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Floating Offshore Wind Turbines (FOWTs) are more prone to suffer from faults and failures than bottom-fixed counterparts due to the severe wind and wave loads typical of deep water sites. In particular, mooring line faults may lead to unacceptably high operation and maintenance costs due to the limited accessibility of FOWTs. Detecting the mooring line faults is therefore critical, but the application of Fault Detection (FD) techniques has not been investigated yet. In this paper, an FD scheme based on a wave-excited linear model is developed to detect in a reliable way critical mooring line faults occurring at the fairlead and anchor ends. To reach the goal, a linear model of the FOWT is obtained by approximating the wave radiation and incident wave forces. Based on this model, an observer is built to predict the rigid rotor and platform dynamics. The FD scheme is thus implemented by comparing the Mahalanobis Distance of the observer prediction error against a probabilistic detection threshold. Numerical simulations in some selected fault scenarios show that the wave-excited linear model can predict the FOWT dynamics with good accuracy. Based on this, the FD scheme capabilities are demonstrated, showing that it is able to effectively detect two critical mooring line faults.