3.5 Change in the highest astronomical tide (δ-HAT)

We also consider a combined tidal variability besides the individual TACs. The M2, S2, K1, and O1 variabilities are summed to produce a combined tidal variability that is compared to MSL (δ-HAT). The δ-HAT is a proxy for the change in the highest astronomical tide, which is estimated by combining the complex time series

#### Figure 2.

M2 TAC relation of detrended absolute tidal amplitude to detrended MSL at Honiara in the Solomon Islands [9]. The green line is a robust linear regression trend, in mmm<sup>1</sup> .

of the yearly analyzed M2, S2, K1, and O1 tides, approximately 75% of the full tidal height. "Complex" means, in this context, that each constituent is considered as a complex number (accounting for both amplitude and phase), the complex vectors are added, and the total amplitude is resolved from the complex sum. The detrended time series of δ-HAT is then compared to the detrended MSL variability. The magnitude of the slope of the regression is the definition of the δ-HAT, and, like the TACs, we report δ-HATs in units of mm m<sup>1</sup> . Theoretically, the four constituents will not be exactly in phase more than once during every 18.6-year nodal cycle, though the constituents may be approximately aligned more often; therefore, this summation provides a suitable proxy for the envelope of possible tidal amplitudes. A detailed description of the step-by-step method, with additional figures showing the intermediate steps in the process, are provided in the supplementary materials of Devlin et al. [8]. The δ-HAT analyses performed for the Atlantic tide gauge stations [10] employed an eight-tide combination, which adds the N2, K2, P1 and Q1 tidal constituents to the δ-HAT sum (Table 2).

#### 3.6 Error analysis and autocorrelation handling

Our approach uses one-year harmonic analyses at a one-month step to yield smooth time-series. However, this approach must be taken with caution, as there may be autocorrelation in the regression due to data overlap. Thus, calculations of regressions and associated statistics (i.e., the p-values) are based on a sub-sampled dataset of one determination per year. The definition of the "year window" used for harmonic analysis may influence the value of the TAC or δ-HAT, i.e. calendar year (Jan–Dec) vs. water year (Oct–Sep). Thus, we use an ensemble of TACs and δ-HATs using 12 distinct year definitions (i.e., Jan–Dec, Feb–Jan, … ). We take the average of this set as the magnitude of the TAC or δ-HAT. For an estimate of the confidence interval of the trend, the interquartile range (middle 50% of the set range) is used. We consider correlations to be significant if they have a p-level of <0.05, the trend is greater than the interquartile range by at least a factor of 1.5, and the magnitude is greater than 10 mm m<sup>1</sup> for individual TACs, and greater than 50 mm m<sup>1</sup> for the <sup>δ</sup>-HATs. Some determinations had unexpected errors or grossly insignificant statistics (especially for the shorter records) that made them


#### Estuaries and Coastal Zones - Dynamics and Response to Environmental Changes


Tidal Evolution Related to Changing Sea Level; Worldwide and Regional Surveys… DOI: http://dx.doi.org/10.5772/intechopen.91061


#### Estuaries and Coastal Zones - Dynamics and Response to Environmental Changes


Tidal Evolution Related to Changing Sea Level; Worldwide and Regional Surveys… DOI: http://dx.doi.org/10.5772/intechopen.91061


The TAC magnitude is determined by the ensemble average of 12 monthly determinations, and the confidence interval on the determined trend is given by the interquartile range (IQR) of the ensemble. Significant determinations are given in bold text. Entries marked with an '\*' indicate locations where analyses failed due to small tidal amplitudes. Units of the TACs and IQRs are in mm m<sup>1</sup> .

#### Table 2.

TAC results for M2 and K1 and δ-HAT results at all Atlantic locations.

unreliable. These were not included in averaging process, though at all locations, a minimum of eight of 12 determinations was required to deem a result significant. For more detailed descriptions of the TAC and δ-HAT determinations, please refer to Devlin et al. [10], and the supplementary material of Devlin et al. [8, 9].
