Conditional multivariate functional PCA for the reconstruction of temperature and salinity profiles partially sampled by deep-diving marine mammals

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We present a statistical method to reconstruct the vertical thermohaline conditions in the Indian Sector of the Southern Ocean, where temperature and salinity profiles are partially sampled by female southern elephant seals. Datasets collected by biologgers provide unprecedented spatial and temporal coverage of ocean conditions. However, the maximum recorded depth varies with the animals' behaviour, offering only a partial view of the vertical environment. Using multivariate functional Principal Component Analysis (PCA), a parametric estimation of the covariance structure and mean function from a set of complete bivariate profiles allows the construction of an eigenfunction basis. By accounting for measurement error variance, partially sampled temperature and salinity profiles can be projected into the eigenspace of the complete profiles through conditional estimation of their functional principal coordinates and then reconstructed over the defined domain. For simulated snippet profiles truncated at depth $z_{\max} = 250$ m and reconstructed over $\mathcal{Z} = [20,500]$ m, reconstruction accuracy increases by 30 % for temperature and 33 % for salinity when incorporating geographical covariates. We then reconstruct ~90,000 incomplete profiles from the multivariate functional PCA of ~10,000 profiles reaching 500 m, covering approximately 3 million km$^2$ around the French subantarctic islands.

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