REMOTE SENS. ENVIRON. 32:17- 33 (1990)

Temperature-Independent Spectral Indices in Thermal Infrared Bands

Frarqois Becker and Zhao-Liang Li Groupement Scientilfique de Te%d&ection Spatial (GSTS), LSlT/ENSPS, Strasbourg, France

I n order to perjorm spectral analysis in the ther- mal infrared bands, temperature-independent ther- mal infrared spectral indices (TlSl) are derived from observable thermal infrared radiances. Being temperature-independent and simply related to spectral emissivities of the observed surface, these indices are as easy to use to pelform spectral analysis as rejlectances in the visible and near infrared spectral domains. Examples of such in- dices are given for TlMS and AVHRR. Several properties of these indices are discussed, particu- larly their sensitivity to spectral emissivity varia- tion, their eficient combination, and the derivation of relative spectral emissivity. These indices are quite complementary to NDVl and can lead to stronger results if used together rather than sepa- rately. For instance, in the examples worked out, these indices are more sensitive to bare soils char- acteristics than NDVl. Furthermore, these indices can be tailored to weight certain bands more heav- ily than others, giving to these indices a wide range of application. They may, for instance, be used for geologic compositional mapping. Practical exam-

Address correspondence to Dr. Fraqois Becker, Groupement

Scientifique de T&dBtection Spatial, LSIT/ENSPS, 7 rue de 1’Uni-

versit6, 6700 Strasbourg, France. Received 23 October 1989; revised 17 April 1990.

0034-4257/ 90 / $3.50

OElseoier Science Publishing Co. Inc., 1990

655 Avenue of the Americas, New York, NY 10010

ples from TlMS and AVHRR data are shown and briefEy discussed with a possible applic 4 tions to the determination from AVHRR data of the sur_hace temperature and sugace emissivity.

INTRODUCTION

The thermal infrared spectral region (8-13 pm> is particularly interesting for earth observation.

1.

2. 3.

4.

This spectral region is one of the most trans- parent region for the atmosphere.

It can be used both night and day.

It contains the maximum thermal emission at

terrestrial temperatures, allowing the infer-

ence of heat budgets and earth response to

solar irradiation. It contains the important reststrahlen bands

which result from interatomic stretching vibra-

tions of silicon and oxygen bound of the sili-

cates (Hunt, 1980), yielding strong spectral variation of the emissivity.

In fact, these channels have already been

widely used in most of the domains of application

of remote sensing. For example, in oceanography sea surface temperature have been measured

(Prabhakara et al., 1974; McClain et al., 1985; Walton, 19881, HCMM (Price, 1986) has been

17

18 Becker and Li

launched to explore the value of day-night tem- perature differences, and in geology, rock of dif- ferent mineral types have been discriminated (Hovis et al., 1968; Vincent and Thomson, 1972; Kahle et al., 1980; Kahle and Goetz, 1983; Gillespie et al., 1986; 1987).

However, one of the most important con- straints, which limited the use and applications of the spectral analysis in the thermal infrared bands, is their dependence on the surface temperature, making difficult the discrimination between ther- mal and spectral effects on thermal infrared radi- ances, as has been again recently recalled by Gillespie et al. (1987). It is therefore of great interest to find a procedure which leads to a spectral analysis which is independent of surface temperature. Several attempts have already been made. For instance, Kahle et al. (1980), using six-channel thermal infrared data from the Bendix 24 channels scanner, and Kahle and Goetz (1983), using TIMS data, showed that it is possible to reduce the correlations with surface temperature and then to separate several rocks and ground- cover types by their spectral emittance differences with a principal-component transformation tech- nique introduced by Soha and Schwartz (1978), or by HSI transformations (Gillespie et al., 1986; 1987). However, all the previous approaches am- plify the spectral emittance differences present in the scene by image processing techniques and are therefore qualitative in nature; on a strong physi- cal basis, they do not separate surface emissivity from surface temperature effects. These methods may therefore be very difficult to use for temporal analysis of the observed phenomena.

In order to fully use the spectral discriminat- ing power of the thermal infrared bands, it is necessary to define indices which are independent of surface temperature and can be used to charac- terize the types of surface, whatever the time and the temperature. In the present paper, we shall describe an approach which eliminates, on a phys- ical basis, the effect of surface temperature, what we can call temperature-independent thermal in- frared spectral indices (TISI). This approach is described in the next section and some of the properties of those indices are discussed in the third section. In the fourth section we present some practical applications of these indices and discuss their interest for future work.

DEFINITION OF THERMAL INFRARED SPECTRAL INDICES INDEPENDENT OF SURFACE TEMPERATURE

Approximate Expression for the Channel Radiances

On the basis of radiative transfer equation, the spectral radiance 1A detected from space at wave- length A for a cloud-free atmosphere under local thermodynamic equilibrium can be written as

I~ = a a * r x + Rat a r (la)

where

T A

R a t a ,~

Ra

in which

= the total spectral transmission of at- mosphere,

= the upwelling atmospheric radiance,

= the spectral radiance observed at ground level given by

= + ( 1 - ( l b )

e a is the spectral emissivity, T s is the surface temperature, Rat a + is the downwelling atmospheric

radiance at the ground level, Ba(T) is the radiation planck function.

In order to achieve reasonable signal to noise ratio, the instruments on board measure outgoing radiation with finite banded radiometers having the characteristic response function fi(A). The channel radiance I i in channel i is therefore a weighted average given by

I, = fy,(~) da (2)

Inserting Eq. (la) into Eq. (2), it is possible to write with a good approximation

I i = R i * T i -}- R a t i ~ , (3)

where R i is the channel radiance observed in channel i at ground level which is given by

R, = < B , ( L ) + ( 1 - e , )R~ . ~ . (4)

In these expressions, the channel quantities R i, ri, Rati "~, ~i, Bi(Ts), and Rau ~ are weighted average of the corresponding spectral quantities defined in the same manner as I i in Eq. (2).

