Two-step harmonic analysis for capturing seasonally-varying amplitudes and phase lags of the predominant tidal constituents
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Abstract: Recent studies have revealed that the predominant tidal constituents have seasonal variations at some locations. However, how to accurately obtain these variations remains a problem for the traditional harmonic analysis (HA) due to the tradeoff between length of time window and resolution of constituents. Therefore, a method named as “two-step HA” is developed in this study, which consists of both long- and short-time-window HA. Through a series of ideal experiments, practical application at two tidal gauges and comparison with the traditional HA, the feasibility and accuracy of the two-step HA are verified: The two-step HA performs better than the traditional HA in estimating monthly amplitudes and phases for the predominant constituents, whether they have seasonal variability or not. In addition to capturing variations of the predominant constituents at tidal gauges, the two-step HA would be useful in investigation of the coherence and incoherence of internal tides.
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Key words:
- tides /
- harmonic analysis /
- seasonal variation /
- tidal gauges
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Figure 1. Monthly amplitudes (blue solid lines) and phase lags (orange solid lines) of the O1 (a), K1 (b), M2 (c) and S2 (d) constituents for traditional HA using a one-month window in IE 1. The dashed line in each subfigure denotes the prescribed values of amplitudes and phase lags.
Figure 2. Monthly amplitudes (blue solid lines) and phase lags (orange solid lines) of the O1 (a), K1 (b), M2 (c) and S2 (d) constituents for the two-step HA in IE 1. Both the prescribed and estimated amplitudes and phase lags are overlapped.
Figure 3. Monthly amplitudes (blue solid lines) and phase lags (orange solid lines) of the O1 (a), K1 (b), M2 (c) and S2 (d) constituents for the two-step HA in IE 4. The dashed lines in b and d denote the prescribed amplitudes and phase lags at the center of each month.
Figure A1. Monthly amplitudes (blue solid lines) and phase lags (orange solid lines) of the O1 (a), K1 (b), M2 (c) and S2 (d) constituents for the two-step HA in IE 2. The dashed lines in b and d denote the prescribed amplitudes and phase lags at the center of each month.
Figure A2. Monthly amplitudes (blue solid lines) and phase lags (orange solid lines) of the O1 (a), K1 (b), M2 (c) and S2 (d) constituents for the two-step HA in IE 3. The dashed lines in b and d denote the prescribed amplitudes and phase lags at the center of each month.
Figure 4. Monthly amplitudes (blue solid lines) and phase lags (orange solid lines) of the O1 (a), K1 (b), M2 (c) and S2 (d) constituents for the two-step HA in IE 5. The dashed lines in each subfigure denote the prescribed amplitudes and phase lags at the center of each month.
Figure 5. Monthly amplitudes of the M2 (a)and S2 (b) constituents at Cuxhaven during 1986–2006, and the 20-year-averaged monthly amplitudes and corresponding STDs of the M2 (c) and S2 (d) constituents. In each subfigure, the blue and orange lines denote the results obtained by the two-step HA and traditional HA using a one-month window, respectively.
Figure 6. Monthly amplitudes of the K1 (a) and O1 (b) constituents at Cuxhaven during 1986–2006, and the 20-year-averaged monthly amplitudes and corresponding STDs of the K1 (c) and O1 (d) constituents. In each subfigure, the blue and orange lines denote the results obtained by the two-step HA and traditional HA using a one-month window, respectively.
Figure A3. Monthly amplitudes of the M2 (a) and S2 (b) constituents at Victoria during 1966–1985, and the 20-year-averaged monthly amplitudes and corresponding STDs of the M2 (c) and S2 (d) constituents. In each subfigure, the blue and orange lines denote the results obtained by the two-step HA and traditional HA using a one-month window, respectively.
Figure A4. Monthly amplitudes of the K1 (a) and O1 (b) constituents at Victoria during 1966–1985， and the 20-year-averaged monthly amplitudes and corresponding STDs of the K1 (c) and O1 (d) constituents. In each subfigure, the blue and orange lines denote the results obtained by the two-step HA and traditional HA using a one-month window, respectively.
Table 1. Alias periods (d) for each pair of constituents corresponding to hourly measurements based on the Rayleigh criterion
O1 P1 K1 N2 M2 S2 K2 Q1 27.6 9.6 9.1 1.0 1.0 0.9 0.9 O1 14.8 13.7 1.0 1.0 0.9 0.9 P1 182.6 1.1 1.1 1.0 1.0 K1 1.1 1.1 1.0 1.0 N2 27.6 9.6 9.1 M2 14.8 13.7 S2 182.6 Note: The unresolved constituent pairs (K1 and P1 as well as K2 and S2) by a one-month record and their alias periods are marked in bold. 
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Table 2. Prescribed amplitudes (cm) and phase lags (°) of the eight constituents in IE 1
Q1 O1 P1 K1 N2 M2 S2 K2 h 14 71 33 100 19 100 47 13 g 310 90 270 250 20 280 100 140
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