into account the standard atmospheric water vapour content for a MLS atmosphere, 2.36 g cm 22, an error on T s of 0.7 K is obtained. When w is measured in situ ow errors can be obtained, as for exampe 0.15 g cm 22 (Esteés, persona communication 2004). In this case, the previous errors are reduced to 0.8 K, 0.3 K, 0.08 K and 0.3 K, respectivey, and for the MLS atmosphere an error of 0.2 K is obtained. Athough the atmospheric parameters are correated and depend on w, sometimes it is interesting to anayse the effect of each parameter independenty, as for exampe when in situ measurements with fied radiometers are made. In this case, the atsurface radiance given by the term e B (T s ) + (12e )L atmq is measured, so ony the downweing radiance must be known. The error on T s due to the uncertainty of the L atmq can be obtained from the foowing equation: e Latm; ðt s Þ~d B Latm; dt B e Latm; ð7þ where d B Latm; : LB ðt s Þ ~ 1{ 1 e LL atm; ð8þ Figure 3 shows the error on T s depending on the and surface emissivity vaue considered. For emissivity vaues between 0.8 and 1, the error on T s is ess than 0.1 K when the uncertainty on the downward radiance is 1% and ess than 0.7 K when the uncertainty is 10%. Moreover, the error on T s aso depends on the surface temperature vaue considered. Anyway, this dependence is negigibe. As an exampe and assuming an uncertainty of 10% on the downward radiance, the error on T s changes from 0.07 K to 0.04 K when the T s vaue changes from 273 K to 373 K (assuming in this case an emissivity of 0.98). In order to obtain a goba view for both dependences, figure 4 shows the error on T s depending on the emissivity and Figure 3. Error on and surface temperature (K) due to the uncertainty on the downward radiance depending on the and surface emissivity vaue considered.

6 1004 J. C. Jiménez-Muñoz and J. A. Sobrino Figure 4. Error on T s and error on downward radiance ratio depending on the and surface emissivity and temperature vaues. A fixed waveength of 11 mm has been considered. temperature vaues for a fixed waveength of 11 mm. The smaer the emissivity and temperature, the greater the temperature retrieva error. 2.2 Noise equivaent deta error The radiance measured by a sensor onboard a sateite is affected by an inherent uncertainty due to eectronic devices invoved in the construction of the sensor. The error on T s due to the at-sensor radiance uncertainty is given by where e Lsensor ðt s Þ~d B Lsensor dt B elat-sensor d B Lsensor : LB ðt s Þ LL at-sensor ~ 1 e t ð9þ ð10þ The resuts obtained with these equations show an error on T s of 0.1 K, 1 K and 10 K for at-sensor radiance uncertainties of 0.1%, 1% and 10%, respectivey. Equation (4) can be used to estimate the equivaence between an uncertainty on radiance and an uncertainty on temperature. As an exampe, an uncertainty of 0.1 K in temperature eads to an error of 0.15% on radiance, whereas an uncertainty of 0.3 K in temperature eads to an error of 0.44% on radiance when the waveength is 11 mm and the surface temperature is 300 K. Most of the therma sensors have noise errors between 0.1 K and 0.3 K, as AVHRR (Advanced Very High Resoution Radiometer), ASTER (Advanced Spaceborne Therma Emission and Refection radiometer), etc., so it is expected to have uncertainties on the at-sensor radiances

7 between 0.1% and 0.5%. According to these resuts, the error on T s due to the NEDT is comprised between 0.1 K and 0.5 K. This error is practicay negigibe for therma sensors with very ow noise errors, as the AATSR (Advanced Aong Track Scanning Radiometer), with a NEDT of 0.05 K. 2.3 Land surface emissivity Error sources on and surface temperature 1005 The infuence of the and surface emissivity uncertainty on T s can be estimated as in the previous cases according to e e ðt s Þ~d B e dt B e ð e Þ ð11þ where d B e : LB "! # ðt s Þ Le ~ 1 L atm: {L at-sensor e 2 zl atm; t ð12þ Here, errors on T s of 0.04 K, 0.4 K and 4 K are obtained when the uncertainty on the and surface emissivity is 0.1%, 1% and 10%, respectivey. The and surface emissivity is the main error source on the LST retrieva (excuding externa factors as a worse caibration of the sensor). Land surface emissivity is typicay retrieved from remote sensing data with an accuracy of 1% (Giespie et a. 1998, Sobrino et a. 2002, etc.), which eads to an error of 0.4 K on the LST retrieva. Emissivity can be measured in situ with an accuracy of 0.5% (Sobrino and Casees 1993, Nerry et a. 1998), which reduces the error on LST to 0.2 K. It shoud be noted that the error on T s depends on the emissivity vaue chosen (and aso on the LST vaue). The resuts mentioned have been obtained assuming an emissivity vaue of 0.98 and T s 5300 K. Figure 5 shows the error on T s for different emissivity vaues when the uncertainty on emissivity is 0.01 and T s 5300 K. Most natura surfaces have emissivity vaues Figure 5. Error on and surface temperature (T s ) depending on the and surface emissivity vaue. An uncertainty on the emissivity of 0.01 and a vaue of 300 K for and surface temperature have been assumed.

