Derivation of split window algorithm and its sensitivity ... - CiteSeerX

JOURNAL OF GEOPHYSICAL RESEARCH, VOL. 106, NO. D19, PAGES 22,655-22,670, OCTOBER 16, 2001

Derivation of split window algorithm and its sensitivity analysis for retrieving land surface temperature from NOAA-advanced very high resolution radiometer data Zhihao Qin1, Giorgio Dall' Olmo2, and Arnon Karnieli Remote Sensing Laboratory, Department of Energy and Environmental Physics, J. Blaustein Institute for Desert Research, Ben Gurion University of the Negev, Sede Boker Campus, Israel

Pedro Berliner Wyler Laboratory for Arid Land Conservation and Development, Department of Dryland Agriculture, J. Blaustein Institute for Desert Research, Ben Gurion University of the Negev, Sede Boker Campus, Israel

Abstract. Retrieval of land surface temperature (LST) from advanced very high resolution radiometer (AVHRR) data is an important methodology in remote sensing. Several split window algorithms have been proposed in last two decades. In this paper we intend to present a better algorithm with less parameters and high accuacry. The algorithm involves only two essential parameters (transmittance and emissivity). The principle and method for the linearization of Planck's radiance equation, the mathematical derivation process of the algorithm, and the method for determining the atmospheric transmittance are discussed with details. Sensitivity analysis of the algorithm has been performed for evaluation of probable LST estimation error due to the possible errors in transmittance and emissivity. Results from the analysis indicate that the proposed algorithm is able to provide an accurate estimation of LST from AVHRR data. Assuming an error of 0.05 in atmospheric transmittance estimate and 0.01 in ground emissivity for the two AVHRR thermal channels, the average LST error with the algorithm is 1.1°C. Two methods have been used to validate the proposed algortihm. Comparison has also been done with the existing 11 algorithms in literature. Results from validation and comparison using the standard atmospheric simulation for various situations and the ground truth data sets demonstrate the applicability of the algorithm. According to the root mean square (RMS) errors of the retrieved LSTs from the measured or assumed LSTs, the proposed algorithm is among the best three. Considering the insignificant RMS error difference among the three, the proposed algorithm is better than the other two because they require more parameters for LST retrieval. Validation with standard atmospheric simulation indicates that this algorithm can achieve the accuacry of 0.25°C in LST retrieval for the case without error in both transmittance and emissivity estimates. The accuary of this algorithm is 1.75°C for the ground truth data set without precise in situ atmospheric water vapor contents. The accuracy increases to 0.24°C for another ground truth data set with precise in situ atmospheric water vapor contents. The much higher accuracy for this data set confirms the appplicability of the proposed algorithm as an alternative for the accurate LST retrieval from AVHRR data.

[Qin and Karnieli, 1999; Vogt, 1996]. Examples of this aspect include the works of Price [1983, 1984], Becker [1987], The extensive requirement of temperature information on Holbo and Luvall [1989], Cooper and Asrar [1989], Ottlé a large scale for environmental studies and management acti- and Vidal-Madjar [1992], Prata [1993 and 1994], Coll et al. vities of the Earth's resources has made the remote sensing of [1994], França and Cracknell [1994], Seguin et al. [1994], land surface temperature (LST) an important issue in recent Choudhury et al. [1995], Calvet and Jullien [1996], etc. decades. Many efforts have been devoted to the establishment The United States National Oceanic and Atmospheric of methodology for retrieving LST from remote sensing data Administration (NOAA) has an on-going operational program of polar-orbiting meteorological satellites with advanced very high resolution radiometer (AVHRR) on board to 1 Now at Department of Land, Air and Water Resources, University monitor the global meteorological change [Cracknell, 1997]. of California, Davis,CA 95616, USA. NOAA-AVHRR has two thermal channels operating in 2 at Center for Advanced Land Management Information wavelengths 10.5-11.3 and 11.5-12.5 µm, respectively. Technologies, University of Nebraska, Lincoln, NE 68588, USA. Though spatial resolution of its High-Resolution Picture Copyright 2001 by the American Geophysical Union Transmission (HRPT) format is relatively low (1.1×1.1 km under nadir), NOAA-AVHRR has the advantages of high Paper number 2000JD900452. revisit time (about two images a day) and easy access (public 0148-0227/01/2000JD900452$09.00

1. Introduction

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domain data at many ground receiving stations). These two features have made it an important source of remote sensing data for monitoring the Earth's resources. The retrieval of LST from NOAA-AVHRR data is achieved mainly through the application of so-called split window technique though other methods such as the one introduced by Reutter et al. [1994] are also in use. Several split window algorithms (SWA) have been developed on the basis of various considerations to the effects of the atmosphere and the emitting surface [Cooper and Asrar, 1989; Sobrino and Caselles, 1991; Kerr et al., 1992]. They are derived from the equation of thermal radiation and its transfer through the atmosphere. However, the effect of the atmosphere is so complex that any treatment is difficult. Therefore different simplifications have been assumed for the derivation, which have led to the establishment of different forms of SWA [Qin and Karnieli, 1999]. If T4 and T5 are the brightness temperatures in channels 4 and 5 of AVHRR data, respectively, which are given by inverting Planck's equation for the radiation received by the sensor, the general form of SWA can be expressed as Ts=T4+A(T4−T5)+B, where Ts represents LST and A and B are the coefficients affected by the atmospheric transmittance and surface emissivity in channels 4 and 5 of AVHRR data. All temperatures in the equation are in degrees of Kelvin. Split window technique was first developed for the estimation of sea surface temperature from AVHRR data [Prabhakara et al., 1974; McMillin, 1975; Deschamps and Phulpin, 1980; McClain et al., 1985; Barton, 1992]. Price [1984] is one of the pioneers applying split window technique for LST retrieval. The ground surface was assumed as a blackbody in his derivation even though he later altered his algorithm by adding a correction term of emissivity. Several modifications to the algorithm of Price [1984] have been published since the mid-1980s. Coll et al. [1994] added the satellite zenith observation θ and surface emissivity εi of channel i into the radiation transfer equation and modified the algorithm into a new form accordingly. Except for transmittance and ground emissivity, this algorithm requires the knowledge of two other atmospheric parameters for the estimation of its coefficients: downward mean atmospheric temperature T↓ai and the parameter γi. On the based of the structural components of ground surface, Sobrino et al. [1991] developed a methodology for atmospheric and emissivity corrections to the brightness temperature of AVHRR data. Ground emissivity, atmospheric transmittance, and two other parameters (water vapor content in atmospheric profile and the parameter stating atmospheric absorption) are directly involved in their algorithm. Through a complicated radiation transfer equation, França and Cracknell [1994] established two atmospheric correction models for retrieving LST from remote sensing data in thermal wavebands: an airborne model for data obtained from low flying altitude and a split window algorithm for AVHRR data. Again, water vapor content and an atmospheric parameter are also required for LST retrieval with this algorithm. The form of these algorithms is the same as the general one mentioned in the previous paragraph, but the calculation of the coefficients A and B is different, owing to different considerations of the atmospheric effect on the radiation transfer through the air. Using a linear relationship between LST and brightness temperature, Ottlé and VidalMadjar [1992] presented SWA as Ts=a0+a1T4+a2T5, in which the coefficients are determined through regression. The

