## 3 Empirical Fatigue Damage Estimation
### Modal Phase Reconstruction  
As is the case with Proper Orthogonal Decomposition, the Modal Phase Reconstruction (MPR) method has the advantage in that mode shapes need not be assumed; they can be estimated empirically from the data. [PERSON] et al. [34] employed MPR to analyze riser response data from CFD simulations. [PERSON] [27] applied MPR to the NDP data sets to analyze the influence of traveling waves on riser response. In this study, the MPR method is employed to analyze the NDP model riser response and to estimate fatigue damage at arbitrary locations along the length of the riser. The general framework for the MPR procedure is briefly presented here. Assume that, at location \(z_{j}\), and at the time instant, \(t_{k}\), the riser response of interest (such as strain), i.e., \(y(z_{j},t_{k})\), may be expressed as follows:  
\[y(z_{j},t_{k})=Re\left\{\sum_{n=1}^{N}e^{i\omega_{n}t_{k}}\phi_{n}(z_{j}) \right\}, \tag{14}\]  
where \(\omega_{n}=2\pi n/(P\Delta t)\) is the \(n\)th circular frequency; \(t_{k}=k\Delta t\) is the \(k\)th time sample; \(N\) is the number of frequency components included in the MPR procedure; \(P\) is the number of discrete time samples available in the record; and \(\Delta t\) is the sampling rate. In addition, \(Re\{\}\) represents the real part of the associated (complex) function. Note that \(\phi_{n}(z_{j})\) is the \(n\)th complex mode coordinate--with real part, \(\phi_{n,Re}(z_{j})\), and imaginary part, \(\phi_{n,Im}(z_{j})\)--at location, \(z_{j}\), which needs to be empirically estimated from the data. Equation (14) may be written in compact form as follows:  
\[y(z_{j},t_{k})=Re\left\{\Delta_{k}\Phi_{j}\right\}, \tag{15}\]  
where \(\Delta_{k}=\left[e^{i\omega_{1}t_{k}},e^{i\omega_{2}t_{k}},\ldots,e^{i\omega_{ N}t_{k}}\right]\), and \(\Phi_{j}=\left[\phi_{1}(z_{j}),\phi_{2}(z_{j}),\ldots,\phi_{N}(z_{j})\right]^{T}\). Then, at location, \(z_{j}\), the riser response recorded at all \(P\) discrete time instants may be expressed as follows:  
\[y(z_{j})=Re\left\{\hat{\Delta}\Phi_{j}\right\}, \tag{16}\]  
where \(y(z_{j})=\left[y(z_{j},t_{1}),y(z_{j},t_{2}),\ldots y(z_{j},t_{P})\right]^{T}\) represents the entire recorded response time series at location, \(z_{j}\); and \(\hat{\Delta}=\left[\Delta_{1},\Delta_{2},\ldots,\Delta_{P}\right]^{T}\) is easily defined given information only on the length and the sampling rate of the record. In Equation (16), it is the mode shape matrix, \(\Phi_{j}\), that needs to be estimated; the MPR method thus defines this system of \(P\) equations and \(2N\) unknowns (since each \(\phi_{n}(z_{j})\) contained in \(\Phi_{j}\) have real and imaginary parts). As long as \(2N\leq P\), this system of equations can be used to solve for \(\Phi_{j}\) in a least-squares sense. As is the case with the POD method, the MPR method only yields empirically estimated mode shape coordinates at those discrete locations where the riser response is measured. If the riser response is to be estimated at a location where no sensor is present, it is necessary to interpolate the real and imaginary parts of the \(N\) mode shapes to the desired location and to, then, reconstruct the response there by accounting for all these \(N\) modes or frequency components. Equation (14), used for the reconstruction, may be rewritten as follows:  
\[y(z_{j},t_{k})=\sum_{n=1}^{N}\left[\phi_{n,Re}(z_{j})\cos(\omega_{n}t_{k})- \phi_{n,Im}(z_{j})\sin(\omega_{n}t_{k})\right]. \tag{17}\]  
To speed up the computation, the measured riser strains are downsampled by a factor of 5, effectively reducing the data sampling frequency from 1200 Hz to 240 Hz, which shrinks the number of time samples, \(P\), in each record, to one-fifth of its original value.