Two-step harmonic analysis for capturing seasonally-varying amplitudes and phase lags of the predominant tidal constituents

Two-step harmonic analysis for capturing seasonally-varying amplitudes and phase lags of the predominant tidal constituents

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Anzhou Cao¹, Zheng Guo²^,^*

Acta Oceanologica Sinica | 2020, 39(7) : 165 - 174

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Acta Oceanologica Sinica | 2020, 39(7): 165-174

• Marine Technology •

Two-step harmonic analysis for capturing seasonally-varying amplitudes and phase lags of the predominant tidal constituents

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Anzhou Cao¹, Zheng Guo²^,^*

Affiliations

¹ Ocean College, Zhejiang University, Zhoushan 316021, China

² Marine Science and Technology College, Zhejiang Ocean University, Zhoushan 316021, China

Published: 2020-07-25 doi: 10.1007/s13131-020-1624-y

Outline

Abstract

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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.

Key words

tides / harmonic analysis / seasonal variation / tidal gauges

Cite this Article

Anzhou Cao, Zheng Guo. Two-step harmonic analysis for capturing seasonally-varying amplitudes and phase lags of the predominant tidal constituents[J]. Acta Oceanologica Sinica, 2020 , 39 (7) : 165 -174 . DOI: 10.1007/s13131-020-1624-y

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1 Introduction

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Tides are one ubiquitous motion in oceans. They are caused by a combination of time-varying gravitational potential of the moon and sun and the centrifugal forces generated as earth rotates about the common center of mass of the earth-moon-sun system (Stewart, 2008). The investigation of tides has a long history. Kelvin, Ferrel, Darwin, Doodson (1921, 1924, 1928) and other great scholars developed and refined the harmonic analysis (HA) method for tidal analysis and prediction (Goldsbrough, 1942; Foreman and Henry, 1989). Up to now, HA has been realized with Fortran and Matlab codes (Godin, 1972; Foreman, 1977; Pawlowicz et al., 2002; Codiga, 2011) and widely used in the analysis of tides and other ocean dynamics related to tides. According to the traditional HA, the surface elevation induced by each constituent is expressed as

(1)

${\xi _n}\left(t \right) = {f_n}{h_n}\cos \left({{\omega _n}t + {u_n} - {g_n}} \right),$

where subscript n denotes different tidal constituents, t is the time, ω is the angular frequency, h and g are the amplitude and phase lag (the well-known harmonic constants), f and u are the nodal factor and angle which vary slowly with a period of 18.61 years. In the conventional wisdom, amplitude and phase lag of each constituent at a certain location would not vary with time.

With the accumulation of water elevation data from global tidal gauges and altimetry satellites, some observations are found to deviate from the conventional wisdom, i.e., the observed tidal amplitudes and phase lags are found to have seasonal variations (Huess and Andersen, 2001; Kang et al., 2002; St-Laurent et al., 2008; Georgas, 2012; Gräwe et al., 2014; Müller et al., 2014), secular changes (Ray, 2006, 2009; Jay, 2009; Müller et al., 2011) and nodal cycles differing from theoretical predictions (Amin, 1985; Ray, 2006; Cherniawsky et al., 2009; Feng et al., 2015). By reviewing previous studies, Woodworth (2010) summarized that a combination of five dynamical and three technical causes could partly account for the long-term changes in tidal amplitudes and phase lags. However, as also indicated by Woodworth (2010), the reasons for seasonal variability in tidal amplitudes and phase lags remain far to be fully understood.

How to accurately obtain seasonally-varying amplitudes and phase lags of the predominant constituents is also a problem for the traditional HA. Because a tradeoff between length of time window and resolution of constituents always exists in the traditional HA, using a long time window yields long-time-averaged amplitudes and phase lags which would differ from the real values of varying amplitudes and phase lags; while a short time window cannot resolve sufficient constituents, so that the unresolved constituents influence the amplitudes and phase lags of the resolved constituents. Therefore, a new way to implement HA is developed in this study. It combines long- and short-time-window HA. To distinguish from the traditional HA, it is named as “two-step HA” for convenience. The feasibility and accuracy of the two-step HA is validated by a series of ideal experiments (IEs) and compared with traditional HA using one-month (St-Laurent et al., 2008; Müller et al., 2014) and three-month (Huess and Andersen, 2001) windows.

