## 3 Empirical Fatigue Damage Estimation
### Hybrid Method: MPR + Modified WWA  
By studying Equation (17), it may be noted that the MPR method decomposes the measured riser response into \(N\) components and, importantly, each component is a single-frequency time series. This suggests that the each single-frequency component may be decomposed further by using the previously discussed modified WWA method. This approach is referred to as a \"hybrid\" method since it combines the MPR and the modified WWA methods. The hybrid method has two advantages when compared with either the MPR or the modified WWA methods. First, unlike the MPR method, the hybrid method does not require interpolation from discrete complex modes to estimate or reconstruct the response at any arbitrary location; it offers the ability to reconstruct the riser response over the entire length of the riser (as a continuous function) despite starting from only a discrete number of measurements. Second, the modified WWA method seeks to represent several frequency components of a wide-band (multi-frequency) response time series with the help of measurements from available sensors; on the other hand, the hybrid method seeks to decompose each MPR mode for a single frequency defined at the same sensors.  
Application of the hybrid method procedure is presented below. First, the MPR approach is followed and the various input strains are represented as in Equation (17). Then, at the location of each input sensor, \(z_{j}\), the real part of the \(n\)th MPR mode, \(\phi_{n,Re}(z_{j})\), may be decomposed using the modified WWA procedure as follows:  
\[\phi_{n,Re}(z_{j})=\sum_{k=1}^{s}\left[a_{k}\varphi_{k}(z_{j})+b_{k}\psi_{k}(z _{j})\right], \tag{18}\]  
where \(\varphi_{k}(z_{j})\) and \(\psi_{k}(z_{j})\) are defined in exactly the same manner as in Equation (10). The real part of the \(n\)th MPR mode at the locations of all \(M\) sensors may be expressed in matrix form (as in Equation (6)), i.e., \(\mathbf{Aw}=\mathbf{d}\), where \(A_{j,2i-1}=\varphi_{i}(z_{j})\) and \(A_{j,2i}=\psi_{i}(z_{j})\), and \(d_{j}=\phi_{n,Re}(z_{j})\). As long as \(2s\leq M\), the modal weights, (\(\mathbf{w}^{T}=\{a_{1},b_{1},a_{2},b_{2}\ldots a_{s},b_{s}\}\)), may be estimated in a least-squares sense, and a spatially continuous function, \(\phi_{n,Re}(z)\), can be derived. Note that exactly the same procedure may be repeated to decompose the imaginary part of the \(n\)th MPR mode, i.e., for \(\phi_{n,Im}\). By iterating this procedure for all the real and imaginary MPR modes, the riser response over its entire span can be reconstructed as a continuous function by this hybrid method. In the present study, where \(M=23\), we use \(s\) equal to 6; the modes in the modified WWA portion of the hybrid method are selected to correspond to frequencies closest to the MPR mode that is being reconstructed according to Equation (18).  
An example using the hybrid method with strain sensor no. 4 as the target sensor is presented in Figure 7, which may be compared directly with Figure 6 based on the MPR procedure.  
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 1.84, which suggests that the fatigue damage is overestimated by 84% when the hybrid (MPR-modified WWA) method is employed with the selected frequency components. Similar results for other choices of the target sensor are discussed in Section 4.