Dilution of precision

Interpretation

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

As a first step in computing DOP, consider the unit vectors from the receiver to satellite ${\displaystyle i}$: $${\displaystyle \begin{array}{cccc}& \left(\frac{x_i-x}{R_i},\frac{y_i-y}{R_i},\frac{z_i-z}{R_i}\right),& R_i& =\sqrt{(x_i-x{)}^2+(y_i-y{)}^2+(z_i-z{)}^2}\end{array}}$$

where ${\displaystyle x,y,z}$ denote the position of the receiver and ${\displaystyle x_i,y_i,z_i}$ denote the position of satellite i. Formulate the matrix, A, which (for 4 pseudorange measurement residual equations) is: $${\displaystyle A=\left[\begin{array}{cccc}\frac{x_1-x}{R_1}& \frac{y_1-y}{R_1}& \frac{z_1-z}{R_1}& 1\\ \frac{x_2-x}{R_2}& \frac{y_2-y}{R_2}& \frac{z_2-z}{R_2}& 1\\ \frac{x_3-x}{R_3}& \frac{y_3-y}{R_3}& \frac{z_3-z}{R_3}& 1\\ \frac{x_4-x}{R_4}& \frac{y_4-y}{R_4}& \frac{z_4-z}{R_4}& 1\end{array}\right]}$$

The first three elements of each row of A are the components of a unit vector from the receiver to the indicated satellite. The last element of each row refers to the partial derivative of pseudorange w.r.t. receiver's clock bias. Formulate the matrix, Q, as the covariance matrix resulting from the least-squares normal matrix: $${\displaystyle Q={\left(A^{\mathsf{T}}A\right)}^{-1}}$$

In general: $${\displaystyle Q={\left(J_{\mathsf{x}}^{\mathsf{T}}{\left(J_d{C}_d{J}_d^{\mathsf{T}}\right)}^{-1}{J}_x\right)}^{-1}}$$

where ${\displaystyle J_x}$ is the Jacobian of the sensor measurement residual equations ${\displaystyle f_i\left(\underset{\_}{x},\underset{\_}{d}\right)=0}$, with respect to the unknowns, ${\displaystyle \underset{\_}{x}}$; ${\displaystyle J_d}$ is the Jacobian of the sensor measurement residual equations with respect to the measured quantities ${\displaystyle \underset{\_}{d}}$, and ${\displaystyle C_d}$ is the correlation matrix for noise in the measured quantities.

For the preceding case of 4 range measurement residual equations: ${\displaystyle \underset{\_}{x}=(x,y,z,\tau {)}^{\mathsf{T}}}$, ${\displaystyle \underset{\_}{d}={\left({\tau }_1,{\tau }_2,{\tau }_3,{\tau }_4\right)}^{\mathsf{T}}}$, ${\displaystyle \tau =ct}$, ${\displaystyle {\tau }_i=c{t}_i}$, ${\displaystyle R_i=|{\tau }_i-\tau |=\sqrt{({\tau }_i-\tau {)}^2}}$, ${\displaystyle f_i\left(\underset{\_}{x},\underset{\_}{d}\right)=\sqrt{(x_i-x{)}^2+(y_i-y{)}^2+(z_i-z{)}^2}-\sqrt{({\tau }_i-\tau {)}^2}}$, ${\displaystyle J_x=A}$, ${\displaystyle J_d=-I}$ and the measurement noises for the different ${\displaystyle {\tau }_i}$ have been assumed to be independent which makes ${\displaystyle C_d=I}$.

