General Description : The geometric system correction is the rotation and the coordinate transformation of the line of sight vectors of detectors to the earth Greenwich coordinate system using only the engineering information from the instrument and the spacecraft to identify the observed points by the detectors. The observed point on the surface is identified by the intersection of the earths surface and an extended line-of-sight vector The engineering information from the instrument and the spacecraft are called the supplementary data and the ancillary data, respectively. The geometric system correction is almost the same for the three subsystems, except for selected numbers of vectors to be transformed. Figure 2-6 shows the geometric system correction flow.

The image data are divided into blocks for both of the cross-track and the along-track directions. The block sizes are as follows.

VNIR bands 1, 2, 3N: 410 x 400 pixels

VNIR band 3B: 500 x 400 pixels

SWIR all bands: 20 x 20 pixels

TIR all bands: 72 x 70 pixels

These values were decided by considering the distortion of optical images on the focal plane in the cross-track direction and spacecraft stability in the along-track direction. The coordinate transformations are carried out only for the line-of-sight vectors of selected detectors. The numbers of the selected detectors are 11, 104 and 11 for VNIR, SWIR and TIR bands, respectively, which correspond to the number of the corner for each block of Level-0 images in the cross-track direction. Dummy detectors will have to be introduced to compensate for and then to completely define the block at the end of the cross-track direction.

The geometric system correction is divided into several parts as follows:

1. The pointing correction
2. The coordinates transformation from Navigation Base Reference of the spacecraft to the Orbital Reference Frame
3. The coordinates transformation from the Orbital Reference Coordinate Frame to the Earth Inertial coordinate Frame
4. The coordinates transformation from the Earth Inertial Coordinate Frame to the Earth Greenwich Coordinate Frame
5. Identification of the intersection of the Earth surface and an extention of the line of sight vector

Pointing Correction : The line of sight vectors in the geometric data base are those for the reference pointing angles (nominal nadir direction). The line of sight vectors are changed using the pointing position and knowledge from the supplementary data. The pointing axes information in the geometric data base are used for the transformation of the line-of-sight vectors due to change in the pointing position.

The line of sight vector changes with the rotation for the pointing axis by an angle of β from S₀ to S as follows.

(2-1)

where

S_0x, S_0y, S_0z : x, y , z components of the line of sight vector S₀ before pointing,

S_x, S_y, S_z : x, y, x components of the line of sight vector S after pointing,

Figure 2-7 shows the relation between the pointing axis and the NBR Coordinate Frame. The angles Dq_yaw and Dq _pitch are the yaw and the pitch rotation angles, respectively, to coalign the X_NBR to the pointing axis.

Figure 2-7 Pointing Axis Vector in NBR Coordinate Frame

 

Spacecraft-to-Orbit Coordinates : The spacecraft coordinates are slightly different from the orbit coordinates. The difference originates from the spacecraft attitude control accuracy and is provided as the attitude angle data in the spacecraft ancillary information. The orbit coordinate system is right-handed and orthogonal. The +z-axis is a line from the spacecraft center of mass to the center of the earth. The +y-axis is a line normal to the z-axis and the spacecraft instantaneous velocity vector (negative orbit normal direction). The x-axis completes the right hand set. This process is carried out by using attitude angles and rates in the ancillary data.

The line of sight vectors in the Spacecraft NBR Coordinate Frame can be converted to the expression in the Orbital Reference Frame using the attitude angle data in the spacecraft ancillary data as follows.

SOR = FSO•yaw FSO•pitch FSO•roll S

(2-6)

where S : the line of sight vector expressed in the NBR Coordinate Frame,
S_OR : the line of sight vector expressed in the Orbit Reference Coordinate Frame,

a_roll, a _pitch , a _yaw : roll, pitch, yaw components of the attitude data, respectively, in the spacecraft ancillary data

Orbit-to-Earth Inertial Coordinates : This process is the coordinate transformation to earth-centered coordinates in inertial space. Two-dimensional array vectors can be obtained by this transformation using the spacecraft movement. The array dimension for one observation depends on each observation period, that is, number of pixels in the along-track direction. This process is carried out by using position and velocity information in the ancillary data. The Precession and the Nutation effects are considered to be the more accurate geolocation data, since the spacecraft position information is based on the mean of the J2000.0 coordinate frame, which is the earth inertial coordinates at noon of January 1st, 2000. The line of sight vectors in the Orbital Reference Coordinate Frame can be converted to the expression in the Earth Inertial Coordinate Frame as follows.

