Some basic information on satellite orbits and celestial mechanics are useful to better understand platform-related distortions� The EO satellites obey the celestial mechanical laws defined by Newton and Kepler for an unperturbed trajectory (Keplerian orbit) and by Gauss and Lagrange for a perturbed trajectory (osculatory orbit; Escobal 1965; Centre National d’Études Spatiales 1980)� A number of perturbations (due to Earth gravity and surface irregularities, atmospheric drag, etc�) slowly change the Keplerian orbit based on the two-body attraction of Newton’s law into an osculatory orbit (Centre National d’Études Spatiales 1980)� Information on orbits is often needed, and different orbital models can be used depending on their utility and required accuracy (Bakker 2000):
To calculate the satellite location on its osculatory orbit in order to compute the
•
Earth coordinates of scanned pixels, requiring high accuracy (submeters) over a small time frame (seconds)
To predict when the satellite will pass over a specific area, requiring low accuracy
•
(kilometers) but over a long time frame (days)
Many orbital models have been developed since 1960 using the same mechanical laws with Gaussian/Lagrangian equations; the differences between the orbital models are mainly in the number and types of perturbations and the techniques to integrate them�
As defined and adapted by the North American Aerospace Defense Command, sim- plified general perturbations (SGPs), SGP4, and the most accurate SGP8 are the orbital models to be used for low- and near-Earth satellites (orbital period less than 225 minutes and altitude less than 6000 km)� Most, if not all, of the civilian EO spacecrafts have near- Earth, retrograde, quasi-circular, quasi-polar, geosynchronous, and sun-synchronous orbits (Figure 8�2)�
Near-Earth orbits (altitude more than 300 km) are high enough to reduce the atmo- spheric drag� Retrograde orbits with 90°–180° inclination are westward-launched orbits, which require extra fuel to compensate for the Earth’s rotation, but they provide the only solution for obtaining sun-synchronous orbits� Quasi-circular orbits avoiding large changes in altitude enable images with similar scales to be acquired, which is desirable for EO� Quasi-polar orbits with 90°–100° inclination enable sensors to image the entire Earth, including most of the poles� Because geosynchronous orbits have a repeating ground track, they have an orbital period that is an integer multiple of the
Earth’s sidereal rotation period� This integer multiple is called the “repeat cycle�” The special case of a geosynchronous orbit that is circular and directly above the equator is called a “geostationary orbit�” The satellite track on the ground, also called the “path,”
is kept fixed within certain limits related to orbit maintenance accuracy� Paths are arti- ficially divided into squared scenes at regular intervals, generating rows� An example of the path–row system is the World Reference System of Landsat satellites� The main advantage is that the satellite follows a fixed pattern on the Earth, which is desirable for operational EO systems� Sun-synchronous orbits enable satellites to pass overhead at the same local solar time and thus to acquire images with almost identical illumina- tion or lighting conditions (Figure 8�2)� Although the comparison of multidate images is easier with a sun-synchronous orbit, this approach has some disadvantages, espe- cially as variations in illumination reveal different structural details� In addition, sun- synchronous orbits require retrograde orbits and strict relationships between orbital parameters (mainly inclination and height), which must be preserved during the satel- lite’s lifetime�
The distortions associated with near-Earth satellites are mainly due to the interaction between the platform and Earth (Earth’s gravity, shape, and movement, generating a quasi-elliptic movement; Escobal 1965; Centre National d’Études Spatiales 1980; Light et al� 1980)� The satellite has six degrees of freedom in space, which can be determined by (Toutin 1983; Kim and Dowman 2006) knowing any one of the following sets of details:
The satellite’s position relative to Earth’s center (three parameters:
• X, Y, Z) and
orientation (a solid angle)
Its position/velocity (left part of Figure 8�3) while the attitude is roll, pitch, yaw, or
•
a specific kind of Euler angles (right part of Figure 8�3)
Z
b
a Z O XX
Y
Y Nadir Nadir
P(t)
G Satellite
trajectory
Geocentric coordinate system Orbital coordinate system Six orbital
parameters (PX,PY,PZ,
VX,VY,VZ)
Three attitude parameters
(R,P,Y)
Satellite trajectory Pitch (rd)
Yaw(rd) Roll(rd) (m/s)V(t)
+ +
+ P(t)(m)
FIgure 8.3
Orbital (left) and attitude (right) parameters defining the orbit and the satellite in its orbit as a function of time�
(Courtesy and copyright Serge Riazanoff, VisioTerra, 2009�)
Its osculatory orbit with six parameters relative to the instantaneous orbit (semi-
•
major axis, eccentricity, inclination, ascending-node longitude, perigee argument, and mean anomaly; Figure 8�4)
Z P t
P t X V t Z
V
yaw pitch yaw
= ( ) =
( ) ; ( )Λ
( ) ;
t Z Y Z X
Λ Λ
yaw
roll= yaw pitch
Depending on the acquisition time (used as the seventh parameter) and the size of the image, all these parameters are time dependent due to orbital perturbations and thus generate a range of nonsystematic distortions� All these nonsystematic and time-dependent distortions are not predictable and must be evaluated for each image from satellite track- ing data or ground-control information or both� Some effects of these distortions include the following:
Platform altitude variation (
• z axis) is the most critical among position parameters�
It can change the pixel spacing in the across-track direction, whereas X and Y parameters generate only a translation in the sensed ground area� However, the altitude variation of the satellite (around 8–10 m/s for near-Earth orbits at around 800-km altitude) is not too significant over the time required to acquire a full scene (5–10 seconds for high- to medium-resolution satellite sensors)� It only represents 1/8000 relative pixel variation� For SPOT-5 panchromatic super mode (12,000 pix- els with 2�5-m spacing), it generates 0�3-m variation in pixel spacing, which cumu- latively generates an error around 2 m at the edge of the image�
Z Satellite
Perigee
Descending node
Center of geocentric system
Ascending node
Node line
Equatorial plane Orbital plane V
N ρ N'
O X
l
Y
Ω ω
FIgure 8.4
Description of a satellite osculatory orbit and its approximation by a Keplerian ellipse� X, Y, Z are the position coordinates in a geocentric frame reference system, I is the orbit inclination, Ω is the longitude of the ascend- ing node (N), ω is the argument of the perigee (P), (ω + v) is the argument of the satellite, and ρ is the distance between the Earth’s center (O) and the satellite�
Platform velocity variations can change the line spacing or create line gaps/over-
•
laps� Variations in spacecraft velocity only cause distortions in the along-track direction�
Platform altitude variation (Figure 8�5) can change the orientation and the shape
•
of VIR images (but it does not affect SAR image geometry)� The roll is the rotation around the flight vector (x axis; right part of Figure 8�3), hence in a “wing down”
direction, its variation causes lateral shifts and scale changes in the across-track direction� The pitch is the vertical rotation of the platform, in the “nose up” plane, and its variation results in scan-line spacing changes� The yaw is the rotation around the vertical axis and its variation changes the orientation of each scanned line, resulting in a skew between the scan lines� Variations in platform altitude can be severe in terms of location on the ground (absolute offset of hundreds of meters) over the time required to scan a full scene (5–10 seconds), and are signifi- cant when the variation is sudden over a few lines (relative offset of tens of meters within milliseconds)