problem of applying this algorithm in the real world is how to obtain a sample of LST for a set of corresponding pixels with scale of up to 1.1×1.1 km. After analyzing the atmospheric and emissivity effects on LST determination, Prata [1993] derived an approximate SWA from the thermal radiance transfer equation with a full consideration of possible factors. This algorithm has the same linear form as that of Ottlé and Vidal-Madjar [1992], but the calculation of its parameters is totally different. Parameters a1 and a2 were determined as the function of ground emissivity and atmospheric transmittance. The calculation of parameter a0 was very complicated, determined by parameters a1 and a2, downwelling atmospheric radiance, Planck's radiance of channel 4 and its derivative, and so on. Owing to the difficulty in estimating atmospheric radiance and calculating the derivative term, the application of this algorithm is also inconvenient. Another important version of SWA is the one developed by Becker and Li [1990] with the form of Ts=A0+P(T4+T5)/2+M(T4T5)/2, where A0, P, and M are coefficients. For the determination of the coefficients they computed the atmospheric effect as constant and gave more attention to the effect of ground emissivity. The assumption of atmospheric effect as constant is not true for most cases. This algorithm was further modified by Wan and Dozier [1996] as a viewing angle dependent method for achieving a higher accuracy of LST retrieval. Most of the existing presentations about SWA do not give a detailed description of their derivation. Some involve a simple derivation but omit many important processes. The incomplete derivation makes the understanding of the existing algorithms, especially the computation of the coefficients, inconvenient and difficult. Moreover, the existing algorithms such as those of Sobrino et al. [1991], Coll et al. [1994], and França and Cracknell [1994] require some parameters that are difficult to estimate in the real world. Besides, sensitivity analysis is usually neglected in many presentations of the methodology. The inclusion of a sensitivity analysis is important in many applications where the knowledge about probable LST estimation error due to possible error in parameter determination is generally desired so that an assessment can be made of the accuracy of the estimate using the desired algorithm. In the current paper we attempt to present a better SWA, which requires only two essential parameters (emissivity and transmittance). Keeping the accuracy of the LST estimate, we avoid the complicated expression of the algorithm and the difficult calculation of its parameters. A complete and detailed description is given to the derivation of the algorithm, which includes the theoretical basis for remote sensing of LST, the thermal radiation transfer through the atmosphere and its computation, the linearization of Planck's radiance function, and the derivation process of the algorithm itself. The focus is also on the determination of atmospheric transmittance and the sensitivity analysis of the algorithm. Finally, we use two methods to validate the algorithm and compare it to the existing 11 algorithms in the literature.

2. Theoretical Basis for the Remote Sensing of LST The theoretical basis for remote sensing of LST is that the total radiance emitted by the ground increases rapidly with temperature. The spectral distribution of the radiance emitted by the ground also depends on wavelength. According to

QIN ET AL.: SPLIT WINDOW ALGORITHM AND SENSITIVITY ANALYSIS Wien’s displacement law on the relationship between spectral radiance and wavelength, for the earth with an ambient temperature of about 300 K, the peak of its spectral radiance occurs at 9.6 µm [Lillesand and Kiefer, 1987; Vogt, 1996]. Though the ozone in the atmosphere absorbs most of the radiance from the ground in 9.4-9.9 µm, the atmospheric window in 10-13 µm is characterized by a minimal loss for the radiance to transfer. Therefore, theoretically, the thermal radiance in relation to the physical or kinetic temperature of the ground can be remotely observed using the sensors operating in 10-13 µm, which have been selected as thermal channels in many remote sensing systems such as NOAAAVHRR and Landsat thematic mapper (TM). The temperature converted from the obtained thermal radiance from the ground surface at the satellite level is called the brightness temperature [Reutter et al., 1994]. Planck’s law links the spectral radiance emitted by a blackbody at a given wavelength and temperature [Chen, 1990; Saraf et al., 1995; Vogt, 1996] in the following way: Bλ (T ) =

λ

5

(e

hc 2 hc / λkT

)

,

(1)

−1

where Bλ(T) is the spectral radiance of the blackbody, generally measured in W m-2 sr-1µm-1, λ is wavelength in meter (1 m=106 µm), h is Planck's constant with h=6.626076 ×10-34 J s, k is Boltzmann's constant with k=1.380658 ×10-23 J K-1, T is temperature in Kelvin, and c is the speed of light with c=2.99792458×108 m s-1. In the absence of atmospheric effects the temperature of ground objects can be theoretically determined, if the emitted spectral radiance Bλ(T) is measured, by inverting Planck’s radiation equation as follows: T=

c2    c  + 1 λ ln  5 1  λ Bλ (T )  

(2)

where c1 and c2 are constants with c1=hc2=5.95522012×10-17 W m2 and c2=hc/k =1.43876869×10-2 m K. However, the remote sensor detecting the spectral radiance is usually mounted on a platform (airplane or satellite) far from the ground. The transmission of the emitted spectral radiance through the atmosphere to the sensor is affected by a number of factors, which makes the LST retrieval from remote sensing data more complicated than just simply inverting Planck's equation. In the thermal region the atmosphere has three important effects on spectral radiation transmission: absorption, upward atmospheric radiance, and bidirectional reflection of the downward atmospheric radiance [França and Cracknell, 1994]. Therefore the spectral radiance reaching the sensors is not only the radiance emitted by the ground and attenuated by atmospheric absorption but also includes the radiance emitted by the atmosphere and the reflected component of the downward atmospheric radiance. At the same time, the thermal characteristics of the ground and different viewing angles of the sensor also have significant effects on the observed radiance from space [Ignatov and Dergileva, 1995; Wan and Dozier, 1996]. For these reasons the brightness temperature can not be considered as a good estimate of LST. The most common method to correct brightness temperature into LST is SWA.

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3. Thermal Radiance Transfer from the Ground to the Remote Sensor The basic idea of SWA can be traced back to the studies of sea surface temperature such as those of Prabhakara et al. [1974] and McMillin [1975] from remote sensing data. For a given temperature there is a linear relationship between the radiance and temperature in the two infrared windows. Since the two thermal channels of AVHRR are close enough to each other but still have different transmittance and emissivity, an equation system describing the thermal radiance reaching the remote sensor in the two channels can be established, which consequently enables the solution of LST. This makes the estimation of LST from the two thermal channels of VAHRR data possible without the need of precise atmospheric temperature and pressure profiles. The derivation of SWA for LST retrieval is based on the thermal radiance of the ground and its transfer from the ground through the atmosphere to the remote sensor. Generally speaking, the ground is not a backbody. Thus ground emissivity has to be considered for computing the thermal radiance emitted by the ground. Atmosphere has important effects on the received radiance at remote sensor level. Considering all these impacts, the general radiance transfer equation [Ottlé and Stoll, 1993] for remote sensing of LST can be formulated as follows: Bi(Ti)=τi(θ)[εiBi(Ts)+(1-εi)Ii↓]+Ii↑,

(3)

where Ts is the LST, Ti is the brightness temperature in channel i, τi(θ) is the atmospheric transmittance in channel i at viewing direction θ (zenith angle from nadir), and εi is the ground emissivity. Bi(Ti) is the radiance received by the sensor, Bi(Ts) is the ground radiance, and Ii↓ and Ii↑ are the downwelling and upwelling atmospheric radiances, respectively. The upwelling atmospheric radiance Ii↑ is usually computed as [França and Cracknell, 1994; Cracknell, 1997] I i↑ =

∫

Z 0

Bi (Tz )

∂τi (θ, z , Z ) , dz ∂z

(4)

where Tz is the atmospheric temperature at altitude z, Z is the altitude of the sensor, and τi(θ,z,Z) represents the upwelling atmospheric transmittance from altitude z to the sensor height Z. Following McMillin [1975], Prata [1993], and Coll et al. [1994], we employ the mean value theorem to express the upwelling atmospheric radiance as follows: Bi (Ta ) =

Z 1 ∂τ (θ, z, Z ) , Bi (Tz ) i dz 1 − τi (θ) ∫ 0 ∂z

(5)

where Ta is the effective mean atmospheric temperature and Bi(Ta) represents the effective mean atmospheric radiance with Ta in channel i. Thus we obtain Ii↑=[1-τi(θ)]Bi(Ta).

