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# Bancroft Method

Fundamentals
Title Bancroft Method
Author(s) J. Sanz Subirana, J.M. Juan Zornoza and M. Hernández-Pajares, Technical University of Catalonia, Spain.
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. ^