## 2 Theoretical Background for Spherical Localization Based on Attenuation of EM Waves
### Underwater Range Sensor Model  
We use a model called the underwater range sensor model (URSM) to estimate the range between receiver and beacons. The URSM describes the path loss attenuation as a function of distance. We use the model that was first introduced in [3]. The beacons transmit an EM signal with constant known power, and the RSS is measured at the receiver position. Based on the RSS, the URSM provides a range estimation between receiver and beacons. In air, the attenuation rate of EM waves is small over distance. This leads to large fluctuations in the RSS due to reverberation and interference. In contrast, EM waves in water attenuate rapidly with distance. Since disturbing effects such as reverberation or interference are less pronounced underwater, the principle of attenuating EM waves can be reliably used to estimate the distance from an EM transmitter to a receiver underwater. In order to map RSS values to range estimates, several techniques have been proposed in the literature. In air, the sensor model relies on the Friis transmission formula, which is commonly used to calculate ranges for a measured received signal strength. However, this model needs to be modified to consider medium parameters, such as temperature and conductivity. [PERSON] et al. [3; 6] developed an EM wave attenuation model for underwater environments. The model is a modified version of the Friis transmission formula, which takes the attenuation constant \(\alpha\) of the plane wave equation into account. The difference between the EM wave power \(S_{\text{R}}\) on the receiver side and the EM wave power \(S_{\text{T}}\) on the transmitter side is the RSS. The RSS as a function of the range \(R_{i}\) between the receiver unit and the \(i\)-th beacon reads:  
\[\text{RSS}_{i}=S_{\text{R},i}-S_{\text{T},i}=-20\,\log_{10}\,R_{i}-20\,R_{i}\, \alpha_{i}\,\log_{10}\mathrm{e}+\Gamma_{i}\left[\mathrm{dBm}\right]\,, \tag{2}\]  
where \(\Gamma_{i}\) is an offset factor representing antenna and environmental influences. The parameters \(\alpha_{i}\) and \(\Gamma_{i}\) can be calculated explicitly if variables such as polarization loss factor, transmitting and receiving antenna gains and the attenuation factor are known. In [3], these parameters are derived for an underwater test tank environment, and the resulting model is validated with experimental data. In our contribution, the model parameters are computed from spatially-distributed RSS measurements by fitting (2) with non-linear least-squares. The range \(R_{i}\) as a function of \(S_{\text{R},i}\) can be obtained from (2):  
\[R_{i}=\frac{1}{\alpha\,\log_{10}\mathrm{e}\,\ln 10}\,W\left\{\alpha\,\log_{10} \mathrm{e}\,\ln 10\,\exp\left[\frac{-\ln 10}{20}\left(S_{\text{R},i}-S_{\text{T},i} \right)-\frac{\Gamma\,\ln 10}{20}\right]\right\}\,, \tag{3}\]  
Figure 3: Spherical localization for determining the receiver position \(\mathbf{p}\) as the intersection of circles in 2D or spheres in 3D of radius \(R_{i}\). The known positions of the beacons are denoted by \(\mathbf{r}_{1}\), \(\mathbf{r}_{2}\) and \(\mathbf{r}_{3}\).  
where \(W(x)\) is the Lambert-W function, which is the inverse function of:  
\[f(x)=x\,\exp(x)\,, \tag{4}\]  
\[f^{-1}(x)=f(y)=W(y)\,. \tag{5}\]  
The procedure of determining the range \(R_{i}\) as a function of signal strength differences is depicted in Figure 4 for one dimension. The beacon emits an EM signal with the known power \(S_{\text{T},i}\), while the receiver measures the signal and determines the channel power \(S_{\text{R},i}\). The URSM allows computing the range between the beacon and receiver with (3).