Schultz et al (1996) carried out observations of diffuse hydrothermal venting from the TAG vent field, Mid-Atlantic Ridge. By deploying a series of MEDUSA instruments, they measured ambient as well as diffuse hydrothermal fluid temperatures, effluent flow rates, and they captured chemical samples for subsequent analysis. A lowpass-filtered version of the ambient and effluent temperature and effluent flow rate is seen below, decimated to a 15 minute sample rate, shown over a period of 100 days starting on September 21, 1994.

Figure follows Jupp (2000) Fig 4.11: MEDUSA data. Th is ambient seawater temperature, Teffk and Tefft are effluent temperature measurements obtained from two different thermocouple types located several cm apart within the instrument's internal flow chamber. v is the effluent flow rate (show here in digitizer counts rather than in calibrated units). Postmortem investigation of the instrument revealed damage to the Tefft sensor, which can explain the divergence between the two temperature sensors starting at about day 20. Note that panels (e) and (f) are close-up views of the Th and v channels, spanning only days 12-18. In this close-up view, one sees that variations in ambient temperatures are semi-diurnal and in-phase with the ocean tide, whereas there is no discernable tidal variation in the effluent flow rate data (v).

Jupp (2000) carried out a (conventional) non-parametric Fourier-based analysis of the tidal variability of these data sets. This can be compared against the parametric non-Fourier Admiralty method employed by the nfTides program distributed on this web site. First, we show the result of a fairly sophisticated (but conventional) non-parametric Fourier analysis that makes use of the Multiple-Window Power Spectrum (MWPS) method, which decomposes the time series into a series of eigenspectra.

Figure 4.12 (from Jupp, 2000) showing non-parametric, Fourier based (MWPS) spectral estimates for 100 days of MEDUSA data at the TAG vent field. Autoregressive prewhitening was used, 104 days of low-pass filtered time series (15 minute sample rate after decimation), comprising 10,000 time series points. The temperature signals were analyzed using MWPS with a time bandwidth product of 5, 10 eigenspectra, and a 96-point (24 hour) autoregressive prewhitening filter. The velocity signal was similarly analyzed, but a higher resolution tie bandwidth product 1, with 2 eigenspectra were calculated.

The power spectrum of the ambient seawater temperature (Th) signal has an ~1/f shape, with one significant spectral at 3.87 cpd, which is consistent with the 2nd harmonic of the M2 tidal component. There are small and somewhat broad spectral peaks that can be identified with the M2 (1.932 cps) and S2 (2.000 cpd) peaks, and with the O1 and K1 diurnal peaks.

The nonparametric spectral analysis of the effluent temperature (Teff) signal produces quite different results. This spectrum falls off with ~1/f**(0.5), with many small and hard to distinguish spectral peaks. Ther is a distinct but broad peak between 1.9-2.0 cpd (semi-diurnal) and another between 0.94-1.13 cpd (diurnal).

The conventional effluent flow rate (v) spectrum is challenging to determine because of the short duration of this time series (the sensor stopped functioning after a short period of time). The best that conventional Fourier-based methods can produce is the suggestion of elevated power levels near 1 and 2 cpd, with possibly greater power in the diurnal than the semi-diurnal band, but this result is hardly conclusive.

Non-Fourier based parametric tidal analysis, as implemented in nfTides, is more tolerant of short, interrupted, noisy time series whose variations are drawn from multiple processes and statistical distributions. The nfTides algorithm was applied by specifying that the 10,000 data points were to be partitioned into 47 overlapping 7-day long (672 point) time series sections, with a 30% overlap between adjoining sections. This provides 47 sets of estimates that were used by nfTides' Jacknife error estimation algorithm to produce confidence limits on the estimates.

