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A, is the surface reflectance of the target object and A the 
mean background reflectance. As far as the optical 
thickness of the atmosphere 7 < 1.0 and surface reflectance 
« 0.5, L, has a linear form with respect to the surface 
reflectance (Kawata, et al, 1988). Furthermore, it is 
reasonable to assume A,-A unless the difference of the 
reflectance between the target and its surrounding area is 
enough large to cause the adjacent effects. 
Eq.(1) is rewritten as 
L.=QA+R 2) 
If the conversion of the count level of a pixel, X, into the 
spectral radiance, Lr, is given by L,= GX + O, where G and 
O are the gain and offset values of the sensor, Eq.(2) is 
expressed as 
X=pA+q 3) 
where p = Q/G and q = (R-O)/G. 
Now, let X; be the count level of a pixel in the image taken 
at the time tland X» the count level of the corresponding 
pixel in the image taken at the time t2. Assuming that the 
surface reflectance of X; is almost the same as that of X», 
the following linear relation is obtained. 
X;-7pXl-4q' (4) 
where p' 7 p(t2)/p(t1) and q' ^ q(t2) - p'q(t1). 
The coefficients, p' and q', only include atmospheric 
parameters and can be estimated if we can identify several 
objects in two different images that have almost same 
surface reflectances. 
The spectral surface reflectance ratio, A(t2)/A(t1), at two 
different times, t1 and t2, is therefore expressed as 
A(Q) 1 X,—q-pq(l) 
AG p Xizqal) 
In Eq. (5) unknown factor is only q(t1). As known from 
Eq.(3), q(t1) is proportional to the additive path radiance 
and is estimated from the minimum count level X; in the 
image taken as reference. It is found that the relative 
atmospheric correction between thye images taken at 
different times is performed by Eq.(5). 
  
(5) 
2.1 Regression Line 
To estimate the coefficients of Eq.(4), we selected the 
surface objects whose reflectances will be unchangeable 
With time to determine the regression line by taking the 
count level of a target in the image on Oct. 10, 1987 as an 
independent variable. Fig.1 shows Landsat TM images in 
the study site taken at Nov. 1991. The study site mainly 
consists of the sea (Sea of Japan; upper-left portion in 
Fig.1), urban area (central portion), forest (upper-right and 
lower-right portion) and rice field (lower-left portion). 
Roads and rivers are also in visible in Fig.1. We chose 
Some pixels corresponding to sea water, urban area, road, 
railway, residential area from the paired images and 
computed the mean count levels for the sampled objects in 
each band. Thus obtained regression line is expressed as 
X3=PX,+Q (6) 
Where P^ and Q' are the estimated values of p' and q', 
385 
respectively. The estimated values of P^ and Q" and the 
correlation coefficients are shown in Table 1. We can see 
from Table 1 that the correlation between X; and X; is very 
high in every spectral band. 
Table 1 Values of Coeffients, P^ and Q^, of the Regression 
Line and Correlation Coefficients 
  
  
TM band _Slope(P’) Intercept (Q’) Correlation 
1 0.597 14.0 0.989 
2 0.580 5.90 0.983 
3 0.676 2.50 0.990 
4 0.752 - 0.03 0.996 
5 0.739 0.11 0.997 
7 0.751 0.15 0.984 
  
2.2 Distribution of Reflectance Ratio 
In order to estimate the spectral reflection ratio given by 
Eq.(5), we carried out the geometrical correction between 
two Landsat TM images by using ground control points. 
After that, we computed the value of reflectance ratio in 
every pixel of paired images by using the values of P^ and 
Q' in Table 1 and minimum count levels in the image of 
Oct, 1987 taken as reference. Fig.2 shows the frequency 
distributions of reflectance ratio at bands 2, 3 and 7. As 
seen from Fig.2, every histogram shows approximately the 
normal distribution with the mean value near the 
reflectance ratio 1. 
3. CHANGE DETECTION 
The spectral reflectance ratio at two diffrent times, t1 and t2, 
is considered to be unity if the surface reflectance of target 
objects is unchangeable with time. In other words, the 
values of the reflectance ratio are different from 1 if the 
reflectance of ground objects under consideration changes 
at two different times. To detect ground objects whose 
surface reflectances have changed, we introduced the 
distance from the value 1 of A(t2)/A(t1). 
In this study we defined the distance D as follows: 
WE 2 
D= Yd V= X (a; 1) 70°) (7) 
i=1 1 i=] 1 
where dj? is the distance at band i, r; = A(t2)/A(t1) at band 
i and G ; is the root mean square error of r; from the 
value 1 in band i. Fig.3 shows the frequency distribution of 
D in the study area. The value of D in Fig.3 is multiplied 
by 10. 
The most difficult problem in detecting the area of change 
by using the distance D is to select the threshold value of D. 
We devised the following two procedures for selecting the 
threshold of D: (1) the threshold selection using sample 
objects that have actually changed at different times and (2) 
the threshold selection using the 2-dim. frequencies of 
distance values in two different spectral bands. 
International Archives of Photogrammetry and Remote Sensing. Vol. XXXI, Part B7. Vienna 1996