We propose a new approach to resolve the isotropic component of the seismic moment tensor and its uncertainty. In linearized inversion problems, where the earthquake or explosive‐source location and origin time are fixed (e.g., assumed to be known), the uncertainty of the moment tensor can be studied through the eigenvalues and the eigenvectors of the design matrix, which allows the representation of the theoretical misfit by means of a 6D error ellipsoid. Because the design matrix depends only on the structural model and receiver source geometry, the analysis can be performed using recorded seismic waveforms, or without. In the nonlinear inversion problems, where the free parameters are eight (e.g., the six elements of the moment tensor, depth, and origin time), we propose a waveform‐inversion scheme in which the trace of the moment tensor varies systematically and the remaining seven free parameters are optimized for each specific value of the trace. In this way, a 1D experimental probability density function of the moment tensor trace is constructed. To demonstrate the applicability of the method, we apply it to two shallow earthquakes (Mw 4.9 and 4.7) with epicenters close to the Columbo volcano, located 20 km northeast of the island of Santorini, Aegean Sea, Greece. We use 15 near‐regional (60–310 km) records at frequencies below 0.1 Hz and two alternative crustal models. We conclude that the main uncertainties are attributed to the crustal model and to the trade‐off between the isotropic component and the source depth.