The paper is organized as follows. The two-step HA is introduced in Section 2. Performances of the two-step HA in a series of IEs are shown in Section 3. In Section 4, the two-step HA is applied to practical scenario to investigate the seasonal variability of the predominant constituents at two tidal gauges. Finally, a discussion and a summary complete the paper in Section 5.

2 Methodology

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Generally, the K₁, O₁, M₂ and S₂ constituents are recognized as the predominant diurnal and semidiurnal constituents in oceans. To accurately capture their seasonally-varying amplitudes and phase lags is the main aim of this study. According to Fang et al. (2004), the Rayleigh criterion for each pair of eight constituents corresponding to hourly measurements is calculated and shown in Table 1. As shown, a record longer than 15 d with hourly intervals is sufficient to separate the K₁, O₁, M₂ and S₂ constituents from most of the other constituents; but it cannot resolve the constituent pair of P₁ and K₁ as well as that of the K₂ and S₂, which leads to incorrect estimations for the K₁ and S₂ (Amin, 1985). Therefore, to obtain accurate estimations for the K₁ and S₂ constituents, we need to firstly extract the P₁ and K₂ from a relatively long record (e.g., a one-year record) according to Table 1. This is the core of the two-step HA and detailed procedure is introduced as follows.

Step one: The whole observation record is divided into several one-year segments and the traditional HA is performed for each one-year segment ξ with the U_Tide Matlab package (Codiga, 2011). For a one-year (common year of 365 d) record, U_Tide would automatically choose 59 constituents. Thereafter, a one-year hindcast η is calculated using 55 of the 59 constituents except the K₁, O₁, M₂ and S₂. Subtracting η from ξ, a one-year record ζ only related to the K₁, O₁, M₂ and S₂ constituents is obtained.

Step two: The one-year record ζ is divided into twelve uniform 30-day segments corresponding to twelve months. For each month except February, the segment lasts from 1 to 30, while the segment of February starts on 31 January and ends on 1 March (29 February) for common (leap) years. Thereafter, the traditional HA is performed for each 30-day segment to obtain the monthly amplitudes and phase lags of the K₁, O₁, M₂ and S₂ constituents.

Note that for the 366-day record of a leap year, U_Tide would automatically choose 67 constituents. Compared with the 365-day record of a common year, 8 more constituents, the S_a, π₁, S₁, ψ₁, H₁, H₂, T₂ and R₂, are automatically resolved. To be consistent with common years, only the record of the first 365 d in leap years is used for analysis. In addition, the T_Tide Matlab package (Pawlowicz et al., 2002) can serve as an alternative to U_Tide in the two-step HA.

3 Validation by IEs

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In this section, a series of IEs are designed to validate the feasibility and accuracy of the two-step HA. In these IEs, a 365-daytime series is synthesized:

(2)

$\zeta \left(t \right) = {S_0} + \sum\limits_{n = 1}^8 {{h_n}\cos \left({{\omega _n}t - {g_n}} \right)} + \varepsilon \left(t \right),$

where S₀=20 cm is the mean water level, ε denotes the measurement error with a maximum value of 5 cm, and n=1,2,...,8 represents the Q₁, O₁, P₁, K₁, N₂, M₂, S₂, K₂ constituents, respectively. Note that the nodal correction and angle are neglected in this study, as they can be obtained through a fitting of the calculated amplitudes and phase lags (Ray, 2006; Feng et al., 2015).

In IE1, all the eight constituents are assumed to have constant amplitudes and phase lags, of which the prescribed values are shown in Table 2. Note that the ratios of prescribed amplitudes among both diurnal and semidiurnal constituents are consistent with those for the equilibrium tide.

