Given the ranges $\rho^{(s)}_{R_x}=\sqrt{(x_{R_x}-X^{(s)})^2+(y_{R_x}-Y^{(s)})^2+(z_{R_x}-Z^{(s)})^2}$ between a receiver $R_x$ and an $s$-th satellite, where $s=1, 2, ... n$, one can construct the design matrix $\textbf{A}$ such as: $$\textbf{A}_{n\times 4}=\begin{pmatrix} \frac{\partial\rho^{(1)}_{R_x}}{\partial x_{R_x}} & \frac{\partial\rho^{(1)}_{R_x}}{\partial y_{R_x}} & \frac{\partial\rho^{(1)}_{R_x}}{\partial z_{R_x}} & 1 \\ \vdots & \vdots & \vdots & \vdots \\ \frac{\partial\rho^{(n)}_{R_x}}{\partial x_{R_x}} & \frac{\partial\rho^{(n)}_{R_x}}{\partial y_{R_x}} & \frac{\partial\rho^{(n)}_{R_x}}{\partial z_{R_x}} & 1 \end{pmatrix} $$ Note that the fourth column in $\textbf{A}$ comes from the temporal coordinate.

If one assumes that different error sources in pseudo-range calculation are uncorrelated, it is then possible to express the covariance matrix of $\textbf{A}$ as: $$Q_{4\times 4}=\sigma^2(A^TA)^{-1}$$ where $\sigma^2$ represents the total user equivalent range error (UERE). Very often one regards $Q\sigma^{-2}$, which can be used to obtain a quantity known as the dilution of precision, $DOP$. $DOP$ is quantified via several components, namely the positional $DOP$ or $PDOP$, the geometric $DOP$ or $GDOP$, horizontal and vertical $DOP$ ($HDOP$ and $VDOP$), etc. $GDOP$ and $PDOP$ can be calculated in a straightforward manner as: $GDOP=\sqrt{Tr\left\lbrace (A^TA)^{-1}\right\rbrace}$ Similarly, $PDOP$ is obtained as square root of the trace of a $3\times 3$ submatrix of $(A^TA)^{-1}$.

These quantities are quite useful, as they can provide an estimate of the navigation error. However, more often than not one ends up with non-geocentric (ECEF) coordinates when trying to track satellites (e.g. often azimuth and elevation angles of a satellite or of a constellation of satellites is known instead). Because of this I would like to know if there is a way to transform matrix $\textbf{A}$ between different coordinate systems (e.g. geodetic, ENU, AER). My initial hunch is that coordinate transforms could somehow be applied, but I am not sure how to do so. Is there a Jacobian-like matrix that could generalize $\textbf{A}$ such transformation? If you are aware of any literature or have some experience with this kind of problems, I would greatly appreciate your input.

  • 1
    $\begingroup$ The process of converting Alt/Az to ECEF won't be linear, so it won't be a simple matrix. The details of converting from ECEF to topocentric Alt/Az are explained in detail in the Explanatory Supplement to the Astronomical Almanac. You can follow the algorithm backwards to do the reverse. Not all of the equations are invertible, but can be approximated using a guess and refined through iteration. $\endgroup$ Oct 4, 2022 at 13:57


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