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.
