The Positioning of SPOT Stereo Images with a Local Orbit
2. Method
The method is developed on the base of collinearity condition equations as shown in equation (1). The sensor positions and attitudes of SPOT imagery generally are approximated as polynomials shown in equation (2) and (3). The polynomials expressed as the functions of time in third and second order. This leads to 21 unknown coefficients. As a result of that, an essential number of ground control points increases and the convergence of the estimating process slow down. To tackle these issues, one of the methods is to use weighted observation equations of the ground control points and the attitudes of imagery. The equations can be expressed as the equations of (4) and (5). Finding suitable weight values for both equations becomes anothernewissue because of very high correlation among the coefficients. In order to reduce the impact of the issue, local orbit of each SPOT image can be estimated.
2.1 Local Orbit
SPOT satellite ephemeris data provides satellite attitude and position b ut their density and accuracy are not good enough to give better result and stability of estimated process. However, the satellite attitude and position of each line of image can be determined with the data provided in SPOT CCT. The determined satellite attitude and position are then expressed as equations (6) that are the function of time. Therefore, a local orbit for a SPOT image can be estimated by using satellite attitude and position of the line of image points that are selected and distributed evenly on the image. For each corresponding image point twelve observation equations are listed. It is not necessary to provide the ground coordinates of the corresponding image points when the local orbit is estimated. The coordinates of ground control points have to be provided while the estimate process is carried out. The local orbit has to satisfy with collinearity condition equation (1) and ground control point equation (4). The
unknown parameters thus are estimated by means of solving equations (1), (4) and (6) with weighted nonlinear least squares method.
where M
1=m
11j(M
i - XL
j)+m
12j(Y
i-YL
j)+m
13j(Z
i-ZL
j)
M
2=m
21j(X
i-XL
j)+m
22j(Y
i-YL
i)+m
23j(Z
i-
ZLj)
M3M31j(X1-XLj)+m32j(Yi-YLj)+m
33j(Z
i-ZL
j)
y
i: Line Number of Image-3000 *13um,
S
y: Scale factor in y direction,
XL
j , YL
j , ZL
j : Sensor position of the j
th line,
X
i , Y
i , Z
i : Coordinates of the i
th ground control point,
f : Focal length,
m
11j ,…, m
33j ˜Rotation matrix of attitude of the j
th line.
