Doppler, range and ellipsoid equations

The following three equations are used regularly throughout the software to compute the point P that corresponds to a certain line and pixel in the master or slave image (see also [7]). Precise orbits are necessary.

- Doppler: The point P at the surface lies perpendicular to the orbit due to zero Doppler processing (otherwise this equation has to be adapted with a slant angle).
- Range: The geometrical distance to P on the surface is equal to the speed of light times the range time.
- Ellipsoid: Force the point to lie on an ellipsoid.

The equations for the point P on the ellipsoid and the satellite S in its orbit are (where x denotes (x,y,z)):

(D43) |

To compute the coordinates of a point P on the ellipsoid, corresponding with line

Next the set of equations is used to solve for P(x,y,z). This is done iteratively by linearization, which requires the derivative of the equations to x and approximate values for the unknowns (the coordinates of the center ( ) given in the SLC leader file, converted to xyz on a sphere).

(D47) |

(D48) |

Solving this exactly determined system of 3 equations yields the next solution and the new values for the unknowns become which are used to compute and . The solution is updated until convergence ( 1e-6 meters).

(D49) |

Where:

contains the observations. (set of equations)

contains the unknowns (coordinates of P). contains the partials (evaluated for previous solution).

To solve for the azimuth time if the coordinates of a point on the ground is known, only the Doppler D.45 equation needs to be used. the derivative with respect to azimuth time of this equation equals

(D50) |

The solution is equal to (use approximate solution to evaluate these expressions).

(D51) |

and

(D52) |

The solution is updated until convergence ( 1e-10 seconds).

The range time is then computed as in equation D.46

(D53) |