8 1006 J. C. Jiménez-Muñoz and J. A. Sobrino between 0.8 and 1 in the region mm, so errors on T s between 0.4 and 0.5 K wi be expected independenty on the emissivity vaue. In this case, the error dependence on the surface temperature has been aso anaysed. Hence, when T s changes from 273 K to 373 K, the error changes between 0.2 K and 0.9 K assuming a fixed emissivity vaue of Both dependences, on temperature and emissivity, are given in figure 6. The T s error rises with increasing surface temperature and emissivity (for a fixed waveength of 11 mm). 3. Effects of aerosos and other gaseous absorbers Athough to retrieve the LST it is usua to assume cear-sky conditions and no aerosos attenuation, the aerosos effect in the therma infrared region is not aways negigibe. Tabe 1 shows the differences between atmospheric transmissivity when the MODTRAN standard aerosos modes are considered with respect to an Figure 6. Error on and surface temperature (T s ) depending on the and surface emissivity and temperature vaues. Tabe 1. Differences between atmospheric transmissivity without incuding the aerosos effect and considering different types of aerosos extinction in the region mm. Aerosos mode t aeroso 2 t no-aeroso (%) Rura extinction, visibiity523 km 1.9 Rura extinction, visibiity55 km 8.1 Navy maritime extinction 0.9 Maritime extinction, visibiity523 km 2.4 Urban extinction, visibiity55 km 9.1 Tropospheric extinction, visibiity550 km 0.2 Radiative fog extinction, visibiity50.5 km 90.7 Desert extinction 0.4

9 atmosphere without aerosos content. The owest difference in transmissivity is 0.2%, which corresponds to the tropospheric extinction with a defaut visibiity of 50 km. In this case the aerosos effect is negigibe and correction is not needed. However, there are great differences between the transmissivity spectrum for an atmosphere with no aerosos content and an atmosphere with fog extinction and a defaut visibiity of 0.5 km. It is cear that in this case the acquisition of a sateite image for remote sensing studies has no sense. In addition to the aerosos effect, other gaseous absorbers aso have an infuence on the LST retrieved. As is we known, in the therma infrared region the main atmospheric absorber is the water vapour. The CO 2 and CH 4 are secondary absorbers, and their variation has not a reevant importance on the LST vaue. As an exampe, et us consider the variation on the CO 2 concentration. The MODTRAN 3.5 code estabishes a defaut vaue of CO 2 of 330 ppmv. However, vaues recommended in 1995 are ppmv. Tabe 2 shows the vaues for the atmospheric parameters and LST when the CO 2 concentration varies from 330 ppmv to 380 ppmv. When the LST vaues obtained are anaysed, an increase of around K is observed when the CO 2 concentration increases 5 ppmv. A tota difference of 0.04 K on LST is obtained when the CO 2 concentration in changed from 330 ppmv to 380 ppmv, so this effect on LST is negigibe. 4. Anguar effects Error sources on and surface temperature 1007 LST retrieved assuming at-nadir view ange is not aways accurate enough, so most sensors have fied of views (FOV) higher than 50u (for exampe, AVHRR and AATSR). In order to anayse the anguar effects, simuations with the MODTRAN 3.5 code have been carried out for seven view anges, from 0u to 60u by steps of 10u. As an exampe, figure 7 iustrates the atmospheric transmissivity differences in % between the vaues obtained for these anges and the ones obtained for a nadir view. The graph shows differences ower than 1% for a view ange of 10u and differences higher than 2% for a view ange of 20u, whereas differences higher than 30% are obtained for view anges of 60u. In order to anayse the error committed on T s when at-nadir view is assumed aong a the FOV of the sensor, the effect of the view ange on water vapour vaues wi be studied. As is we known, the reation between the atmospheric water vapour at certain view ange (h) and the one at nadir view (0u) is Tabe 2. Effect of the CO 2 concentration on the atmospheric parameters and on the retrieved and surface temperature from equation (1) at 11 mm. CO 2 (ppmv) t L atmq (W m 22 sr 21 mm 21 ) L atmq (W m 22 sr 21 mm 21 ) T s (K)

