## 3 Empirical Fatigue Damage Estimation
### Proper Orthogonal Decomposition
Using the first thirteen POD modes that together preserve 99% of the total field energy, the strain time series can be reconstructed at the location of the target sensor (strain sensor no. 4). Based on the first thirteen POD modes, the PSD for CF strain at each input logger and the summation of all these PSDs at each frequency are plotted in Figure 5a,b, respectively. Figure 5c shows that the reconstructed CF strain PSD (red dashed line) at the target sensor includes contributions from the first, third, and fifth harmonics, and matches the PSD based on measurements (blue solid line) reasonably well. In Figure 5d, RMS values of CF curvature at the twenty-three input sensor locations based directly on measurements are indicated by blue circles; at the location of the target sensor, the POD-based interpolation (indicated by the blue line which also shows estimated RMS values at other locations nearby) is very close to the RMS value (red cross) obtained from the measurements. Recall that POD modes are discrete along the spatial direction, i.e., modal coordinates are only available at locations of the input sensors. If the location of interest (here, the location of the target sensor) is one where no sensor is assumed to be available, an interpolation of the POD-based modal coordinates between nearby sensors is needed. In this case, the target sensor lies between sensor nos. 3 and 5, and the modal coordinates at the target may be interpolated quite effectively with a local third-order polynomial fit using the POD-based modal coordinates at four nearby sensors (i.e., sensor nos. 2, 3, 5, and 6) mode by mode. Then, the strain time series at the target location can be reconstructed using the interpolated modal coordinates based on Equation (13). Similarly, the riser response at any location, within the spread of the sensors, may be estimated by such piecewise interpolation and by employing a subset of the POD modes. The variability factor, representing the ratio of the estimated damage to that based directly on measurements, at the target sensor (i.e., sensor no. 4), is 1.05, which suggests that the fatigue damage is overestimated by 5% when the POD method is employed with thirteen POD modes. Similar results for other choices of the target sensor are discussed in Section 4. Note that other interpolation techniques with a different number of sensors may also be employed for interpolating the discrete POD modes. In addition to the use of third-order polynomial interpolation, rational function interpolation [30; 31], and trigonometric/Fourier interpolations [32] were tested on simulated data sets as well as on NDP data sets. Results showed that, for the sensor arrangement of the NDP data sets, the rational function interpolation often resulted in artificial poles while the trigonometric function interpolation led to sharp wiggles sometimes. Both of those interpolation options led to large errors; in contrast, the third-order polynomial interpolation is easy to use, relatively stable, and almost always led to reasonable results. Thus, in this study, the third-order polynomial interpolation technique is selected when working with both the POD method and another empirical method (Modal Phase Reconstruction) that is described in Section 3.4. In closing, we note that an alternative and related approach to POD is to use dynamic mode decomposition (DMD) [33], which is a proven method for extracting spatio-temporal coherent structures from data. DMD is different from POD in that the former seeks to represent data sequences by orthogonalizing them in time (i.e., by isolating distinct frequencies in the data), while the latter achieves decomposition of data sequences by orthogonalizing in space. Furthermore, DMD is applied directly to the data, while POD is applied to second-order statistics of the data.