## 2 Materials and Methods
### Simulation Procedure
The Far-Superficial versus Far-Deep configurations indicate (relative) distances: \"deep\" denotes a distance from the origin (located at the anterior commisure) <6 cm and \"superficial\" >6.5 cm. The distance between dipoles is evaluated as \"far\" if the relative distance to the other dipoles exceeds 8 cm. Figure 3: Ground Truth 2 with two fixed, one moving dipole. Figure 2: Ground Truth 1 with three fixed dipoles. As an additional check for robustness of source localization, noise sources were added as a background activity. These were modeled using pink noise, also called 1/f noise, and created by scaling the amplitude spectrum of random white Gaussian noise with the factor1/f using the Fourier transform and its inverse. After generating these noise sources, the forward problem is construed:  
\[Y=LX+e, \tag{4}\]  
where \(Y\) denotes the scalp-recorded potentials, \(X\) represents the electrical sources in the brain (the dipoles), \"\(e\)\" is measurement noise (electrode noise) and \(L\) is the head volume conductor model (also called the leadfield matrix). The leadfield matrix determines how the activity flows from dipoles to electrodes. In this work, the New York Head model [23] was used. The pink noise and the source activity are then projected onto the scalp, after which they are summed:  
\[Y^{brain}(t)=\gamma\times\frac{Y^{active}(t)}{\big{|}\big{|}Y^{active}(t) \big{|}\big{|}_{FRO}}+(1-\gamma)\times\frac{Y^{noise}}{\big{|}\big{|}Y^{noise }(t)\big{|}\big{|}_{FRO}} \tag{5}\]  
\(Y^{active}\) and \(Y^{noise}\) refer to the scalp-projected source signals and pink noise activity, respectively; both are scaled by dividing them by their Frobenius norm ( \(|\mid Y^{active}(t)\mid\mid_{FRO}\), \(|\mid\)\(Y^{active}(t)\mid\mid_{FRO}\)). The Signal-to-Noise Ratio (SNR) is computed for all dipoles simultaneously and set to 0.9 (\(\gamma=0.9\)). Next, white noise (spatially and temporally uncorrelated activity) is added to \(Y^{brain}\) to simulate electrode noise, resulting in Equation (6) where \(Y^{measurement}\) represents the simulated EEG signal. Again \(\gamma=0.9\) is imposed as Signal-to-Noise Ratio:  
\[Y^{measurement}(t)=0.9\times\frac{Y^{brain}(t)}{\big{|}\big{|}Y^{brain}(t) \big{|}\big{|}_{FRO}}+0.1\times\frac{Y^{meas\_noise}}{\big{|}\big{|}Y^{meas \_noise}(t)\big{|}\big{|}_{FRO}} \tag{6}\]  
Afterwards, the simulated scalp-EEG data are source-reconstructed using exact low-resolution brain electromagnetic tomography (eLORETA) [24]. There have also been improvements to eLORETA, such as Sparse eLORETA, which uses a masking approach to improve the source localization density [25]. The eLORETA method is a discrete, three-dimensional (3D), linear, weighted minimum norm inverse solution [24]. In the absence of noise, an exact zero-error localization accuracy can be obtained with eLORETA, but this does not hold for noisy data, as was shown in a study comparing both scenarios [26]. The MATLAB implementation of the eLORETA algorithm (mkfilt_eloreta2.m) from which spatial filters are obtained was developed by [PERSON] and is available in the MEG/EEG Toolbox of Hamburg (METH; [[https://www.uke.de/english/departments-institutes/institutes/neurophysiology-and-pathophysiology/research/research-groups/index.html](https://www.uke.de/english/departments-institutes/institutes/neurophysiology-and-pathophysiology/research/research-groups/index.html)]([https://www.uke.de/english/departments-institutes/institutes/neurophysiology-and-pathophysiology/research/research-groups/index.html](https://www.uke.de/english/departments-institutes/institutes/neurophysiology-and-pathophysiology/research/research-groups/index.html)), accessed  
Figure 4: Locations (MNI-coordinates) of the dipoles the Ground Truths have in common. Left: Far–Superficial fixed dipoles. Right: Far–Deep fixed dipoles. on 22 December 2021).