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Dilution of Precision Metrics

ACONS computes Dilution of Precision (DOP) values after each epoch using the measurement geometry matrix. This page describes the underlying equations implemented in src/reporting.py.

Geometry matrix

For every pseudorange-like measurement, the simulator forms a row of the geometry matrix

\[ \mathbf{H} = \begin{bmatrix} \hat{\mathbf{u}}_1^\top & 1 \\ \hat{\mathbf{u}}_2^\top & 1 \\ \vdots & \vdots \\ \hat{\mathbf{u}}_m^\top & 1 \end{bmatrix}, \]

where \(\hat{\mathbf{u}}_i\) is the line-of-sight unit vector from the user to satellite \(i\), and the final column captures the partial derivative with respect to clock bias. If fewer than four satellites are available, or \(\mathbf{H}\) is rank-deficient, ACONS reports infinite DOP values.

Normal matrix

The geometry quality is encoded in

\[ \mathbf{Q} = (\mathbf{H}^\top\mathbf{H})^{-1}, \]

whose diagonal elements correspond to the variance factors of the 3D position components and the clock bias. Let \(Q_{xx}, Q_{yy}, Q_{zz}\) denote the diagonal entries associated with \(x\), \(y\), \(z\), and \(Q_{bb}\) the clock-bias entry.

DOP formulas

The simulator writes the following metrics into dop_timeseries.csv:

\[ \mathrm{GDOP} = \sqrt{\operatorname{tr}(\mathbf{Q})}, \]
\[ \mathrm{PDOP} = \sqrt{Q_{xx}+Q_{yy}+Q_{zz}}, \]
\[ \mathrm{TDOP} = \sqrt{Q_{bb}}, \]
\[ \mathrm{HDOP} = \sqrt{Q_{xx}+Q_{yy}}, \]
\[ \mathrm{VDOP} = \sqrt{Q_{zz}}. \]

Height-constrained HDOP (HHDOP)

When an external height source is available (e.g., DEM or altimeter), ACONS also computes a height-constrained HDOP (HHDOP). The geometry matrix is expressed in a local ENU frame:

\[ \mathbf{H}_{\mathrm{ENU}} = \begin{bmatrix} u_{E_1} & u_{N_1} & u_{U_1} & 1 \\ \vdots & \vdots & \vdots & \vdots \\ u_{E_m} & u_{N_m} & u_{U_m} & 1 \end{bmatrix}, \]

and the height constraint is enforced by appending a row

\[ \begin{bmatrix} 0 & 0 & 1 & 0 \end{bmatrix}, \]

so that the augmented matrix is

\[ \mathbf{H}_h = \begin{bmatrix} \mathbf{H}_{\mathrm{ENU}} \\ 0 & 0 & 1 & 0 \end{bmatrix}, \]

and

\[ \mathbf{Q}_h = (\mathbf{H}_h^\top\mathbf{H}_h)^{-1}. \]

The HHDOP metric is then

\[ \mathrm{HHDOP} = \sqrt{Q_{h,EE}+Q_{h,NN}}. \]

These figures quantify the positional and temporal geometry of the visible satellite set. Lower DOP values indicate better-conditioned geometry and typically lead to smaller positioning covariance once measurement noise is applied.