3.1 Tidal analysis

Recent studies have developed a reliable methodology to analyze tidal variability related to MSL variability [7–11]. These methods have been applied to many tide gauge locations worldwide, with a twofold approach. The first technique involves analyzing individual tidal constituents, while the second involves the consideration of the combination of multiple tides. For any individual tide gauge, water levels are typically recorded hourly as a continuous time series. Harmonic analysis of this data yields individual time series of multiple tidal constituents, each corresponding to an individual component of astronomical motion of the Sun and Moon, and their cointeractions. The largest parts of the tidal energy concentrate in the once-a-day (diurnal) and twice-a day (semidiurnal) frequency bands, with several closely spaced tidal constituents being important in each band. In practice, however, only a small number of these contain most of the tidal energy. For the purposes of our discussions of past studies we will only need to mention a few. The major twicedaily (semidiurnal) tide due to the Moon is denoted M2, and the twice-daily tide due to the Sun is denoted S2. Two important lunisolar interaction tides that define the once-a-day (diurnal) tides are denoted K1 and O1. Most of the past analyses only consider these four components, however, some of the locations considered (in the Atlantic Ocean) also consider two more semidiurnal components, denoted N2 and K2, and two more diurnal components, P1 and Q1 (Table 1).

#### 3.2 Tidal admittance calculations

Investigations of tidal trends are carried out using a tidal admittance method. An admittance is the unit-less ratio of an observed tidal constituent to the corresponding tidal constituent in the astronomical tide generating force (ATGF) expressed as a potential, V, divided by the acceleration due to gravity, g, to yield a quantity, Zpot(t) = V/g, with units of length that can be compared to tidal elevations Zobs(t) on a constituent by constituent basis. Because nodal and other low-frequency astronomical variability is present with similar strength in both the observed tidal record and in V, its effects are eliminated in the yearly analyzed admittance time series. Yearly tidal harmonic analyses are performed at monthly time steps on both the observed tidal records and the hourly Zpot(t) at the same location, using the r\_t\_tide MATLAB package [58], a robust analysis suite based on t\_tide [59]. The tidal potential is determined based on the methods of Cartwright and Tayler [24], and Cartwright and Edden [60]. The result from a single harmonic analysis of Zobs(t) or Zpot(t) determines an amplitude, A, and phase, θ, at the central time of the analysis window for each tidal constituent, with error estimates. Analyzing the entire tide gauge record produces time-series of amplitude and phase. From amplitude A(t) and phase θ(t) time series, one can construct complex amplitudes Z(t):

$$\mathbf{Z}(t) = A(t)e^{i\theta(t)}\tag{8}$$








> Table 1.

Tidal anomaly correlations (TACs) in the Pacific for M2, K1 and δ-HAT, along with 95% confidence limits.

Time-series of tidal admittance amplitude (A) and phase lag (P) for a constituent are formed using Eqs. (9) and (10):

$$\mathbf{A}(t) = abs|\frac{Z\_{obs}(t)}{Z\_{pot}(t)}|\tag{9}$$

$$\mathbf{P}(t) = \theta\_{obs}(t) - \theta\_{pot}(t) \tag{10}$$

The harmonic analysis that generates the As and Ps also provides an MSL timeseries. For each resultant dataset (MSL, A and P), the mean and trend are removed from the time series, to allow direct comparison of their co-variability around the trend. Applying trend removal also reduces the effects of land motions (e.g., glacial isostatic adjustment (GIA), subsidence, and tectonic effects. All of these are assumed linear on the time scale of tidal records) that occur on longer time scales, whereas we are concerned with short-term variability.
