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Bancroft Method

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FundamentalsFundamentals
Title Bancroft Method
Author(s) J. Sanz Subirana, J.M. Juan Zornoza and M. Hernández-Pajares, Technical University of Catalonia, Spain.
Level Advanced
Year of Publication 2011

The Bancroft method allows obtaining a direct solution of the receiver position and the clock offset, without requesting any "a priori" knowledge for the receiver location.


Raising and resolution

Let the prefit-residual of satellite-, computed from equation (1)


after removing all model terms not needing the a priory knowledge of the receiver position:[footnotes 1]


Thence, neglecting the tropospheric and ionospheric terms, as well as the multipath and receiver noise, the equation (3)

can be written as:


Developing the previous equation (4), it follows:


Then, calling and considering the inner product of Lorentz [footnotes 2] the previous equation (5) can be expressed in a more compact way as:


The former equation can be raised for every satellite (or prefit-residual ).


If four measurements are available, thence, the following matrix can be written, containing all the available information on satellite coordinates and pseudoranges (every row corresponds to a satellite):


Then, calling:


The four equations for pseudorange can be expressed as:


from where:


Then, taking into account the following equality

,


and that

,


from the former expression (10), one obtains:


The previous expression (13) is a quadratic equation in (note that matrix and the vector are also known) and provides two solutions, that introduced in expression (10) provides the searched solution:

.

The other solution is far from the earth surface.

Generalisation to the case of -measurements:

If more than four observations are available, the matrix is not square. However, multiplying by , one obtains (Least Squares solution):


where:


and then:


Notes

  1. ^ The tropospheric and ionospheric terms, and , can not be included, because the need to consider the satellite-receiver ray. Off course, after an initial computation of the receiver coordinates, the method could be iterated using the ionospheric and tropospheric corrections to improve the solution.
  2. ^