3.1. Determine the Numbers of IPs
Different IP numbers represent oscillations at different time scales in the EHA method [
30]. If the number of IPs is set equal to 1, the oscillation is a constant. If it is set equal to 0, the oscillation represents the long-term trend of the data. Moreover, the long-term trend consists of linear and nonlinear trends, which are decided by a parameter in the S_TIDE MATLAB toolbox. When the number of IPs is equal to 0, the result is the linear trend of data if the parameter is equal to 2. Additionally, if the parameter is equal to 3, the oscillation represents the nonlinear trend. It can be found that the S_TIDE is a powerful tool to analyze time-series data. Furthermore, it should be realized that the number of IPs plays a vital role in the results of the EHA. Either fewer or more IPs will affect the outcome of the EHA method. Small IPs will introduce large root-mean-square error (
RMSE) and large IPs will cause the overfitting and fictitious conclusion. As a result, a primary and vital procedure is to develop a reasonable evaluation criterion on the number of IPs.
In this paper, three parameters are used to determine the number of IPs. The first is the
RMSE between the hindcasts obtained by the EHA and the primary data. It can be expressed as follows:
where
represents the estimated value of the tidal amplitude or phase by the EHA and
represents the real tidal amplitude or phase.
N is the length of
.
The second is the 95% confidence interval. The shorter confidence interval can give rise to a higher accuracy result based on the same confidence level. Therefore, the length of the 95% confidence interval can lead to a restriction for the result of EHA.
The third is the signal-to-noise ratio (
SNR). It can be calculated as follows:
where
H represents the time-varying amplitude
H(
t), and
represents the 95% confidence interval for
H(
t). It is credible when the
SNR is greater than 2 [
33].
A series of sensitivity experiments are implemented to select the optimum number of IPs. The N2 tide is affected by 18.61- and 8.85-year cycles while 2N2 tides are mainly influenced by 8.85- and 4.42-year cycles. These two tides can be analyzed for these two cycles by the EHA method simultaneously. Then, the process of determining the numbers of IPs can be revealed distinctly by following analysis taking the 8.85-year cycle of N2 tide at Boston as an example.
Table 3 and
Table 4 display the
SNR and
RMSE for the 8.85-year cycle of N
2 tide at Boston, respectively. The IP1 and IP2 represent the numbers of IPs for the 18.61- and the 8.85-year cycle of N
2 tide, respectively. The numbers of IP1 and IP2 are greater than 1 because the amplitudes of those cycles (18.61 and 8.85) change with time. In other words, the amplitudes are constant in the case of IPs equal to 1, which turns into the traditional method (Equation (1)) and are not the objective of this paper (Equation (4)). However, it is noteworthy that the number of IPs is set equal to 1 for the 8.85-year cycle when the EHA method is used to analyze the 2N
2 tide because the major influence is introduced by the 4.42-year cycle.
The values of SNR and RMSE both reduce gradually with the increase of IP2 when the number of IP1 is fixed. The traditional standard that the SNR is greater than 2 is deemed to be credible. However, a more stringent criterion is applied in this paper: the last number of IP2 can keep the reduced tendency of SNR and the corresponding SNR value must be greater than 2.
According to the above stringent criterion, the applicable numbers of IP1 and IP2 are selected in
Table 3 (bold). They are unique for the number of IP1 equal to 2, 4, 5, 7, and 8, respectively. However, there are two applicable choices for IP2 when IP1 = 3 and IP1 = 6, respectively. We may apply the 2nd criterion in determining the number of IPs, which is the length of the 95% confidence interval. Taking IP1 = 3 as an example, the corresponding values of
RMSE and the length of the 95% confidence interval in the case of IP2 equal to 10 and 16 are listed in
Table 5. Taking IP2 = 2 as a reference, it shows that the
RMSE reduces by 4.8%, and the length increases by 84.6% when IP2 raises from 2 to 10. The
RMSE reduces by 22.6%, and the length increases by 506.7% when IP2 raises from 2 to 16. Thus, 10 is the best choice for IP2 when IP1 = 3. According to the above method, 9 is the best choice for IP2 when IP1 = 6. The IP2 could be determined when IP1 ranges from 2 to 8 by the same method. Then, all the
RMSE values listed in
Table 4 are compared to each other, and 4 and 16 are the best-fit numbers for IP1 and IP2, respectively.