The results of nfTides analysis can be summarized in the following table (which can be assembled from data obtained in the output files of nfTides) - where Hj is the amplitude and gj is the phase of the jth Admiralty basis function. We use Jupp's (2000) calculations of the tidal potential and the ocean tides at this site, from the ETGTAB and CSR codes, respectively:

TAG H1 (M2A) g1 (M2A) H2 (S2A) g2 (S2A) H3 (K1A) g3 (K1A) H4 (O1A) g4 (O1A)

potential (m2/s2) 2.28 88 deg 1.06 89 deg 1.23 45 deg 0.90 45 deg

ocean tide (m) 0.15 326 deg 0.07 1 deg 0.01 190 deg 0.03 228 deg

Th (deg C) 0.139 233 deg 0.094 273 deg 0.087 71 deg 0.042 101 deg

Teffk (deg C) 0.156 192 deg 0.074 256 deg 0.059 67 deg 0.006 75 deg

Tefft (deg C) 0.159 191 deg 0.102 245 deg 0.035 135 deg 0.018 237 deg

v (digitizer counts) 2.238 245 deg 2166 335 deg 2320 182 deg 1097 357 deg


These values may be plotted in phasor format, with magnitudes represented by vectors rotated from vertical (in-phase with variations recorded on the prime meridian) by the number of degrees indicated in the table.

Figure showing the harmonic coefficients estimated by the nfTides algorithm for MEDUSA data at TAG (Teffk and Tefft being effluent temperature, v being effluent flow rate). Those accustomed to conventional spectral representation will likely be confused by this display. Rather than having a display of spectral power at each of a series of N/2-1 discrete frequency bins, as is the case for conventional spectral analysis, using the Adminralty method we have decomposed the time series into a) a drift term, which was removed from the signal and b) 4 complex valued Adminralty basis functions that represent the diurnal and semi-diurnal variations. The phasor diagrams above display this highly condensed form of the tidal content of these signals. For example, panel (g) represents the Teffk effluent temperature channels, with M2A and S2A spectral components represented by vectors whose magnitude matches that in the table above, and whose angle is the phase lag in degrees relative to the prime meridian, as also seen in the table above. At the end of each vector are a pair of vertical and horizontal error bars, which have been calculated for each vector using the Jacknife method. One can immediately see that there is a signficant semi-diurnal modulation of both effluent temperature channels, but that the diurnal variation is not statistically significant.

The absence of error bars on the phasor diagram for effluent flow rate (v) is unfortunate, but a consequence of the short duration of the data from that channel, which was insufficient to divide the time series into a signifcant number of overlapping time series sections. This made it impossible to obtain reasonable estimates of the confidence limits on these parameters. Rather, a Bayesian drift removal term was applied to this time series as part of the pre-processing step, and then a single estimate of semi-diurnal and diurnal tidal variability was obtained. While the diurnal component (K1A and O1A) appears to dominate the flow rate tidal variabililty, the lack of confidence limits must be noted.

What inferences can be drawn from this analysis? Some additional information is required. nfTides analysis of the ocean tides and tidal potentials allows us to see that the M2A components of effluent temperature lags the ocean tides by ~225 degrees, so that the effluent temperature peaks in the interval between low tide and the next high tide. The poroelastic theory of tidal modulation of seafloor hydrothermal systems (Jupp & Schultz, 2004) produced a prediction for TAG that is compatible with this observation, and that was used to constrain the thickness of the poroelastic hydrothermal circulation layer beneath the hydrothermal field. Such a conclusion, based on small confidence limits on the phase lag between ocean (or Earth) tides and the observed time series, would not be possible using conventional spectral estimation methods, since phase estimates with such small confidence limits would not be possible from such analyses.

Why is this the case? The reason comes from several factors. First, we use statistically robust methods that are tolerant of noise sources and other processes that are not drawn from the same statistical distribution as the tidally modulated processes. More substantially, where the Fourier transform maps N real valued time samples into N/2-1 complex valued frequency bins (plus the DC or mean value of the time series), each of which has infinite variance, which can then only lead to finite variance estimates if these are section or band-averaged, the algorithm in nfTides partitions the information in the same N time series samples into only 8 degrees of freedom, i.e. the real and imaginary (or amplitude and phase) of the two semi-diurnal and two diurnal Admiralty basis functions. Such a large number of degrees of freedom, N, being used to determine such a small number of degrees of freedom (always 8) will naturally lead to smaller confidence limits. To a considerable degree, that is the secret to success of parametric methods for analysis of time series whose behaviour can reasonable be attributable to tidal modulation.