Data Analysis Tutorial : Characterizing horizontal currents

3.3 Characterizing horizontal currents

This topic is worthy of a whole book on its own. The main interest here is to compare two ways of characterizing currents that rotate in different directions in the horizontal plane. Consider currents measured at two levels:

$\begin{equation} \label{eq:tideW} W(1)=(1+\epsilon )e^{i\omega t}\quad W(2)=e^{-i\omega t} \end{equation}$

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where $\omega$ represents the tidal frequency, for instance $M2$ . Note the importance of having $\epsilon \ne 0$ , or else the EOF analysis returns garbage. A tidal analysis of these currents (using t_tide) yields the expected result that one time series rotates in the positive direction in time while the other rotates in the opposite sense. Aside from a straightforward time domain analysis of the four scalar time series, it is possible to compute the principal values of the complex time series $W$ , or of the Hilbert transform of each component: $H(\Re {W(1)}$ , $H(\Im {W(1)}$ , $H(\Re {W(2)}$ , $H(\Im {W(2)}$ . The results are illustrated in Fig. , generated by the script test_tidal_currents.

The two eigenvalues of the $cov(W)$ matrix are equal, and the vectors, illustrated in the top left hand frame of Fig. , reflect the fact that each bin corresponds to a different mode, independent of the orientation of the coordinate system, and is consistent with the fact that the sense of rotation is opposite. The advantage of this analysis is that the different sense of rotation is clearly identified, although two modes are required to represent the full data set. In contrast the Hilbert transform analysis includes all the variance in a single mode. The amplitude is nearly uniform in depth (for small $\epsilon$ ). The phase distribution depends on the orientation of the coordinate system. In the original system (blue line), the $x$ velocities have the same phase ( $\pi /2$ ) for both bins, whereas the $y$ components are $\pi$ out of phase, consistent with the original definition of $W$ . When the coordinate system is rotated by $\pi /4$ (green line), the phase results are more difficult to interpret, although this analysis provides a much more compact representation of the data, since all the variance is represented by a single mode.

Given that both methods have advantages, both ought to be used to give a full description of the observations.