(6)

The downwelling atmospheric radiance is generally viewed from a hemispherical direction and hence can be computed as [França and Cracknell, 1994] I i↓ = 2 ∫

π /2 0

0

∫φ B (T ) i

z

∂τ 'i (θ ' , z,0) cos θ ' sin θ ' dz dθ ' , ∂z

(7)

where θ' is the downwelling direction of atmospheric

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QIN ET AL.: SPLIT WINDOW ALGORITHM AND SENSITIVITY ANALYSIS Table 1. Underestimate of Ts (°C) by Approximating B4(Ta↓) with B4(Ta) For ε4=0.96 and τ4=0.6

For ε4=0.98 and τ4=0.8

Ta↓-Ta °C

At Ts=10°C

At Ts=30°C

At Ts=50°C

At Ts=10°C

At Ts=30°C

At Ts=50°C

2

0.0119

0.0099

0.0086

0.0054

0.0045

0.0039

3

0.0178

0.0149

0.0128

0.0081

0.0068

0.0058

5

0.0297

0.0249

0.0214

0.0135

0.0113

0.0097

radiance, φ denotes the top of the atmosphere, and τ'i(θ',z,0) represents the downwelling atmospheric transmittance from altitude z to the ground surface. According to França and Cracknell [1994], it is rational to assume that ∂τ'i(θ',z,0) =∂τi(θ,z,Z) for the thin layers of the whole atmosphere. Based on this assumption, application of the mean value theorem to (7) gives

I i↓ = 2 ∫

π /2 0

[1 − τ i (θ )]B i (T a↓ ) cos θ ' sin θ ' dθ ' ,

(8)

where Ta↓ is the downward effective mean atmospheric temperature. The integration term of this equation can be solved as

2∫

π/2 0

cos θ' sin θ' dθ' = (sin θ' ) 2

π/2 0

= 1.

(9)

Substituting the solution into (8), we obtain Ii↓=(1-τi(θ))Bi(Ta↓).

the underestimate of Ts. The magnitude of underestimating Bi(Ts) and Bi(Ta) in (11) for the approximation depends on their coefficients in the equation, i.e., εiτi for Bi(Ts) and (1τi)[1+τi(1-εi)] for Bi(Ta). We consider three cases of |Ta-Ta↓| and two cases of εi and τi for the simulation to the two AVHRR thermal channels, which gives the results shown in Tables 1 and 2. The underestimates of Ts for all cases are quite small. For |Ta-Ta↓|=5°C, ε4=0.96, and τ4=0.6 the approximation of B4(Ta↓) with B4(Ta) can only lead to the underestimate of Ts, 0.0297°C at Ts=10°C and 0.0214°C at Ts=50°C. The underestimates of Ts are even smaller for ε4=0.98 and τ4=0.8 (Table 1). A similar effect by approximating B5(Ta↓) with B5(Ta) can also be seen in Table 2. Therefore we can conclude that the approximation of Bi(Ta↓) with Bi(Ta) will have an insignificant effect on the estimate of Ts from (11). With this approximation the observed radiance of NOAA-AVHRR can be expressed as Bi(Ti)=εiτi(θ)Bi(Ts)+[1-τi(θ)][1+(1-εi)τi(θ)]Bi(Ta).

(10)

(12)

Application of (12) to the thermal channels 4 and 5 of AVHRR gives an equation system as follows:

Thus (3) can be rewritten as Bi(Ti)=εiτi(θ)Bi(Ts)+[1-τi(θ)](1-εi)τi(θ)Bi(Ta↓) +[1-τi(θ)]Bi(Ta).

B4(T4)=ε4τ4(θ)B4(Ts) (11)

In order to solve (11) for LST, we need to analyze the possible effect of the difference between Ta↓ and Ta on Ts for AVHRR. Because of the vertical heterogeneity of the atmosphere, the upwelling atmospheric radiance is generally greater than the down-welling one. Consequently, Bi(Ta) is greater than Bi(Ta↓), or Ta>Ta↓. Under the clear sky, the difference between Ta and Ta↓ is usually within 5°C, i.e., |TaTa↓|Bi(Ta↓), the approximation of Bi(Ta↓) with Bi(Ta) will lead to the underestimate of both Bi(Ts) and Bi(Ta) in (11) for a fixed Bi(Ti). Consequently, it will lead to

+[1-τ4(θ)][1+(1-ε4)τ4(θ)]B4(Ta),

(13a)

B5(T5)=ε5τ5(θ)B5(Ts) +[1-τ5(θ)][1+(1-ε5)τ5(θ)]B5(Ta).

(13b)

On the basisd of (13a) and (13b), a split window algorithm can be derived under several assumptions and simplifications to the function of Planck's radiance in the equation.

4. Linearization of Planck's Radiance Function In order to develop a SWA from (12), we need to apply the method of linearizing Planck's radiance function, which allows us to directly relate the radiance to the temperature in the corresponding channels. It is a critical step in deriving the algorithm. The linearization of Planck's radiance function can be done through the application of Taylor's expansion. As indicated in Figure 1a, the relationship between Planck's radiance and temperature in the two AVHRR channels is close to linear for a given wavelength. In this case, keeping the first two terms

Table 2. Underestimate of Ts (°C) by Approximating B5(Ta↓) with B5(Ta) For ε5=0.964 and τ5=0.57

For ε5=0.984 and τ5=0.77

Ta↓-Ta °C

At Ts=10°C

At Ts=30°C

At Ts=50°C

At Ts=10°C

At Ts=30°C

At Ts=50°C

2

0.0101

0.0156

0.0262

0.0045

0.0069

0.0116

3

0.0087

0.0134

0.0225

0.0038

0.0059

0.0099

5

0.0076

0.0118

0.0198

0.0034

0.0052

0.0087

QIN ET AL.: SPLIT WINDOW ALGORITHM AND SENSITIVITY ANALYSIS

For the parameter Li Slater [1980] and Price [1984] suggested the following approximation:

9

(a)

8

Ch4

Li=Ti/ni .

7

Ch5

B4(T4)

Radiance

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França and Cracknell [1994] gave the values of ni for channels 4 and 5 as constant with n4=4.592 and n5=4.12636, respectively, for the temperature range 280-320 K. Actually, we find that ni is not a constant but changes obviously with temperature. Regression of ni against T gives

B5(T5)

6 5 4

(17)

B5(T4)

3

T4

T5

n4=8.81614-0.01438T4,

2 0

5

10

15

20

25

30

35

40

45

50

55

60

65

R2=0.99711

70

SEE=0.0116.

(18a)

SEE=0.00936.

(18b)

Temperature

n5=7.96727-0.01284T5,

95

R2=0.99765

(b)

90

L5

Parameter Li

85 80

In order to evaluate the relationship between Li and temperature Ti, we plot it against temperature T in Figure 1b, which shows a strong linearity between them. Correlating Li to Ti for the general range 0-50°C of Ti, we obtain

L4

75 70

L4=-62.239281+0.430589T4,

65

R2=0.9991

60 55 0

5

10

15

20

25

30

35

40

45

50

55

60

65

70

of Taylor's expansion can achieve a high accuracy of approximation. This is the general way to approximate Planck’s function for SWA derivation [Price, 1983; França and Cracknell, 1994; Coll et al., 1994]. Following this method, we have Bi(Tj)=Bi(T)+(Tj-T)∂Bi(T)/∂T=(Li+Tj-T)∂Bi(T)/∂T, (14) where i refers to channel 4 or 5 and Tj refers to the temperatures of channel 4 (j=4), channel 5 (j=5), land surface (j=s), and air (j=a). The parameter Li is given as Li= Bi(T)/[∂Bi(T)/∂T],

(15)

in which Li has the dimension of temperature in Kelvin. The physical meaning of Taylor's expansion in this case, as illustrated in Figure 1a, is to express the radiance of Bi(Tj) in terms of the radiance Bi(T) with a fixed temperature T, which is usually defined as T4 [França and Cracknell, 1994; Prata, 1993]. For a moderate departure T-Tj≤10 K, the accuracy is better than 1% for 10-13 µm within the temperature range 270-320 K [Prata, 1993]. Thus expansion of B5(T5) for T5 in this way gives B5(T5)=B5(T4)+(T5-T4)∂B5(T4)/∂T =(L5+T5-T4)∂B5(T4)/∂T.

