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== Dilution of precision {{anchor|gdop}} ==
{{main articlesee|Dilution of precision (navigation)}}
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=== Computation of geometric dilution of precision ===
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The concept of geometric dilution of precision was introduced in the section, ''error sources and analysis''. Computations were provided to show how PDOP was used and how it affected the receiver position error standard deviation.
When all visible GPS satellites are close together in the sky (i.e., small angular separation), the DOP values are high; when far apart, the DOP values are low. Conceptually, satellites that are close together cannot provide as much information as satellites that are widely separated. Low DOP values represent a better GPS positional accuracy due to the wider angular separation between the satellites used to calculate GPS receiver position. HDOP, VDOP, PDOP and TDOP are respectively Horizontal, Vertical, Position (3-D) and Time Dilution of Precision.
Figure 3.1 Dilution of Precision of Navstar GPS data from the U.S. Coast Guard provide a graphical indication of how geometry affect accuracy.<ref>{{Cite web |last=<!--Staff writer(s); no by-line.--> |date=September 1996 |title=NAVSTAR GPS User Equipment Introduction |url=http://www.navcen.uscg.gov/pubs/gps/gpsuser/gpsuser.pdf |access-date=July 5, 2014 |website=US Coast guard navigation center |publisher=US Coast Guard}}</ref>
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,</math> and <math>\ z</math> denote the position of the receiver and <math>\ x_i, y_i,</math> and <math>\ 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 any other convenient system. Formulate the matrix ''A'' as:
:<math>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}</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}
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}
</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>\begin{align}
PDOP &= \sqrt{d_x^2 + d_y^2 + d_z^2} \\
TDOP &= \sqrt{d_{t}^2} = |d_{t}| \\
GDOP &= \sqrt{PDOP^2 + TDOP^2}
\end{align}</math>
in agreement with [https://web.archive.org/web/20141122153439/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} = |d_z|</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 equations for computing geometric dilution of precision ===
The equations for computing the geometric dilution of precision terms have been described in the previous section. This section describes the derivation of these equations. The method used here is similar to that used in [https://books.google.com/books?id=lvI1a5J_4ewC&pg=PA474 "Global Positioning System (preview) by Parkinson and Spiker"]
Consider the position error vector, <math>\mathbf{e}</math>, defined as the vector from the intersection of the four sphere surfaces corresponding to the pseudoranges to the true position of the receiver.<math>\mathbf{e} = e_x\hat{x} + e_y\hat{y} + e_z\hat{z} </math> where bold denotes a vector and <math>\hat{x}</math>, <math>\hat{y}</math>, and <math>\hat{z}</math> denote unit vectors along the x, y, and z axes respectively. Let <math>\ e_t</math> denote the time error, the true time minus the receiver indicated time. Assume that the mean value of the three components of <math>\mathbf {e}</math> and <math>\ e_t</math> are zero.
:<math>A
\begin{bmatrix}
e_x \\ e_y \\ e_z \\ e_t
\end{bmatrix} =
\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}
\begin{bmatrix}
e_x \\ e_y \\ e_z \\ e_t
\end{bmatrix} =
\begin{bmatrix}
e_1 \\ e_2 \\ e_3 \\ e_4
\end{bmatrix}
\ (1)</math>
where <math>e_1</math>, <math>e_2</math>, <math>e_3</math>, and <math>e_4</math> are the errors in pseudoranges 1 through 4 respectively. This equation comes from linearizing [[Global Positioning System#Multidimensional Newton-Raphson calculations|the Newton-Raphson equation]] relating pseudoranges to receiver position, satellite positions, and receiver clock errors. Multiplying both sides by <math>A^{-1}</math> there results
:<math>
\begin{bmatrix}
e_x \\ e_y \\ e_z \\ e_t
\end{bmatrix} =
A^{-1}
\begin{bmatrix}
e_1 \\ e_2 \\ e_3 \\ e_4
\end{bmatrix} \ (2)</math> .
