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22. Geocoding: Geometry

The location of the (i,j) pixel in a given image can be derived from knowledge of the sensor position and velocity. More precisely, the location of the antenna phase centre in an Earth referenced coordinate system is required.
The target location is determined by the simultaneous solution of three equations:
1. range equation
2. Doppler equation
3. Earth model equation

The figure shows the geocentrical coordinate system illustrating a graphical solution for the pixel location equations.

The range equation is given by

where:
R s sensor position vector
R t target position vector
For a given cross-track pixel number j in the slant range image, the range to the j th pixel is

where ΔN represents an initial offset in complex pixels (relative to the start of the sampling window) in the processed data set. This offset, which is nominally 0, is required for pixel location in subswath processing applications, or for a design where the processor steps into the data set an initial number of pixels to compensate for the range walk migration.

The Doppler equation is given by:

where:
λ radar wavelength
fDC Doppler centroid frequency
Vs sensor (antenna phase centre) velocity
Vt target velocity
The target velocity can be determined from the target position by:

where ωe is the Earth's rotational velocity vector. The Doppler centroid in the Doppler equation is the value of the used in the azimuth reference function to form the given pixel. An offset between the value of fDC in the reference function and the true causes the target to be displaced in azimuth according to

where
ΔfDC is the difference between the true fDC and the reference fDC
fκ is the Doppler rate used in the reference function
and Vsw is the magnitude of the swath velocity

To compensate for this displacement, when performing the target location, the identical fDC used in the reference function to form the pixel should be used in the Doppler equation. The exception to this rule is if an ambiguous fDC is used in the reference function. That is, if the true fDC is offset from the reference fDC by more than +/- fp /2. In this case, the pixel shift will be according to the Doppler offset between the reference fDC and the Doppler centroid of the ambiguous Doppler spectrum, resulting in a pixel location error of

where m is the number of PRFs the reference fDC is offset from its true value (i.e., the azimuth ambiguity number).
The third equation is the Earth model equation. An oblate ellipsoid can be used to model the Earth's shape as follows:

where Re is the radius of the Earth at the equator, h is the local target elevation relative to the assumed model, and Rp, the polar radius, is given by

where f is the flattening factor of the ellipsoid. If a topographic map of the area imaged is used to determine h, the Earth model parameters should match those used to produce the map. Otherwise, a mean sea level model can be used.

The target location as given by {xt, yt, zt} is determined from the simultaneous solution of the Range, Doppler and Earth model equations for the three unknown target position parameters.
This is illustrated in the figure; it shows the Earth (geoid) surface intersected by a plane whose position is given by the Doppler centroid equation. This intersection, a line of constant Doppler, is then intersected by the slant range vector at a given point, the target location. The left-right ambiguity is resolved by knowledge of the sensor's pointing direction.

The accuracy of this location procedure (assuming an ambiguous was not used in the processing) depends on the accuracy of the sensor position and velocity vectors, the measurement accuracy of the pulse delay time, and knowledge of the target height relative to the assumed Earth model.
The location does not require attitude sensor information. The cross-track target position is established by the sampling window, independent of the antenna footprint location (which does depend on the roll angle). Similarly, the azimuth squint angle, or aspect angle resulting from yaw and pitch of the platform, is determined by the Doppler centroid of the echo, which is estimated using a clutterlock technique.
Thus the SAR pixel location is inherently more accurate than that of optical sensors, since the attitude sensor calibration accuracy does not contribute to the image pixel location error.