(16a)

Similarly, we obtain the following for other terms:

R2=0.9993

(16b)

B4(Ts)=(L4+Ts-T4)∂B4(T4)/∂T,

(16c)

B4(Ta)=(L4+Ta-T4)∂B4(T4)/∂T,

(16d)

B5(Ts)=(L5+Ts-T4)∂B5(T4)/∂T,

(16e)

B5(Ta)=(L5+Ta-T4)∂B5(T4)/∂T.

(16f)

SEE=0.1860.

(19b)

2

These extremely high R values (their T-tests are statistically significant at 99%) indicate that the approximation of Li as a linear function of temperature is very successful. Higher accuracy or less standard error of estimation (SEE) can be achieved to correlate Li to Ti for a smaller range of Ti (Table 3). Thus we approximate the parameter Li in the following derivation as Li=ai+biTi .

(20)

Generally speaking, the coefficients ai and bi can be given from (19) as follows: a4=-62.23928 and b4=0.43059 for channel 4 and a5=-66.54067 and b5=0.46585 for channel 5. For more precise computation the equations in Table 3 can be used. In summary, using the approximations of (16) and (20), we can derive a SWA involving only emissivity and transmittance from (13).

5. Derivation of Split Window Algorithm for AVHRR Data The derivation of our SWA is based on (13). For simplification we define

Table 3. Approximating Parameter Li for Different Ti Ranges Ti Ranges 0°-30°C

B4(T4)=(L4+T4-T4)∂B4(T4)/∂T=L4∂B4(T4)/∂T,

(19a)

L5=-66.540666+0.465845T5,

Temperature

Figure 1. Changes of (a) Planck's radiance and (b) parameter Li with temperature.

SEE=0.1893.

0°-40°C 10°-40°C 20°-50°C

Equations for Ti, K

R2

SEE

L4=-58.156179+0.416385T4 L5=-62.425258+0.454559T5 L4=-60.222601+0.423580T4 L5=-64.409228+0.458456T5 L4=-61.666695+0.428371T4 L5=-65.887933+0.463334T5 L4=-66.209596+0.443386T4 L5=-70.610099+0.478971T5

0.99967 0.99928 0.99899 0.99959 0.99905 0.99985 0.99911 0.99953

0.09937 0.11235 0.16328 0.11212 0.12230 0.05213 0.12220 0.09612

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QIN ET AL.: SPLIT WINDOW ALGORITHM AND SENSITIVITY ANALYSIS Ci=εiτi(θ),

(21)

Di=[1-τi(θ)][1+(1-εi)τi(θ)].

(22)

the atmospheric column. This changes from day to day and along the day. Thus this algorithm provides space for a relatively dynamic determination of the required coefficients.

Thus (13) can be rewritten as B4(T4)=C4B4(Ts)+D4B4(Ta),

(23a)

B5(T5)=C5B5(Ts)+D5B5(Ta).

(23b)

Substitution of Planck’s function with (16) gives L4∂B4(T4)/∂T=C4(L4+Ts-T4)∂B4(T4)/∂T +D4(L4+Ta-T4)∂B4(T4)/∂T,

(24a)

(L5+T5-T4)∂B5(T4)/∂T=C5(L5+Ts-T4)∂B5(T4)/∂T +D5(L5+Ta-T4)∂B5(T4)/∂T.

(24b)

Elimination of the term ∂B4(T4)/∂T from (24a) and ∂B5(T4)/∂T from (24b) leads to L4=C4(L4+Ts-T4)+D4(L4+Ta-T4),

(25a)

L5+T5-T4=C5(L5+Ts-T4)+D5(L5+Ta-T4).

(25b)

Elimination of Ta from (25a) and (25b) gives D5L4-D4(L5+T5-T4)=D5C4(L4+Ts-T4)-D4C5(L5+Ts-T4) +D5D4(L4-T4)-D4D5(L5-T4).

(26)

Reorganizing (26) to express Ts in terms of T4 and T5, we obtain our SWA in the general form as follows: Ts=T4+A(T4-T5)+B,

(27)

where the parameters are defined as A=D4/E0,

(28)

B=E1L4-E2L5,

(29)

E1=D5(1-C4- D4)/E0,

(30)

E2=D4(1-C5-D5)/E0,

(31)

E0=D5C4- D4C5.

(32)

6. Determination of Atmospheric Transmittance Atmospheric transmittance is a critical parameter that affects the accuracy of LST retrieval using split window algorithm. The thermal radiance is attenuated on its way to the remote sensor. Transmittance depicts the magnitude of the attenuation of the radiance transferring through the atmosphere. It varies with wavelength and viewing angle. Many atmospheric constituents such as water vapor, O3, CO2, and other gases have impacts on the thermal atmospheric transmittance. However, the contents of O3, CO2, and other gases are relatively stable in the atmosphere. Accordingly, their impacts can be assumed as constant and simulated by standard atmospheric profiles. On the contrary, water vapor content is highly variable. Thus the variation of atmospheric transmittance strongly depends on the dynamics of water vapor content in the profile. Consequently, many split window algorithms relate the determination of atmospheric transmittance to the change of water vapor content while assuming other impacts as constant [Sobrino et al., 1991; França and Cracknell, 1994; Coll et al., 1994]. 6.1. Impact of Water Vapor on Atmospheric Transmittance Because of many technical difficulties, atmospheric transmittance is usually not available at in situ satellite passes.

Since Li is temperature dependent, we may continue the derivation for simplification. Replacing Li with (20), we have B=E1(a4+b4T4)-E2(a5+b5T5).

(33)

Reorganization of (33) gives B=E1b4T4-E2b5T5+E1a4-E2a5.

(34) Water vapor

Substituting into (27) and reorganizing the terms, we get a new form of SWA as follows: Ts=A0+A1T4-A2T5,

0.04

(35)

(b)

where the coefficients A0, A1, and A2 are defined as A0=E1a4-E2a5,

(36a)

A1=1+A+E1b4,

(36b)

A2=A+E2b5.

(36c)

Trans. difference

0.03

Ch 5 0.02

0.01

Ch 4

0

The algorithm in (35) relates the determination of the three important coefficients A0, A1, and A2 to the atmospheric transmittance, ground emissivity, and viewing angle. Viewing angle is known for a specific image. Ground emissivity may be estimated from AVHRR data using the methodology given by Sobrino et al. [2001] and Li and Becker [1993]. The atmospheric transmittance in the thermal window is generally estimated from water vapor content in

-0.01 0

5

10

15

20

25

30

35

Zenith angle

Figure 2. (a) Atmospheric transmittance as a function of water vapor content and (b) the difference τ4(10)-τ4(θ) as a function of the zenith angle of viewing (ZAV).

QIN ET AL.: SPLIT WINDOW ALGORITHM AND SENSITIVITY ANALYSIS Generally, the most practical way to determine the atmospheric transmittance is through simulation with local atmospheric conditions, especially water vapor content. Simulation of the relationship between atmospheric transmittance and water vapor content can be done through atmospheric modeling programs such as LOWTRAN and MODTRAN. Here we use the LOWTRAN 7 to determine this relationship. Two atmospheric profiles are used for this simulation: summer and winter. For summer we assume the air temperature near surface to be equal to 30°C and for winter to be 18°C, which represents the case of air temperature change in our study area but could be extended to the low- to middle-latitude region of the Earth. Viewing angle is also important in the simulation, and for this reason we select 10° from nadir because very few images have a viewing angle smaller than 10° from nadir. The results of transmit-tance simulation for the summer profile are plotted in Figure 2a, which shows that atmospheric transmittance does change with water vapor content in the two AVHRR thermal channels. Atmospheric transmittance in these two channels is above 0.90 for water vapor < 1 g cm-2 while it is below 0.5 for water vapor >5 g cm-2. For a small range of water vapor content the relationship between water vapor content and atmospheric content may be approximated as a linear equation even though the whole one is a curve as described by França and Cracknell [1994] and Sobrino et al. [1991]. In order to have a simple accurate estimation of transmittance, we divide the whole curve into several parts and determine their regressions. The results are given in Table 4. Squared correlation coefficients (R2) of the transmittance estimation equations given in Table 4 are very high (>0.995), which indicates that the atmospheric transmittance has strongly linear relationship with water content for the narrow