Temperature-Independent Spectral Indices 1 9

The channel brightness temperature T i and surface brightness temperature T~i in channel i are respectively defined from I i and R i as

I i = B i (T i ) and R i = Bi(Tgi ). (5)

From these equations, it is also possible to define the weighted average wavelength aio of channel i (which depends on T) by approximating B i ( T ) by a Planck function at wavelength Aio according to

B i ( r ) -- Ba io (T ) . (6)

In order to determine the radiance or the brightness temperature measured on the surface from the space measurements, Eq. (3) shows that atmospheric corrections have to be performed. A number of methods have been developed with a single infrared channel (see, for instance, Scott and Chedin, 1981; Kneizys et al., 1983; Price, 1983; Susskind et al., 1984; Chedin et al., 1985). In this paper, we used the Lowtran-6 atmospheric path radiance model developed at the Air Force Cambridge Geophysics Laboratory (Kneizys et al., 1983).

Since we look for a thermal infrared index which is independent of surface temperature, we must introduce a dependence of the channel radi- ance on the surface temperature which can be easily handled in ratioing procedure. For modest temperature variations around a reference temper- ature T o, it has been shown by Hardy et al. (1934) and Slater (1980) that the Planek function Ba(T) can be written with a good approximation as

Ba(T) --- a a ( T o ) T "*'r°) (7)

Assuming that, for T = T o, the approximate expression (7) and its derivative are respectively equal to the exact Planck function and its deriva- tive, expressions for n,(T o) and aa(T 0) are easily

derived and read as

na ( ro )= ~ 1+ e x p ( C 2 / a r o ) - i '

B,(T0) rg (To), (8b)

with C 2 = 1.4388 × 104/~mK Therefore, from Eqs. (5) and (7) the channel

radiance R i can be approximated around a refer- enee temperature Tg o by

R, =- aiT~",', (9a)

where a i and n i are channel constants which depend on Tg o. These constants have been deter- mined using a least squares fit method in order to solve the equation

B,( T~,) = <T~i ' . (9b)

Expression (9a) is somehow a channel extension of the approximation (7) which turns out to be quite accurate as checked in Table 1 for Channels 3, 4, and 5 of TIMS and of AVHRR radiometers (de- fined in the fourth section) using the spectral response f i(a) respectively given by Palluconi et al. (1985) for TIMS and by C. Ottl6 (private communication) for AVHRR. [As emphasized by F. Palluconi (private communication), those func- tions fi(A) of TIMS may present some changes in actual use of TIMS.] The values of the root mean square errors ATrm s and maximum errors ATmR x resulting from approximation (9a) shown in Table 1 are sufficiently small to validate the concept for a temperature ranging from 285 K to 325 K.

From expression (6) approximate expressions for n i and a i may be derived also by replacing )t by aio in Eqs. (8a) and (8b). It is therefore possi- ble to compute ni and oq using these expressions [or graphs given by Price (1989)]. Results are

Table 1. Comparison between Results of Approximate Formula (9b) (Columns Having Symbol 1) and Results of Formulae (8a) and (8b) (Columns Having Symbol 2) for a Temperature Ranging from 285 K to 325 K for Channels 3, 4, and 5 of TIMS and Channels of 3, 4, and 5 of AVHRR

h o ni ai ATrm, ATmax

Channel (Ix m) 1 2 1 2 1 2 1 2

3 9.210 5.162 5.153 1.3693 × 10- is 1.4445 X 10- is 0.2 0.3 0.4 0.6 4 9.895 4.816 4.808 1.1385 X 10-17 1.1977 X 10-17 0.2 0.3 0.4 0.6 5 10.687 4.474 4.468 9.1632 X i0-17 9.5009 X 10-17 0.2 0.3 0.4 0.6 3 3.733 1 2 . 5 9 7 1 2 . 6 3 7 3.7306 X 10 -a9 3.0000 X 10 -39 0.2 0.3 0.4 0.7 4 10.746 4,445 4.445 1.0878 X 10-16 1.0924 X 10- m 0.2 0.3 0.4 0.7 5 11.790 4.067 4.075 1.0630×10 - i s 1.0190× 10 -15 0.2 0.3 0.4 0.7

2 0 Becker and Li

compared with the previous ones on Table 1. Both results are similar.

Using the approximation (9a), it is therefore possible to rewrite expression (4) as

R i = e e i T ; ' / = e i a , T ~ " ' + ( 1 - e , ) R ~ t , ~ . (10)

Two-Channel Temperature-Independent Thermal Infrared Spectral Indices

If, for each channel radiance R~, the atmospheric reflected radiance could be neglected, expression (10) could be written

Ri

o r

= e iee iL" ' (11a)

T~;'= e ,_~.T"' . ( l i b )

From these expressions it is very simple to obtain temperature-independent thermal infrared indices with two channels i and j by ratioing the two-channel radiances or brightness temperatures with adequately chosen powers a~ and a j, namely,

R? "' "~ E i O~ i o, o, (12a)

I 8 j OLj

or

with

£lii l t i ( l I

TO,,,, ,,, (12b) ~j 6j-

a i n i - a j n j = 0. (13)

(The ratios generally used until now were simple ratios with a~ = % = 1, leading to a temperature- dependent index.)

The effect of the reflected atmospheric radi- ance is, in fact, very small as will be discussed below. However, neglecting this effect would in- duce on the emissivity deduced from these ratios a relative error given by

Ae, (1 - ei)R~,ti ~. <B,(L) (14)

For 77~ = 300 K and the Channel 5 of TIMS data which we are analyzing, the typical values of these errors according to the variation of emissiv- ity are given in Table 2.

