Noise-dependent localization error. The mean squared localization error err() over 100 simulations has been determined depending on the noise level and on the different number of multipole moments used. The curves are plotted up to the noise level, where all simulations still produced a stable localization result. We used the inner moments up to the 3rd order , , plotted in curves 3χ and the inner moments up to the 4th order , , , plotted in curves 4χ. The outer moments which were used to model the disturbing fields were none (curves χ0), 2nd order moments (curves χ2: homogeneous fields), 2nd and 3rd order moments (curves χ3: homogeneous and gradient fields), and 2nd to 4th order moments (curves χ4: external fields up to 2nd order). The simplest disturbing field to model is a homogeneous field having index χ2. The dipole field of a dipole with a strength of 20 Amm2 at position (x, y, z) = (0, 0, -300 mm) is superimposed by white, Gaussian distributed noise, which is given in fT and as the signal to noise ratio (SNR).