International Archives of the Photogrammetry, Remote Sensing and Spatial Information Sciences, Vol XXXV, Part B3. Istanbul 2004 
  
In order to acquire a higher accuracy of the geocoded image, 
three different geometric corrections method were performed 
and tested. They are 2-D polynomial method, rational function 
method and rigorous model method. It was found that the 
rigorous model method, proposed by Toutin, 2000, gave the 
best result on the geometric correction of the IKONOS image 
with a root mean square error approximately one pixel (4m). 
3. TOPOGRAPHIC CORRECTIONS 
In this study, three topographic correction methods were tested 
and compared. They are the Cosine Correction, Minnaert 
Correction and a Normalization Method. 
3.1 Cosine Correction 
In this method, the surface is assumed to have Lambertian 
behaviour, i.e. to be a perfect diffuse reflector, having the same 
amount of reflectance in all view directions. Thus, the 
Lambertian correction function attempts to correct only for 
differences in illumination caused by the orientation of the 
surface (Jones et al., 1988). 
For the Lambertian assumption, the most widely used 
correction is this cosine method using the equation (1), (2) and 
(3), proposed by Teillet et al. (1982), 
cos i = cos E cos Z + sin E sin Z cos( A, — As) (1) 
Ln(4)- L(A)/ cos i (2) 
L, = L, cosz/cosi (3) 
where L = radiance 
Z = solar zenith angle 
Ly7 radiance for horizontal surface 
Ly = radiance observed over the inclined terrain 
i = incidence angle with respect to surface normal 
E = slope inclination 
Z = solar zenith angle 
A, = solar azimuth 
As 7 surface aspect of the slope angle 
Although the Lambertian assumption is simple and convenient 
for topographic correction, there is a recognised problem in the 
corrected images. Thus when correcting the topographic effect 
under a Lambertian surface assumption, images tended to be 
over-corrected, with slopes facing away from the sun appearing 
brighter than sun-facing slopes due to diffuse sunlight being 
relatively more influential on the shady slope (Jones et al., 
1988). Therefore, non-Lambertian topographic correction 
method has been developed. 
3.2 Minnaert Correction 
In 1980, Smith ef al. introduced an empirical photometric 
function, the Minnaert constant, to test the Lambertian 
assumption for surfaces. The Minnaert function was developed 
in 1941, and has been used for photometric analysis of lunar 
surfaces (Justice and Holben, 1979). 
In the study by Smith ef al. (1980), the Minnaert constant, k, 
was derived by first linearizing the equation below: 
642 
k(A k(2)-1 
L(à,e) = Lncos ( Vy eds (a) e 
(4) 
where L=radiance 
/=wavelength 
e=slope angle 
L, = radiance when i=0 
k = Minnaert constant 
i = incidence angle 
ER nA 
Lcose - Lncos icos e (5) 
After linearization, equation (2) becomes: 
log (L cos e) = log Ln + k log (cos i cos e) (6) 
Now, we can obtain the regression value of k using equation 
(3), from the linear form of y=mx+c 
where x = log (cos icos e) 
y =log (; COS e) 
c=logln 
The value of the Minnaert constant lies between 0 and 1. It is 
used to describe the roughness of the surface. When the surface 
has Lambertian behaviour, the value of the Minnaert constant 
is 1. Otherwise, it is less than 1. 
After the Minnaert constant, k, is determined, a backwards 
radiance correction transformation model can be developed. 
Ln = L(cos e) Kcos' i cos" e) (7) 
If we compare this method with the non-Lambertian cosine 
method mathematically, it can be seen that the Minnaert 
constant, k, is used to weaken the power of topographic 
correction. In other words, it is used to describe the roughness 
of the surface of the terrain. As a result, the problem of over- 
correction in the area facing away from the sun can be solved. 
3.3 Normalization Method 
The normalization method used here is modified from the two- 
stage normalization proposed by Civco, 1989, and consists of 
two stages. In the first stage, shaded relief models, 
corresponding to the solar illumination conditions at the time 
of the satellite image are computed using the DEM data. This 
requires the input of the solar azimuth and altitude provided by 
the metadata of the satellite image. The resulting shaded relief 
model would have values between 0 and 255. 
After the model is created, a transformation of each of the 
original bands of the satellite image is performed to derive 
topographically normalized images using equation (8). 
(mx) | 
Hk 
àDN,, » DN,; * | DNajjx C (8)