## 1 Introduction  
Current velocity flow fields on marine risers can lead to the formation and shedding of vortices downstream; fluctuations in resulting hydrodynamic pressures can cause such risers to experience sustained vortex-induced vibration (VIV), which can lead to fatigue damage. Accumulated fatigue damage may cause riser failure, a shortened service life, and even ocean pollution. To prevent such catastrophic consequences, it is useful to be able to accurately estimate the rate of fatigue damage accumulation and the expected service life of a marine riser at the design stage, or to measure and monitor the accumulated fatigue damage and predict the remaining life for an installed riser. In an effort to achieve this ambitious goal, several marine riser monitoring campaigns have been undertaken over the past decade [1]. Riser response measures (such as strains and/or accelerations) are measured at discrete locations along the length of a riser. Often, ocean current profiles are also measured at a nearby location over the period of the riser monitoring campaign. Such in situ full-scale measurements of riser VIV response are extremely valuable in the study of riser VIV response and the estimation of fatigue damage. In contrast to the use of riser response measurements obtained from monitoring campaigns, given a current profile and the physical properties of a riser, its response and fatigue damage may be estimated using available semi-empirical computer programs. The estimated riser response, though, with many of these tools, often only accounts for energy of the first harmonic at the Strouhal frequency (vortex-shedding frequency) while neglecting or filtering out higher harmonics (i.e., the response at frequencies that are multiples of the Strouhal frequency) which can result in fatigue damage rate being underestimated by a factor as large as 40 [2]. Some recent studies have presented approaches to incorporate higher harmonics in fatigue damage estimation (see, for example, [PERSON] and [PERSON] [3] and [PERSON] et al. [4]). The general idea is to estimate the ratio of the higher harmonics to the first harmonic from the recorded riser response, and then to use that ratio to modify the estimated fatigue damage rate caused by the first harmonic alone. In addition to the use of semi-empirical computer programs, there have been numerous investigations that have focused on estimating the VIV response and fatigue damage of risers using computational fluid dynamics (CFD) simulations [5; 6; 7; 8]. The greatest challenge in estimating fatigue damage using CFD simulations lies in the imprecision associated with the use of Reynolds Averaged Navier-Stokes models employed in these simulations and the resulting errors for such flows. For example, [PERSON] et al. [9] showed that, in turbulent flows with flow separation and recirculation, the predictions based on such models are highly uncertain. [PERSON] et al. [10] showed that predictions using such Reynolds-averaged Navier-Stokes (RANS) models for marine risers lead to large errors in many cases with complex turbulent flows. In the present study, several empirical methods are employed to estimate the VIV response and fatigue damage over the entire length of a riser using responses measurements at only a small number of discrete locations. This work was earlier discussed very briefly in a shorter article by the authors [11]. Here, we introduce useful analytical insights and extensions to the methodology introduced in the earlier shorter work. Unlike semi-empirical or analytical VIV prediction software, empirical methods only require data (i.e., the measured riser response); no information or very little information on the current profile and on the physical and hydrodynamic properties of the riser is required. Five empirical methods, referred to as WWA (Weighted Waveform Analysis), modified WWA, POD (Proper Orthogonal Decomposition), MPR (Modal Phase Reconstruction), and a hybrid method which combines MPR and modified WWA, are studied.