Table 4. Relationship between Transmittance and Water Vapor Content in the Atmosphere w g cm-2

Estimation Equations

R2

SEE

0.99425 0.99716 0.99746 0.99879 0.99999 0.99947 0.99954 0.99747

0.002266 0.002501 0.002602 0.002486 0.000825 0.002246 0.001269 0.002821

0.99469 0.99679 0.99817 0.99948 0.99998 0.99934 0.99969 0.99821 0.99907 0.99649

0.002215 0.002896 0.002749 0.001973 0.000262 0.002184 0.000966 0.002214 0.001424 0.002195

Summer Profile 0.4-1.6 1.6-3.0 3.0-5.0 5.0-6.4

τ4(10)=0.979160−0.062918w τ5(10)=0.968144−0.098942w τ4(10)=1.035378−0.097514w τ5(10)=1.026468−0.135133w τ4(10)=1.098068−0.118847w τ5(10)=1.034865−0.139598w τ4(10)=1.060569−0.111773w τ5(10)=0.866450−0.105842w Winter Profile

0.4-1.6 1.6-3.0 3.0-4.4 4.4-5.4 5.4-6.4

τ4(10)=0.983311−0.072444w τ5(10)=0.981868−0.121979w τ4(10)=1.058059−0.121354w τ5(10)=1.048364−0.163678w τ4(10)=1.111348−0.140080w τ5(10)=1.033785−0.160330w τ4(10)=1.071156−0.131166w τ5(10)=0.879292−0.125053w τ4(10)=0.964106−0.111430w τ5(10)=0.681518−0.088431w

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ranges. SEE illustrates that the estimation with these equations can reach an accuracy of high up to ≤0.003, which is much higher than that estimated by a parabolic equation proposed by França and Cracknell [994] and Sobrino et al. [1991]. Using the parabolic equation to fit the curve, we find that SEE is high up to 0.01015-0.01481 for water vapor range 0.4-6 g cm-2. This error together with possible water vapor content measuring error may cause an obvious overall error in LST estimation, as illustrated in section 7. The regression coefficients of the equations in Table 4 indicate that transmittance change is within a range of 0.006-0.0164 for a small change (0.1 g cm-2) of water vapor content. This means that the maximum of the possible error of transmittance estimation is less than 0.033 for a probable water vapor content error 0.2 g cm-2. Usually, the in situ measurement of atmospheric water vapor content by means of the sunphotometer (CIMEL, Paris, France) such as the one on the roof of our laboratory can meet this measuring accuracy for transmittance estimation. 6.2. Viewing Angle and Atmospheric Transmittance The viewing direction of the remote sensor determines the pathway of the radiance through the atmosphere to reach the sensor. Zenith angle of viewing (ZAV, also denoted as θ) describes the viewing direction of the sensor from the nadir. Even though the possible error of LST retrieval with split window algorithm is negligible at a high viewing angle such as ≥75° [Wan and Dozier, 1996], we still want to analyze the effect of ZAV on atmospheric transmittance. This is because many images have a viewing angle of less than 75° (θ >15°). The span of ZAV for the analysis is up to 35° from nadir. Atmospheric water vapor content is selected to be at the typical level of 2.0 g cm-2 when transmittance is in the range of 0.7-0.85. Simulation with LOWTRAN 7 indicates that the atmospheric transmittance for the two NOAA-AVHRR thermal channels decreases rapidly with ZAV. This is because the path of radiance transfer through the atmosphere increases with ZAV. Atmospheric transmittance difference from the transmittance with θ=10°, i.e. τi(10)-τi(θ), is within 0.04 for ZAV less than 35°. Specifically, the difference is about 0.02 for θ=25°, which may result in an average LST estimation error of about 0.15°C according to sensitivity analysis given in section 7. Because of the heterogeneous atmospheric conditions in each layer, the relationship between atmospheric transmittance decrease and ZAV increase is not linear but parabolic. Figure 2b clearly shows that the difference of atmospheric transmittance with θ=10° increases rapidly with ZAV. This feature implies that ZAV has to be considered in atmospheric transmittance estimation for reaching a relatively accurate LST retrieval from the images with a ZAV of greater than 20°. Using a parabolic function to fit the change of transmittance difference with ZAV, we obtain the following ZAV correction equations for AVHRR channels 4 and 5: ∆τ4(θ)=(-2.3994×10-3)+(2.2976×10-5)θ2, R2=0.998487

SEE=0.00043.

(37a)

∆τ5(θ)=(-3.2766×10-3)+(3.1454×10-5)θ2, R2=0.998729

SEE=0.00054.

(37b)

where ∆τ4(θ) and ∆τ5(θ) are atmospheric transmittance

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QIN ET AL.: SPLIT WINDOW ALGORITHM AND SENSITIVITY ANALYSIS

differences of AVHRR channels 4 and 5 and θ is ZAV in degrees. Therefore for an image with θ we can estimate the transmittance in AVHRR channels 4 and 5 as follows:

1.6 1.4

(a)

(38a)

τ5(θ)=τ5(10)-∆τ5(θ),

(38b)

where τ4(10) and τ5(10) are the atmospheric transmittances of channels 4 and 5 at ZAV of 10°, given by the equations listed in Table 4.

LST error (C)

1.2

τ4(θ)=τ4(10)-∆τ4(θ),

1

Trans4 error 0.005 Trans4 error 0.01 Trans4 error 0.02 Trans4 error 0.03 Trans4 error 0.05

0.8 0.6 0.4 0.2 0

7. Sensitivity Analysis of the Split Window Algorithm

0.402 0.454 0.509 0.551 0.607 0.649 0.704 0.757 0.805

0.905 0.952

0.81

0.878 0.939

1.8 1.6 1.4

LST error (C)

Though the estimation of atmospheric transmittance can reach a high accuracy using the above method, there still may be some errors due to possible effects of other neglected factors such as the variation of other gases in the atmosphere and/or the accuracy of the water vapor measurement itself. Besides, because of many difficulties, the estimation of ground emissivity always carries some unknown errors [Humes et al., 1994; Becker and Li, 1990]. In order to evaluate the impact of these possible errors on LST estimation, we need to perform the sensitivity analysis of the split window algorithm. For convenience of expression we compute the probable LST estimation error δTs as follows: δTs=Ts(x+δx)−Ts(x),

0.85

Trans. in ch. 4

1.2 1

Trans4&5 error 0.005

(b)

Trans4&5 error 0.01 Trans4&5 error 0.02 Trans4&5 error 0.03 Trans4&5 error 0.05

0.8 0.6 0.4 0.2 0 0.321 0.372 0.428 0.471 0.531 0.577 0.638 0.699 0.756

Trans. in ch. 4&5

(39)

where x is the variable that sensitivity analysis orients to (transmittance or emissivity), δx is possible error of the variable x, and Ts(x+δx) and Ts(x) are the LST simulated by our algorithm (35) for x+δx and x, respectively. 7.1. Sensitivity Analysis to Atmospheric Transmittance The sensitivity analysis of the algorithm to transmittance is executed on the basis of several conditions. First of all, we have to assume a ground emissivity and brightness temperature for the two channels. For the analysis we use assumption of average ground emissivity 0.97 and follow the method created by Sobrino and Caselles [1991] to determine the emissivity of the two channels as ε4=0.967 and ε5=0.971. For brightness temperature we assume that T4 is greater than T5 and that their difference is T4-T5=0.7. For most natural surfaces of the Earth, ground emissivity is between 0.95 and 0.98 [Humes et al., 1994]. As shown in Figure 3, the T4-T5= 0.7 assumption is rational for most cases. Then we have to

T4-T5 (C)

Figure 3. Histogram of T4-T5, computed from the IsraelSinai (Egypt) peninsula image acquired on July 18, 1998.

