Inverse methods can be used to estimate wave excitation forces on offshore structures from measured dynamic responses, such as accelerations. Force estimation is particularly challenging at low frequencies, where offshore structures are typically excited by waves and where the frequency response function (FRF) is ill-conditioned, leading to amplification of measurement noise. This study investigates how measurement noise propagates into the estimated forces and examines the effectiveness of Tikhonov regularization in mitigating this effect. Theoretical expressions for the variance of the force estimates are derived, demonstrating that measurement noise results in error amplification proportional to the inverse FRF, producing large estimation errors at low frequencies. Regularization is introduced to reduce variance, but it also introduces bias to the estimated force. Constant and frequency-dependent regularization strategies are investigated. The trade-off between variance and bias is investigated, showing that while regularization stabilizes the inverse problem, it can introduce significant distortions in the estimated force spectrum, particularly at low frequencies. Since offshore structures are often loaded by waves with low frequencies, these findings underscore the importance of carefully selecting regularization parameters when applying inverse force estimation techniques to marine structures.