“chap02” — 2004/2/6 — page 33 — #1 Chapter 2 Land surface temperature retrieval techniques and applications Case of the AVHRR Yann H. Kerr, Jean Pierre Lagouarde, Françoise Nerry and Catherine Ottlé 2.1 Introduction Except for solar irradiance components, most of the fluxes at the surface/atmosphere interface can only be parameterized through the use of surface temperature. Land surface temperature (LST) can play either a direct role, such as when estimating long wave fluxes, or indirectly as when esti- mating latent and sensible heat fluxes. Moreover, many other applications rely on the knowledge of LST (geology, hydrology, vegetation monitoring, global circulation models – GCM). Consequently, for many studies, it is crucial to have access to reliable estimates of surface temperature over large spatial and temporal scales. As it is practically impossible to obtain such information from ground- based measurements, the use of satellite measurements in the thermal infrared appears to be very attractive since they can give access to global and uniform (i.e. with the same sensor and measurement characteristics) estimates of surface temperature. As a matter of fact, satellite thermal infrared sensors measure a radiance, which can be translated into top-of- the-atmosphere brightness temperature. If the sensor is designed to work in a part of the spectrum where the atmosphere is almost transparent (e.g. 10.5–12.5 µm), access to surface temperature would seem to be an easy task. It is not generally the case however, due to the fact that the atmosphere, even though almost transparent, still has a non-negligible effect. Moreover, the surface emissivity is almost always unknown when land surfaces are not black or even grey bodies (i.e. the emissivity is not unity and may also be frequency dependent). In summary, with satellites, we have a means of deriving spatial and temporal values of surface temperature, provided we can perform accurate atmospheric corrections and account for the surface emissivity. Since thermal infrared data have been available, several approaches have been developed to infer surface temperature. The first problem to be solved is to translate the satellite radiance into surface brightness temperature. After calibration and conversion of radiance into temperature using inverse “chap02” — 2004/2/6 — page 34 — #2 34 Yann H. Kerr et al. Planck’s law, it is necessary to account for the atmospheric contribution. It is then necessary to transform surface brightness temperature into surface tem- perature, and thus to take into account emissivity, and directional effects. Actually, the problem is slightly more complicated as atmospheric, emissiv- ity, and directional effects are coupled and these modulating factors cannot be approached independently. The rationale here is to establish which are the most relevant factors. The goal of this chapter is to give an overview of existing methods to retrieve surface temperature. Based on the existing space system we will assume that we have access to two thermal infrared channels around 11 and 12 µm. The practical aspects will be done with data from the Advanced Very High Resolution Radiometer (AVHRR) on board the National Oceanic and Atmospheric Administration (NOAA) polar orbiting satellites. The different issues and possible solutions will then be presented. Finally, several examples of uses of surface temperatures will be briefly delineated. In the following, we will not consider data calibration issues and assume that we have access to accurately calibrated top-of-the-atmosphere brightness temperatures. In the second part, we will consider potential and/or proven applications of LST with associated problems. 