Figure 1 displays the monthly amplitudes and phase lags of the four predominant constituents calculated with the traditional HA using a 30-day window. Note that the constituents are automatically determined by U_Tide, and only the amplitudes and phase lags of the O₁, K₁, M₂ and S₂ constituents are shown. The amplitudes and phase lags of the O₁ and M₂ constituents are close to the prescribed values, but there still exist small deviations. The mean absolute errors (MAEs) between the O₁ and M₂ amplitudes and their prescribed values are 1.6 and 0.4 cm, respectively. Note that these deviations are not caused by the measurement error ε, because these deviations still exist even if we remove the measurement error from the one-year time series (Eq.(2)). In other words, these deviations are related to the choice of the 30-day window. When a longer time window (e.g. one year) is used, these deviations decrease significantly and even vanish if the measurement error is not taken into consideration. For the K₁ and S₂ constituents, unrealistic semiannual cycles are found in their amplitudes and phase lags, which are caused by the unresolved P₁ and K₂ constituents explained above. If the monthly K₁(S₂) amplitudes (Figs 1b and d) are fitted using the following function:

(3)

$A\left(t \right) = {A_0} + {A_s}\cos \left({\frac{{2{\text π} }}{{{T_s}}}t - \phi } \right),$

where T_s equals to half of a year, the calculated A₀ and A_s are close to the prescribed K₁ (S₂) and P₁ (K₂) amplitudes, respectively. The MAEs between the K₁ and S₂ amplitudes and their prescribed values are 20.6 and 8.2 cm, respectively. In addition, the traditional HA using a three-month window leads to a similar result with slightly smaller MAEs, which are not shown.

In contrast to the traditional HA, the two-step HA provides a more accurate estimation. As shown in Fig. 2, the monthly amplitudes and phase lags of the four predominant constituents are almost the same as the prescribed values. The unrealistic semiannual cycles (Figs 1b and d) are absent from the amplitudes and phase lags of the K₁ and S₂ constituents, suggesting that the two-step HA can indeed resolve the K₁ and S₂ constituents from the P₁ and K₂, respectively. The MAEs between the O₁, K₁, M₂ and S₂ amplitudes and their prescribed values are all less than 0.1 cm, which are at least one order of magnitude smaller than those using the traditional HA. Combining these results, it can be concluded that the two-step HA can provide an accurate estimation for the amplitudes and phase lags of the predominant constituents and perform better than the traditional HA using one- or three-month windows.

As the two-step HA can resolve the K₁ and S₂ constituents from the P₁ and K₂, its performances on estimating seasonally-varying amplitudes and phase lags are examined in the following IEs. In IE2, the K₁and S₂ constituents are assumed to have seasonal variability whereas the amplitudes and phase lags of the other six constituents are constant as listed in Table 2. The amplitudes and phase lags of the K₁ and S₂ constituents in IE2 are prescribed as

(4)

${h_4}\left(t \right) = 100 + 10\cos \left({\frac{{2{\text π} }}{{{T_a}}}t} \right)\;\;\left({{\rm{cm}}} \right),$

(5)

${g_4}\left(t \right) = 250 + 10\cos \left({\frac{{2{\text π} }}{{{T_a}}}t + \frac{{\text π} }{2}} \right)\;\;\left({^ \circ } \right),$

(6)

${h_7}\left(t \right) = 47 + 4.7\cos \left({\frac{{2{\text π} }}{{{T_a}}}t - \frac{{\text π} }{2}} \right)\;\;\left({{\rm{cm}}} \right),$

(7)

${g_7}\left(t \right) = 100 + 10\cos \left({\frac{{2{\text π} }}{{{T_a}}}t} \right)\;\;\left({^ \circ } \right),$

where T_a equals to 365 d. In IE3, the P₁ and K₂ constituents are assumed to have seasonal variability whereas the other six constituents remain invariant (Table 2). The amplitudes and phase lags of the P₁ and K₂ constituents in IE3 are prescribed as

(8)

${h_3}\left(t \right) = 33 + 3.3\cos \left({\frac{{2{\text π} }}{{{T_a}}}t} \right)\;\;\left({{\rm{cm}}} \right),$

(9)

${g_3}\left(t \right) = 270 + 10\cos \left({\frac{{2{\text π} }}{{{T_a}}}t + \frac{{\text π} }{2}} \right)\;\;\left({^ \circ } \right),$

(10)

${h_8}\left(t \right) = 13 + 1.3\cos \left({\frac{{2{\text π} }}{{{T_a}}}t - \frac{{\text π} }{2}} \right)\;\;\left({{\rm{cm}}} \right),$

(11)

${g_8}\left(t \right) = 140 + 10\cos \left({\frac{{2{\text π} }}{{{T_a}}}t} \right)\;\;\left({^ \circ } \right).$

