Tikhonov regularization & the L-curve

Because the inverse problem is ill-conditioned, distributed solutions require regularization: a penalty that trades data fit against a stability constraint on the solution. Tikhonov (L2) regularization adds a term, lambda times the squared source norm, to the fitting cost. The regularization parameter lambda sets the balance — too small and noise is amplified into wild source estimates; too large and the solution is over-smoothed and biased toward the prior.

Choosing lambda is a practical crux. Common data-driven strategies set it from the estimated measurement SNR, or use the L-curve (the log-log plot of solution norm versus residual norm, whose corner marks a compromise) or generalized cross-validation. Depth weighting is a related device that rescales the lead field before regularization to reduce the superficial bias of the L2 prior. No choice is objectively correct; the regularization parameter is a modelling decision that shapes the result.