Therefore, in order to take care of this effect to first order, we rewrite Eqs. (11) as

I1 i R i = e i a ( r ~ C,

Table 2. h n p a c t of A t m o s p h e r i c Ref lec ted Radiat ion on C h a n n e l Emiss iv i ty D e t e r m i n a t i o n

e 1.0 0.95 0.90 0.8.5 0.80

Ae/~" 0 0.017 0.036 0.057 0.081

o r

where

T".i=eiT]"iCi

n a t i ~, C, = 1+/3, * ( 1 - e~)/e , with/3~ = B , ( L ) .

a i a~ This leads to a correction Factor C i / C , - in ex- pressions (12a) and (12b).

Since there are two unknowns, namely, a~ and a; for one equation [Eq. (13)], several solutions can be obtained from these equation. One particu- lar obvious solution is

a i = 1 / n i and a; = 1 / n a .

This solution has the advantage to lead to the simple ratio of brightness temperature, but leads to complicated powers to both emissivities. We propose to introduce a solution with one of the powers equal to 1, namely, a i = 1 and therefore, from Eq. (13), aj = n , / n ; = ni j . We therefore de- fine a temperature-independent thermal infrared spectral index TISIi.s by

"' R , R , TISIia = oe, , • _ _ = Mij • _ _ (15a)

RT,, RT,,

o r

TISI,, = (15b)

From Eqs. (12a) or (12b), this index would be equal to

TISIEi; = e , / e j '~,, (16)

which is obviously independent of Z,.. Taking account of the reflected atmospheric

radiance leads obviously to

TISI 0 = TISIE o * C 0 (17)

with

% = c , /C;". (18)

Theoretically, C, i depends on emissivity, at- mospheric condition, and surface temperature, be- ing strictly equal to 1 when Z~ --+ c¢ or when e i =

Temperature-Independent Spectral Indices 21

T a : 3 0 2 K 0 TISf r l TISIE

1 . . . . . . . . . . . . . . . . . . . . . . . . .

.981

.9s~ •941

92~ 92

. 8 8 1

.s62

.e42 s21

. 8 • , . . . , . , . . . . . , . . . . . . . . .

300 302 304 306 308 310 312 314 316 318 320 322 324 Ts

Figure 1. Comparison between the variation with suriCaee temperature T~ of the TISI obtained from simulated data using formula (19b) and of the asymptotic value TISIE ob- tained using formula (24). This figure shows clearly that TISI is a constant and very close to TISIE.

ej = 1. TISIE is therefore the asymptotic value of TISI. However, it is worth noting that this varia- tion is very small and can be neglected with good accuracy as it will be shown below (see Fig. 1). Therefore, assuming that Cij is independent of surface temperature and is equal to 1 makes good approximations for a large range of emissivity and surface temperature.

These TISI are indeed not unique and all the TISI may be obtained by raising one of them to any arbitrary power. As an example, had we cho- sen a i = l / n i and a . = l / n . , we would have

J 1 d obtained TISI~ = (TISI~j) ( /" ' ) .

N-Channel Temperature-Independent Thermal Infrared Spectral Indices

As it will be discussed below, it is very interesting to extend the TISI to more than two spectral channels. For N channels, it is straightforward to extend formulae (15) and (13) by giving the ak's algebraic values (which means that the ak's can take on negative values). In that case the TISI, can be defined as

N

T I S I = I--[ Taknk (19a) ~ak k = l

o r

N

TISI = M 1-I R~ k (19b) k = l

with N

E a , . , = 0, (20) k = l

and M a constant given by

N N

M = 1-I Bk(r~)-"'-~ I-I Ol/ak" (21) k = l k = l

One should notice that at least one of the ak's is necessarily negative in order to fulfill Eq. (20), because all the nk's are positive.

With the same reasoning as in the previous subsection, TISI 's are equal to

N

TISI = C * I--I e~, ~, (22) k = l

where C is given by generalizing equation (18), which reads

N

C = 1--I c~, . (23) k = l

Similarly, from (22), we define TISIE by an expression generalizing (16), namely,

N

TISIE = I--[ e~ ~, (24) k = l

which means that (22) can be rewritten in the same form as expression (17).

As previously discussed and as will be shown below, C is generally very close to 1 and is almost independent of surface temperature T s which means that

TISI ~ TISIE.

Here again, Eq. (20) has many solutions for the ak's, even if we choose one of the powers to be equal to 1. This flexibility is very interesting be- cause it allows weighting certain bands more heav- ily than others and therefore tailoring the sensibil- ity of these TISIs as will be discussed in the next section. Two solutions are displayed as examples in Table 3 for three channels. For instance, for TIMS, we impose for the first solution a 3 = 1 and a z = 1, while we impose a 4 = 1 and a z = 1 for the second solution.

It should be noted that, though the values of n i obtained by the two methods discussed in this section and given in Table 1 are different, the coefficients for constructing the TISIs are the same whatever the methods.

22 Becker and Li

Table 3. E x a m p l e s of T w o S imp le So lu t ions of P o w e r s a k D e t e r m i n e d U s i n g Eq . (20) for C h a n n e l s 3, 4, a n d 5 of T I M S and C h a n n e l s 3, 4, a n d 5 o f A V H R R

Solutions Parameters TIMS A VHRR

1 ¢t :~ 1 l

TISI I a ~ - 2 - I a : , 1 - 2

2 a~ - 1.8 1 TISI 2 a ~ I -:3.75

a 5 1 I

Validation of the Concept of TISI--Effect of Reflexion

It is easy to check that TISIs are actually indepen- dent of the surface temperature. For this valida- tion, we considered a situation actually observed during HAPEX (Andr6 et al., 1986) at the Castel- nau test site with the radiometer TIMS. The ac- tual channel emissivities (e l = 0.853, e 2 = 0.856, e 3 = 0.866, e 4 = 0.933, e s = 0.946, e~ = 0.946) have been computed with formula (2) using the spectral values of emissivity measured by N e r o et al. (1988) lbr bare soil at the Castelnau site. From these values, we computed the actual TISI from definition (19) with the ak's given in Table 3 and the channel radiance calculated from expression (4) for the spectral response of the radiometer TIMS. In order to calculate these radiances and to check the impact of T s on TISI, we inserted in expression (4), for each channel, the channel emis- sivity given above and the downwelling radiance computed with Lowtran 6 using the actual atmo- spheric radiosoundings measured during HAPEX experiment (Brutsaert, private comnmnication) and we varied the surface temperature Z~.- The actual dependence of TISI on the surface temperature is displayed in Figure 1. The line with small circles corresponds to the actual TISI values, while the line with small squares is the value of TISIE which does not depend on T~ as it should be.

