In marine sciences, time series are often nonlinear and nonstationary. Their analysis faces new challenges and thus requires the implementation of adequate and specific methods. We use the Hilbert-Huang Transform (HHT) for the spectral analysis of high frequency sampled time series in near shore waters of the Réunion island, located in the Indian Ocean 700 km east of Madagascar. We focus particularly on automatic measurements of sea level fluctuations, temperature records and current data sets at four different stations in the island. We look at the contribution of different Intrinsic Mode Functions (IMFs) obtained by the Empirical Mode Decomposition (EMD) and also compare the Hilbert spectra with the wavelet spectra. The inertial wave and several low-frequency tidal waves are identified by the application of EMD. Furthermore, the authors investigate the cross-correlations between data at the different stations. Wavelet coherence and EMD based Time Dependent Intrinsic Correlation (TDIC) analyses are applied to consider the correlation between two nonstationary time series. By TDIC analysis, it was concluded that the high frequency modes have small correlation; whereas the trends are perfectly correlated. The results obtained by wavelet coherence are very similar, thus confirming that both approaches could be used for identification of main properties of marine environmental time series. The methodologies presented in this paper are general and thus can be applied on other time series from the environmental and oceanic sciences, where time series are complex with fluctuations over a large range of different spatial and temporal scales, from seconds to thousands of years.