Adv. Space Res. Vol. 22, No. 5, pp. 655-667, 1998 01998 COSPAR. Published by Elsevier Science Ltd. All rinhts reserved

PII: SO273-1177(97)01127-7

Printed in -&eat Britain 0273-l 177f98 $19.00 + 0.00

THERMAL INERTIA MAPPING FROM NOAA-AVHRR DATA

J. A. Sobrino, M. H. El Kharraz, J. Cuenca and N. Raissouni

Department of Thermodyamics, Faculty of Physics, University of Valencia. C/Vr. Moliner, 50. 46100 Bwjassot, Spain

ABSTRACT

This paper presents a method to retrieve thermal inertia from NOAA-AVHRR data. The method, based on

Xue and Cracknell’s model (1992), involves using the phase angle information of the diurnal surface

temperature change. The method needs three different NOAA overpasses of the same area throughout a

single diurnal cycle. To apply this method we present a methodology that takes into account satellite

calibrations, viewing geometry and correction of atmospheric effects. The advantage is that it does not need

other data than the ones supplied by the satellite. Finally, we present the preliminary results obtained using

the proposed method and the methodology for AVHRR data over the Jberian Peninsula.

INTRODUCTION

01998 COSPAR. Published by Elsevier Science Ltd.

The thermal inertia, P (with peculiar unit, TlU, 1 TIU=l J m- 2 s -li2 K-l), is defined as a measure of the

resistance offered by materials to change their temperature. For a homogeneous material, this amount can

be expressed by:

P= G (1)

where K is the thermal conductivity (J m-1 s-1 K-t), p is the density (kg m-3) and c is the specific heat

(J kg-* K-1) of the material. Thermal inertia is the most important property which governs surface

temperature variations. High thermal inertia values lead to small changes in temperature, for a given transfer

of heat, while low thermal inertia values lead to large changes in temperature for the same transfer of heat

(Pratt and Ellyett, 1978). So, the thermal inertia is a measure of the thermal response to the temperature

variations and commands the temperatures that the soil surface may reach under the effect of diurnal or

655

656 J. A. Sobrino er al.

annual variations of soil heat flux. Knowing the thermal inertia of the terrestrial materials is hence necessary

to understand how the surface temperatures are indicators of the environmental conditions, such as the soil

humidity and its lithological composition. Therefore this is of great interest to many studies (hydrological,

geological, etc.).

Remote sensing provides a unique tool for estimating the thermal inertia in large areas. With the object of

mapping the thermal inertia of the soil surface, the experimental thermal satellite HCMM (Heat Capacity

Mapping Mission) was launched on April 26th, 1978, introducing a new era in geologic exploration. The

general&d theory to calculate thermal inertia from global remote sensing data was given by Price (1977).

This theory is difficult to apply because it needs to know a set of physical variables (wind speed, air

humidity, etc...), which are not readily found over areas of thousands of square kilometers, as it is

appropriate for remote sensing. Therefore, the model has been applied to HCMM and NOAA data in a

simplified way, considering a constant value for the quotient of the variation of the energy flux to the

atmosphere and the variation of flux into the ground, for example setting a=1 in Eq. (8) of Price (1977).

This approach provides the apparent thermal inertia which should not be used in regions having variability

in surface moisture (Price, 1985).

In recent years, a simple thermal inertia model was developed by Xue and Cracknell (1992). This model (in

the present paper we refer to it as XC) involves the use of the phase angle information of the diurnal

temperature change. Using this model, a map of real thermal inertia can be obtained, without needing to

know in advance the value of the parameter B, which, as it will be shown below, represents the

linearization coefficient of the linearized boundary condition from the dynamic energy balance equation at

the ground surface. The model requires field measurement of one parameter for the calculation of real

thermal inertia: this is the time of maximum surface temperature in the daytime, which is obtained from a

meteorological station.

In this context, the objective of the present paper is to propose an alternative procedure that allows to obtain

the real thermal inertia from remote sensing data alone, so it does not require field measurements. This

method, based in Xue and Cracknell’s model (1992), uses three surface temperatures taken at different

hours on the same day, which can be obtained from NOAA satellite overpasses.