In IE4, the P₁, K₁, S₂, K₂ constituents are assumed to be seasonally-varying and their amplitudes and phase lags are prescribed as Eqs (4–11) while the Q₁, O₁, N₂ and M₂ constituents remain invariant (Table 2). In IE 5, all the amplitudes and phase lags of the eight constituents are seasonally-varying. Amplitudes and phase lags of the P₁, K₁, S₂, K₂ constituents in IE 5 are the same as those in IE 4, and those of the Q₁, O₁, N₂ and M₂ are prescribed as follows:

(12)

${h_1}\left(t \right) = 14 + 1.4\cos \left({\frac{{2{\text π} }}{{{T_a}}}t} \right)\;\;\left({{\rm{cm}}} \right),$

(13)

${g_1}\left(t \right) = 310 + 10\cos \left({\frac{{2{\text π} }}{{{T_a}}}t + \frac{{\text π} }{2}} \right)\;\;\left({^ \circ } \right),$

(14)

${h_2}\left(t \right) = 71 + 7.1\cos \left({\frac{{2{\text π} }}{{{T_a}}}t} \right)\;\;\left({{\rm{cm}}} \right),$

(15)

${g_2}\left(t \right) = 90 + 10\cos \left({\frac{{2{\text π} }}{{{T_a}}}t + \frac{{\text π}}{2}} \right)\;\;\left({^ \circ } \right),$

(16)

${h_5}\left(t \right) = 19 + 1.9\cos \left({\frac{{2{\text π} }}{{{T_a}}}t - \frac{{\text π} }{2}} \right)\;\;\left({{\rm{cm}}} \right),$

(17)

${g_5}\left(t \right) = 20 + 10\cos \left({\frac{{2{\text π} }}{{{T_a}}}t} \right)\;\;\left({^ \circ } \right),$

(18)

${h_6}\left(t \right) = 100 + 10\cos \left({\frac{{2{\text π} }}{{{T_a}}}t - \frac{{\text π} }{2}} \right)\;\;\left({{\rm{cm}}} \right),$

(19)

${g_6}\left(t \right) = 280 + 10\cos \left({\frac{{2{\text π} }}{{{T_a}}}t} \right)\;\;\left({^ \circ } \right).$

Figures A1, A2 (in the Appendix) and 3 show the amplitudes and phase lags of the four predominant constituents estimated by the two-step HA in IEs 2–4, respectively. In IE 2, both the seasonally-varying amplitudes and phase lags of the K₁ and S₂ and constant amplitudes and phase lags of the O₁ and M₂ are successfully captured by the two-step HA (Fig. A1). The MAEs between the O₁, K₁, M₂ and S₂ amplitudes and their prescribed values are 0.2, 0.4, 0.2 and 0.3 cm, respectively. Although these MAEs are a little larger than those in IE1 using the two-step HA (less than 0.1 cm), they are still smaller than the MAEs for the traditional HA using a 30-day window (0.4–1.6 cm). These results indicate that if the P₁ and K₂ constituents are invariant, the seasonally-varying amplitudes and phases of the K₁ and S₂ can be accurately obtained through the two-step HA. When the P₁ and K₂ constituents are varying (IEs 3 and 4), the varying tendencies of the K₁ and S₂ can still be captured by the two-step HA, but the deviations in the K₁ and S₂ amplitudes and phase lags are increased (Figs A2 and 3). The MAEs between the O₁, K₁, M₂ and S₂ amplitudes and their prescribed values are 0.1, 3.0, 0.1 and 1.1 cm in IE 3 and 0.1, 2.9, 0.2 and 1.1 cm in IE 4. In these two IEs, MAEs of the O₁ and M₂ constituents are comparable to those in IE 2, while MAEs of the K₁ and S₂ are approximate one order of magnitude larger than those in IE 2. This result indicates the difficulty in accurately estimating the amplitudes and phase lags of the K₁ and S₂ with varying P₁ and K₂ constituents. To our knowledge, there is no other approach that can solve the problem with high accuracy. Combining the results shown in Figs A2 and 3 and the calculated MAEs, we believe that performances of the two-step HA in IEs 3 and 4 are acceptable, because (1) the amplitudes and phase lags estimated by the two-step HA generally capture the variation of the prescribed K₁ and S₂, and (2) MAEs of the K₁ and S₂ are smaller than their amplitudes as well as the maximum value of measurement error (5 cm).