10 1008 J. C. Jiménez-Muñoz and J. A. Sobrino Figure 7. Anguar effect on transmissivity (10 12 mm). given by: wðhþ~ w ð 00 Þ cosh [Dw~wð00Þ 1 cosh {1 where Dw is the difference between w(h) and w(0u). Assuming the difference Dw as the error on w, e(w), due to anguar effects and taking into account equations (2), (4) and (6), it is possibe to obtain the foowing equation: et ð s Þ~kwð0 0 Þ 1 cosh {1 ð14þ where e(t s ) is the error on T s due to anguar effects and k is a magnitude which depends on severa parameters as waveength, temperature, emissivity, etc. In fact, k is given by d B w dt B (see 3.1). For the conditions considered in the paper (MLS atmosphere, T s 5300 K, e50.98 and mm), a vaue of k is obtained. The resuts obtained according to equation (14) are represented in figure 8, in which a typica vaue of w52.36 g cm 22 for an MLS atmosphere has been considered. This figure shows that for view anges ower than 25u, the error on T s is ower than 1 K. As an exampe, for a view ange of 55u, the error on T s is higher than 7 K. For the moment, homogeneous and fat surfaces have been considered. However, anguar effects can be aso noticed for heterogeneous and rough surfaces pixes (Sobrino et a. 1990), which is a more compex situation and wi not be treated in this paper. It shoud be mentioned that the errors due to the and surface emissivity anguar dependence must be aso taken into account. Most natura surfaces show emissivity differences between nadir and certain ange view higher or equa to 0.01 for view anges higher than 30u (Sobrino and Cuenca 1999). These differences wi ead to errors on T s equa to or higher than 0.4 K. As an exampe, for a view ange of 55u the emissivity difference with respect to the nadir view for water is 0.04, which eads to an error on T s higher than 1 K (see 3.3). ð13þ

11 Error sources on and surface temperature 1009 Figure 8. Error on and surface temperature (T s ) when anguar effects are not considered, i.e. when at-nadir vaues are assumed for the whoe fied of view (FOV) of the sensor, versus the view ange. 5. Waveength uncertainty In order to obtain LST from the radiative transfer equation, it is necessary to invert the Panck s aw. For this purpose, a vaue of waveength is required. The goa of this section is to anayse the error on LST obtained with the Panck s aw due to the waveength uncertainty. This error can be cacuated using the foowing equations: where d T is given by e ðt s Þ~d T eðþ d T : LT s L ~ g 1g 2 {g 3 g 4 g 2 2 ð15þ ð16þ and the functions g i (i51,4) are given by g 1 :{ c 2 2 c 1 g 2 :n 5 z1 B ð17aþ ð17bþ g 3 : {5c 1 c 1 zb 6 g 4 : c 2 ð17cþ ð17dþ

12 1010 J. C. Jiménez-Muñoz and J. A. Sobrino with B the radiance emitted by a backbody at temperature T s, c ?10 8 W mm 4 m 22 sr 21 and c mm K( and T s are given in mm and K, respectivey). Assuming an uncertainty on the waveength of 0.1 mm and different surface temperatures, the resuts shown in figure 9 are obtained. The foowing ideas can be extracted from this graph. i) The error on T s decreases when the waveength increases unti a certain infexion point, from which the error on T s increases when the waveength increases. This infexion point is 10.6 mm, 9.7 mm and 9.0 mm for a T s vaue of 273 K, 300 K and 323 K, respectivey. Therefore, the waveength vaue of this infexion point decreases when the T s increases. ii) iii) At the infexion point, the error on T s is negigibe. As an exampe, at 11 mm a waveength uncertainty of 0.1 mm produces an error on T s of 0.09 K, 0.4 K and 0.6 K when T s is 273 K, 300 K and 323 K, respectivey. These resuts iustrate the importance of choosing an appropriate waveength in order to invert the Panck s aw. Normay, there are two options: (1) the centra waveength or (2) the effective waveength, according to the channe fiter function of the sensor. These two vaues of waveength can differ significanty, so errors on T s higher than a haf of a degree can be committed. 6. Bandpass and FWHM effects The sensor onboard a sateite measures with finite banded radiometers having a characteristic response function. This fact introduces an error on the LST retrieved. If B (T s ) is the radiance emitted by a backbody at temperature T s and f() is the response function or fiter functions of the sensor, the radiance measured by the sensor,b (T s ). sensor is given by Figure 9. Error on and surface temperature due to an uncertainty of 0.1 mm on the waveength.