THEORY

To obtain the surface thermal inertia we start with the diffusion equation, which is given by

a ax2 a2T(x,t) _ aT$,t)

(2)

Thermal Inertia Mapping 657

where T(x,t) is the temperature at depth x below the surface and time t and a is the thermal diffusivity of

the half-space. For simplicity, in Eq. (2), we have assumed one-dimensional periodic heating of a uniform

half-space (a region bounded by a plane on its upper side and extending downward to infinity) of constant

thermal properties. To solve Eq. (2) the Fourier series method is used, which is commonly applied to this

type of problem, subject to the following boundary conditions (Watson, 1975):

and

_ iWx,t) K+xzu = (1-A) St-j Ct cosZ’- R? +R& - H-LE.

T(x,t) is finite as x + 00 ,

(3)

(4)

where A is the surface albedo, So the solar constant, 1375 W m-2, Ct the atmospheric transmittance,

typically of order 0.75 for vertical viewing,

cos Z’= cos d cos Z - sin d[sin @ cos 6 sin(ot) - cos 41 sin 6 co& sin 6 sin h cos(ot)] (5)

cos Z = sin 6 sin h + cos 6 cos h cos(ot), (6)

d is the surface slope angle measured downward from the horizontal, 6 is the solar declination, h is the

local latitude, Z is the local zenith angle for a smooth surface, Z’ is the local zenith angle for an inclined

surface, t is the time measured from local noon, o is the Earth’s rotation frequency 0=(21c/l day)

= 7.27 10-5 s-l, R?’ is the earth emitted radiation, RJ is the downward longwave sky radiation, H is the

sensible heat flux to the atmosphere and LE is the latent heat flux to the atmosphere. The first term of the

right-hand side of Eq. (3) is the absorbed short-wavelength flux coming from the sun (modulated by the

atmospheric transmission and surface albedo) which is the dominant energy flux causing temperature

changes at the Earth’s surface.

The mathematical difference in complexity resulting from replacing the expressions for the fluxes by a

simple linear form with surface temperature is very great and leads to the expression:

- Kvix=u = (1-A) So Ct cosZ’-[Ac+BT(O,t)], (7)

where & and B are the linearization coefficients obtained from the dynamic energy balance equation -see

Eq.(3)- at the ground surface. Pratt et al. (1980) and Xue and Cracknell (1992) have shown that the

advantages of this linearization far outweight their limitations.

Thus, the solution of Eq (2) subject to the boundary conditions, Eqs. (4) and (7), can be mathematically

expressed for the surface temperature at depth x as (Xue and Cracknell, 1992):

658 J. A. Sobrino e1 al.

T(x,t) = - 2 + (1-A) So Ct X=1 An exp(-k&ix)cos(nmt-k&Gx-6n)

wnP2+GBP+B2

where

ko= E ;,

S, = arctan(

(8)

(9)

(10)

Al = 1 sin6 sinh sinv + & co& cash [sin (21@+2~] n=2,3,... (11)

A n

= 2sinS sinh sin(ny) + 2 cos6 cash nrt 7t(n2- 1)

[n sin (nv) cosv - cos(n\lr) sin v] (12)

yr=arccos (tg6 tgh) (13)

From Pq. (12) we know that An tends to become zero quickly with increasing values of n. Price (1977)

compared the surface temperature obtained with all terms of E!q. (8) with the first term of the series and

concluded that the first term in the series is dominant. Therefore, for the purpose of this study we only used

the first term of the Fourier series. Then, the equation giving the surface temperature (when x=0) is:

T(O,t)= - ++ (1-A) SOCt A1 cos(ot-61)

(14)

However, it is not easy to extract P from Eq. (14) because the relationship existing between P and B must

be found out. The best solution is writing Eq. (14) for two different times, tl and t2, that gives the best

approximate description of the diurnal amplitude of surface temperature, for example around 2:30 a.m. and

2:30 p.m. local time respectively, then subtracting both equations, one can obtain:

AT = (1-A) So Ct Al cos(ot~-6~)-cos(ot~-~~)

where AT =Tz(O,tz)-Tl(O,tl).

Now, defining b like b= B , and

therefore,

consequently, one obtains:

B_ (1-A) So Ct

AT&

Thermal Inertia Mapping

b QG&)

-l-tan(61)

A1 [cos(ot&) - cos(otl-&)I

659

06)

(17)

Here the main problem to determinate the thermal inertia is 61. In the work of Xue and Cracknell (1992)

this was solved by maximising Eq.( 14) i.e., 61,0tniax. Now, the problem is to know the time at which the

maximum surface temperature occurs (t max)~ This is not an easy problem because this parameter changes

with the type of the surface and its knowledge requires field measurements.

