Fitting a ballistic trajectory to noisy data where both spacial and temporal domains observations are noisy

Fitting a curve to noisy data is somewhat trivial. However it generally assumes that data abscissa is fixed, and the error is computed on the ordinate.

In my setup, I have 3D spacial observations of ballistic trajectories (that I model with a simple parabola), but the observations time are also noisy.

Therefore, I have to estimate the initial position $x_0$, $y_0$, $z_0$ and initial speed $v_{x_0}$, $v_{y_0}$, $v_{z_0}$, based on 4D (noisy) observations $(X_i, Y_i, Z_i, T_i), i\in[0,N]$, such that they fit the following model:

$$\left\{\begin{align}x(t) &= x_0 + v_{x_0}t\\y(t) &= y_0 + v_{y_0}t\\z(t) &= z_0 + v_{z_0}t- \frac{g}{2} t^2\\\end{align}\right.$$

with $t$ monotonically increasing with $i$.

I'm not sure how to formulate such optimization problem because I have $6$ parameters to estimate, but also $4N$ variables with only $3N$ equations… My intuition tells me there's only one single parabola that minimizes the error (MSE for example), but I can't formulate the problem.

Hint.

Define an error function like

$$ E(x_0,y_0,z_0,v_{x_0},v_{y_0},v_{z_0})= \sum_{k}(x_k-x_0-v_{x_0}t_k)^2+(y_k-y_0-v_{y_0}t_k)^2+(z_k-z_0-v_{z_0}t_k-\frac g2 t_k^2)^2 $$

so the minimum can easily be obtained by solving

$$ \nabla E = 0 $$