13 Error sources on and surface temperature 1011 Ð B ðt s Þf ðþd SB ðt s ÞT sensor ~ Ð f ðþd ð18þ T s is obtained from,b (T s ). sensor by inversion of the Panck s aw using the effective waveength cacuated with the foowing expression: Ð f ðþd effective ~ ð19þ f ðþd To anayse the error committed on this process, et us consider a backbody at a temperature of 300 K and an idea fiter function simiar to a Gaussian with a fuwidth haf-maximum (FWHM) vaue of 1 mm and centred at 11 mm (Jiménez-Muñoz and Sobrino 2003). When the radiance measured by the radiometer is cacuated using equation (18) and T s is retrieved using the effective waveength given by equation (19), a vaue of K is obtained. This resut shows a difference of 0.15 K with regard to the rea vaue of T s (300 K). This difference depends on the T s vaue. So, if we choose a vaue of 273 K and 323 K for T s, a difference of 0.17 K and 0.12 K is obtained, respectivey. These differences aso depend on the waveength considered and on the FWHM of the fiter function, as is shown in tabe 3. When the FWHM increases, the error on the T s increase aso, whie when the waveength increases the error decreases. It shoud be noted that this error can be sometimes avoided if the fiter functions of the sensor are known. However, in some cases the fiter functions are not avaiabe or, even if they are avaiabe, most authors use the effective waveength in order to simpify the cacuus. Anyway, this effect cannot be avoided when at-sensor vaues are used. More detais about bandpass effects can be found in Richter and Co (2002). 7. Tota error on LST Once the error sources on the LST retrieva from therma infrared remote sensing have been anaysed, we are in the position of estimating a tota error. For this purpose, the foowing equation is used: rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi X et ð s Þ~ ðt s Þ ð20þ where e i (T s ) is the error on T s due to the different factors discussed: transmissivity, i e 2 i Tabe 3. Error on and surface temperature (in K) due to bandpass effects (T s 5300 K). (mm) FWHM50.5 (mm) FWHM51 (mm) FWHM52 (mm)

14 1012 J. C. Jiménez-Muñoz and J. A. Sobrino upward and downward radiances, emissivity, anguar effects, etc. In order to obtain a vaue for e(t s ) et us consider a typica situation: surface temperature 300 K, emissivity 0.98, waveength 11 mm, FWHM 1 mm, a midatitude summer atmosphere (w52.36 g cm 22 ), cear sky conditions, no aerosos and nadir view. Assuming an uncertainty on water vapour of 0.15 g cm 22,aNEDT of 0.1 K, an uncertainty of 1% on emissivity and an uncertainty of 0.1 mm on waveength, the appication of equation (15) eads to an error on T s of 0.6 K. If a NEDT of 0.3 K is considered, then the error on T s is 0.8 K. When an uncertainty of 0.5 g cm 22 instead of 0.15 g cm 22 for the atmospheric water vapour is considered, the error on T s is 0.9 K and 1 K for NEDT of 0.1 K and 0.3 K, respectivey. Taking into account the error on T s ony due to the emissivity and water vapour uncertainties, vaues of 0.5 K and 0.8 K are obtained assuming an uncertainty on the emissivity of 1% and 0.15 g cm 22 and 0.5 g cm 22 for water vapour, respectivey. When an uncertainty on emissivity of 0.5% is considered, errors between 0.3 K and 0.7 K are obtained. As has been shown in this exampe, the most important contribution corresponds to the atmospheric correction and the emissivity effect. Hence, at east an error on T s between 0.3 K and 0.8 K is obtained, so it is difficut to retrieve the LST with an accuracy ower than these vaues uness accurate in situ vaues for emissivity and atmospheric parameters were avaiabe. 8. Concusions The LST retrieved by therma infrared remote sensing techniques is one of the most used parameters for environmenta studies. For this purpose, severa agorithms can be used, as spit-window or dua-channe, dua-ange, etc. These agorithms retrieve LST with a certain error, so they are obtained by an approximation to the radiative transfer equation, simuation procedures and statistica fits. However, the use of the radiative transfer equation itsef does not guarantee an accurate vaue of LST, even if the atmospheric correction is free of errors. Ony the uncertainty on the and surface emissivity eads to an error on the LST of 0.4 K, athough this error is reduced to 0.2 K when in situ vaues are considered. The atmospheric correction introduces errors on the LST retrieva of 0.2 K or 0.7 K depending if in situ or remote sensing data are used, respectivey. So, in optima conditions and when in situ data are avaiabe, a minimum error of 0.3 K is obtained, whereas when remote sensing data are considered a minimum error of 0.8 K is expected. When other error sources such as noise error, bandpass effects and waveength indetermination are considered, then an accuracy for the LST retrieva between 0.5 and 0.9 K is obtained. Acknowedgments We wish to thank the European Union (EAGLE, project SST3-CT ) and the Ministerio de Ciencia y Tecnoogía (project REN /CLI) for the financia support. This work has been carried out whie Juan C. Jiménez was in receipt of a grant from the Universitat de Vaència. References ABREU, L.W. and ANDERSON, G.P. (Eds), 1996, The MODTRAN 2/3 Report and LOWTRAN 7 MODEL, Modtran Report, Contract F C-0132, Hanscom, MA: USA.

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