2.1.1 Theoretical background Without unnecessary details, we will now give the very basic concepts necessary to define the problem. Role of the atmosphere The energy going through an elementary solid angle per unit time and unit wavelength can be written as (Chandrasekhar 1960): dI λ /ds = (I λ + B λ )τ λ (2.1) where I λ is the intensity of radiation at wavelength λ passing through an absorbing and emitting layer, s is the path length, B λ is the blackbody emission of the layer given by the Planck function, and τ λ is the optical depth. After integrating equation (2.1) along the complete path between the surface and the top of the atmosphere, we have: I λ ( θ ) = ε λ (θ)B λ (T s )τ λ ( θ ) + R atm↑ (θ) + (1 − ε λ (θ))R atm↓ τ λ ( θ ) “chap02” — 2004/2/6 — page 35 — #3 Land surface temperature retrieval techniques 35 with R atm↑ =  h 0 B λ ( T(z) ) ∂τ λ ( θ, z ) ∂z dz (2.2) R atm↓ = 1 π  2π 0  π/2 0  0 ∞ B λ ( T(z) ) ∂τ λ ( θ, z ) ∂z dz sin(θ i ) cos(θ i ) dθ i dφ i where τ λ ( θ ) is the total directional transmission defined by τ λ ( θ ) = exp   h 0 α ( λ, z ) cos ( θ ) e(z) dz  and α is the absorption coefficient for water vapor and e the water vapor concentration. The first term of equation (2.2) is related to the surface contribution, the second to the atmospheric contribution along the upward path, and the third to the atmospheric contribution along the downward path, reflected by the surface and attenuated along the upward path. Equation (2.2) indicates that it is necessary to take into account the atmo- spheric effects and advantage to make measurements in a spectral region where the atmospheric contribution is as small as possible. In most cases the satellite-borne sensors are designed to work in one of the thermal region “atmospheric windows” (10.5–12.5 µm in this chapter). When this condi- tion is met, the first term of equation (2.2) will be least affected, while the relative importance of the second term will be very variable depending upon meteorological conditions (thin cirrus will have a significant influence, for instance). The role of the third term is related to the surface characteristics: the larger the emissivity, the smaller the contribution. In this section, we neglect scattering in the atmosphere, this effect being small when visibility is higher than 5 km (McClatchey et al. 1971). We also neglect the influences of carbon dioxide (CO 2 ) and ozone (O 3 ), as they are much smaller than the effect of water vapor. However, the simulations take these effects into account (MODTRAN). Finally, we assume that the char- acteristics of the sensor (normalized response function, calibration) are well known and perfectly taken into account during the data calibration. It is worthwhile to note, however, that usually the modulation transfer func- tion (MTF) of the sensor is not perfectly taken into account and that future systems would greatly benefit from an improved MTF. Radiance temperature relationship This section is again basic, but allows defining the terms given in some of the presented algorithms. Digital counts recorded by the radiometer are first con- verted into radiances and subsequently into brightness temperature values. For this, calibration is performed, giving the radiance in channel i : I i . “chap02” — 2004/2/6 — page 36 — #4 36 Yann H. Kerr et al. The radiance I i is related to the brightness temperature T B,i through the integration over the channel bandwidth [λ 1 , λ 2 ] of Planck’s black- body function for the temperature T B,i weighted by the sensor’s normalized response: I i (T B,i ) =  λ 2 λ 1 f λ,i 2hc 2 λ 5 exp  (hc/λκT B,i ) − 1  dλ (2.3) where h is Planck’s constant, c is the velocity of light, and κ is Boltzman’s constant. Consequently, the brightness temperature T B,i can be retrieved from equation (2.3) either through a look-up table relating radiances to brightness temperatures, or by defining a central wavelength λ i for each channel: I i (T B,i ) = 2hc 2 λ 5 i exp  (hc/λ i κT B,i ) − 1  It should be noted that the central wavelength λ i is temperature dependent and is usually defined by the temperature range. This section presented the main concepts necessary to compute a surface brightness temperature provided the sensor characteristics, the atmosphere, and the surface emissivity are all perfectly known. We subsequently will detail the evaluation of atmospheric and emissivity effects. 