In the present work, we obtain the phase difference by rewriting Eq. (14) for three different times of passes

tl, t2 and t3 for which there are three surface temperatures Tl, T2 and T3 respectively. In this way, the

difference between T(O,tl)-T(O&) and T(O&)-T(O,tg) can be obtained, then the two expressions are

divided to obtain the phase difference 61 according to the following expression:

tan + (Tz--Cd [cos(otl)-cos(Ot2)l-(Tl-Tz) bsW)-cos(m)l (Tl-T2) [sin(ot2)-sin(ot3)]-(T2-T3) [sin(otl)-sin(ot2>]

(18)

To obtain Eq. (18) we have considered that & and B are constant in time for the considered day. The

importance of this equation is that it gives the phase angle information 61 as a function of the surface

temperature differences and the times of the three satellite overpasses, and therefore it is not necessary to

know the time of the maximum diurnal surface temperature as suggested by Xue and Cracknell (1992).

Model Performance

Aiming to evaluate our model, as a previous step to its application to satellite data, a temperature data

measured in situ was used. The measurements were carried out using an infrared thermometer (EVEREST-

4OOOA), which operates in the spectral band of 8-14 l.trn, mounted at the meteorological station of

Mollerusa (Lleida, Spain), three meters above the soil surface with an IFOV of 15”. The thermometer took

a reading every 10 seconds and stored the mean of each minute in the datalogger. These measurements

were taken from the 14& to the 27th of June 1995 over a bare soil (see Figure 1). The temperature values

present different behaviour following the meteorological changes for each day.

I. A. Sobrino et al.

i

165 167 168 169 170 171 172 173 174 175 176

DOY

Fig. 1. Temperature values as function of the time for all the days of measurements

From these measurements, we present a comparison between the proposed model (Eq. 17) and the Price’s

model, i.e. obtained setting a=1 in Eq. (8) of Price (1977), and XC’s model (Xue and Cracknell, 1992).

This was achieved using the temperature data given in Table 1. This Table also gives the maximum

temperatures and their times for each day of measurements, which are needed to apply XC. The calculated

thermal inertia values are plotted in Figure 2. The most important aspect observed in this figure is that XC

and Eq. (17) overlapped most of the days of measurements. However, it should be remembered that our

model needs only data supplied by the satellite, which makes it more operative. On the contrary, Price’s

model presents the lowest thermal inertia values, due to the fact that it gives the apparent thermal inertia.

Because of the presence of clouds, DOYs (day of year) 172 and 175 present low thermal amplitude values

(T3-TI in Table 1) and therefore higher thermal inertia values (see Fig. 1).

. . . ensrtrvrtv Anal- The exposed method to obtain the thermal inertia is affected by different sources of

error of different nature. In order to evaluate these errors, a sensitivity analysis was carried out. The errors

in the thermal inertia were obtained applying the error theory to Eq. (17).

We have to notice that the error estimation was done taking into account that A1 is known, which is not

difficult if we consider that the first Fourier series coefficient depends on regional latitude and solar

declination, which values can be known with precision in the study area. The error in the temperature

difference between the two satellite passes, @AT), was taken equal to 2K, i.e., we assumed that the error

committed in the surface lemperature determination with a split-window method for the channels 4 and 5 of

NOAA-AVHRR is about 1.4 K and the absolute error in albedo, o(A), was assumed to be according to

Saunders (1990) of 10%. The others parameters which are necessary to evaluate the previous thermal

inertia expressions, as the surface temperatures at the time of satellite passes were obtained from data

ThermalInertiaMapping 661

measured in situ in Mollerusa on the DOY 168. In this way we have obtained a(Model)= 50 TIU. This

result makes clear the good performance that presents the model. Otherwise, the error in P is similar to the

obtained for XC formulation when the time of maximum is known with precision.

Table 1. Times that Correspond to Maximum (tmax> and Minimum (trrr& Temperatures as

well as the Maximum and Minimum Temperatures Tmax and Trr,in. T1, T2 and T3 are the

Temperatures at 2:30,7:30 and 14:30 GMT Respectively.