Figure 4. Probable land surface temperature (LST) estimation error with transmittance due to possible transmittance error in (a) channel 4 and (b) both channels 4 and 5. assume a temperature range for the analysis. Considered the possible LST change of the Earth, 0°-70°C is selected as the range of brightness temperature T4 change. The result computed by the algorithm indicates that the LST estimation error is almost independent with temperature change. For δτ4=0.005 the δTs only changes 0.0052°C, within the temperature range 0°-70°C, from 0.132°C at 0°C to 0.137°C at 70°C. This small change is negligible in practice. Thus the average δTs for the range 0°-70°C is used to represent the LST estimation error in the following analysis. Figure 4a plots the change of probable δTs against transmittance due to separate transmittance errors in channels 4 and 5, respectively. As shown in Figure 4a, the probable LST estimation error is 0.210°-0.361°C for δτ4=0.01 when τ4 is within the range of 0.805-0.905. The LST error may reach to 0.393°-0.623°C for δτ4=0.02 and 0.695°-1.098°C for δτ4= 0.05 in the same transmittance range. However, the error is below 0.2°C when channel 4 transmittance is below 0.9 and δτ4 is less then 0.005. A similar possible LST estimation error can be seen for the same small change of transmittance in channel 5. Because transmittance is estimated by water vapor content in the atmosphere, the separate error of channel 4 or 5 transmittance is almost impossible. Instead, the transmittance error occurs simultaneously in both channels 4 and 5. Figure 4b shows the possible LST estimation error due to the possible transmittance error simultaneously occurring in both channels 4 and 5. Comparing Figure 4b with Figure 4a, one can find that the possible error of LST estimation due to the simultaneous transmittance error of channels 4 and 5 is much lower than the separate one (Figure 4a). This is because the simultaneous transmittance error of channels 4 and 5 is in

QIN ET AL.: SPLIT WINDOW ALGORITHM AND SENSITIVITY ANALYSIS 1

Trans4=0.905

LST error (C)

0.8

Trans4=0.85 Trans4=0.805

0.6

Trans4=0.704

(a)

0.4

0.2

0 -2

-1.5

-1

-0.5

0

0.5

1

1.5

2

1

1.5

2

1.5

2

1.5

2

T4-T5 (C) 0.7

Trans5=0.851

0.6

LST error (C)

Trans5=0.797 0.5

Trans5=0.706 0.4

(b)

0.3 0.2 0.1 0 -2

-1.5

-1

-0.5

0

0.5

T4-T5 (C) 0.5

Trans4=0.905 Trans5=0.851 Trans4=0.850 Trans5=0.769

0.4

LST error (C)

Trans4=0.805 Trans4=0.706 Trans4=0.704 Trans5=0.572

0.3

(c)

0.2

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0.851 (see Figure 2a). The error is 0.136°-0.297°C for δτ4=δτ5 =0.02 and 0.345°-0.748°C for δτ4=δτ5=0.05 in the same transmittance. The above analysis is based on the assumption of T4T5=0.7. As shown in Figure 3, the difference of brightness temperature T4-T5 is actually in the range of ±3.5°C with above 95% pixels concentrating within ±2°C and above 60% within ±1°C. Therefore the above analysis based on the assumption T4-T5=0.7°C can only represent the average case. In order to evaluate the sensitivity of LST estimation to transmittance, we need to consider a wider range of brightness temperature difference. Figure 5 shows the change of possible LST estimation error with T4-T5 change. When T4-T5 is small, δTs is small in all cases, δTs increases rapidly with T4-T5. Separate transmittance error in channels 4 and 5 causes much higher LST estimation error than does the simultaneous one. Specifically, for a small δτ4=0.005 the δTs might be up to above 0.7°C for τ4≥0.85 and T4-T5≥2°C (Figure 5a). A similar LST error is seen in the transmittance error of channel 5 (Figure 5b). However, the error can be lower than 0.4°C within T4-T5≤2°C for δτ4=δτ5=0.01 (Figure 5c). Higher LST error changes with brightness temperature difference are seen in Figure 5d for δτ4=δτ5=0.05. This indicates that the accurate estimation of transmittance such as with an error ≤0.03 is the essential prerequisite for an accurate retrieval of LST from AVHRR data. Another implication of the result is that the maximal δTs occurs at the pixels with great T4-T5. This conclusion can help us to precisely evaluate the accuracy of LST estimation in a pixel scale when split window algorithm is applied to the real world. When T4-T5 is small such as less than 0.5°C, the application of the split window algorithm still can produce a very accurate LST estimation in spite of a big transmittance error.

0.1

0 -2

-1.5

-1

-0.5

0

0.5

1

T4-T5 (C) 2.5

Trans4=0.905 Trans5=0.851 Trans4=0.850 Trans5=0.769

LST error (C)

2

Trans4=0.805 Trans4=0.706 Trans4=0.704 Trans5=0.572

1.5

(d)

1

0.5

0 -2

-1.5

-1

-0.5

0

0.5

1

T4-T5 (C)

Figure 5. Probable LST estimation error with brightness temperature difference due to possible transmittance errors (a) δτ4=0.01, (b) δτ5=0.01, (c) δτ4=δτ5=0.01, and (d) δτ4=δτ5 =0.05. accordance with the relative identical change of transmittance in the two channels, which can be seen from the regression coefficients listed in Table 4. Therefore, for the simultaneous transmittance error δτ4=δτ5=0.01 the possible LST estimation is 0.068°-0.148°C (Figure 4b) when the average transmittance of channels 4 and 5 is between 0.756 and 0.878, which corresponds to τ4 in the range of 0.805-0.905 and τ5 in 0.706-

Emissivity error

Figure 6. Average LST estimation error due to (a) transmittance error and (b) emissivity error.

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The change of the average LST estimation error with possible transmittance error is shown in Figure 6a. This average LST error is computed under T4-T5=1°C with transmittance range 0.70-0.90 and emissivity range 0.95-0.98. As indicated in Figure 6a, a separate transmittance error 0.05 in channel 4 or 5 may cause an average LST estimation error of up to 0.6°-0.8°C, while the simultaneous error can only produce an average δTs of about 0.4°C. Moreover, the relationship between average δTs and simultaneous transmittance error is almost linear. The average δTs increases from 0.079°C for simultaneous transmittance error δτ4=δτ5=0.01 through 0.16°C for δτ4=δτ5=0.01 and 0.23°C for δτ4=δτ5=0.03 to 0.4°C for δτ4=δτ5=0.05 (Figure 6a). Generally, the measurement of water vapor content for transmittance estimation can reach the accuracy of less than 0.2 g cm-2. As mentioned above, the possible estimation error of transmit-tance due to the measurement error of water vapor content in the atmosphere is generally ≤0.035.

Emissivity in ch. 4

7.2. Sensitivity Analysis to Ground Emissivity The probable error of LST retrieval due to emissivity error has been discussed in various papers such as those of Becker [1987], Li and Becker [1993], and Coll et al. [1994]. On the basis of their algorithm, Li and Becker [1993] estimated the impact of ground emissivity error on probable LST error as a function of average emissivity error δε and emissivity difference error δ(ε4-ε5) between channels 4 and 5. They concluded that the probable LST estimation error may be as large as up to 1.6 K for δε=δ(ε4-ε5)=0.01. Thus a good estimation of ground emissivity is the basis for an accurate estimation of LST from AVHRR data.

Emissivity in ch. 4&5

Temperature (C)

Figure 7. Probable LST estimation error due to simultaneous emissivity errors in both channels 4 and 5, illustrating the change of δTs against (a) emissivity and (b) temperature.