All the most suitable numbers for IPs in analyzing 8.85-year cycles of N
2 tide and 4.42-year cycles of 2N
2 tide at 4 stations are listed in
Table 6. Meanwhile, the comparisons between the observational and hindcast amplitudes by the EHA methods of the N
2 and 2N
2 tide at four stations are exhibited in
Figure 2 and
Figure 3, respectively. These hindcasts are obtained with the best-fit numbers of IPs listed in
Table 6. Their amplitudes agree well with the observations except for extreme values, which may be caused by weather factors and tidal resonance. The errors between hindcasts and observations are small and reasonable. However, hindcasts at Halifax (
Figure 3d) show a large difference with observations. This phenomenon implies that the number of IP3 = 1 in analyzing the 2N
2 tide is inapplicable at Halifax. In other words, the amplitude of the 8.85-year cycle is not a constant and its variation affects the water level remarkably at Halifax. Taking the numbers of IP3 for the 8.85-year cycle and IP4 for the 4.42-year cycle to be equal to 8 and 35, respectively, the results would improve remarkably (
Figure 4 vs.
Figure 3d). It is noted that the long-term trend shown in
Figure 5 is not affected by the number of IPs. The numbers of IP3 and IP4 listed in
Table 6 at Halifax are applied in
Figure 6b and
Figure 7b to keep the consistency of the method.
3.2. Long-Term Trend of N2 and 2N2 Tidal Amplitudes
As Ray and Foster [
22] pointed out, the nuisance flooding, which is only triggered by high spring tides, has become more frequent since 2011 at Boston. Consequently, much attention should be paid to secular variation in water levels, as well as the long-term trend of tidal components.
Figure 5 displays the long-term nonlinear trends of the N
2 (blue dash line) and 2N
2 (red solid line) tidal amplitudes at Boston, Eastport, Portland, and Halifax, respectively. As indicated in
Section 3.1, the number of IPs is set equal to 0 and the parameter is equal to 3 to obtain the long-term nonlinear trend of N
2 and 2N
2 tidal amplitudes by the S_TIDE.
The N2 and 2N2 tidal amplitudes kept growing for almost one century at Boston; however, the trends of these two tides have reversed since the 1980s at Halifax. The trends are more complex at Eastport and Portland. The N2 tidal amplitude increases in the exponential profile at Eastport, while the amplitude of 2N2 tide started to decrease from the 1980s. At Portland, the trend of the N2 tidal amplitude keeps growing all the time, and the 2N2 tidal amplitude increased until it reached a constant value in the 1980s.
In summary, for the four stations, the N
2 tidal amplitudes have increased in almost one century except for Halifax and the 2N
2 tidal amplitudes have decreased in the recent 30 or 40 years except for Boston. These two tides’ amplitudes kept growing for one century only at Boston. This may account partially for the event that the nuisance flooding has become frequent at Boston by Ray and Foster [
22].
3.3. The 8.85- and 4.42-Year Cycles
The changing amplitudes of the 8.85-year cycle of N
2 tide and the 4.42-year cycle of 2N
2 tide are displayed in
Figure 6 for the four stations of Boston, Eastport, Portland, and Halifax. The variations in the amplitudes of the 8.85- and 4.42-year cycles are the largest at Eastport among the four stations, and the least at Halifax. As mentioned in
Section 2.1, the water level record at Halifax is selected for contrasting the influence of topography on the 8.85- and 4.42-year cycles. Halifax and Eastport are located at an approximate latitude and the most striking difference is the topography, and these two cycles at these two stations are compared in
Figure 7. An obvious same phase can be seen in the 8.85-year cycle at Eastport and Halifax from the middle 1930s to nowadays, and this phenomenon appears in the 4.42-year cycle from the middle 1930s to the middle 1980s. It is demonstrated that the topography does not affect the phase of these cycles. However, the amplitudes of these cycles are greatly influenced by the topography and resonance.
The amplitude of the 8.85-year cycle has an approximately 20-year periodic oscillation at Eastport and Halifax with a consistent phase (
Figure 6a). The 20-year oscillation appears at Boston too, but it lags that at Eastport and Halifax. At Portland, the 20-year oscillation has emerged since the 1950s, and the phase was in step with those at Eastport and Halifax between the 1990s and the 2010s. The variation in the 4.42-year cycle amplitude of 2N
2 tide is more complex than that in the 8.85-year cycle of N
2 tide. The phenomenon of 20-year oscillation also appeared from the 1940s to the 2010s at Eastport (
Figure 6b). From the early 1950s to the 1970s, there was a 10-year oscillation at Boston. Then, the phase adjusted to 20 years. At Portland, there was no obvious oscillation until the 1970s. Afterwards, the phases kept in step with that at Eastport and Boston. At Halifax, the quasi 20-year oscillation also appeared from the 1940s to the 1980s, which kept in step with that at Eastport.
The 95% confidence intervals for the 8.85-and 4.42-year cycles are presented in
Figure 8. The confidence intervals are relatively large at the beginning and end of the time series for both cycles, of which the 8.85-year cycle is larger than those of the 4.42-year cycle. Nevertheless, they are all reasonable and demonstrate the validness of variations in the amplitude by the EHA method.