Dilution of Precision in Single Point Positioning: 18-Geom-B2 Satellite Navigation, May 2016, Q2

Question 2: Dilution of Precision in Single Point Positioning (15 marks)

Question text not reproduced: the examination questions are © Engineers and Geoscientists BC. Open the official past paper (linked at the top of this page) to read the question, then follow the worked solution below.

Given / setting. A single receiver observes L1 C/A pseudoranges to $n\ge 4$ satellites and solves for its position and clock offset by least squares. The Dilution of Precision (DOP) factors describe how the geometry of the visible satellites amplifies the ranging error into position error, independently of the magnitude of that ranging error.

Find. The measurement (design) model, the cofactor matrix, and the definitions of GDOP, PDOP, HDOP, VDOP and TDOP with all notation explained.

Satellite geometry and DOP: a well-spread constellation (left) yields a well-conditioned AᵀA and low DOP; clustered satellites (right) give a near-singular matrix and high DOP. DOP amplifies the ranging error (UERE) into position error.

Measurement model. The L1 C/A pseudorange to satellite $i$ is $P^i = \rho^i + c\,dt + \varepsilon^i$, where $\rho^i=\lVert \mathbf{x}^i-\mathbf{x}\rVert$ is the geometric range, $\mathbf{x}=(X,Y,Z)$ is the unknown receiver position, $dt$ the receiver clock offset, $c$ the speed of light and $\varepsilon^i$ the residual error. Linearizing about an approximate position $\mathbf{x}_0$ gives one row of the design matrix per satellite:

$$\mathbf{A}=\begin{bmatrix} -e_X^1 & -e_Y^1 & -e_Z^1 & 1\\ \vdots & \vdots & \vdots & \vdots\\ -e_X^n & -e_Y^n & -e_Z^n & 1\end{bmatrix},$$

where $(e_X^i,e_Y^i,e_Z^i)$ are the direction cosines of the unit line-of-sight vector from the receiver to satellite $i$, and the fourth column of ones corresponds to the receiver clock unknown $c\,dt$. The unknown-correction vector is $\delta\mathbf{u}=[\delta X,\delta Y,\delta Z,\;c\,\delta t]^{\mathsf T}$.

Cofactor matrix. Assuming uncorrelated pseudoranges of equal unit variance, the covariance of the estimated unknowns is proportional to the cofactor matrix

$$\mathbf{Q}=(\mathbf{A}^{\mathsf T}\mathbf{A})^{-1}=\begin{bmatrix} q_{X} & & & \\ & q_{Y} & & \\ & & q_{Z} & \\ & & & q_{t}\end{bmatrix}\;(\text{diagonal shown}).$$

Rotating the position part of $\mathbf{Q}$ into the local east–north–up (ENU) frame gives $q_{E},q_{N},q_{U}$. The DOP values are square roots of sums of these diagonal cofactors:

$$\text{GDOP}=\sqrt{q_E+q_N+q_U+q_t},\qquad \text{PDOP}=\sqrt{q_E+q_N+q_U},$$ $$\text{HDOP}=\sqrt{q_E+q_N},\qquad \text{VDOP}=\sqrt{q_U},\qquad \text{TDOP}=\sqrt{q_t}.$$

Definitions. GDOP (geometric) scales the combined position-and-time error; PDOP (position) the 3-D position error; HDOP (horizontal) the 2-D east–north error; VDOP (vertical) the height error; TDOP (time) the clock error. They combine as $\text{GDOP}^2=\text{PDOP}^2+\text{TDOP}^2$ and $\text{PDOP}^2=\text{HDOP}^2+\text{VDOP}^2$. The predicted position error follows from the User-Equivalent Range Error (UERE): $\sigma_{\text{pos}}=\text{DOP}\times\sigma_{\text{UERE}}$.

Physically, a well-spread geometry (satellites low and distributed all around the sky, plus one high) makes $\mathbf{A}^{\mathsf T}\mathbf{A}$ well-conditioned and yields small DOP (good); clustered satellites make the matrix nearly singular, inflating $\mathbf{Q}$ and the DOP (poor). A PDOP of ≤ 3–4 is generally considered good for surveying.

Concept & Theory

Topic. Propagation of ranging error into position error through satellite geometry.

Key relations. $\mathbf{Q}=(\mathbf{A}^{\mathsf T}\mathbf{A})^{-1}$; $\text{DOP}=\sqrt{\text{trace of the relevant diagonal block of }\mathbf{Q}}$; $\sigma=\text{DOP}\cdot\sigma_{\text{UERE}}$; identities $\text{GDOP}^2=\text{PDOP}^2+\text{TDOP}^2$, $\text{PDOP}^2=\text{HDOP}^2+\text{VDOP}^2$.

Why this works. DOP is purely a function of the unit line-of-sight geometry, so it can be predicted from an almanac before going to the field, letting survey windows be planned when geometry is strong. It cleanly separates the (hardware/atmosphere-driven) UERE from the (geometry-driven) amplification factor.

Common pitfall. Forgetting the fourth (clock) column, which distinguishes GDOP/TDOP from a pure trilateration; and confusing DOP (unitless amplifier) with actual position accuracy, which also needs the UERE.

Source. Leick, GPS Satellite Surveying, ch. 6; Kaplan & Hegarty, ch. 7.

Question 2 of 8: Dilution of Precision in Single Point Positioning

Question 2: Dilution of Precision in Single Point Positioning (15 marks)