Amplification In order to amplify the variation of TISI with Z~, we introduce the amplified TISI by

ATISI = (TISI - V) * A, (25)

where A is an amplification factor and V is a value close to TISI. Figure 2 shows the amplification of TISI displayed in Figure 1, with V = TISIE = 0.941 and A = 100. Figure 2 confirms that the values of TISI and TISIE are close, that the

1 3 V=TISIE A=100

1 2 5

115

11

~ I 05

1

95 9

85

8 300"3;2'3& "a&'3&'3;0"3;2 '3;4"3;6 'a;8"3~0'3~2"324

Ts

Figure 2. Ampl i f ica t ion of T I S I I o f F i g u r e 1 a c c o r d i n g to formula (25) with V = TISIE = 0.941 and A = 100.

difference between these values goes to zero when T~. increases, and that the variation of TISI with T~. is vel y small (of the order of 0.4%), ranging from 0.953 to 0.949. From expressions (22) and (24), this difference is given by

T I S I - TISIE = TISIE(C - 1),

showing that (C - 1 ) varies from 0.013 tcr 0.009. We can conclude that:

i. TISIs are almost independent of surface tem- perature and therefore that they will have about the same spectral properties as TISIE.

ii. The difference between TISI and TISIE is small (of the order of 1%) and is even smaller when surface temperature increases, the atmo- sphere is clearer, and emissivities approach 1.

iii. The TISI are very easy to compute directly from the channel radiances and do not need any knowledge of the spectral emissivities nor of the surface temperature.

SOME PROPERTIES OF TISI

Tuning the Weight of the Channel Emissivities: Interest of TISI with More than Two Channels

The power a k introduced in the definition of TISI has an interesting physical meaning. They indicate the weight of a channel in the variation of TISI with this channel emissivity as it is easily seen. We obtain by differentiating expression (24)

N A/~k ATISI = y , at- e~. TISI k = J

Assuming that in an image all the channel emissivities but e k are constant, it is easy to show

Temperature-Independent Spectral Indices 23

that a k is equal to the relative variation of TISI with e k and is therefore the sensibility of TISI to emissivity in channel k.

Using a two-channel TISI, one obtains

ATISI Aei Ae;

TISI = a~ e~ + aj e---j-. '

with ain i + a~nj = 0, i.e., a i / a j = - n j / n i. This means that the relative weight of these

emissivities is imposed by the physics. It is not possible in that case to tune one-channel stronger than the other one. Contrariwise, if one uses a three-channel TISI, then one obtains

A(TISI) Ae i A % AE k = + a . - - + a k , TISI ai e~ J e~ e k

with a~n~ + a~nj + akn k = O. In that case, it is possible to choose one of the

ak's to be equal to 1 and to choose the ratio of two ak'S. If, for instance, e k is almost constant on the image, then Ae k = 0 and the corresponding TISIs are equivalent to a two-channel index for which it is possible to choose arbitrarily the ratio a i / % . Examples are shown in Table 3 for TIMS and AVHRR. For instance, for TIMS indices, solution 1 amplifies the sensitivity to e 4 whereas solution 2 amplifies the sensitivity to e 3.

These indices can therefore be tailored to weight certain bands more heavily than others as will be shown in the next section. This gives these indices a wide range of applications.

The Temperature Independence of TISI Increases Its Sensibility to Emissivity Variation

The temperature independence of TISI increases its sensibility with respect to surface characteristic variations. This can be quantitatively estimated on a simple example with two channels. For instance, the variation with temperature of the simple ratio R34 = R ( T g 3 ) / R ( T g 4) proposed by Vincent and Thomson (1972) and discussed by Gillespie et al. (1987) can be compared with that of TISIa4 de- fined by expression (15a). These variations are quite different as it is seen in Figure 3. The range of variation of the ratio is about 5% for the range of value actually observed for the surface tempera- ture. This range is very high compared with the variation of TISI which has been shown to be of the order of 0.4%. The indices TISI increase

0 TISI [ ] Ratio .95 . . . . . . . . . . . . . . . .

.945

.94

.93[

.93

.925

.92

.915

.91

.905

.9 285 290 295 300 305 310 315 320 325

TS

Figure 3. Comparison between the variations with surface temperature of TISI34 and of the ratio R(Tg3) / R(Tg4).

therefore the sensibility of discrimination of rela- tive emissivity by a factor of 5-10, which is of great interest.

A quantitative comparison can be done by calculating the relative variations of the ratio R34 and of TISI34 from their definitions [see formulae (l la), (16), and Table 1). This is left to the reader.

TISI Algebra-Extended TISI

Any product or ratio of temperature-independent index given by (19) is also a temperature-indepen- dent index having all the properties of equation (19). The demonstration is straightforward and is left to the reader. For instance, for TIMS indices given in Table 3 and formula (16) with ni~ = n i / n j , it is easy to show that

TISI~ .s • TISI 2 = (TISI54) 2's.

On the other hand, the sum or difference of TISIs is also a temperature-independent index, but does not have the simple form (19). By exten- sion, we shall also call them TISI. For instance,

E 3 E 4 TISI35 +TISI45 = e~-- ~ + e~4---l~.

This index is indeed independent of surface tem- perature, and is the analogous to the sum or difference of reflectances widely used in visible and near infrared domains.

In conclusion, since any function of TISI is also independent of temperature, it can be called a temperature-independent thermal index.

