## 3 Empirical Fatigue Damage Estimation
### Modified Weighted Waveform Analysis  
By introducing a cosine term to complement the sine term for each frequency component in the WWA method, what results is a modified WWA procedure that can better account for the effect of traveling waves in the riser response as well as for local curvature changes at boundaries and other discontinuities. In lieu of Equation (3), we now have:  
\[\begin{split}& x(z,t)=\sum_{i=1}^{N}\left[a_{i}(t)\varphi_{i}(z)+b_{ i}(t)\psi_{i}(z)\right],\\ &\varphi_{i}(z)=\sin\left(\frac{n_{i}\pi z}{L}\right);\psi_{i}(z )=\cos\left(\frac{n_{i}\pi z}{L}\right),\end{split} \tag{10}\]  
where \(a_{i}(t)\) and \(b_{i}(t)\) are modal weights associated with the assumed sine and cosine mode shapes, \(\varphi_{i}(z)\) and \(\psi_{i}(z)\), respectively. Strains are computed in a similar manner as with the WWA method.  
As with the WWA procedure, given curvature measurements at \(M\) logger locations, a system of equations in matrix form results (exactly as in Equation (6)), but with modified terms--i.e., \(A_{j,2i-1}=\varphi_{i}^{\prime\prime}(z_{j})\) and \(A_{j,2i}=\psi_{i}^{\prime\prime}(z_{j})\). This is a linear system of \(M\) equations with \(2N\) weights to be estimated. At any instant of time, \(t\), as long as \(2N\leq M\), the modal weights, (\(\mathbf{w}^{T}=\{a_{1},b_{1},a_{2},b_{2}\ldots a_{N},b_{N}\}\)), may again be solved for in a least-squares sense using Equation (7). The riser response such as CF bending strain, at any location, \(z\), may then be reconstructed. Though the modified WWA procedure can better describe traveling waves in the riser response as well as local curvature variations (than is possible with the WWA method), the smaller number of modes that may be assumed given the same suite of measurements represents a trade-off.  
It should be noted that the modified WWA procedure presented here is similar to the spatial Fourier decomposition with \"full reconstruction\" that is employed by [PERSON] [26; 27]; however, one key difference is that these cited studies use sequentially ordered modes, while the modified WWA method is based on the selection of important physically excited (energetic) modes that are not, in general, sequential.  
With the modified WWA method, where twenty-three input strain sensors are available (i.e., \(M=23\)), careful selection of six modes (i.e., \(N=6\)) based on generally non-consecutive frequencies corresponding to peaks in the CF strain power spectra provided reconstructed strain time series with reasonable accuracy. Figure 4 presents results for the modified WWA method which are analogous to those presented for the WWA method in Figure 3. The PSD for CF strain at each input logger and the summation of all these PSDs at each frequency are plotted in Figure 3 and 4b, respectively. The six modes selected in a similar manner to the procedure used with the WWA method include modes spanning the first to the fifth harmonics of the input signals. Results summarizing the reconstructed CF strain PSD and the reconstructed RMS curvature at the locations of the various sensors are presented in Figure 4c,d, respectively. The results suggest that, at the target sensor, the PSD of the reconstructed CF strain matches the corresponding PSD based on actual strain measurements quite well; in addition, the RMS values of the reconstructed curvatures match measured values reasonably well at all the sensor locations.  
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 1.62, which suggests that the fatigue damage is overestimated by 62% with the modified WWA method. Similar results for other choices of the target sensor are discussed in Section 4.