176 54.8 1 6:39 1 15.2 1 20.2 1 16.7 1 52.7 ]

OJ.Price q xc A Model 55001 k

5000-

4500. s F 4000.

.g 3500.

2 3000. 7

E 2500. &?z 2000-

1500-

1000.

500 a I t 164 166 166 170 172 174 176 176

L)ays

Fig. 2. Comparison between the proposed model (Model), Price’s model (J. Price) and

Xue and Cracknell’s model (XC) for the period of measurements.

662

MEmODoLmY

1. A. Sobrino er al.

The objective of this section is to propose a simple operational procedure to determinate thermal inertia

from NOAA-AVHRR data. The flowchart of the methodology is shown in Figure 3.

I t

11 P

Fig. 3. Flowchart illustrating the procedure to compute thermal inertia.

ow Model

There are many operative algorithms that permit the land surface temperature (LST) determination from

remote sensing. Our objective is to adapt the structure of one of the existent algorithms to calculate thermal

differences, AT, between two passes of the sensor over the same area. To reach this goal the model

developed by Sobrino et al. (1996) was used. The model starts from the radiative transfer equation for an

atmosphere in thermodynamic equilibrium, which proportionates the radiance Ii measured from space in

channel i with zenith angle 8 as a sum of three terms: 1) the radiation emitted by surface and attenuated by

atmosphere, 2) the radiation emitted by the atmosphere towards the sensor, and 3) the down welling

radiation emitted by the atmosphere that incides over the surface and is reflected towards the sensor,

according to:

& Bi(Tio) = EiaBi(Ts) Ziu+ Raue’I’ +Ri(ref) Zie (19)

In Eq. (19) all quantities refer to a spectral integration over the band width of channel i. Bi is the Planck’s

function, Tie is the radiometric temperature measured at satellite level at zenith observation angle 8, Eie is

the ground surface emissivity at zenith observation angle 8, Bi(Ts) is the radiance measured if the surface

Thermal Inertia Mapping 663

were a blackbody with the surface temperature Ts, Ziu is the total atmospheric path transmittance at the

zenith angle 8, Rauu? is the atmospheric upwelling radiance at the zenith angle 8, which is given by the

mean value theorem, according to Rauut=(l-zie) Bi(T& (Sobrino et aZ., 1991) where Tg is the mean

temperature of the atmosphere between the surface and the sensor, being Ri(ref) the reflected atmospheric

radiance given by Sobrino et al. (1996), according to Ri(ref)=(l-aiu)(l-Q3)Bi(T&, where is the total

atmospheric transmittance at 8=53’. So, the single-channel correction method, may be written as:

Bi(T+ Ei Bi(Ts) Ziu + [(l-Tie) + (l-&i) (l-2353) Tie] Bi(T& (20)

However the inconvenient of Eq. (20) is the necessity of atmospheric radiosoundings in the study area,

which are rather the exception. Nevertheless, this problem can be avoided if two radiative temperatures are

involved in the process of estimating the LST. According to this, Eq. (20) is rewritten for two channels of

interest, AVHRR Channels 4 and 5 inside the window, then solving between both equations for Bi(Ts) and

using a first-order Taylor series expansion of the Planck function around Ti and Ti, we have;

Ts=Tq+A(Tq-T$-Bu+(l-&)Bl-A&B2 (21)

where T4 and Ts are, respectively, the brightness temperatures measured in channels 4 and 5 of AVHRR,

&=(4+&5)/2 is the mean emissivity, A&=(4-&5) is the spectral emissivity difference, and A, Bo, B1 and B2

are the split-window coefficients given by (Sobrino et aZ.,1996);

A l-24 =- Q-25

(22)

RF A (l-75) GA-Tas) (23)

Bl- (1-2424(53)) (T4-T5) + 24(53) L4 (24-25)

(24)

B2--t5ml (25)

where Ta4 and Ta5 are respectively, the mean atmospheric temperatures in the channels of interest,

L4=B(T4)/[aB(T)/aT]T4 is a parameter with a dimension of temperature, that may be obtained adjusting

Planck’s function, as B(T)aTn, with q a radiometric parameter that depends on the channel and the

considered temperature range. In particular, in the temperature range between 260K and 320K we obtained

1~44.673 and q+.26O for the channels 4 (1 lpm) and 5 (12pm) of the AVHRR. An example of validity

of the structure of Eq. (21) for determining the surface temperature from the data provided by the sensor

AVHRR, may be obtained by simulation using LOWTRAN-7 (Kneizys et aZ., 1988). This program gives

the values of the atmospheric transmissivity, as well as the different temperatures included in Eq. (21).