2.1.2 The AVHRR data The NOAA meteorological polar orbitors are sun synchronous satellites whose altitude is nominally 825 km. They carry a scanning radiometer: the AVHRR. We will consider hereafter the case of the AVHRR/2 onboard the NOAA satellites, since only this version of the AVHRR has two different thermal infrared channels. The AVHRR/2 has five channels in the short wave (red and near-infrared), mid-infrared, and thermal infrared. The AVHRR/2 field of view is of ± 55 ◦ , which enables the system to view almost any point of the Earth’s surface twice a day (ascending and descending orbits). Nominally (i.e. without considering the drift of the satellite), the overpass time is around 2 pm local solar time. Even though a given point of the surface is viewed every day, it must be noted that it will be viewed at different viewing angles on subsequent days, with the viewing conditions being approximately repeated only every 9 days. It is worth mentioning that a second satellite operates simultaneously, but on a different orbit (overpass time around 7 am local solar time). The AVHRR/2 spectral bands are: 0.58–0.68, 0.725–1.1, 3.55–3.93, 10.30–11.30, and 11.50–12.50 µm. Nadir resolution is of the order of 1.1 km. Algorithms using the 3.7-µm channel will not be discussed here, “chap02” — 2004/2/6 — page 37 — #5 Land surface temperature retrieval techniques 37 since they can only be efficiently used at night due to the reflected solar sig- nal (the directional reflectance in channel 3 is not well known) and since this channel can be saturated during daytime over some areas. The first two bands are in the short-wave part of the spectrum and are widely used to derive the Normalized Difference Vegetation Index (NDVI), which is the ratio of the difference to the sum of the reflectances ρ 1 and ρ 2 : NDVI = ρ 2 − ρ 1 ρ 2 + ρ 1 (2.4) It has been shown (e.g. Tucker and Sellers 1986) that this ratio can be used to monitor biophysical properties of vegetation such as the Leaf Area Index (LAI) – which is the total area of the leaves per unit area – and pho- tosynthetic capacity. However, the NDVI can only be used to quantify the vegetation LAI when the LAI does not exceed 3–5 due to a saturation effect. Another use of the short-wave channels is in estimating the vegetation frac- tional cover by using another index, the Modified Soil Adjusted Vegetation Index (MSAVI) (Chehbouni et al. 1994; Qiet al. 1994), which is insensitive to soil reflectances, but has to be computed from surface reflectances (hence requiring atmospheric corrections): MSAVI = ρ 2 − ρ 1 ρ 2 + ρ 1 + L (2.5) with L = 1 − 2γ NDVI ( ρ 2 − γρ 1 ) and where γ is the bare soil slope (γ = 1.06). We will tentatively use either the NDVI or the MSAVI to quantify the vegetation cover (i.e. the ratio between bare soil and vegetation). Finally, another vegetation index the Global Environment Monitoring Index (GEMI) (Pinty and Verstraete 1992) is: GEMI = η(1 − 0.25η) − (ρ 1 − 0.125)/(1 − ρ 1 ) (2.6) with η =[2(ρ 2 − ρ 1 ) + 1.5ρ 2 + 0.5ρ 1 )/(ρ 1 + ρ 2 + 0.5)] This index is rather insensitive to the atmosphere (Leprieur et al. 1996), but very sensitive to surface reflectances. Hence, we propose to use it as a surrogate for finer cloud discrimination. Actually, partial cloud cover might not be readily visible using conventional methods (especially due to partial cloud cover and to cirrus clouds) when, according to our experience gained from HAPEX-SAHEL (Prince et al. 1995), the GEMI has a tendency to “chap02” — 2004/2/6 — page 38 — #6 38 Yann H. Kerr et al. show partially covered pixels. However the method will require some sort of thresholding which is delicate to implement on an operational scheme. Other indices do exist and it might be interesting to check whether some of them would not prove more interesting. Thermal band calibration is rather straightforward. The sensor views ther- mistances and deep space that gives the calibration curve (Kidwell 1986). Non-linearities can be taken into account (Brown 1985). The procedure is simple and reliable even though some questions were recently raised con- cerning the “hot target” blackbodies. Consequently, deriving brightness temperatures at the top of the atmosphere is relatively simple and reliable. The problem we will study now is the atmospheric correction procedure. The errors induced by the atmospheric contribution will be especially large for hot surfaces with humid atmospheres. 