## 3 Empirical Fatigue Damage Estimation
### Weighted Waveform Analysis  
Weighted Waveform Analysis (WWA) is a computational procedure that is widely used to analyze and reconstruct the response over the entire length of a riser from measurements at a limited number of sensors [13; 23]. Assume that the riser displacement, \(x\), at location, \(z\), and time, \(t\), may be expressed approximately as a weighted sum of \(N\) assumed modes:  
\[x(z,t)=\sum_{i=1}^{N}w_{i}(t)\varphi_{i}(z), \tag{3}\]  
where this is assumed by selecting \(N\) (not necessarily consecutive) modes. In addition, \(\varphi_{i}(z)\) represents the \(n_{i}^{\text{th}}\) mode shape of the displacement, while \(w_{i}(t)\) represents a time-varying modal weight to be applied with the \(n_{i}^{\text{th}}\) mode shape. Note that \(i=1,2,\ldots,N\), are integer indices associated with the selected mode set which include the \(n_{1}^{\text{th}}\), \(n_{2}^{\text{th}}\),..., and \(n_{N}^{\text{th}}\) modes; again, these selected modes are not necessarily sequentially ordered. In addition, the \"mode\" in the empirical methods discussed here refers to a deformed shape of the riser associated with dominant energy in the power spectral density (PSD) of the CF strain at the logger locations. The bending strain, which equals the product of the riser outer radius, \(R\), and the local curvature, \(x^{\prime\prime}\), may be expressed as follows:  
\[\varepsilon(z,t)=Rx^{\prime\prime}(z,t)=\sum_{i=1}^{N}Rw_{i}(t)\varphi_{i}^{ \prime\prime}(z). \tag{4}\]  
For the NDP model riser with constant tension and pinned-pinned boundary conditions, it is assumed that the mode shapes may be represented as sinusoidal functions:  
\[\varphi_{i}(z)=\sin\left(\frac{n_{i}\pi z}{L}\right);\quad\varphi_{i}^{\prime \prime}(z)=-\left(\frac{n_{i}\pi}{L}\right)^{2}\sin\left(\frac{n_{i}\pi z}{L }\right), \tag{5}\]  
where \(\varphi_{i}(z)\) and \(\varphi_{i}^{\prime\prime}(z)\) are the \(n_{i}^{\text{th}}\) mode shape displacement and curvature, respectively, and \(L\) is the length of the riser. Given strain measurements or, equivalently, curvature measurements at \(M\) logger locations, \(z_{j}\) (where \(j=1\) to \(M\)), WWA requires solution of a system of equations in matrix form:  
\[\mathbf{Aw}=\mathbf{d}, \tag{6}\]  
where the \(M\times N\) matrix, \(\mathbf{A}\), comprises curvatures of the assumed mode shapes at all the logger locations and the vector, \(\mathbf{d}\), represents riser curvatures and is formed using the measured strains at all loggers--i.e., \(A_{j,i}=\varphi_{i}^{\prime\prime}(z_{j})\) and \(d_{j}=\varepsilon_{j}(t)/R=\varepsilon(z_{j},t)/R\). Equation (6) is a linear system involving \(M\) equations, in which \(N\) weights (\(\mathbf{w}^{T}=\{w_{1},w_{2},...w_{N}\}\)) need to be estimated. At any instant of time, \(t\), as long as \(N\leq M\), the modal weights vector, \(\mathbf{w}\), may be solved for in a least-squares sense:  
\[\mathbf{w}(t)=\left(\mathbf{A}^{T}\mathbf{A}\right)^{-1}\mathbf{A}^{T} \mathbf{d}(t). \tag{7}\]  
In the present study, where twenty-three strain sensors are available, it is found that careful selection of the modes based on frequencies corresponding to peaks in the CF strain power spectra generally provides good WWA-based reconstructed strain time series at target sensors. In our studies, twelve modes (i.e., \(N=12\)) was generally a good choice. The procedure for the selection of the \(N\) modes is important and it is briefly described next. Given the CF strain time series, \(\varepsilon(z_{j},t)\), measured at each of the twenty-three locations, \(z_{j}\) (where \(j=1,2,3,5,\ldots,24\); i.e., assume that the target sensor no. 4 does not exist), its power spectral density (PSD), \(PSD_{j}(f)\), describes the energy distribution by frequency of the riser response at each of those locations. As illustrated in Figure 3a, the twenty-three input strain power spectra indicate very similar frequency content; all these spectra show the presence of the first and third harmonics in the response (the fifth harmonic is also visible if the PSDs were plotted using a log scale).