Relative Spectral Emissivities

The TISI indices are also very useful to obtain from satellite or aircraft relative spectral emissivi-

24 Becker and Li

ties, i.e., knowing the downwelling atmospheric radiances in all channels and the emissivity e,. in the reference channel r. Using two channel in- dices, one gets, from expressions (16) and (17), for each channel emissivity e~,

TISIi~ (26a) ltil Ei ---~ Er Ci r ,

where er is the reference emissivity in channel r and

f i r 1-/3~

8"'r" *TISI,r]

(26b)

In that case, it is important to include the reflected atmospheric radiation. The error intro- duced in the emissivity by assuming Cir =-1 is given by

Aei ACir E i Cir

Actual values obtained with TIMS data taking the emissivity in Channel 5 as a reference (e 5 = 0.90) are given in Table 4.

PRACTICAL USES A N D EXAMPLES FOR TIMS A N D AVHRR

The temperature-independent thermal infrared in- dices, TISIs, have many applications. We just give here some examples from TIMS and AVHRR data. In the thermal infrared bands, AVHRR has three channels, 3, 4, and 5 respectively, defined by the spectral bands 3.55-3.93/xm, 10.5-11.5/xm, and 11.5-12.5 /~m, whereas TIMS (Palluconi and Meeks, 1985) is a six-channel scanner with the following spectral bands: 8.2-8.6/xm, 8.6-9.0/xm, 9.0-9.4 /zm, 9.6-10.2 /~m, 10.3-11.3 /xm and 11.3-11.7/xm.

Table 4. Effect of Cir on the Relative Emissivity Retrieved from TISIi~ Using Formula (26a) with e 5 = 0.900 as Refer- ence Emissivity

Channel Relative Emissivity

3 4 5

Cir = 1 0.804 0.886 0.900 Actual C~r 0.778 0.876 0.900

Determination of TISI for TIMS and AVHRR

Atmospheric Corrections In order to obtain TISI from space or aircraft measurements, it is necessary to perform atmo- spheric correction in order to transform the chan- nel brightness temperature T i or channel radiance I i measured on board into the channel surface brightness temperature T~i or channel radiance R i at ground level as required by both expres- sions (19).

Since we need to correct for many pixels, it is not possible to compute the atmospheric cor- rections pixel by pixel. We therefore used the atmospheric corrections method based on the lin- earization of both the planck's radiation and the radiative transfer equation (3) which reads as a linear relation between a brightness temperature measured on the ground, T~i, and the brightness temperature Ti, observed on board in channel i (see Becket, 1987):

( AiW ) A i W O i T~i = 1 + cos(/9) Ti cos(/9) L + - - oos(/9) '

(27a)

where

7". = an effective air temperature,

A i= a weighted absorption factor in channel i,

W = the weighted water vapor content,

/9 = the zenith angle,

D i = a residual term which contains the effects of absorption due to the aerosol and the other gases, and the residues of the linear approxi- mations.

In fact, Eq. (27a) can be rewritten as

( ai ) bi Tei = 1 + cos(/9) T~ cos(/9)' (27b)

where a i and b~, which are constant for a given atmospheric state, have been determined by a least squares method using the data simulated by the Lowtran-6 model with the actual radiosound- ings over the Lubbon site for a temperature rang- ing from 295 K to 335 K. Such a method is well known and has been used for instance by Price (1983).

The values of a i and b i for Channels 3, 4, and 5 of TIMS and for the actual atmospheric condi-

Temperature-Independent Spectral Indices 25

Table 5. Accuracy of Retr ieved Brightness Surface Tempera- ture Tg i Using Approximate Formula (27b) a

m a x

Channel a i b i (°K) ATzi (°K) AT~i (°K)

3 0.1617 46.962 0.10 0.20 4 0.1760 51.138 0.10 0.14 5 0.1679 48.864 0.09 0.14

"Coefficients a i and b i determined using a least squares fit method for data simulated by Lowtran-6 with the actual radiosound- ings, for a temperature ranging from 295 K to 335 K for Channels 3, 4, and 5 of TIMS.

not performed, the errors on TISI would be of the order of 2% for the example given in Table 6. Using, however, a standard atmosphere for correc- tion reduces this error to 0.4%, showing, at least on that example, that TISI concept can be used with standard atmospheric correction or with the atmospheric parameters retrieved from the satel- lite onboard soundings if radiosoundings are not available.

tion are given in Table 5. This table shows that approximation (27b) is quite accurate for the situa- tion considered since ATg i (rms) is of the order 0.1 K. It should be noted here that the calibration is not good for TIMS instrument as mentioned by F. Palluconi (private communication), but this does not strongly affect the band to band brightness temperatures.

Once atmospheric corrections are accom- plished, we have the Tg i or R i for all channels, and we can compute the indices of interest using any of the expression (19). As discussed previ- ously, the dominant terms in atmospheric correc- tion are the upwelling radiation [Rat i ~ in Eq. (3)] and the transmission ~'i. If these corrections were

Choice of the Channel k As mentioned above, the thermal infrared spectral index depends not only upon the individual chan- nel emissivity, but also upon the combination of channels. Since the spectral dependence of emis- sivities is characteristic of surface properties, it is possible to seek for the indices which are the most sensitive to a given peculiarity of the surface by choosing the most sensitive combination of chan- nels and then determining the correspondent val- ues of a k from Eq. (20).

Vicker and Lyon (1967) have derived spectral emissivity signatures from laboratory reflectance measurements of various rock types in the 8-13 /zm region; they pointed 9ut that silicate rocks

Table 6. Illustration of Effect of Upwell ing Atmospheric Radiation on TISI a

T I S I E 0.941

Angle 0 ° 40 °

T, (°K) 300 310 320 300 310 320 TISI b 0.954 0.955 0.956 0.956 0.958 0.959 TISI T 0.941 0.941 0.941 0.941 0.941 0.941 TISI~ 0.939 0.940 0.942 0.938 0.940 0.942

QTISII, is obtained with the channel brightness temperature T i (i.e., without atmospheric correction). TISI r and TISI~ are the TISI obtained with the channel brightness temperature Tai obtained respectively using the actual radiosoundings and the atmospheric standard model (midlatitude summer in Lowtran-6).