2.1.3 Practical satellite-based methods Problem 1: atmospheric profile method A primary method to perform atmospheric corrections is to use a radiative transfer model coupled with a characterization of atmospheric structure. The characterization can be made from “standard values” such as climato- logical means, but this characterization is bound to introduce large errors due to the spatial and temporal variability of the atmosphere. It has also been suggested to use indirect methods, such as the use of a reference target (typically a large water body) of known and uniform temperatures (which is another challenge), to assess atmospheric contribution, assuming that the atmosphere characteristics will not change spatially, which, obviously, is not the general case. Moreover, this method relies on only one measure- ment, which is a “cold” reference in the case of a water body, when at least two are necessary (hot and cold as the lower levels of the atmosphere are affected by surface temperature). It is thus necessary to use more accurate characterizations of the atmo- sphere. Several methods have been used to assess the pressure, temperature, and humidity (PTU) profiles of the atmosphere. The most evident being to use radiosoundings. The PTU profile can then be used as an input to a radiative transfer code such as the “4A” (Scott and Chedin 1981), LOWTRAN (Kneizys et al. 1983 and subsequent updates), MODTRAN, or even WINDOW (Price 1983). This approach can give very satisfactory results, provided the radiosoundings are synchronous and collocated with the satellite measurement. Otherwise, large errors can be introduced (up to 10 K as shown by Cooper and Asrar 1989). Moreover, the use of radiosound- ings is hampered by the insufficient density of the network in some areas (3, for example, for the whole Sudanian Sahel), by the timing (usually 12:00 UT and, in some cases, for 00:00 UT), which is not the satellite overpass “chap02” — 2004/2/6 — page 39 — #7 Land surface temperature retrieval techniques 39 time, by the poor representativity in some cases (e.g. near the coast in arid areas), and by the difficulty to access the data in a timely fashion or in digital form. Generally, ground-based radiosoundings do not really fit our needs. In this study, we, nevertheless, relied heavily on atmospheric profiles and RT codes for assessing the different methods. An alternative to radiosoundings is to use atmospheric profiles derived from satellite measurements (Susskind et al. 1984; Chedin et al. 1985), but in this case the inversion algorithms are time consuming and very com- plex. Moreover, the existing sounders do not have the capacity to accurately retrieve profiles near the surface, where most of the atmospheric water vapor is located. Large errors may result from the resolution (30 km) and related surface emissivity variability within the pixel as shown by Ottlé and Stoll (1993). Such methods are thus not yet relevant, but they will need to be investigated further when we enter the Earth Observing System (EOS) era since the NASA and ESA polar platform will carry more sophisticated sound- ing instruments (AIRS, Atmospheric InfraRed Sounder; IASI, Interféromètre Atmosphérique de Sondage dans l’Infra-rouge). The simultaneous use of such a sensor coupled with (MODIS) MODerate resolution Imaging Spectro- radiometer or (MERIS) MEdium Resolution Imaging Spectrometer should allow us to derive accurate surface brightness temperatures. Another possibility is to use the output of meteorological forecasting mod- els. Actually this is the most appropriate method for the time being. The reanalyses are global and available one a roughly 1 × 1 ◦ grid every 6 h. Crude interpolation might be sufficient to derive accurate enough estimates of the integrated water content to be used with differential absorption meth- ods. For a radiative code correction however, the reanalyses will not be accurate enough, and more importantly, they are available only at UT times (usually 0, 6, 12, 18) posing temporal interpolation