Dilution of Precision (GPS) - Computation of DOP Values

Computation of DOP Values

As a first step in computing DOP, consider the unit vectors from the receiver to satellite i: where and where and denote the position of the receiver and and denote the position of satellite i. Formulate the matrix, A, as:

$A = \begin{bmatrix} \frac {(x_1- x)} {R_1} & \frac {(y_1-y)} {R_1} & \frac {(z_1-z)} {R_1} & -1 \\ \frac {(x_2- x)} {R_2} & \frac {(y_2-y)} {R_2} & \frac {(z_2-z)} {R_2} & -1 \\ \frac {(x_3- x)} {R_3} & \frac {(y_3-y)} {R_3} & \frac {(z_3-z)} {R_3} & -1 \\ \frac {(x_4- x)} {R_4} & \frac {(y_4-y)} {R_4} & \frac {(z_4-z)} {R_4} & -1 \end{bmatrix}$

The first three elements of each row of A are the components of a unit vector from the receiver to the indicated satellite. If the elements in the fourth column are c which denotes the speed of light then the factor is always 1. If the elements in the fourth column are -1 then the factor is calculated properly. Formulate the matrix, Q, as:

$Q = \left (A^T A \right )^{-1}$

This computation is in accordance with Section 1.4.2 of Principles of Satellite Positioning where the weighting matrix, P, has been set to the identity matrix.

The elements of Q are designated as:

$Q = \begin{bmatrix} d_x^2 & d_{xy}^2 & d_{xz}^2 & d_{xt}^2 \\ d_{xy}^2 & d_{y}^2 & d_{yz}^2 & d_{yt}^2 \\ d_{xz}^2 & d_{yz}^2 & d_{z}^2 & d_{zt}^2 \\ d_{xt}^2 & d_{yt}^2 & d_{zt}^2 & d_{t}^2 \end{bmatrix}$

The Greek letter is used quite often where we have used d. However the elements of Q 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:

$\begin{align} PDOP &= \sqrt{d_x^2 + d_y^2 + d_z^2}\\ TDOP &= \sqrt{d_{t}^2}\\ GDOP &= \sqrt{PDOP^2 + TDOP^2}\\ \end{align}$

in agreement with Section 1.4.9 of Principles of Satellite Positioning.

The horizontal dilution of precision, and the vertical dilution of precision, 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.

Read more about this topic: Dilution Of Precision (GPS)

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