Nonlinear complex principal component analysis of nearshore bathymetry

Spatially and temporally extensive nearshore bathymetric datasets have recently been analyzed with complex principal component analysis (CPCA) and a nonlinear generalisation of PCA known as NLPCA.cir to extract propagating spatial patterns that constitute most to the dominant lower-dimensional structure in the datasets. Both of these methods, however, do not fully capture the nonlinear features in the datasets. NLPCA.cir is restricted to extracting only nonlinear phase information (missing the amplitude variability), whereas CPCA captures both the amplitude and phase information but only linearly. A new Neural Network method called the nonlinear CPCA (NLCPCA) overcomes the deficiency of both the CPCA and NLPCA.cir, and captures both the phase and amplitude nonlinearly. In this study we examine the applicability of the NLCPCA, NLPCA.cir and CPCA methods to bathymetry data at Egmond (Netherlands), Hasaki Coast (Japan), and Duck (North Carolina, USA). All 3 sites are characterized by sandbars that have multiannual lifetimes and behave in an interannual quasi-periodic offshore directed manner. A cycle comprises bar birth in the inner nearshore, followed by up to several years of net offshore migration and final disappearance in the outer nearshore zone. At Duck, the underlying low-dimensional data structure was found to have only linear phase and amplitude variability and is well modelled by the CPCA. At Egmond, the data has a notable nonlinear phase variability (that is, rather peaked bars and wide shallow bar troughs) and is well modelled by the NLPCA.cir. At Hasaki, the data structure displays not only nonlinear phase variability but also amplitude variability, and the NLCPCA method is needed to model Hasaki well. In any propagating phenomena, it is difficult to know the structure of the data in advance as to which of one of the three mehthods should be used. In this study the simplest model representing well the data structure at Duck, Egmond and Hasaki is the CPCA, NLPCA.cir and NLCPCA, respectively. To avoid choosing a model, the generalized NLCPCA model can be used for all 3 cases.