issues. Consequently, based on existing systems and ancillary data, we will focus here on alternative methods, which, even though less accurate theoretically, have the advantage of being suitable for global applications and can be run “operationally,” without sophisticated ancillary data. We have mainly investigated the differential absorption method (the so called Split Window Techniques, SWT). The differential absorption method: background When two channels, or more, corresponding to different atmospheric trans- missions, are available, it is possible to use the differential absorption to estimate the atmospheric contribution to the signal. This method was first suggested by Anding and Kauth (1970) and put in its now “classical” form by Prabhakara et al. (1974). It has been since adapted and tested successfully with AVHRR data, mainly over sea surfaces (Njoku 1985). Its general name is the SWT. The SWT has been tested mainly for Sea Surface Temperature “chap02” — 2004/2/6 — page 40 — #8 40 Yann H. Kerr et al. (SST) retrievals. Some comparisons over land surfaces have also been done, but with varying degrees of success (Price 1984; Lagouarde and Kerr 1985; Cooper and Asrar 1989). The SWT relies on the different absorption characteristics of the atmo- sphere within two different but close wavelengths. The algorithm consists simply of a linear combination of the thermal channels, which gives a sur- face temperature pseudo-corrected for the atmospheric contribution. For the AVHRR/2 the equation is of the type: T s = a 0 + a 1 T 10.8 + a 2 T 11.9 (2.7) with a 1 + a 2 = 1 (2.8) where T 10.8 and T 11.9 are the brightness temperatures at the top of the atmo- sphere in the two infrared bands. The a i coefficients are estimated using various methods depending on the authors. The SWT is now used opera- tionally over oceans with a claimed accuracy of 0.7 K (McClain et al. 1985: AVHRR data). We note that an alternative method, consisting in using different view angles can be used. It can rely on measurements made by two different satellites (Becker 1982; Chedin et al. 1982), or the same satellite provided it can view along track with two different angles. The Along Track Scanning Radiometer (ATSR) on board ERS-1 satisfies this dual viewing angle and differential absorption technique simultaneously (Eccles et al. 1989; ESA 1989; Prata et al. 1990). Even though the SWT works satisfactorily over sea surfaces, when used directly as developed for SST over land surfaces, the errors can reach 6 K (Lagouarde and Kerr 1985). This is mainly due to the fact that the assump- tions made for the SWT over sea surfaces are not applicable for land surfaces. We are now going to study the SWT assumptions and describe why this is so. We will then describe the main methods currently proposed for land surface temperature estimation. PROBLEM 2: ROLE OF THE EMISSIVITY The SWT has been developed for sea surfaces. It is a simplified way to take into account atmospheric effects, and thus relies on a number of assumptions 1 the surface is lambertian; 2 the surface temperature is close to the temperature in the lower layers of the atmosphere, the latter varying slowly (Planck’s law linearization); 3 the surface temperature does not exceed 305 K; such as (see Becker 1987 for a more detailed analysis): “chap02” — 2004/2/6 — page 41 — #9 Land surface temperature retrieval techniques 41 4 absorption in the atmosphere is small and occurs essentially in the lower layers; 5 the surface emissivity is very stable spatially and close to unity; 6 the emissivities ε 10.8 and ε 11.9 are almost identical and ε 10.8 >ε 11.9 . It is obvious that these conditions are not usually met over land surfaces, hence the problems encountered when using the SST-SWT over land surfaces. Nevertheless, provided we accept a somewhat reduced accuracy, the SWT could be adapted to land surface temperature retrieval. In the specific case of the AVHRR, several limitations linked to the instrument itself are to be considered: 1 the sensor saturates for temperatures higher than 320K; 2 the ascending node time may drift; 3 due to its large scanning angle, the