The intersection of all circles of equal altitude is performed numerically. The result of the measurement process previously outlined produces:
From each pair altitude/GP, it is possible to define a circle of equal altitude on the surface of the Earth. As a two-dimensional circle in three dimensional space. The important point is that, if we parameterize the Earth surface in longitude/latitude coordinates, the circles of equal altitudes are not two dimensional circles. Instead, they are the loci of solutions of [7]:
where is a point in latitude/longitude space,
is
the GP of Sun in this space and
is the measured altitude.
If we have pairs altitude/GP, we have a series of
equations in
two unknowns. We set the following non-linear least squares problem:
The minimization starts with a grid of ``presumed positions.'' The user defines a rectangular region in terms of latitude/longitude and spacings for a grid that covers a region on Earth. The size of the initial grid is arbitrary; the larger the grid, the longer the minimization will take.
If we have a large number of measurements and a large grid, the minimization
involved will be impractical. Instead, we use the following procedure instead.
Initially set to 1 and minimize (2) using only the
first pair altitude/GP. For each point in
the grid, minimize numerically equation (2). This
produces a list of solutions. Solutions that are too close are combined.
The remaining solutions will serve as seeds for the next minimization process.
The minimization step is repeated
times, each time including a new
measurement and combining solutions that are too close.
The remaining solutions are the result of the whole procedure.