*3.2. Input Parameters*

The distribution functions of the updated model parameters (available from step one) are used in step two as input for the uncertainty quantification procedure. The stochastic variables reflect both the aleatory and epistemic uncertainties, which constitute the updated *Xd* and *Xl* uncertainties.

#### *3.3. Uncertainty Propagation*

The effect of the updated model parameters on the fatigue damage accumulation is established by a Monte Carlo uncertainty propagation method [52], as indicated in step three in Figure 2. Based on the uncertainty in the input parameters (i.e., the distribution functions of the updated numerical model parameters), we obtain the distribution of fatigue damage, hence quantifying the uncertainties in fatigue damage due to the updated model parameters. The uncertainty quantification procedure is described next. The aim is to express the uncertainty as a stochastic variable multiplied to the fatigue stress ranges.

The uncertainty in fatigue damage accumulation due to an uncertain parameter, *α<sup>j</sup>* ∈ *α*, can be quantified by simulating *n* realizations from this parameter's distribution function and calculating the corresponding fatigue damage. When calculating fatigue damage, the remaining parameters are assumed to be deterministic. Moreover, the fatigue damage is calculated assuming one sea state parameter. In this way, the introduced uncertainty is solely governed by the variability of *αj*, hence quantifying this parameter's contribution to the fatigue damage accumulation uncertainty. For example, a distribution function of updated soil stiffness implies structural dynamics uncertainty, while a distribution function of an updated inertia coefficient in Morison's equation implies loading uncertainty.

Among a number of uncertainty quantification methods [53], a Bayesian framework [54] is recommended by a number of standard committees, for example, IEC and Joint Committee on Structural Safety (JCSS), due to its sound theoretical basis and wide range of applicability. However, a main challenge in the Bayesian framework is the requirement of a prior distribution on the parameters to be quantified. In the context of offshore wind uncertainties, information on prior distributions is not available in the background documents for the above mentioned standards and committees. Consequently, in the proposed framework, we implemented a simplified method where we start with the uncertainty modeling consistent with the design standard of wind turbines [40], and subsequently we quantify the uncertain parameters already included in (1) using the maximum likelihood method.

Assuming the fatigue damage, modeled as a stochastic variable depending on the uncertain parameter *<sup>α</sup>j*, is normally distributed, *<sup>D</sup>*(*αj*) ∼ N - *μDj* , *σ*<sup>2</sup> *Dj* , the fatigue damage distribution (mean value *μDj* and standard deviation *σDj* ) can be found through the maximum likelihood method, where the likelihood is defined as

$$L\left(\mu\_{D\_{j}},\sigma\_{D\_{j}}\right) = \prod\_{i=1}^{n} \frac{1}{\sqrt{2\pi}\sigma\_{D\_{j}}} \exp\left(-\frac{1}{2}\left(\frac{D\_{i}-\mu\_{D\_{j}}}{\sigma\_{D\_{j}}}\right)^{2}\right),\tag{2}$$

with *Di* being the fatigue damage associated with the *i*th realization of *α<sup>j</sup>* computed based on the updated structural model contained in the digital twin.

The log-likelihood function becomes

$$\ln L\left(\mu\_{D\_j,}\sigma\_{D\_j}\right) = -n\ln\left(\sqrt{2\pi}\sigma\_{D\_j}\right) - \sum\_{i=1}^{n} \frac{1}{2} \left(\frac{D\_i - \mu\_{D\_j}}{\sigma\_{D\_j}}\right)^2,\tag{3}$$

and the optimal parameters are found to be

$$\underset{\mu\_{D\_{\dot{j}'}},\sigma\_{D\_{\dot{j}}}}{\text{argmax}} \ln L\left(\mu\_{D\_{\dot{j}'}},\sigma\_{D\_{\dot{j}}}\right). \tag{4}$$