This formula for Q arises from applying best linear unbiased estimation to a linearized version of the sensor measurement residual equations about the current solution ${\displaystyle \mathrm{\Delta }\underset{\_}{x}=-Q\ast \left(J_x^{\mathsf{T}}{\left(J_d{C}_d{J}_d^{\mathsf{T}}\right)}^{-1}f\right)}$, except in the case of B.L.U.E. ${\displaystyle C_d}$ is a noise covariance matrix rather than the noise correlation matrix used in DOP, and the reason DOP makes this substitution is to obtain a relative error. When ${\displaystyle C_d}$ is a noise covariance matrix, ${\displaystyle Q}$ is an estimate of the matrix of covariance of noise in the unknowns due to the noise in the measured quantities. It is the estimate obtained by the first-order second moment (F.O.S.M.) uncertainty quantification technique which was state of the art in the 1980s. In order for the F.O.S.M. theory to be strictly applicable, either the input noise distributions need to be Gaussian or the measurement noise standard deviations need to be small relative to rate of change in the output near the solution. In this context, the second criteria is typically the one that is satisfied.

This (i.e. for the 4 time of arrival/range measurement residual equations) computation is in accordance with [6] where the weighting matrix, ${\displaystyle P={\left(J_d{C}_d{J}_d^{\mathsf{T}}\right)}^{-1}}$ happens to simplify down to the identity matrix.

Note that P only simplifies down to the identity matrix because all the sensor measurement residual equations are time of arrival (pseudo range) equations. In other cases, for example when trying to locate someone broadcasting on an international distress frequency, ${\displaystyle P}$ would not simplify down to the identity matrix and in that case there would be a "frequency DOP" or FDOP component either in addition to or in place of the TDOP component. (Regarding "in place of the TDOP component": Since the clocks on the legacy International Cospas-Sarsat Programme LEO satellites are much less accurate than GPS clocks, discarding their time measurements would actually increase the geolocation solution accuracy.)

The elements of ${\displaystyle Q}$ are designated as:

$${\displaystyle Q=\left[\begin{array}{cccc}{\sigma }_x^2& {\sigma }_{xy}& {\sigma }_{xz}& {\sigma }_{xt}\\ {\sigma }_{xy}& {\sigma }_y^2& {\sigma }_{yz}& {\sigma }_{yt}\\ {\sigma }_{xz}& {\sigma }_{yz}& {\sigma }_z^2& {\sigma }_{zt}\\ {\sigma }_{xt}& {\sigma }_{yt}& {\sigma }_{zt}& {\sigma }_t^2\end{array}\right]}$$

PDOP, TDOP, and GDOP are given by: $${\displaystyle \begin{array}{cc}\mathrm{PDOP}& =\sqrt{{\sigma }_x^2+{\sigma }_y^2+{\sigma }_z^2}\\ \mathrm{TDOP}& =\sqrt{{\sigma }_t^2}\\ \mathrm{GDOP}& =\sqrt{{\mathrm{PDOP}}^2+{\mathrm{TDOP}}^2}\\ & =\sqrt{\mathrm{tr}Q}\end{array}}$$

Notice GDOP is the square root of the trace of the ${\displaystyle Q}$ matrix.

The horizontal and vertical dilution of precision, $${\displaystyle \begin{array}{cc}\mathrm{HDOP}& =\sqrt{{\sigma }_n^2+{\sigma }_e^2}\\ \mathrm{VDOP}& =\sqrt{{\sigma }_u^2}\end{array},}$$ are both dependent on the coordinate system used. To correspond to the local east-north-up coordinate system,

EDOP^2 x x x x NDOP^2 x x x x VDOP^2 x x x x TDOP^2

and the derived dilutions: $${\displaystyle \begin{array}{cc}\mathrm{GDOP}& =\sqrt{{\mathrm{EDOP}}^2+{\mathrm{NDOP}}^2+{\mathrm{VDOP}}^2+{\mathrm{TDOP}}^2}\\ \mathrm{HDOP}& =\sqrt{{\mathrm{EDOP}}^2+{\mathrm{NDOP}}^2}\\ \mathrm{PDOP}& =\sqrt{{\mathrm{EDOP}}^2+{\mathrm{NDOP}}^2+{\mathrm{VDOP}}^2}\end{array}}$$