This study is concerned with the estimation of long-term fatigue damage for a floating offshore wind turbine. With the ultimate goal of efficient evaluation of fatigue limit states for floating offshore wind turbine systems, a detailed computational framework is introduced and used to develop a surrogate model using Gaussian process regression. The surrogate model, at first, relies only on a small subset of representative sea states and, then, is supplemented by the evaluation of additional sea states that leads to efficient convergence and accurate prediction of fatigue damage. A 5-MW offshore wind turbine supported by a semi-submersible floating platform is selected to demonstrate the proposed framework. The fore-aft bending moment at the turbine tower base and the fairlead tension in the windward mooring line are used for evaluation. Metocean data provide information on joint statistics of the wind and wave along with their relative likelihoods for the installation site in the Mediterranean Sea, near the coast of Sicily. \textcolor{black}{A coupled frequency-domain model} provides needed power spectra for the desired response processes. The proposed approach offers an efficient and accurate alternative to the exhaustive evaluation of a larger number of sea states and, as such, avoids excessive response simulations.
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