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The International Archives of the Photogrammetry, Remote Sensing and Spatial Information Sciences. Vol. XXXVII. Part B7. Beijing 2008 
, , r p o‘ ( d !,y)0( d !, |n»J,) 
= ^ d ' —^ -——— 
P» ;, (d /.;) 
i - dA u 
(6) 
where the first integral term, similar to the integral term in 
equation (3), represents the parameter retrieval constraint from 
the observational data at time t, and the second integral terms 
represent parameter retrieval constraints from the observational 
data at time /+1. 
Assume that model parameters, observation variables and the a 
priori information on the model parameters are Gaussian, then, 
the first integral term in equation (6) can be written analogously 
as follow. 
biophysical variables at time t using the observational 
information at times t-N, ... , /-1, t, f+1,..., t+Ntogether. 
3.2 Radiative transfer model 
Radiative transfer models describe the relationship between 
canopy characteristics and reflectance, and many of them have 
been developed to obtain land surface biophysical parameters 
(Kuusk, 1994; Jacquemoud, 1992; Liang, 1993). In our 
parameter retrieval, the Markov chain reflectance model 
(MCRM) developed by Kuusk (Kuusk, 1995; Kuusk, 2001) is 
chosen as the forward model to simulate the canopy reflectance. 
This model incorporates the Markov properties of stand 
geometry into an analytical multispectral canopy reflectance 
model, which makes the model more flexible and more 
applicable. The MCRM can calculate the angular distribution of 
the canopy reflectance for a given solar direction from 400 to 
2500 nm. The inputs of the forward MCRM are summarized in 
Table 1. 
p D , (d' y )é?(d ,,|m ) 
= f dû’ ‘ J — 
Jd - ' /V/d'.) 
= a exp f ~(g(m',) - d'f ) T C D ‘ (g(m',) - d'f ) 
(7) 
And the second integral term is 
f i p D „,(d' + ;^(d;;;im' y ) 
R(m ,.) = f ,dd'; — J — 
•' ^ •' ^ D;rJ (d;:;) 
r \r im; + ;)l (8) 
*L d <\L m ü „ ,h»i, ’ g «im;,> 
1m I m 'ij) 
where | m' ; ) is a transition probability that the 
parameter is m '*J at time /+1 given the parameter was m ' at 
time t and is related to process models which describe the 
relationship between m'* 1 and m ' . 
Then, equation (6) can be written as follow. 
(mi,y) = kp M (m)T(m'u)R(m' l j ) (9) 
Parameters 
Value range 
Unit 
Solar zenith angle 
0~90 
Degree 
Relative azimuth angle 
0-180 
Degree 
Viewing zenith angle 
0-90 
Degree 
Angstrom turbidity factor 
0.1-0.5 
Ratio of leaf dimension and canopy 
0.02-0.4 
height 
Markov parameter 
0.4-1.0 
Factor for refraction index 
0.7-1.2 
Eccentricity of the leaf angle 
0.0-4.5 
Degree 
distribution 
Mean leaf angle of the elliptical 
0.0-90.0 
Degree 
LAD 
Leaf specific weight 
100 
g/m 2 
Chlorophyll AB content 
0.3-0.8 
%of 
Leaf water content 
100-200 
SLW 
%of 
Leaf dry matter content 
95-100 
SLW 
%of 
Leaf structure parameter 
1.0-3.0 
SLW 
’Weight of the first Price function 
0.05-0.4 
‘Weight of the second Price 
-0.1-0.1 
function 
Weight of the third Price function 
-0.05~0.05 
Weight of the forth Price function 
-0.05-0.05 
‘Leaf area index 
0-10 
m 2 /m 2 
* free parameters 
Apply logarithm to both sides of the equation (9), then 
Table 1. The parameters needed to run the MCRM 
3.3 Process model 
COSTim'.j) = -|(m - m'f f C D ‘ (m' y - )- 
i(g(m'y) - d';f ) T C D ' (g(m[.) - d'f ) + 
Fisher (Fisher, 1994(a); Fisher, 1994(b)) used an empirical 
statistical model to describe the temporal NDVI profile of 
agriculture crops. The model is a double logistic function to 
describe the NDVI profile. In our study, the double logistic 
model, shown in formula (11), is used to describe the seasonal 
LAI trajectory. 
Equation (10) is the cost function to retrieve the canopy 
biophysical variables at pixel (i,j) using the observational 
information at time t and time /+1. Similar method can be 
applied to derive the cost function to retrieve the canopy 
LAI(0-t) = vb + 
av 
1 + exp (-c(/-p)) 
av + vb- ve 
1 + exp (-d(t-q)) 
(11)