sensor views simultaneously points whose local solar time are quite different (almost 2 h from one end of the scan to the other); 4 for two successive overpasses, the sensor views a given point at different angles and at a different solar time. Thus, over heterogeneous areas, angular effects are bound to exist between subsequent acquisitions. The perturbing effects on the SWT when used over land surfaces are mainly the following: 1 the surface spectral emissivity is a priori unknown and different from unity; 2 spatial variability of the emissivity can be high; 3 surface temperature may have high spatial variability at scales smaller than the resolution of the AVHRR; 4 a strong difference between air and surface temperature may exist. We are now going to analyze the influence of the emissivity on the split window algorithms. EMISSIVITY INDUCED ERRORS Land surface emissivity has two characteristics whose effects are negative in terms of retrieval accuracy: 1 The spectral emissivity in the band 10.3–12.5 µm is not equal to 0.99 but presents such a spectral variability (Buettner and Kern 1965; Fuchs and Tanner 1966; Fuchs et al. 1967; Salisbury and D’Aria 1992, 1994) that integrated values over the AVHRR thermal channels might range from 0.92 to 1. “chap02” — 2004/2/6 — page 42 — #10 42 Yann H. Kerr et al. 2 The spectral emissivity is generally constant over the two AVHRR chan- nels. It has been shown by Becker (1987) that if we assumed that the spectral emissivities ε 10.8 and ε 11.9 were equal to 1 when they are actually different from one another and different from one, the error T induced by such an assumption on the retrieved surface temperature using the SWT could be written (Becker 1987): T = 50 1 − ε ε − 300 ε ε (2.9) where ε = ε 10.8 + ε 11.9 2 and ε = ε 10.8 − ε 11.9 (2.10) The difference ε, when positive, reduces the errors since the second term compensates the first in equation (2.9). This case occurs for water and vegetation. Moreover, the closer to 1 is ε the smaller will be the errors on the retrieved surface temperature T s . In conclusion, the classical SWT will give good results over water, slightly less over fully vegetated areas, and poor results on dry bare soil. It is thus necessary to know the two spectral emissivities to accurately derive surface temperature, which gives us a system of two measurements for three unknowns. The problem is thus a priori not solvable. Several authors thus proposed local SWT with coefficients a i being functions of the surface, atmosphere and view angles, and derived from either exact knowledge of the emissivity or empirically. 2.2 Review of existing algorithms 2.2.1 Algorithms that do not satisfy either accuracy or global applicability requirements We assume here that the surface temperature algorithm has to be appli- cable nearly globally and limit ourselves to AVHRR-type data. We have not considered algorithms requiring both night and day data, since it not practical in many areas (the probability of having regularly successive night and day cloud free acquisitions proving to be very small) and algorithms using variance/covariance methods to infer atmospheric variations (Ottlé et al. 1998) as there is still some controversy on the global efficiency of such methods. The accuracy goal is 1.5 K. Empirical methods The first way to approach atmospheric corrections is to use methods quali- fied here as “empirical.” Using bodies of known temperature (oceans, water [...]... following formulations, each retrieve temperature T having a subscript recalling the author’s initials: Becker and Li (1990): Tbl = 1 .27 4 + T4 + T5 (1 + 0.15616 ε1bl − 0.4 82 ε2bl ) 2 + T4 − T 5 (6 .26 + 3.98 ε1bl + 38.33 ε2bl ) 2 (2. 19) Becker and Li 1990 (modified by Sobrino et al 1994): Tbls = 1.737 + T4 + T5 (1 + 0.00305 ε1bl − 0.376 ε2bl ) 2 + T4 − T5 (5.17 + 21 .44 ε1bl + 30.67 ε2bl ) 2 (2. 20) Prata... “chap 02 — 20 04 /2/ 6 — page 55 — #23 56 Yann H Kerr et al 368.955 360 Tui 340 Tuiri Tuisi 320 Tusi 28 3.8 72 300 0 20 40 60 80 100 1 12 i Figure 2. 2 Different results obtained with emissivities of bare soil (red), or emissivities given in Table 2. 