Temperature (C)

Figure 8. Probable LST estimation error due to emissivity errors in channel 4, illustrating the change of δ Ts against (a) emissivity and (b) temperature. The sensitivity of our algorithm to ground emissivity variation is shown in Figures 7 and 8. Several simulations have been done for the sensitivity analysis: probable LST estimation error against the change in emissivity and its difference in the two channels as well as in the change of brightness temperature and its difference in the two channels. All the analyses are oriented to possible emissivity error 0.001, 0.002, 0.003, and 0.005. Figure 7a shows that probable LST estimation error is not very sensitive to the change in ground emissivity but does depend on the possible emissivity error. For a possible emissivity error of 0.001 in both channels 4 and 5 the LST error is less than 0.01°C (ranging from 0.081°C for ε5=0.901 to 0.065°C for ε5=0.991). The change of LST error against ground emissivity is linear, with a decrease rate of 0.0156. Moreover, the probable δTs is also linearly increasing with the possible δε. For an emissivity error of 0.005 the LST error is 5 times the LST error for an emissivity error of 0.001. Therefore, for δε=0.10 the probable δTs is expected to be 0.65°-0.81°C for the emissivity ranging from 0.991to 0.901 (Figure 7a). The LST error is not sensitive to the variation of ground emissivity difference between channels 4 and 5. The probable δTs changes little against ε4-ε5. This implies that emissivity difference error has no significant effect on the accuracy of LST estimation. However, the LST error increases steadily with the emissivity error. For an emissivity error 0.001 the δTs may be 0.07°C, and the error increases to 0.35°C for the emissivity error 0.005. The probable LST estimation error due to emissivity error is slightly dependent on temperature level (Figure 7b). However, the change of δTs in temperature range 0°-70°C is

QIN ET AL.: SPLIT WINDOW ALGORITHM AND SENSITIVITY ANALYSIS within 0.03°C for every error 0.001 of emissivity. Thus, for an emissivity error 0.005 the range of δTs is 0.15°C, changing from 0.27°C at temperature level 0°C to 0.42°C at the level 70°C. Analysis indicates that the probable LST estimation error also has little dependence on the brightness temperature difference T4-T5. Within the most possible range of T4T5=2°C, δTs only increases 0.022°C, which is quite small. The impact of ground emissivity error separately in channels 4 and 5 has also been analyzed on the probable LST estimation error, and the results for channel 4 are shown in Figure 8. An obviously greater LST error due to the separate emissivity error can be seen in Figure 8, compared to the LST error due to the simultaneous one shown in Figure 7. For δε4=0.005 the possible δTs would be about 0.8°C at emissivity 0.96 (Figure 8a) and temperature 35°C (Figure 8b). Ground emissivity difference ε4-ε5 has little effect on the change of δTs for a fixed emissivity error, but the brightness temperature differenceT4-T5 has some effects (about 0.2°C). However, the LST error due to separate emissivity error is always less than 1°C for an emissivity error of up to 0.005. Because the estimation of ground emissivity is simultaneously applied to both channels, an emissivity error of 0.005 rarely appears to be only in one channel. Instead, it is possible for it to appear in both channels. Therefore we can conclude that the accuracy of the algorithm to the probable error in emissivity is quite high for the general purposes of LST estimation. For a comprehensive evaluation of the sensitivity of the algorithm to ground emissivity, the change of average LST estimation error is plotted against emissivity error in Figure 6b. The condition for computing the average LST estimation error is the same as that of Figure 6a. Separate emissivity error in channels 4 and 5 causes much higher average LST error than does the simultaneous one. If emissivity in channel 4 has an error of 0.01, the possible average δTs may be up to 1.72°C. The δTs is only 1.03°C for δε5=0.01. As mentioned above, a separate error in channel 4 or 5 rarely exists, and instead the error usually occurs in a simultaneous way for the two channels. More fortunately, the average LST estimation error is much lower for a simultaneous emissivity error in both channels 4 and 5 than it is for a separate one (Figure 6b). For δε4=δε5=0.01 the average δTs is 0.708°C. This error is about a half of that estimated by Li and Becker [1993].

8. Validation of the Algorithm The above sensitivity analysis aims to assess the effect of possible errors in parameter estimation on the LST retrieval with the algorithm. Validation is also necessary in order to understand how well the retrieved LST with the algorithm matches the actual one in the real world. Two methods were used to validate the algorithm: standard atmospheric simulation and ground truth data sets provided by Prata [1994]. 8.1. Validation through Standard Atmospheric Simulation Since there are many difficulties in conducting the in situ ground truth measurements at the satellite pass, standard atmospheric simulation with such programs as LOWTRAN, MODTRAN, or 6S is an alternative way for validating split window algorithm in remote sensing [Sobrino et al., 1991]. In this study we use LOWTRAN 7 to simulate the required parameters for the validation.

22665

The principles and procedures involved in the validation are as follows. The atmospheric simulation program LOWTRAN 7 is used to simulate the thermal radiance reaching the remote sensor at the satellite level for the input profile data with the known ground thermal properties (LST and emissivity). The required atmospheric quantities such as transmittance for remote sensing of LST can also be computed from the output of the simulation with the program [Kneizys et al., 1988]. The simulated total radiance is then converted, using (2), into the brightness temperature for the two channels of AVHRR, from which the LST can be estimated with split window algorithms. Comparison of the assumed LST used for the simulation with the retrieved one from the brightness temperature enables us to examine the accuracy of various split window algorithms for the true AVHRR data in the real world. A number of situations were designed for the validation. Four land surface temperatures (20°C, 30°C, 40°C, and 50°C) with four correspondent air temperatures (18°C, 23°C, 30°C, and 38°C) near the surface (at 2 m height) were arbitrarily assumed for the simulation. Seven atmospheric profiles were used: USA1976, the tropical 15° N, the subtropical 30° N July and January, and the midlatitude 45° N July and January. For any combination of these temperatures and profiles, five cases of total atmospheric water vapor content (1 g cm-2, 2 g cm-2, 2.5 g cm-2, 3 g cm-2, and 3.5 g cm-2) were considered. Since most natural surfaces of the Earth have an emissivity of 0.95-0.985, four cases of mean ground emissivity (ε=0.951, 0.961, 0.965, 0.971, and 0.981) were used for the simulation. Our algorithm was compared with the 11 proposed split window algorithms found in the literature for the validation. They are the algorithms of Price [1984], Becker and Li [1990], Coll et al. [1994], Sobrino et al. [1991], Prata [1993], França and Cracknell [1994], Ulivieri et al. [1996], Vidal [1991], Prata and Platt [1991], Kerr et al. [1992], and Ottlé and Vidal-Madjar [1992]. Because of the volume limitation of the current study, a detailed presentation of the computation, validation, and comparison of all the algorithms for all above situations will be given in another article. Here we mainly present the results for the midlatitude 45° N summer (July) atmosphere with two emissivity cases (0.96 and 0.97). Table 5 gives the simulation outputs of LOWTRAN 7 for the two cases of total water vapor content of the atmosphere. Comparison of our algorithm with others is given in Tables 6 for two cases of total water vapor content of the atmosphere. The root mean square (RMS) error computed as [Σ(Ts′- Ts)2/N]1/2, where N is the number of samples for the computation, in Table 6 indicates the mean difference of the retrieved LST (denoted as Ts′ in Table 6) from the assumed one for the simulation (denoted as Ts). Table 6 indicates that our algorithm and that of Sobrino et al. [1991] have very low RMS errors for both emissivity cases. The RMS errors of these two algorithms are less than 0.3°C, which confirms the accuracy of these two algorithms in LST retrieval. The validation results for other atmospheres and other cases of total water content and emissivity also demonstrate the generally better performance of these two algorithms. However, our algorithm is better than that of Sobrino et al. [1991] in real application because it requires only two parameters for LST retrieval while the algorithm of Sobrino et al. [1991] needs more parameters.

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QIN ET AL.: SPLIT WINDOW ALGORITHM AND SENSITIVITY ANALYSIS Table 5. Simulation Outputs with LOWTRAN for Midlatitude 45° N Atmospherea Ts °C

ε=0.961

Ta0 °C

T4, K

ε=0.971

T5, K

T4, K

τ4

τ5

Ta K

290.106 298.774 308.071 317.523

0.817 0.830 0.843 0.853

0.723 0.741 0.760 0.768

285.686 290.317 296.801 304.210

289.250 297.209 306.193 315.683

0.691 0.715 0.740 0.759

0.555 0.587 0.619 0.650

285.686 290.317 296.801 304.210

T5, K

w=2 g cm-2 20 30 40 50

18 23 30 38

290.021 299.051 308.500 318.114

289.710 298.320 307.560 316.968

290.476 299.561 309.062 318.725 w=3 g cm-2

20 30 40 50

18 23 30 38 a

289.561 298.069 307.303 316.818

289.002 296.903 305.830 315.260

289.898 298.465 307.756 317.323

Ta0 is air temperature at the surface (2 m height). Ta is mean atmospheric temperature.