Transposing both sides:
:<math>
\begin{bmatrix}
e_x & e_y & e_z & e_t
\end{bmatrix} =
\begin{bmatrix}
e_1 & e_2 & e_3 & e_4
\end{bmatrix}\left (A^{-1} \right )^T \ (3)</math> .
Post multiplying the matrices on both sides of equation (2) by the corresponding matrices in equation (3), there results
:<math>
\begin{bmatrix}
e_x \\ e_y \\ e_z \\ e_t
\end{bmatrix}
\begin{bmatrix}
e_x & e_y & e_z & e_t
\end{bmatrix} =
A^{-1}
\begin{bmatrix}
e_1 \\ e_2 \\ e_3 \\ e_4
\end{bmatrix}
\begin{bmatrix}
e_1 & e_2 & e_3 & e_4
\end{bmatrix}\left (A^{-1} \right )^T \ (4)
</math> .
Taking the expected value of both sides and taking the non-random matrices outside the expectation operator, E, there results:
:<math>E
\left (\begin{bmatrix}
e_x \\ e_y \\ e_z \\ e_t
\end{bmatrix}
\begin{bmatrix}
e_x & e_y & e_z & e_t
\end{bmatrix} \right ) =
A^{-1} E
\left (\begin{bmatrix}
e_1 \\ e_2 \\ e_3 \\ e_4
\end{bmatrix}
\begin{bmatrix}
e_1 & e_2 & e_3 & e_4
\end{bmatrix} \right )
\left (A^{-1} \right )^T \ (5)
</math>
Assuming the pseudorange errors are uncorrelated and have the same variance, the covariance matrix on the right side can be expressed as a scalar times the identity matrix. Thus
:<math>
\begin{bmatrix}
\sigma_x^2 & \sigma_{xy}^2 & \sigma_{xz}^2 & \sigma_{xt}^2 \\
\sigma_{xy}^2 & \sigma_{y}^2 & \sigma_{yz}^2 & \sigma_{yt}^2 \\
\sigma_{xz}^2 & \sigma_{yz}^2 & \sigma_{z}^2 & \sigma_{zt}^2 \\
\sigma_{xt}^2 & \sigma_{yt}^2 & \sigma_{zt}^2 & \sigma_{t}^2
\end{bmatrix} = \sigma_R^2 \ A^{-1} \left (A^{-1} \right )^T =
\sigma_R^2 \ \left (A^T A \right )^{-1} \ (6)</math>
since <math>\ A^{-1} \left (A^{-1} \right )^T \left (A^T A \right ) = I </math>
Note: <math>\left( A^{-1} \right)^T = \left( A^{T} \right)^{-1},</math> since <math>I = \left( A A^{-1} \right)^T = \left( A^{-1}\right)^T A^T</math>
Substituting for <math>\left( A^T A \right)^{-1} = Q</math> there follows
:<math>
\begin{bmatrix}
\sigma_x^2 & \sigma_{xy}^2 & \sigma_{xz}^2 & \sigma_{xt}^2 \\
\sigma_{xy}^2 & \sigma_{y}^2 & \sigma_{yz}^2 & \sigma_{yt}^2 \\
\sigma_{xz}^2 & \sigma_{yz}^2 & \sigma_{z}^2 & \sigma_{zt}^2 \\
\sigma_{xt}^2 & \sigma_{yt}^2 & \sigma_{zt}^2 & \sigma_{t}^2
\end{bmatrix} = \sigma_R^2
\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} \ (7)
</math>
From equation (7), it follows that the variances of indicated receiver position and time are
:<math>\sigma_{rc}^2 = \sigma_x^2 + \sigma_y^2 + \sigma_z^2 = \sigma_R^2\left(d_x^2 + d_y^2 + d_z^2\right) = PDOP^2 \sigma_R^2</math> and
:<math>\sigma_t^2 = \sigma_R^2 d_t^2 = TDOP^2 \sigma_R^2</math>
The remaining position and time error variance terms follow in a straightforward manner.
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