1 (dotted lines) The x-axis represents the ground data number (arbitrary), the y-axis is the temperature in K (see Colour Plate I) of emissivity corresponding,... neighboring pixels for which the surface temperature changes measurably and have the same surface emissivity Over land, these conditions are rarely satisfied and the varying surface emissivities must be accounted “chap 02 — 20 04 /2/ 6 — page 57 — #25 58 Yann H Kerr et al for Sobrino et al (1993) obtained the following equation for this evaluation: R 12, 11 ( Tij )11 = ( Tij ) 12 + τ 12 ε11 ε 12 F11 − ε11 ε 12 F 12. .. 1.189 2. 360 1.730 0.305 −0.1 72 1.940 2. 385 −0.591 0.637 0.941 1.618 1.550 1 .25 9 0. 520 0. 720 1 .28 8 1.468 0.936 0.9997 0.9993 0.9993 0.9981 0.9987 0.9996 0.99 92 0.9978 0.9983 0.9989 1. 027 0 1.0358 1.0814 1.05 92 1.0467 1.0 023 1.0041 1. 025 0 1.0553 1. 020 3 −5.786 1 .26 9 1.654 0 .25 5 1.8 62 2. 191 −1.450 0.081 0.5 82 0. 626 4.099 1.108 1. 123 1.074 1.015 1.089 0.638 0.883 1.0 52 1.745 0.9778 0.9990 0.9989 0.9990 0.9987... − 2. 45 + 40 + T0 ε4 ε5 ε4 (2. 21) Prata and Platt (1991) (modified by Caselles et al 1997): Tppc = 3.46 2. 46 1−ε T4 − T5 + 40 ε ε ε (2. 22) Prata and Platt (1991) (modified by Sobrino et al 1994): Tpps = 3.56 T4 − T0 T 5 − T0 1 − ε4 − 2. 61 + 30.7 + T0 ε4 ε5 ε4 “chap 02 — 20 04 /2/ 6 — page 47 — #15 (2. 23) 48 Yann H Kerr et al Price (1984): Tp = T4 + 3.33 (T4 − T5 ) 5.5 − ε4 + 0.75 T5 4.5 ε (2. 24) ε (2. 25)... Sobrino et al 1994): Tps = T4 + 2. 79 (T4 − T5 ) 7.6 − ε4 + 0 .26 T5 6.6 Ulivieri and Cannizzaro (1985): Tu0 = T4 + 3 (T4 − T5 ) + 51.57 − 52. 45 ε (2. 26) Ulivieri et al (19 92) : Tu = T4 + 1.8 (T4 − T5 ) + 48(1 − ε) − 75 ε (2. 27) Ulivieri et al (19 92) (modified by Sobrino et al 1994): Tus = T4 + 2. 76 (T4 − T5 ) + 38.6(1 − ε) − 96.0 ε (2. 28) Vidal (1991): Tav = T4 + 2. 78 (T4 − T5 ) + 50 ε1bl − 300 ε3v (2. 29)... is given in Prata and Platt (1991) Several authors have developed modified algorithms, notably “chap 02 — 20 04 /2/ 6 — page 46 — #14 Land surface temperature retrieval techniques 47 Sobrino et al (1994) In summary, we have a large number of algorithms that are described below: With the following definitions: ε = ε4 − ε5 , ε= ε4 + ε5 2 1−ε ε ε = (ε )2 ε = ε (2. 15) ε1bl = (2. 16) ε2bl (2. 17) ε3v (2. 18) we... and McMillin (1990), Harris and Mason (19 92) , and Sobrino et al (1993), R 12, 11 (which is the ratio of the transmittances in these two channels) is related to the water vapor (Wv ) in the atmospheric column by an inverse relationship, provided the total transmittance due to the other atmospheric gases is assumed constant: R 12, 11 = K1 Wv + K2 “chap 02 — 20 04 /2/ 6 — page 58 — #26 (2. 50) Land surface temperature... emissivity Otherwise, the method is inefficient and taking an average value for the emissivity could prove just as reliable, provided snow- and ice-covered regions can be identified and proper emissivity values estimated “chap 02 — 20 04 /2/ 6 — page 56 — #24 Land surface temperature retrieval techniques 57 2. 4 Water vapor retrieval 2. 4.1 Introduction: existing sources In the 10– 12 µm spectral region, water vapor... and surface emissivity errors (5% on Wv ; varying on ε, depending on the emissivity value, as explained in the section on “Sensitivity analysis to surface emissivity”), has been estimated for the 20 algorithms This error is defined by the following equation: Ts = δTs δε ε+ δTs δw w “chap 02 — 20 04 /2/ 6 — page 66 — #34 (2. 52) Land surface temperature retrieval techniques 67 The results are presented in . vegetation). Finally, another vegetation index the Global Environment Monitoring Index (GEMI) (Pinty and Verstraete 19 92) is: GEMI = η(1 − 0 .25 η) − (ρ 1 − 0. 125 )/(1 − ρ 1 ) (2. 6) with η = [2( ρ 2 − ρ 1 ). 1997): T ppc =  3.46 ε T 4 − 2. 46 ε T 5 + 40 1 − ε ε  (2. 22) Prata and Platt (1991) (modified by Sobrino et al. 1994): T pps =  3.56 T 4 − T 0 ε 4 − 2. 61 T 5 − T 0 ε 5 + 30.7 1 − ε 4 ε 4 + T 0  (2. 23) “chap 02 — 20 04 /2/ 6. Salisbury and D’Aria 19 92, 1994) that integrated values over the AVHRR thermal channels might range from 0. 92 to 1. “chap 02 — 20 04 /2/ 6 — page 42 — #10 42 Yann H. Kerr et al. 2 The spectral emissivity