The results in Table 6 also indicate that the algorithms of Becker and Li [1990], Kerr et al. [1992], Prata [1993], Price [1984], and Prata and Platt [1991] also work well, in general. The computation of Prata [1993] is complicated and required not only transmittance and emissivity but also ∆I↓ (down-welling radiance difference between the two channels) and the derivative of Planck's function. If the atmospheric transmit-tance is not available, the algorithms of Becker and Li [1990] and Kerr et al. [1992] may be an alternative for

LST retrieval, followed by those of Price [1984] and Prata and Platt [1991], because these algorithms do not consider the variation of transmittance.

8.2. Validation with Ground Truth Data It is very difficult to obtain the in situ ground truth measurements of LST matching the pixel scale (1.1×1.1 km

Table 6. Comparison of Various Algorithms with LST Estimation Error for Various Situationsa SWAs 1 2 3 4 5 6 7 8 9 10 11 12 a

ε

Ts′-Ts in °C for w=2 g cm-2

Ts′-Ts in °C for w=3 g cm-2

Ts=20°C Ts=30°C Ts=40°C Ts=50°C Ts=20°C Ts=30°C Ts=40°C Ts=50°C

RMS error °C

0.961 0.971 0.961 0.971 0.961 0.971 0.961 0.971 0.961 0.971 0.961 0.971 0.961 0.971 0.961 0.971 0.961 0.971 0.961 0.971 0.961 0.971 0.961

0.094 0.078 -1.223 -0.570 0.137 0.269 -1.042 -0.904 -0.114 -0.114 0.469 0.363 -0.005 -0.050 -0.455 -0.373 1.508 2.128 -0.097 -0.095 0.080 0.663 1.056

0.159 0.135 -0.753 -0.064 0.307 0.460 -1.336 -1.124 -0.126 -0.113 0.426 0.334 -0.139 -0.167 -0.667 -0.539 1.708 2.369 0.366 0.304 0.015 0.642 1.190

0.195 0.166 -0.574 0.153 0.343 0.518 -1.476 -1.206 -0.173 -0.144 0.358 0.284 -0.276 -0.283 -0.842 -0.669 1.737 2.44 0.722 0.598 -0.087 0.584 1.286

0.205 0.174 -0.241 0.549 0.535 0.752 -1.408 -1.066 -0.249 -0.199 0.249 0.201 -0.469 -0.438 -0.859 -0.628 1.922 2.687 1.241 1.071 -0.032 0.698 1.543

0.126 0.116 -0.851 -0.218 0.326 0.414 -1.108 -1.019 -0.300 -0.253 0.677 0.539 -0.101 -0.12 -0.468 -0.45 1.738 2.323 0.059 0.014 0.154 0.683 1.131

0.234 0.214 -0.278 0.411 0.462 0.585 -1.385 -1.174 -0.213 -0.169 0.653 0.534 -0.192 -0.196 -0.867 -0.79 1.934 2.579 0.456 0.365 -0.033 0.556 1.141

0.293 0.266 0.016 0.758 0.539 0.697 -1.373 -1.075 -0.197 -0.147 0.587 0.491 -0.322 -0.303 -1.08 -0.947 2.022 2.722 0.841 0.700 -0.138 0.506 1.234

0.328 0.296 -0.156 0.614 0.315 0.490 -1.625 -1.286 -0.228 -0.168 0.524 0.444 -0.407 -0.38 -1.413 -1.240 1.772 2.505 0.948 0.739 -0.441 0.24 1.125

0.217 0.193 0.639 0.477 0.391 0.542 1.355 1.112 0.208 0.169 0.511 0.415 0.241 0.236 0.882 0.752 1.799 2.476 0.709 0.589 0.178 0.588 1.221

0.971

1.650

1.830

1.97

2.287

1.671

1.742

1.891

1.820

1.867

1, our algorithm; 2, Price [1984]; 3, Becker and Li [1990]; 4, Coll et al. [1994]; 5, Sobrino et al. [1991]; 6, Prata [1993]; 7, França and Cracknell [1994]; 8, Ulivieri et al. [1996]; 9, Vidal [1991]; 10, Prata and Platt [1991]; 11, Kerr et al. [1992]; 12, Ottlé and Vidal-Madjar [1992]. Ts is the assumed land surface temperature (LST) for the simulation, and Ts' is the retrieved LST. SWAs, split window algorithms.

QIN ET AL.: SPLIT WINDOW ALGORITHM AND SENSITIVITY ANALYSIS at nadir) of AVHRR data at the satellite pass for the validation of algorithms. Generally speaking, LST varies from point to point on the ground, and ground measurement is generally point measurement. It is a problem to obtain the measured LST matching the pixel scale of AVHRR data. On the other hand, AVHRR observes the ground at different angles, and precisely locating the pixel of the measured ground in AVHRR data especially night images is also a problem. In addition to these difficulties, ground emissivity and the in situ atmospheric conditions have also to be known for the validation. Since there are so many difficulties in obtaining ground truth data, validation using this method is almost unfeasible. However, Prata [1994] overcame many of these difficulties and managed to provide a data set of the two Australian semiarid sites for the period of 1990-1992. The first site (35°12′S, 142°36′E) was a wheat field near the town of Walpeup, about 350 km to the northwest of Melbourne. The second site (23°24′S, 145°18′E) was a pastureland near the town of Hay in New South Wales. Prata [1994] estimated the ground LST (Tg) and vegetation LST (Tv) of the two sites from a number of measurements and provided the corresponding brightness temperatures T4 and T5 of the two AVHRR thermal channels for each good image during the period. He also provided the mean emissivity of the soils of the sites. However, his data set does not contain the required in situ water vapor contents for each measurement. Instead, only the monthly mean water vapor contents are available in the data set. Unfortunately, Prata [1994] did not directly give the mixed LST (Ts) of his measurements for the corresponding brightness temperatures. Instead, he presented Tg and Tv. In order to use this data set for validation, first we have to estimate Ts from Tg and Tv. In the Walpeup site the field was not always covered with wheat crop. It was fallow in some months, when we use Ts=Tg. For the wheat growing period we estimate Ts=PvTv+(1-Pv)Tg, in which Pv is vegetation cover rate. As Prata [1994] pointed out, the wheat crop did not fully cover the ground even though it was in mature. Therefore we arbitrarily use Pv=0.8 for the wheat growing period. The same method is used to estimate Ts in Hay site. Ground emissivity and atmospheric transmittance are generally required for algorithms to retrieve LST from AVHRR data. The available monthly water vapor content was used to estimate the atmospheric transmittance for all the

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measurements of that month. This is not good for a precise LST retrieval but is better than using a constant transmittance for the whole data set. The soil in the Walpeup site is mainly sandy soil. Prata [1994] gave it the mean emissivity of 0.955 for 11 µm. He also gave the emissivity of growing wheat as 0.976 for the same wavelength. Since the central wavelength of AVHRR channel 4 is 10.9 µm, we use ε4=0.955 for the bare ground and ε4=0.976 for the wheat growing period at the Walpeup site. For AVHRR channel 5 we use ε5=ε4+0.004. The Hay site is a vegetation site and Prata [1994] gave it the mixed emissivities of 0.978 and 0.982 for 11 and 12 µm, respectively, which serve as ε4 and ε5 in the validation. Viewing angle is also required for LST estimation. Unfortunately, Prata [1994] also did not present the viewing angle for the corresponding AVHRR data. Since the water vapor content in the atmosphere was very low (