Appendix A. The method of Matsumoto's[1991] 
to decompose mixed data; The Maximum 
Likelihood Estimation 
The spectral reflectance(I) of K bands 
from the mixel which contains of H landcover 
categories is expressed like the Eq. 1; it is 
the multiplication of the mean vector(M) of 
the spectral reflectance from the pure pixels 
of each landcover category and the mixture 
ratios(A) of landcovers in the mixel. 
Yor VA Arkaden Abk Anbieter Eq.1 
where, 
I=(1 1 Sw... In) 
t 
Asa rag Pin 8 ) 
My7 M12 ++ 0000000 mig 
M==1"""" qp EN 
9922229222292 
Mxri Mxrz2 00000000 0° Mxyu 
Now, suppose the reflectance(Iij)of the 
band (i) on the landcover category (j) has a 
normal distribution with mean M*ij and 
variance (2. Then, the mean vector (Mi) of 
the spectral reflectance of the band (i) is 
expressed like Eq.3. The variance of the 
spectral reflectance is expressed like Eq.4. 
INA, =— IM >= Z7 
à a. TTT dT m um m MN UN NN m m man Eq.3 
: + + * * 
2 dz 
o°.= * S. * 
PEE Trees = 
. 2 2 2 
S diag(04 ,0, , «eie en) 
Then, the likelihood;P(Ii), which the 
reflectance data of the band. i may be 
observed, is expressed like the Eq. 5. 
Supposing that the spectral reflectance is 
independent among bands, the total likelihood; 
P(I), which the spectral  reflectance 
Ii,I2,....,IK may be observed, is expressed 
like Eq. 6 and 7. 
_ exp = (IM) Trey) 
o,*4 (2m) 
  
P(I,) ----Bq.5 
International Archives of Photogrammetry and Remote Sensing. Vol. XXXI, Part B3. Vienna 1996 
N 
P(I)-II P(ri) dide 6 
i=1 
O — — In { P(I) ) T----Eq.7 
k 
Yiu aL) Pare 0 | (hep, 0%. fie) 
j=1 
EG 
Note: 
H: Number of landcovers 
K: Number of bands of image data 
i: Band No. (1 to H) 
j: Landcover ID. 
ai: Area ratio of landcover[h] in the IFOV 
of pixel[i,j] (unit: IFOV) 
The mixture ratios of landcovers in a 
mixel can be computed by obtaining A, which 
minimizes @ under the  conditon which 
expressed in the Eq.8. This is the method to 
decompose mixed data using the Maximum 
Likelihood Estimation developed by Matsumoto 
in 1990.