.98.

.96.

.941

.921 .9. .~>

.~ .88. E uJ .86.

.84.

.82.

Lubbon site Oa ng

<>d + f

.",h

.8

Channel

Figure 4. The variation of surface relative channel emissivi - ties obtained from TIMS data at Lubbon test site when e~ = 0.96: g ,h , f ) bare soil surface samples; a,d) vegetation covers and forest samples.

26 Becker and Li

.95 .9

=,. ,85

• .8 '~,

, 7

. 6

1 . . . . . . . . . . . . . . . . . . . . . . . . . .

/ , o . . . . oooo . . . . . . . ~ #+ / o 6,,Kzel~lllill a ~ ~ o / ,°°......¢I'++ +++

4- o oO ++

-, :v • +~ Os l c

a . n m , . ~ . a c t l ~ a W ~ r+ e l e h m 2

. . - , , . , • . • , , , , , , , . . , . . , , . , , ,

3.2 3 4 3.G 3,8 4 4 .2 4,4 4.G 4.8 5 5 2 5 .4 5 G

(a)

1 * ' . . . . . ' _ _ ' _ - 2 ~ ' ' 2 . - ~ '

. _ _ ! , ~ .

- ~ . 7 5 q ~ t~E zx tx o

o~ o 0$1C. . 7 t x a

o o

,¢. Loe3s G i:~o L~ Ss nd

55 7 " ~ ' ,; ' i~ ' i ', ' G ' & " ~:~ ' ,~

(b)

Figure 5. Absolute spectral emissivity in thermal infrared region for four mineral types measured in laboratory by Nerry et al. (1988): a) in the 3-5.5/zm region; b) in the 8-13.5/~m region. In these figures: (o) sic, the SiC powder with cali- brated granulometry of 45/zm; (e) Lehm2, a variety of loess, surface agricultural soil; (+) Loess, the same kind of soil as Lehm2, but with more fine grain (< 2/zm); (zx) Sand is the quartz sand (SiO 2) material and low carbonate content with granulometry around 100/zm,

have a broad minimum in emissivity between 8 /xm and 11 /~m, the position of which depends upon the rock composition and texture and is due to the so-called reststrahlen effect. Using the method proposed in Becker and Li (1990) and assuming that the emissivity in Channel 5 has a value close to the value measured by Nerry et al. (1988), i.e., e 5 = 0.96, we have derived the surface emissivity of various part of the HAPEX test site from the TIMS data; the results are shown in Figure 4. Similarly, Nerry et al. (1988) and Nerry (1988) showed from Laboratory measurements of various soil samples that the spectral emissivity in the band 3-5 .5 /xm, which contains Channel 3 of AVHRR, is generally much more specific of vari- ous materials than in the 8-12 ~ m band (Chan- nels 4 and 5 of AVHRR) as illustrated in Figure 5. Therefore, it is interesting to use Channels 3, 4, and 5 of TIMS and Channels 3, 4, and 5 of AVHRR to discriminate among rocks, lands, and soils.

Calculation of the Indices The indices which we shall briefly use as examples are then calculated by computing the power a k as explained in the second section and making a particular choice in order to weigh either Channel 3 or Channel 4 for TIMS. They are displayed in Table 3.

TIMS Image from HAPEX

The spectral radiometer TIMS has been flown on board NASA C-130 during HAPEX (Andr6 et al., 1986) in the southwestern France in summer of 1986. Figure 6 presents an aerial infrared photo- graph of Lubbon test site provided by T. Schmugge (private communication) taken si- multaneously with TIMS data. This test site is a large clearing in the northeast part of the Landes pine forest; it provides a large plot of sandy soil, artificially drained, and cultivated with oats and corn. The index TISI 1 has been computed (see Table 3) over this same area and is displayed in Figure 7.

Since e:~ << e~ [br bare soils and e.~-= e~ for vegetation (see Fig. 4), bare soils correspond to low TISI l values and are displayed in white, while vegetated areas with high TISI 1 appear in dark. From this image various types of bare surface may be distinguished, each grey tone in the image presents either different types of surface or dif- ferent humidity states of the same surface. It is worth noticing, as expected, that TISIs are more sensitive to bare soil characteristics than to vegeta- tion. The dynamics of TISIs over bare soils range from 0.70 to 0.99 while over vegetation it ranges only from 0.98 to 1.01. This shows the interesting complementarities of TISI and NDVI. Further- more, one cheeks easily in Figure 8 displaying the bidimensional histogram TISI(T s) that TISI is not directly correlated to T~ as expected. Here the general behavior of this histogram corresponds to the fact that at noon, the time of acquisition of the image, the surface temperature is generally smaller for vegetation area than for bare soils. Here again, there is an interesting complementarity with NDVI.

AVHRR Data The index TISI45 strongly amplifed [using expres- sion (25)] has been calculated from the night data

Temperature-Independent Spectral Indices 27

Figure 6, The aerial infrared pho- tograph of the Lubbon test site.

350

MINIMITM NUMBER OF PIXELS

:

0

1

I0

30

ii

!iii

i i!? iiiiill

iiiiiiiiiiiiiiiiiiiiii!ii

6O

[00

500

3c

33

340

330

320

310

300

890

~80

3i7:

0.

8 01

9 "T

~

I I

1.0

TIS

I1

Fig

ure

8.

Illu

stra

tion

o[

corr

elat

ions

bet

wee

n T

ISI

l an

d su

rfac

e te

mpe

ratu

re

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The

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ghtn

ess

tem

pera

ture

Tg

5 in

Cha

nnel

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as b

een

mtu

ally

use

d in

stea

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sinc

e e 5

is

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rall

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rges

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issi

vity

and

is

alm

ost

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tant

for

all

par

ts o

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age.

