Context:
Let's consider distance measurements in two dimensions between $N$ targets $t_n$ and $P$ sources $s_p$, with $n = 0, 1, ..., N-1$ and $p = 0, 1, ..., P-1$.
The sources position, $(p_{\mathrm{s},\mathrm{x},p},\ p_{\mathrm{s},\mathrm{y},p})$, are known.
The targets position, $(p_{\mathrm{t},\mathrm{x},n}+{\Delta p}_{\mathrm{t},\mathrm{x},n}, \
p_{\mathrm{t},\mathrm{y},n}+{\Delta p}_{\mathrm{t},\mathrm{y},n})$, contain an known part and an unknown part.
The square of the distance measured between the $n$th target and the $p$th source is $$d_{n,p}^2 = ({\Delta p}_{\mathrm{t},\mathrm{x},n} + p_{\mathrm{t},\mathrm{x},n} - p_{\mathrm{s},\mathrm{x},p})^2 + ({\Delta p}_{\mathrm{t},\mathrm{y},n} + p_{\mathrm{t},\mathrm{y},n} - p_{\mathrm{s},\mathrm{y},p})^2. \tag{1}$$
Using differential measurements (e.g. subtracting the distance obtained with the first source), we get $$ \begin{align*} d_{n,p}^2 - d_{n,0}^2 &= ({\Delta p}_{\mathrm{t},\mathrm{x},n} + p_{\mathrm{t},\mathrm{x},n} - p_{\mathrm{s},\mathrm{x},p})^2 - ({\Delta p}_{\mathrm{t},\mathrm{x},n} + p_{\mathrm{t},\mathrm{x},n} - p_{\mathrm{s},\mathrm{x},0})^2 + \mathrm{same \ with \ y} \\ &= 2{\Delta p}_{\mathrm{t},\mathrm{x},n} (p_{\mathrm{s},\mathrm{x},0} - p_{\mathrm{s},\mathrm{x},p})+ 2 p_{\mathrm{t},\mathrm{x},n} (p_{\mathrm{s},\mathrm{x},0} - p_{\mathrm{s},\mathrm{x},p}) + p_{\mathrm{s},\mathrm{x},0}^2 - p_{\mathrm{s},\mathrm{x},0}^2 + \mathrm{same \ with \ y}. \end{align*} \tag{2} $$
Thus, we obtain linear equations, which can be solved easily once put in matrix form. To get the $2N$ unknowns (${\Delta p}_{\mathrm{t},\mathrm{x},n}$ and ${\Delta p}_{\mathrm{t},\mathrm{y},n}$), $2N$ equations are needed, i.e. 3 sources are needed.
Question:
My question is for the case where there is also an unknown error on the source position. In this case, the square of the distance measured between the $n$th target and the $p$th source is
$$d_{n,p}^2 = ({\Delta p}_{\mathrm{t},\mathrm{x},n} - {\Delta p}_{\mathrm{s},\mathrm{x},p} + p_{\mathrm{t},\mathrm{x},n} - p_{\mathrm{s},\mathrm{x},p})^2 + ({\Delta p}_{\mathrm{t},\mathrm{y},n} - {\Delta p}_{\mathrm{s},\mathrm{y},p} + p_{\mathrm{t},\mathrm{y},n} - p_{\mathrm{s},\mathrm{y},p})^2. \tag{3}$$
Using differential measurements again, we get $$ \begin{align*} d_{n,p}^2 - d_{n,0}^2 &= ({\Delta p}_{\mathrm{t},\mathrm{x},n} - {\Delta p}_{\mathrm{s},\mathrm{x},p} + p_{\mathrm{t},\mathrm{x},n} - p_{\mathrm{s},\mathrm{x},p})^2 - ({\Delta p}_{\mathrm{t},\mathrm{x},n} - {\Delta p}_{\mathrm{s},\mathrm{x},0} + p_{\mathrm{t},\mathrm{x},n} - p_{\mathrm{s},\mathrm{x},0})^2 + \mathrm{same \ with \ y} \\ &= {\Delta p}_{\mathrm{s},\mathrm{x},p}^2 - {\Delta p}_{\mathrm{s},\mathrm{x},0}^2 + \mathrm{other \ things}. \end{align*} \tag{4} $$
Here, we don't get linear equations.
Is there a way to use a similar trick as the first case to obtain linear equations?
Any tip?