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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