Figure 4 illustrates the amplitudes and phase lags of the four predominant constituents estimated by the two-step HA in IE 5. Although all the eight prescribed constituents are seasonally-varying, the prescribed O₁ and M₂ constituents can be accurately obtained with MAEs of 0.4 and 0.7 cm in amplitudes; the MAEs between the K₁ and S₂ amplitudes and their prescribed values are 2.9 and 1.0 cm, which are comparable to those in IEs 3 and 4 and also acceptable. This result suggests that except the P₁ and K₂ constituents, variations of other constituents almost have no effects on the estimation of the K₁ and S₂, when using the two-step HA.

Combining all the results of IEs, it can be concluded that the two-step HA is a useful tool to estimate the seasonally-varying amplitudes and phase lags of the predominant constituents. Compared with the traditional HA using a short time (e.g., one-month) window, the two-step HA yields more accurate results and avoids the occurrence of unrealistic semiannual cycles caused by the unresolved constituents. In the next section, we will show the performance of the two-step HA dealing with real scenario.

4 Practical applications

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Müller et al. (2014) have shown that the M₂ constituent exhibits apparent seasonality at Cuxhaven, Germany and Victoria, Canada. In this study, the two-step HA is applied to the water elevation records at the two tidal gauges. To be consistent with Müller et al. (2014), the records at Cuxhaven from 1986 to 2005 and those at Victoria from 1966 to 1985 are used, which are obtained from the University of Hawaii Sea Level Center (UHSLC, https://uhslc.soest.hawaii.edu/data/, Caldwell et al., 2015).

Figure 5 shows the monthly M₂ amplitudes at Cuxhaven using both the two-step and traditional HA. As a comparison, the monthly S₂ amplitudes are also displayed. On the whole, the monthly M₂ amplitudes using the two-step HA are generally consistent with those using the traditional HA. This is reasonable because the M₂ constituent is able to be resolved with a short record (Table 1). However, a little difference still exists between the monthly M₂ amplitudes using the two-step and traditional HA at some time. As a result, the 20-year-averaged monthly M₂ amplitudes obtained by the two-step HA have slightly smaller seasonal variation than those obtained by the traditional HA (Fig. 5c). At the same time, the 20-year-averaged monthly M₂ amplitudes obtained by the two-step HA have smaller standard deviations (STDs) than those obtained by the traditional HA (Fig. 5c). Combining this result and those in IEs, we believe that the two-step HA yields a more accurate estimation. In other words, the traditional HA using a 30-day window may slightly overestimate the seasonal variation in the M₂amplitude. In contrast to the M₂ constituent, the S₂ amplitudes obtained with the two-step HA are almost invariant throughout the year with small STDs (Fig. 5d). Whereas the traditional HA using a 30-day window yields unrealistic semiannual cycles with large STDs, which is accounted for by the unresolved K₂ constituent. Similar result is found for the K₁ and O₁ constituents at the tidal gauge (Fig. 6). The O₁ constituent obtained by the two-step and traditional HA is almost the same, whereas the traditional HA using a 30-day window yields unrealistic semiannual cycles for the K₁ constituent due to the unresolved P₁ constituent. Combining these results, it can be concluded that the two-step HA can yield a good estimation of the monthly amplitudes and phase lags for the predominant constituents, whether they exhibit seasonal variability or not. Similar results are found at the tidal gauge of Victoria, which are shown in Figs A3 and 4 in Appendix.

Müller et al. (2014) explored the dynamics behind the seasonal variation of the M₂ constituent based on an ocean circulation and tide model. They found that the seasonal change in stratification on the continental shelf and frictional effect between sea-ice and the surface ocean layer cause the seasonal variation of the M₂ constituent. However, results of this study show that there is no apparent seasonal variability for the S₂, K₁ and O₁ constituents. Why do these dynamical processes not influence the S₂, K₁ and O₁ constituents? It needs further exploration.

