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We now take on the task of how to compute the dilution of precision terms. As a first step in computing DOP, consider the unit vector from the receiver to satellite i with components <math>\frac{(x_i- x)}{R_i}</math>, <math>\frac {(y_i-y)}{R_i}</math>, and <math>\frac {(z_i-z)}{R_i}</math> where the distance from receiver to the satellite, <math>\ R_i </math>, is given by:
:<math>R_i\,=\,\sqrt{(x_i- x)^2 + (y_i-y)^2 + (z_i-z)^2}</math>
where <math>\ x, y, and\ z</math> denote the position of the receiver and <math>\ x_i, y_i, and\ z_i</math> denote the position of satellite ''i''. These ''x'', ''y'', and ''z'' components may be components in a North, East, Down coordinate system a South, East, Up coordinate system or other convenient system. Formulate the matrix ''A'' as:
:<math>A =
\begin{bmatrix}
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\end{bmatrix}</math>
The first three elements of each row of ''A'' are the components of a unit vector from the receiver to the indicated satellite. The elements in the fourth column are c where c denotes the speed of light. Formulate the matrix, ''Q'', as
:<math> Q = \left (A^T A \right )^{-1}
</math>
This computation is in accordance with Chapter 11 of The global positioning system by Parkinson and Spilker where the weighting matrix, ''P'', has been set to the identity matrix. The elements of the ''Q'' matrix are designated as:<ref>Parkinson (1996)</ref>
:<math>Q =
\begin{bmatrix}
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</math>
The Greek letter <math>\ \sigma</math> is used quite often where we have used ''d''. However the elements of the ''Q'' matrix do not represent variances and covariances as they are defined in probability and statistics. Instead they are strictly geometric terms. Therefore d as in dilution of precision is used. PDOP, TDOP and GDOP are given by
:<math>PDOP = \sqrt{d_x^2 + d_y^2 + d_z^2}</math>,
:<math>\ TDOP = \sqrt{d_{t}^2} = |d_{t}|\ </math>, and
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in agreement with [http://www.gmat.unsw.edu.au/snap/gps/gps_survey/chap1/149.htm "Section 1.4.9 of PRINCIPLES OF SATELLITE POSITIONING"].
The horizontal dilution of precision, <math> HDOP = \sqrt{d_x^2 + d_y^2}</math>, and the vertical dilution of precision, <math>\ VDOP = \sqrt{d_{z}^2}</math>, are both dependent on the coordinate system used. To correspond to the local horizon plane and the local vertical, ''x'', ''y'', and ''z'' should denote positions in either a North, East, Down coordinate system or a South, East, Up coordinate system.
=== Derivation of DOP equations ===
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