## 3 Empirical Fatigue Damage Estimation
### Proper Orthogonal Decomposition  
The WWA approach presented requires a priori assumed mode shapes based on knowledge of the physical properties of the riser; the modified WWA approach is directly related to these modes by introducing cosine function counterparts to each sine function. For the NDP model riser, the mass and tension force are almost constant over its length; hence, it might be reasonable to assume sinusoidal functions for the mode shapes as was done with the WWA method used in this study. For real drilling risers, the mass and the tension force often vary spatially due, for example, to the presence of buoyancy units. In such cases, riser mode shapes will usually deviate from simple sinusoidal functions and can be difficult to estimate accurately. The error in assumed mode shapes affects the accuracy of approaches such as the WWA method and in resulting estimations of response and fatigue damage. In the following, Proper Orthogonal Decomposition (POD) is proposed as an alternative empirical procedure for riser VIV response and fatigue analysis. With POD, empirical mode shapes are estimated from the data alone and do not rely on assumed mode shapes nor on knowledge of the physical properties of the riser. POD is useful for extracting energetic spatial \"modes\" or patterns of variation of any physical phenomenon that is represented by a high-dimensional spatio-temporal stochastic field (such as the suite of riser strain time series from multiple sensors that we have here). The application of POD for the analysis of riser VIV response may be found in the literature (see, for example, [PERSON] [28] or [PERSON] and [PERSON] [29]). Given a suite of strain time series measured at \(M\) locations, \(\mathbf{V}(t)=\left\{v_{1}(t),v_{2}(t),\ldots,v_{M}(t)\right\}^{T}\), one can establish a \(M\times M\) covariance matrix, \(\mathbf{C}_{v}\), from the strain time series, \(\mathbf{V}(t)\). By solving an eigenvalue problem, one can diagonalize \(\mathbf{C}_{v}\) so as to obtain the matrix, \(\Lambda\), as follows:  
Figure 4: The modified WWA procedure applied with twenty-three input sensors (sensor no. 4 is the target) for the NDP2350 (sheared current) data set: (**a**) PSDs of the strains measured at the twenty-three sensors; (**b**) summation of the strain PSDs at each frequency and identification of the selected modes; (**c**) strain PSD at the target sensor, reconstructed vs. measured; and (**d**) RMS curvatures, reconstructed vs. measured. \[\Phi^{T}\mathbf{C}_{v}\Phi=\Lambda;\quad\mathbf{C}_{v}\Phi=\Phi\Lambda. \tag{11}\]  
Solution of the eigenvalue problem yields eigenvalues, \(\Lambda=diag\{\lambda_{1},\lambda_{2},\ldots,\lambda_{M}\}\) (where \(\lambda_{1}>\lambda_{2}>\ldots>\lambda_{M}\)) and associated eigenvectors, \(\Phi=\{\phi_{1},\phi_{2},\ldots,\phi_{M}\}\). It is possible to rewrite the original \(M\) correlated time series, \(\mathbf{V}(t)\), in terms of uncorrelated scalar subprocesses, \(\mathbf{U}(t)=\{u_{1}(t),u_{2}(t),\ldots,u_{M}(t)\}^{T}\), such that  
\[\mathbf{V}(t)=\Phi\mathbf{U}(t)=\sum_{j=1}^{M}\phi_{j}u_{j}(t), \tag{12}\]  
where \(\phi_{j}\) represents the \(j\)th POD mode shape corresponding to the \(j\)th scalar subprocess, \(u_{j}(t)\). The energy associated with \(u_{j}(t)\) is described in terms of the corresponding eigenvalue, \(\lambda_{j}\). A reduced-order representation of the strain time series, \(\mathbf{\hat{V}}(t)\), may be obtained by including only the first \(N\) POD modes and associated generalized coordinates:  
\[\mathbf{\hat{V}}(t)=\sum_{j=1}^{N}\phi_{j}u_{j}(t),\qquad N<M. \tag{13}\]  
In the present study, strains are measured at twenty-three locations, i.e., \(M=23\). Following the procedure as outlined above, we can thus obtain twenty-three POD mode shapes, \(\phi_{j}\), and then decompose the original twenty-three strain time series into twenty-three uncorrelated POD scalar subprocesses, \(u_{j}(t)\), which when scaled by the POD mode shapes reconstruct all the measurements in space and time.