The

num

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the

grey

~c

ales

are

the

num

bers

of

pixe

l w

hich

fal

l in

tha

t ca

se.

Temperature-Independent Spectral Indices 29

4 - 5

6 - 9

10 - 1 4

15 - 1~

Figure 9. The coded image of amplification of TISI with Chan - nels 4 and 5 of AVHRR obtained from the data acquired at the night.

of AVHRR of 16 June 1986. From the spectral variations of ex, as displayed in Figure 5, it would have been more interesting to compute TISI r Unfortunately, we could not use AVHRR TISI 1 because data from Channel 3 of AVHRR are not quantitatively available since this channel is con- taminated with a persistent periodic noise from an unidentified source and we have not been able to quantitatively remove this noise even with the most sophisticated methods (Warren, 1989).

In order to reduce the noise introduced with such an amplification, the pixels have been aver- aged by squares of 9 pixels. This averaging proce- dure reduces the already small dynamics and sev- eral features appearing in the resulting image may be due to remaining noise. However, this noise is not principally due to the averaging procedure introduced in the calculation of TISIs, as it can easily be shown by a simple calculation.

If TISI is the TISI averaged over the 9 pixels, it is still related to the averaged pixel brightness

temperature Tg k in channel k by expression

N

TISI ~ I-I T~ * nk with an accuracy of k = l

-= 1%, independent of surface temperature.

This averaged TISI is displayed in Figure 9.

Despite the noise and low dynamics, we can rec- ognize, however, the coast and clouds coded in white and black. The deep-grey zone is not easily understood; it may be a hazy area. Similarly the dark zone has no simple meaning, since it does not correspond exactly to the Landes forest area. This image confirms the small spectral variations of e between Channels 4 and 5 and the various struc- ture observed should not be interpreted too seri- ously. This image is displayed here just to show what can be done with Channels 4 and 5 of AVHRR, but certainly TISI 1 would be more inter- esting because emissivities have larger spectral variations in Channel 3. For comparison the image of Channel 4 of AVHRR on the same area is shown on Figure 10.

Another interesting application of TISI is to retrieve the exact characteristics of clouds using directly the radiances measured by radiometers on board without performing atmospheric corrections, since in that case the transmittance from the top of clouds to radiometers is very close to unity and the reflexion of downward atmospheric radiances can be neglected. This implies that C = 1 and that the radiances measured by radiometers are almost equal to those emitted by clouds. TISI would be therefore useful to perform cloud classifications independently of their temperature (i.e., altitude).

30 Becker and Li

NOAA 9 16/6//86 C H A N N E L 4

VALEUR DE DEBUT DE CLASSE :

368 398 411 4 1 6 421 4 2 5

iii?~iii!i!~::i::::ii::i::i::~ : :z: ~ ii ~J:.

431 4 4 1

STRASBOURG 20 / ' 09 /89 14H 13M 165

Figure 10. The image of Channe l 4 of AVHRR on the same area as the image of Figure 8.

Application to the Determination of e3 and T,

Another possible application of TISI with AVHRR could be the direct determination of surface tem- perature and spectral emissivities. Unfortunately, this is theoretically possible only if emissivity ra- tios do not vary between day and night. In that case, the scheme is the following:

Using night data Tgn, it is possible to compute TISI 1 (night) referred to as TISI1. [see Table 3, expression (5), and formula (19b)]:

B3(Tg3,,) TISI 1. = M *

B 4 ( T ¢ 4 n ) * Bs(Tgs.)2 = TISIE. * C,,,

where TISIE. and C. are the values of TISIE and

C given respectively by formula (24) and (23) with their corresponding night values.

From day data, assuming that the reflexion of solar flux is Lambertian in Channel 3 of AVHRR, it is possible to write at ground level

B~(T~3,,) = e3B3(Ld) + (1 -- e3)Rat 3

(I- + - - Re3 cos 0s (28)

where Re3 and 0~ are respectively the global solar irradiation and angle of irradiation. From the def- inition of TISI 1

B3(Tg3d) TISI1d = M * (29a)

B4(T~4d)* Bs(Tgsd) 2'

Temperature-Independent Spectral Indices 31

and inserting expression (28) into (29a),. one gets

TISI~ = TISIE a * C,1

+ M * ( (1-e3) /Tr}RG3c°sOs (29b)

R4(rg4 ), Rs(rgs,,) If emissivity ratios do not vary from day to night (T IS IEd=TISIE a) and assuming that C,t= C a, the first term on the right-hand side of expression (29b) is nothing but TISI n. Solving (29b) for ( 1 - e3), one gets

Tr * (TISI a - TISI n) B4(Tg4,1 ) * Bs(T~s,t) 2 1 - e 3 = M * Rc; 3 *cos 0~.

(30) Everything on the right-hand side of Eq. (30) is measurable from space, leading to a possible esti- mation of ( 1 - ea ) . (Noticing, however, that the error on (TISI , t -TISI .) may be large.) Then, from formula (28), T s can be estimated. Using Channels 4 and 5, e 4 and e 5 are then easily derived once T s is known.

Many other applications may be done such as classifications with various TISI, and correlation between TISI and NDVI or Ts, etc.

