## 3 Empirical Fatigue Damage Estimation  
The NDP model riser, densely instrumented with twenty-four strain sensors in the CF direction, allows us to test different empirical methods by the use of \"leave-one-out\" cross-validation where estimations are compared with measurements. According to this cross-validation, we first select any one sensor (among the original twenty-four sensors) as the \"target\" sensor and set it aside; we use measurements of the remaining twenty-three sensors (termed \"input\" sensors) with the empirical method under consideration to estimate strain time series at the location of the target sensor. Next, we assess the fatigue damage rate at the location of the target sensor from the estimated strain and compare it with the \"true\" value, which was calculated from the strains directly measured by the target sensor (but never used by the empirical method). This process is repeated by selecting a new sensor as target, one at a time, until all the twenty-four sensors are thus considered. These iterative steps are followed for each empirical method. A parameter, termed variability factor, \(V\), is defined as the ratio between the fatigue damage rate calculated using the estimated riser strains to the rate based on direct measurements at the target sensor. With a sufficient number of realizations of the variability factor, its probability distribution quantifies the variability in the estimated fatigue damage which is directly associated with the imperfect aspects of the empirical reconstruction method and the incompleteness or sparseness of the measurements. The variability factor, \(V\), is calculated as:  
\[V=\frac{d_{e}}{d_{t}}=\frac{n_{e}S_{e}^{b}}{n_{t}S_{t}^{b}}, \tag{2}\]  
where the estimated fatigue damage, \(d_{e}\), estimated equivalent stress range, \(S_{e}\), and estimated number of cycles, \(n_{e}\), are calculated based on the estimated/reconstructed stress time series at a location of interest. Similarly, \(d_{t}\), \(S_{t}\), and \(n_{t}\) represent, respectively, the fatigue damage, equivalent stress range, and number of cycles based directly on the true/measured stress time series at the same location. As illustrated in Equations (1) and (2), the fatigue damage is proportional to the \(b\)th power (typically, \(b=3\) or \(4\)) of the amplitude of the stress/strain and, thus, the variability factor is a better indicator of the inaccuracy in the empirically reconstructed riser response compared with any ratio based on the  
Figure 2: Measured CF and IL fatigue damage rates (per year) for the four NDP data sets.  
strain or displacement (e.g., the ratio of the RMS values of strain or displacement amplitude). This is the main reason for choosing the ratio of fatigue damage over other options, such as the ratio of RMS values of strains (strain-RMS ratio) or the ratio of RMS values of displacements (displacement-RMS ratio) to evaluate the different proposed empirical methods. An example presented in Section 4 will be used to further address this subject.  
In this section, the theoretical formulation for each proposed empirical method is presented very briefly. Then, an example involving the use of a linearly sheared current data set, NDP2350, while making use of strain sensor no. 4 at the location, \(z/L=0.11\) (\(z\) is the spatial coordinate along the riser axis and \(L\) is the riser length) as the target sensor, is illustrated. In Sections 4 and 5, fatigue damage rates estimated by the use of the different empirical methods at various locations (including the location of sensor no. 4) for all the four NDP data sets are presented. Based on these results, the advantages and disadvantages of each empirical method, and some basic guidelines on selecting the most appropriate empirical method to suit a specific situation are discussed in Sections 7 and 8.