	SEI = FOI SOR	(2-10)
where	SOR : the line of sight vector expressed in the Orbit Reference Coordinate Frame, SEI : the line of sight vector expressed in the Eartt Inertial Coordinate Frame,

	FOI = (Tx Ty Tz )	(2-11)
Tx Ty Tz : unit vector components of x, y and z axes of the Orbital Coordinate Frame expressed in the Earth Inertial Coordinate Frame and defined as

	Tx = Ty x Tz Ty = unit (-R x V ) Tz = unit (-R )	(2-12)
R ,V : the spacecraft position and velocity vectors expressed in the Earth Inertial Frame

For more accurate calculation the Precession matrix P and the Nutation matrix N shall be applied to the line of sight vector S_EI in the Earth Inertial Coordinate Frame.

Earth Inertial-to-Earth Fixed coordinates : This process is the coordinate transformation to the earth centered and earth-fixed coordinates, and carried out by using the earth rotation values calculated from the time information in the ancillary data. The UTC time, which is provided from the spacecraft, is converted to the UT1 to calculate the exact earth rotation angle.

The line of sight vectors in the Earth Inertial Coordinate Frame can be converted to the expression in the Earth Fixed Coordinate Frame as follows.

	SEF= FIF SEI	(2-13)
where	SEI : the line of sight vector expressed in the Eartt Inertial Coordinate Frame, SEF : the line of sight vector expressed in the Eartt Fixed Coordinate Frame,

		(2-14)
	qg : Greenwich true sidereal hour angle.

Eath Surface Identification : The observation point is identified from the intersection of the earth surface and an extension of the line-of-sight vector. The WGS-84 is used as the earth surface model.

The observing earth surface can be identified calculating the crossing point between the extension line of the LOS vector and the earth surface. The extension line of the LOS vector can be expressed as follows.

	x = X + SEF•x r y = Y + SEF•y r z = Z + SEF•z r	(2-15)
where	SEF•x , SEF•y , SEF•z : x, y, z components of the LOS vector SEF in the Earth Fixed Coordinate Frame, X, Y, Z : x, y, z components of the Spacecraft position vector in the Earth Fixed Coordinate Frame r : parameter.

The earth surface can be expressed as follows.

	(x 2 + y 2)/a 2 + z 2/b 2 = 1	(2-16)
where	a = 6378136m (Earth radius at equator--WGS-84), b = a (1 - f ) (Earth radius at pole----WGS-84), f = 1/298.2572 .	(2-17)

The intersection can be calculated from eqs.(2-15) and (2-16). When the observing point is expressed as Px, Py, and Pz , the geocentric latitude y and the longitude l can be expressed as follows.

	y = tan-1{Pz /(Px2 + Py2)1/2}	(2-18)
	l = tan-1(Py /Px )	(2-19)

System Correction Accuracy : Table 2-2 shows the pixel geolocation knowledge as a result of the geometric system correction considering both the spacecraft and the instrument contributions. Total ASTER pixel geolocation knowledge is decided by the spacecraft position knowledge, the spacecraft pointing knowledge and ASTER pointing knowledge. Only the pixel geolocation knowledge of VNIR is considered, since the SWIR and TIR bands will be coregistered to VNIR band 2 as a reference band in the Level-1 processing .

*1: Three non-optimal 9 minute TDRS contacts per orbit, GJM2 Geopotential ( 30 x 30), solar flux of 175, 5 % Cd error, TDRS ephemeris error of 75 meters. Two star trackers, rigid body/low frequency pointing knowledge error removed.
*2: Slightly larger than RSS of two values (Spacecraft and ASTER instrument), because of unallocated margin.