IEEE TRANSACTIONS ON GEOSCIENCE AND REMOTE SENSING, VOL. GE-24, NO. 3, MAY 1986 A Stable Iterative Procedure to Obtain Soil Surface Parameters and Fluxes from Satellite Data MARCEL RAFFY AND FRANCOIS BECKER, MEMBER, IEEE Abstract-Two general methods recently proposed to obtain the thermal inertia, the sensible heat flux, and the evapotranspiration flux from satellite data are summarized. In these methods, stable algo- rithms were used, but the stabilization coefficients presented were em- pirical. In this paper, it is shown that optimal coefficients exist and can lead to an iterative process to calculate the thermal inertia and the fluxes with improved accuracy despite of errors of measurements. Some applications using in situ data and theoretically simulated data are given as an example. I. INTRODUCTION FROM THE RADIANCE of the soil, it is possible to obtain the spectral emissivity, albedo, solar incoming flux, and the surface temperature. The present task is to obtain from these parameters the thermal inertia, evapo- transpiration flux, and the sensible heat flux by a stable numerical method. Many methods have been proposed to determine some of these soil characteristics [4], [9], [10], [17]-[20], but these works need an a priori knowledge of aerodynamic resistances. In a previous paper [14], we defined two general pro- cedures and showed, as is the case with most inverse problems, that they were not stable with respect to error of measurements in the input data. We proposed a stabi- lization procedure for these methods that requires the in- troduction of empirical coefficients that are crucial for the accuracy of the results. We will show in this paper that optimal coefficients exist and we will derive an iterative algorithm to calculate the surface fluxes and the thermal inertia. Section II recalls briefly the basis of the method pro- posed. Section III gives some indications about the effect of the errors of modeling on the calculated fluxes. The existence of the optimal stabilization coefficients and their value is explained in Section IV, as is the derived iterative algorithm. Numerical results obtained from simulated sit- uations and in situ experiments are presented in Section V. Manuscript received September 12, 1985; revised December 4, 1985. This work was done under the collaboration of the CNRS, the CNES, and IBM/France, and was supported by the CNRS through ATP Teldedtection. The authors are with the Groupement Scientifique T6leddtection Spatial, Universite Louis Pasteur de Strasbourg, GSTS, 23 rue du Loess, B.P. 20, 67037 Strasbourg, France. IEEE Log Number 8607725. II. COMPUTATION OF THE FLUXES AND THE THERMAL INERTIA TAKING INTO ACCOUNT ERRORS IN THE MEASUREMENTS A. Hypothesis and Methods We focus our attention on one pixel. Generally such a pixel corresponds to an IFOV which may have a diameter of several kilometers and shows, therefore, a very heter- ogeneous surface. Such a pixel is referred to as an "het- erogeneous pixel." It represents the brightness tempera- ture of an area that does not have a well-defined surface thermodynamic temperature. The interpretation of the brightness temperature of such a pixel and its relationship with the various local thermodynamic surface tempera- tures on the terrain are very complicated questions (cf. [3], [14]) that will not be addressed by this paper. We shall therefore assume all along this paper that the pixels that are considered are "homogeneous," i.e., represent the brightness temperature of a homogeneous area in the terrain, having a well-defined thermodynamic surface temperature. Furthermore, we assume, following Tab- bagh [21] and Raffy [13], that the horizontal fluxes can be neglected. Then we are led to study a homogeneous layer of depth L, for which we assume that the volumetric moisture profile is constant in time during one day and that the temperature at depth L is constant. Thus, the problem addressed by this paper is to deter- mine the best approximation of the thermal inertia, evapo- transpiration flux (LE) and sensible heat flux (H) for this homogeneous media, the input data being an interpolated time profile of the surface temperature Tmeas(t), the air temperature, at 2 m, Ta(t), the horizontal wind speed Ua(t), at 2 m, and the net radiation flux RN (t). The temperature profile T(x, t) for x E [0, L] and t > 0 satisfies AT a2T c -- K --- = 0 Cat -Kx2= T(O, t) = Tmeas(t), T(L, t) = TL, T(x, 0) = To(x) xe]0,L[, t> 0 t > 0 (P) t > 0 x E[0, L] in which c is the constant heat capacity, K the constant heat conductivity, and TL the temperature at the depth L. It is shown in [14] that a knowledge of the initial profile 0196-2892/86/0500-0327$01.00 © 1986 IEEE 327