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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)