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## Motivation for the method

adam — Thu, 09/27/2012 - 04:14

**Motivation**: Conventional analysis of hydrothermal time series make use of periodogram (windowed or otherwise), or multitaper spectral analysis methods. These methods employ the Fourier Transform, which decomposes the time series into a set of sines and cosines of varying frequencies. The analyst typically generates a power spectrum (or if carrying out multivariate analysis, cross-spectra and coherences), and examines the resulting spectra for peaks. Statistically proficient analysts would then test those peaks for statistical significance using a variety of established methods. Most methods employed implicitly assume that the error terms are drawn from a normal (gaussian) distribution, an assumption that is more often violated in practise than upheld.

There are several potentially serious shortcomings to this approach.

First, in order to fully resolve all extant tidal variations (for both ocean and earth tides) requires long time series. Tides are caused by gravitational forces that depend on the relative positions of the Earth, Sun and Moon, a pattern that repeats every 18 years. Indeed, the Tamura tidal potential catalog (Tamura, 1987) employs 1200 tidal components.

In the absence of long hydrothermal time series capable of resolving every spectral peaks that can be attributed to tidal forces, the conventional method is to apply Fourier analysis to substantially shorter time series (typically days, weeks or in only exceptional cases, months in length), and to assume that the data window recorded represented a stationary sample, i.e. once where the pattern of time series variations within the window endlessly repeats itself in the next (unrecorded) window of the same length, and for all subsequent windows, and also for the preceeding (unrecorded) window of the same length, and for all preceeding windows. Of course, in highly dynamic systems such as hydrothermal settings, and in real world tidally-modulated systems, such an assumption is a nonsense. The nonstationary nature of the tidal modulation, the hydrothermal variability, and the many other processes underway make it necessary to adapt a non-Fourier based method to extracting the tidal variability, and to do so in a manner that is tolerant of the many processes that modulate the time series.

As an alternative to conventional Fourier analysis, oceanographers often employ the Harmonic Method, which directly fits a set of sinusoids to the time series at established tidal frequencies. Where conventional Fourier analysis is limited (through the Fast Fourier, or Discrete Fourier Transform) to employing sines and cosines of discrete frequency content that is determined by the sample rate of the digitizer used to sample the time series (i.e. the time step between adjacent sample points), Harmonic Analysis fits exclusively tidal frequencies, i.e. finding the best fitting (in a least squares sense) sine functions at up to 1200 discrete tidal frequencies. The Harmonic Method, as its close relative the Fourier Transform, is a poor means of analyzing relatively short time series for tidal content, since genreally speaking an unobtainably long time series is required to estiamte the many harmonic coefficients at a sufficient number of frequencies to provide a meaningful and unbiased tidal decomposition.

Whatever transofrmation are employed to estimate tidal variability, it is necessary to accommodate the non-tidal causes of variation in a manner that does not lead to bias in estimation of the tidal components. For this latter point, we make use of "robust statistics". That is, in contrast to conventional methods generally employed in the analysis of hydrothermal time series, we do not assume that there is only one process at work that modulates the time series, nor do we assume that that process, or any process, is drawn from a normal distribution. Rather we build a parametric model for the tidal components based on the four complex valued Admiralty basis functions, and we estimate the fit to those basis functions by allowing the error terms to be non-gaussian through the use of Huber weighting. Interested readers may wish to read Chave, et al (1987), On the robust estimation of power spectra, coherences, and transfer functions, Journal of Geophysical Research, Vol. 92, No. B1, pp. 633-648, for an overview of the algorithm we have adopted for robust estimation of the aforementioned parameters.

Our estimates of the standard errors are based on a Jacknife estimate, where we divide the time series into a set of finite length windows, and examine the variability of the estimates from one window to the next n order to build up the error estimate. We also add one additional term to the parametric estimate of time series variability - a non-tidal "drift term", which is based on Bayesian statistics. Most hydrothermal time series are riddled with longer-term drift problems. Sensors placed into hydrothermal flow experience chemical, thermal shock and also biofouling, all of which can impose drift unrelated to the primary signals under study. We provide a means to estimate and remove such drift as part of the estimation of the Admiralty (condensed form) tidal basis functions. Finally, we use the basis functions to generate a time series containing only the tidal terms, which can be compared to the original time series. In this way the drift terms as well as the non-tidal components of the time series may be examined by comparison.

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