5 Discussion and summary

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Recent studies have shown that amplitudes and phase lags of the predominant constituents exhibit seasonal variations at some locations (Huess and Andersen, 2001; Kang et al., 2002; St-Laurent et al., 2008; Georgas, 2012; Gräwe et al., 2014; Müller et al., 2014). However, how to accurately obtain these variations remains a problem for the traditional HA due to the tradeoff between the length of time window and resolution of constituents. A new way to implement HA, named as “two-step HA” to distinguish from the traditional HA, is developed to capture these variations, which consists of both long- and short-time-window HA. Through a series of IEs, practical application to water elevation records at two tidal gauges as well as comparison with the traditional HA, the feasibility and accuracy of the two-step HA is verified: The two-step HA can provide a good estimation of monthly amplitudes and phase lags for the predominant constituents whether they have seasonal variability or not. The two-step HA also performs better than the traditional HA using one- or three-month windows: First, the two-step HA leads to smaller deviations than the traditional HA; Second, the traditional HA using one- or three-month windows causes unrealistic semiannual cycles in the amplitudes and phase lags of the K₁ and S₂ constituents due to the influence of the unresolved P₁ and K₂, whereas the two-step HA avoids it.

Several methods have been proposed to capture the variations of the predominant constituents. For instance, the tidal admittances, which are the complex ratios of the harmonic constants of the observed tide versus the equilibrium tide, have been used to investigate the tidal seasonality (Devlin et al., 2018). Based on the traditional HA, an interpolation method and an independent point (IP) scheme, Jin et al. (2018) and Guo et al. (2018) put forward two methods to capture variations of harmonic parameters. Jin et al. (2018) adopted the cubic spline interpolation while Guo et al. (2018) employed the linear interpolation. Each method has its own advantages. The two-step HA introduced in this study is easy to realize based on existing Matlab packages such as U_tide and T_tide. It calculates harmonic constants with temporal changes while avoids introducing unrealistic fluctuations.

The two-step HA introduced in this study is validated as a useful tool to capture varying amplitudes and phase lags of the predominant constituents. In the near future, we will apply it to water elevation records at global tidal gauges and provide a global map of seasonal variability for the predominant constituents, which we hope to be helpful to understanding the dynamics lying behind. In addition, we will also apply the two-step HA to current observations to investigate the variations of internal tides, because mesoscale eddies with lifecycles of months to years can scatter the internal tidal energy (Dunphy and Lamb, 2014) so as to change the current ellipses of internal tides over a period. This is meaningful and important when investigating the coherence and incoherence of internal tides (Nash et al., 2012; Xu et al., 2013; Cao et al., 2017).

Acknowledgements

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We thank UHSLC for providing water elevation records at the two tidal gauges used in this study.

Funding

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The National Natural Science Foundation of China under contract No. 41806012.

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Year 2020 volume 39 Issue 7

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Article Info

doi: 10.1007/s13131-020-1624-y

Receive Date：2019-10-05
Online Date：2026-03-31
Published：2020-07-25

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History

Received：2019-10-05
Accepted：2019-11-19

Funding

The National Natural Science Foundation of China under contract No. 41806012.

Affiliations

¹ Ocean College, Zhejiang University, Zhoushan 316021, China

² Marine Science and Technology College, Zhejiang Ocean University, Zhoushan 316021, China

Corresponding:

* E-mail: guozheng-gz@163.com

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表12种不同金属材料的力学参数

科 Family	属数 Number of genus	种数 Number of species	占总种数比例 Percentage of total species (%)	属 Genus	种数 Number of species	占总种数比例 Percentage of total species (%)
鹅膏菌科Amanitaceae	2	11	5.26	鹅膏菌属 Amanita	10	4.78
小菇科 Mycenaceae	2	12	5.74	丝盖伞属 Inocybe	5	2.39
多孔菌科 Polyporaceae	8	14	6.70	蜡蘑属 Laccaria	5	2.39
红菇科 Russulaceae	3	23	11.00	小皮伞属 Marasmius	6	2.87
				小菇属 Mycena	11	5.26
				光柄菇属 Pluteus	5	2.39
				红菇属 Russula	17	8.13
				栓菌属 Trametes	5	2.39

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TxT

	O₁	P₁	K₁	N₂	M₂	S₂	K₂
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.
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

	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

Fig. 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.

Fig. 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.

Fig. 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.

Fig. 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.

Fig. 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.

Fig. 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.

Fig. 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.

Fig. 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.

Fig. 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.

Fig. 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.

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