CONCLUSIONS

Spectral analysis in the thermal infrared bands has not been performed for a long time because the spatial variability of surface temperature was hid- ding the spectral variations of emissivities. We have shown how to eliminate, on a physical basis, the effects of surface temperature, opening the way to a clean spectral analysis in the thermal infrared bands. More precisely, we have shown that:

i. It is possible to define thermal infrared spec- tral indices which are independent of surface temperature to a very good accuracy and are simply related to emissivities again with a good accuracy. We called them TISIs (better names may be found!).

ii. These indices can be accurately computed with very simple expressions. Examples have been given in Table 3 for TIMS and AVHRR. A general method valid for any spectral radiome- ter has been presented for deriving these in- dices.

iii. There are several possible TISIs for a given combination of channels, allowing an interest- ing freedom for amplifying a particular chan- nel as wanted. Those indices can therefore be tailored to weight interesting spectral bands more heavily than others.

iv. TISIs may be constructed with any number of channels (except one) of a spectral radiometer. However, all these constructions are not inde- pendent and examples have been given. Func- tions of TISIs are themselves TISIs, leading to an interesting TISI algebra.

v. Variations of TISI may be amplified until sig- nificant sensibility is achieved.

vi. For a mixed pixel, heterogeneous in tempera- ture, it is still possible to define a TISI with a good approximation from the spectral emissiv- ity spatially averaged.

vii. The use of TISI increases the accuracy and the sensitivity of the spectral analysis in the ther- mal infrared bands. It allows, as in the visible, muhitemporal analysis with the particular in- terest of day-night comparisons.

viii. Relative spectral emissivities can be easily de- rived from the TISIs.

These TISI give therefore the opportunity to perform spectral analysis and classification as eas- ily as in the visible and near infrared spectral domains, with the advantage that day and night images may be considered. We have given only two simple examples of applications, one with TIMS data over HAPEX, the other one with AVHRR night data. The image from TIMS data rises the interest of TISI for analyzing bare or partially covered soils, where it discriminates many characteristics and among them probably surface moisture because the emissivity is very sensitive to moisture.

Channel 3 of AVHRR is particularly interest- ing because day-night comparison of these data may be used to determine absolute spectral emis- sivities and surface temperature if TISIEs are the same for day and night.

TISI may also be useful for quantitative pho- tointerpretation and soil maps, because it can be used to reduce the N thermal infrared channels into three types of TISIs which may be particu- larly interesting combination of these channels, and HSI or false-color image can be made from these three selected TISIs.

32 Becker and Li

In the examples discussed, it appears that TISI is probably more appropriate than NDVI for monitoring bare soil or partially covered surfaces. It should be therefore quite interesting to compare NDVI and TISI; similarly it should be instructive to correlate TISI with NDVI and surface tempera- ture T~ and to perform spectral analysis combining visible, near infrared, and thermal infrared chan- nels. Such possibilities will be explored in. future works.

We wish to thank Professor M. P. StoU for constructive discus- sions, Drs. John C. Price and F. Palluconi for very interesting and stimulating comments, J. Hirsch for assistance, and Drs. C. Ottl~ and C. Tarrieu for providing the AVHRR and TIMS data.

REFERENCES

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Becker, F. (1987), The impact of spectral emissivity on the measurement of land surface temperature from a satellite, Int. J. Remote Sens. 10:1509-1522.

Becker, F., and Li, Zhao-Liang (1990), Towards a local split window method over land surface, Int, J. Remote Sens., 3:369-393.

Chedin, A., Scott, N. A., Wahiche, C., and Moulinier, P., (1985), The improved initialisation inversion method: a high resolution physical method for temperature retrievals from the Tiros-N series, J. Clim. Appl. Meteorol. 24:124-143.

Gillespie, A. R., Kahle, A. B., and Walker, R. E. (1986), Color enhancement of highly correlated images. I. Decorrelation and HSI contrast stretches, Remote Sens. Environ. 20:209-235.

Gillespie, A. R., Kahle, A. B., and Walker, R. E. (1987), Color enhancement of highly correlated images. II. Channel ratio and "chromaticity" transformation techniques, Re- mote Sens. Environ. 22:343-365.

Hardy, J. D., and Muschenkeim, C. (1934), The radiation of heat from the human body, IV the emission, reflexion and transmission of infrared radiation by the human skin, J. Clin. Invest. 13:817-834.

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Hunt, G. R. (1980), Electromagnetic radiation: The communi- cation link in remote sensing, in Remote Sensing Geology (B. S. Siegal and A. R. Gillespie, Eds.), pp. 5-45.

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Kneizys, F. X., Shettle, E. P., Gallery, W. O., Chetwynd, J. H., Jr., Abreu, L. W., Selby, J. E. A., Clough, S. A., and Fenn, R. W. (1983), Atmospheric transmittance/radiance: computer code Lowtran 6, Optical Physics Division, U.S. Air Force Geophysics Laboratory, Hanscom AFB, MA, AFGL-TR-83-0187.

McClain, E., Paul, William, Pichel, G., and Charles, C. Walton (1985), Comparative performance of AVHRR-Based multichannel sea surface temperatures, J. Geophys. Res., 20:11587-11601.

Nerry, F. (1988), Contribution h l'~tude exp~rimentale et thdorique de l'~missivitd et de la r~flectivitd des milieux naturels terrestres dans rinfrarouge thermique dans le contexte de la t~l~d~tection spatiale, Ph.D. thesis, Univer- sity of Strasbourg 1, France.

Nerry, F., Labed, J., and Stoll, M. P. (1988), Essai de deter- mination absolue des signatures speetrales de sols nus Proceedings of the 4th international colloquium on spec- tral signatures of objects in remote sensing held at Aussois, France, 18-22 January 1988, ESA SP-287, April 1988, pp. 185-188.

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Price, J. C. (1986), Remote sensing in the thermal infrared, Remote Sens. Rev. 2:187-196.

Price, J. C. (1989), Quantitative aspects of remote sensing in the thermal infrared, in Theory and Applications of Optical

Temperature-Independent Spectral Indices 33

Remote Sensing (Ghassem Asarr, Ed.), Wiley, New York, pp. 578-603.

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Vicker, R. S., and Lyon, R. J. P. (1967), Infrared sensing from spacecraft: A geological interpretation, in Thermophysics of Spacecraft and Planetary Bodies-- Radiation Properties of Solids and the Electromagnetic Radiation Environment in Space, (O. B. Heller, Ed.).

Vincent, R. K., and Thomson, F. J. (1972), Rock-type discrimi- nation from ratioed infrared scanner images of Pisgah Crater, California, Science, Vol. 175, pp. 986-988.

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