Dilution of precision (GPS)

Dilution of precision (DOP) or geometric dilution of precision (GDOP) is a GPS term used in geomatics engineering to describe the geometric strength of satellite configuration on GPS accuracy.

Introduction
When visible GPS satellites are close together in the sky, the geometry is said to be weak and the DOP value is high; when far apart, the geometry is strong and the DOP value is low. Thus a low DOP value represents a better GPS positional accuracy due to the wider angular separation between the satellites used to calculate a GPS unit's position. Other factors that can increase the effective DOP are obstructions such as nearby mountains or buildings. DOP can be expressed as a number of separate measurements. HDOP, VDOP, PDOP, and TDOP are respectively horizontal, vertical, positional (3D), and temporal dilution of precision. They follow mathematically from the positions of the usable satellites. GPS receivers allow the display of these positions (skyplot) as well as the DOP values.

The term can also be applied to other location systems that employ several geographical spaced sites. It can occur in electronic-counter-counter-measures (electronic warfare) when computing the location of enemy emitters (radar jammers and radio communications devices). Using such an interferometry technique can provide certain geometric layout where there are degrees of freedom that cannot be accounted for due to inadequate configurations.

The effect of geometry of the satellites on position error is called geometric dilution of precision and it is roughly interpreted as ratio of position error to the range error. Now lets imagine that a tetrahedron is formed by lines joining the four satellites and receivers. The larger the volume of the tetrahedron, the better the value of GDOP; the smaller the volume of the tetrahedron, the worse the value of GDOP will be. Similarly, the greater the number of satellites, the better the value of GDOP.

Meaning of DOP Values
The DOP factors are functions of the diagonal elements of the covariance matrix of the parameters, expressed either in a global or a local geodetic frame.

Computation of DOP Values
As a first step in computing DOP, consider the unit vectors from the receiver to satellite i: $$\scriptstyle \left(\frac {(x_i-x)} {R_i}, \frac {(y_i-y)} {R_i}, \frac {(z_i-z)} {R_i}\right)$$ where $$\scriptstyle R_i= \sqrt{(x_i- x)^2 + (y_i-y)^2 + (z_i-z)^2}$$ and where $$\scriptstyle\ x,\ y$$ and $$\scriptstyle \ z$$ denote the position of the receiver and $$\scriptstyle \ x_i, y_i$$ and $$\scriptstyle \ z_i$$ 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} & c \\ \frac {(x_2- x)} {R_2} & \frac {(y_2-y)} {R_2} & \frac {(z_2-z)} {R_2} & c \\ \frac {(x_3- x)} {R_3} & \frac {(y_3-y)} {R_3} & \frac {(z_3-z)} {R_3} & c \\ \frac {(x_4- x)} {R_4} & \frac {(y_4-y)} {R_4} & \frac {(z_4-z)} {R_4} & c \end{bmatrix} $$ The first three elements of each row of A are the components of a unit vector from the receiver to the indicated satellite. The elements in the fourth column are c where c denotes the speed of light. Formulate the matrix, Q, as:
 * $$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} $$ The Greek letter $$\scriptstyle \sigma$$ 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, $$\scriptstyle HDOP = \sqrt{d_x^2 + d_y^2}$$, and the vertical dilution of precision, $$\scriptstyle \ VDOP = \sqrt{d_{z}^2} $$, 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.