Calibration

What remains to be shown is how we can obtain values for $x_0, y_0, z_0$, and $C_x^\ast, C_y^\ast, C_z^\ast$. Since the values $x_0, y_0, z_0$ are uncritical for the achieved accuracy, we assume they are measured directly. The $C_\ast^\ast$ values, however, are very critical for the accuracy as was shown with the example at the beginning of Section 4.2. Therefore we have to perform a calibration.

For each of the three lighthouses we have to determine values for the four variables $C^b, C^\beta, C^\gamma, C^\delta$. For this, we place the observer at known locations $(d_i, h_i)$ and obtain the respective $\alpha_i$ using equation 1. Doing so for at least four locations and using equation 9, we obtain the following linear equation system in $C^b, C^\beta, C^\gamma, C^\delta$:

$$\begin{array}{rcl} 2 d_1 \sin \frac{\alpha_1}{2} & = & C^b +... ...2 + h_4^2}C^\beta + h_4 C^\gamma + d_4 C^\delta \\ \end{array}$$ (12)

As with the other equation systems, this system does not necessarily have a solution, since the parameters are only approximations obtained by measurements. Again, MMSE methods can be used to obtain approximations for the $C^\ast$. If the system has a solution, it can also be obtained by Gaussian elimination. For this, the $d_i$ and $h_i$ have to fulfill certain requirements. One simple rule of thumb is that both the $d_i$ and the $h_i$ should be pairwise distinct.

Note that calibration has to be performed only once for each base station (assuming that the system is stable enough and needs not be recalibrated) and is independent of the receiver nodes. Therefore, calibration can be performed using a more powerful receiver device than the limited Smart Dust node. As explained in Section 4.1, the base station broadcasts these calibration parameters to the Smart Dust nodes, which use them to compute their location using Equation System 10.