664 J. A. Sobrino et al.

From these data, Sobrino et al., (1996) obtained a standard deviation of 0.4 K and a maximum error in

temperature of 1.6K, for a set of 10800 different situations (60 radiosoundings x 5 surface temperatures x

36 different combinations of emissivities).

For a sensor with channels at visible and near-infrared wavelengths, as it is the case for AVHRR, the broad

band bidirectional reflectance p(z, 8,4) is given by:

P(G 09 $I= a1 p1(z7 8, Q) + 02 p2(z, 0, $1 (26)

where p1 and p2 are the filtered bidirectional surface reflectances for the channels 1 and 2 of AVHRR

respectively, where z, 8 and 0 are respectively the solar zenith angle, the satellite zenith angle and the

relative azimuth angle between the sun and the satellite at the surface, and 01 and 02 am the factors which

weight each reflectance by the received solar radiation in each channel. These factors were assumed to be

equal to 0.5 for both channels (Saunders, 1990). In the present work, the atmospheric correction of p1 and

p2 in the visible and the near infrared respectively was carried out using SMAC (Simplified Model for

Atmospheric Correction) (Rahman and Dedieu, 1994).

Finally, it should be noticed that in order to obtain the thermal inertia, the images must be overlapped at

nocturnal and diurnal times of passes. This may be achieved using orbital methods, from the orbit

modelling described by satellite or by using ground control points. In order to work with correct data, the

pixels contaminated by clouds should be eliminated, for example with a threshold method as the proposed

by Saunders and Kriebel(1988).

APPLICATION

It should be noticed that for the estimation of the thermal inertia, the emissivity and total water vapour can

be assumed to be constant during one day. This is a further advantage of the methodology because it uses

only temperature differences.

We have applied the methodology developed to the whole Iberian Peninsula. An application of this

methodology is shown in Figure 4 that has been obtained using three passes of NOAA satellites acquired at

2:40 h, 13:OO h and 18:26 h GMT on 23rd July 1995. In this Figure, brighter areas represent smaller

thermal inertia values, darker areas represent higher values. The North and Northwest of the Peninsula

(GaIicia, Asturias, etc.), that ate the most vegetated areas, show the highest values of thermal inertia (more

than 2000 TIU). The semiarid regions in the Southeast of the Iberian Peninsula (mainly areas of Almerfa

and Murcia) have the smallest values of thermal inertia (less than 1000 TIU). The thermal inertia values

between 1000 and 2000 TIU correspond to the most extended regions (Castilla, South of Portugal and

Thermal Inertia Mapping 665

Andalucia). These are the most sensitive regions to the seasonal and climatic changes, and at this time they

were affected by severe drought. Otherwise, these thermal inertia values are similar to the given by the

bibliography for sandy (-1000 TIU) and clayey soils (~1800 TIU), see for instance Janza (1975).

Fig. 4. Thermal inertia mapping of the Iberiti Peninsula using the model developed in the paper.

666

CONCLUSIONS

J. A. Sobrino et al.

In this work, a model for estimating the real thermal inertia using NOAA-AVHRR data has been proposed

and applied to the Iberian Peninsula. The main advantage of the suggested model is that it represents a

simple operational model that only needs satellite data. The model is based on the use of three consecutive

temperatures over the same area which permits obtaining the phase difference. The application of the model

to experimental data has given consistent results.

ACKNOWLEDGEMENTS

The authors wish to express their gratitude to the Air Force Geophysics Laboratory (Massachusetts, USA)

for supplying the LOWTRAN-7 computer code. We also thank M. Ibaflez for providing the AVHRR

images of 1995 and the Mollerusa data. This work has been carried out while M. H. El Kharraz is granted

by the Instituto de CooperaciQ con el Mundo Arabe (ICMA) of the Ministerio de Asuntos Exteriores. The

authors wish to thank to the Project (CL1951815CO2-01) of the Programa National de1 Clima

(MOPU/CICyT, Spain) for financial support. The referees and editorial suggestions are also

acknowledged.

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