# Extracting tidal frequencies using multivariate harmonic analysis of sea level height time series

## Abstract

This contribution is seen as a first attempt to extract the tidal frequencies using a multivariate spectral analysis method applied to multiple time series of tide-gauge records. The existing methods are either physics-based in which the ephemeris of Moon, Sun and other planets are used, or are observation-based in which univariate analysis methods—Fourier and wavelet for instance—are applied to tidal observations. The existence of many long tide-gauge records around the world allows one to use tidal observations and extract the main tidal constituents for which efficient multivariate methods are to be developed. This contribution applies the multivariate least-squares harmonic estimation (LS-HE) to the tidal time series of the UK tide-gauge stations. The first 413 harmonics of the tidal constituents and their nonlinear components are provided using the multivariate LS-HE. A few observations of the research are highlighted: (1) the multivariate analysis takes information of multiple time series into account in an optimal least- squares sense, and thus the tidal frequencies have higher detection power compared to the univariate analysis. (2) Dominant tidal frequencies range from the long-term signals to the sixth-diurnal species interval. Higher frequencies have negligible effects. (3) The most important tidal constituents (the first 50 frequencies) ordered from their amplitudes range from 212 cm (M2) to 1 cm (OQ2) for the data set considered. There are signals in this list that are not available in the 145 main tidal frequencies of the literature. (4) Tide predictions using different lists of tidal frequencies on five different data sets around the world are compared. The prediction results using the first significant 50 constituents provided promising results on these locations of the world.

### Keywords

Least-squares harmonic estimation (LS-HE) Multivariate tidal time series analysis Tidal frequencies Tide prediction## 1 Introduction

Tidal analysis and prediction have for long been an important issue for different applications such as safe navigation and hydrographic surveys. Because the tide is a periodic phenomenon, it can be modeled by a series of periodic functions such as sinusoidal ones. A reliable tidal analysis and prediction requires a reliable knowledge on the (main) tidal frequencies. Different tidal frequencies have been listed by many researchers based on the tidal theory. They usually expand the tide generating potential harmonically using major planets (e.g., Moon and Sun) ephemeris through different methods. We may at least refer to Doodson (1921, 1954), Cartwright and Tayler (1971), Cartwright and Edden (1973), Büllesfeld (1985), Xi (1987, 1989), Tamura (1987, 1995); Hartmann and Wenzel (1994, 1995), Roosbeek (1996) and Kudryavtsev (2004). These studies are all physics-based because no tidal observations were used. The methods assume that the tidal frequencies are known but their amplitudes are unknown.

To extract the tidal frequencies, many studies have analyzed sea level height with different methods such as the Fourier and wavelet. Flinchem and Jay (2000) and Jay and Kukulka (2003) considered tide times series to be non-stationary and introduced the continuous wavelet transform (CWT) method, a complementary to harmonic analysis and Fourier methods, to extract tidal information. Ducarme et al. (2006a) used a method based on the maximum likelihood—it is the Akaike Information Criterion (AIC) method (Sakamoto et al. 1986)—to find non-tidal components in tidal residues obtained from reduction of all estimated tides through the program VAV (Venedikov et al. 2001, 2003, 2005). Pytharouli and Stiros (2012) applied spectral analysis to the time series of the astronomical tide (smoothed tide time series) based on the NormPeriod code (Pytharouli and Stiros 2008). Capuano et al. (2011) adopted independent component analysis (ICA) (Hyvarinen et al. 2001) to obtain nonlinear independent tidal constituents.

As an observation-based method, this study is also based on tidal observations of which the frequencies, the amplitudes, and the phases are assumed unknown to be estimated. We aim to estimate the tidal frequencies based on a mathematical and statistical approach, namely the least-squares harmonic estimation (LS-HE) developed by Amiri-Simkooei et al. (2007) and Amiri-Simkooei (2007). LS-HE has already been used by Mousavian and Mashhadi-Hossainali (2012) to extract the tidal frequencies in which at most only 17 tidal constituents were extracted using a univariate analysis. We now apply the multivariate formulation of LS-HE developed by Amiri-Simkooei and Asgari (2012) to multiple time series. After successful applications of LS-HE and its multivariate formulation to many GNSS data series such as Amiri-Simkooei et al. (2007), Amiri-Simkooei and Tiberius (2007), Amiri-Simkooei and Asgari (2012), Sharifi et al. (2012), Sharifi and Sam Khaniani (2013) and Mousavian and Mashhadi-Hossainali (2013) in which different periodic patterns were identified in the GNSS series, we now consider another application of LS-HE in the geodetic community, namely, tidal time series analysis. The time series employed in this contribution concern the tide-gauge records (heights) obtained from the UK tide-gauge stations sampled at the rate of 15 min.

Apart from the above-mentioned methods such as the Fourier method, wavelet method and VAV—they are all univariate analysis methods—this contribution presents the multivariate LS-HE method. An important advantage of this method over the observation-based and/or the physics-based methods is that it enables one to detect the main shallow water tidal constituents (non-linear tides) using common-mode signals in multiple series. In this study, we managed to find a considerable number of major (Appendix B, Table 4) and minor (Appendix B, Table 5) shallow water components, and investigated their effects on tidal predictions.

The available tide-gauge data sets over the last century around the world make valuable data that need to be properly processed in the presence of modern computing techniques. In addition, proper analysis methods are to be developed for estimating the tidal frequencies and amplitudes in an appropriate manner. As an efficient method, this contribution uses the LS-HE method in its univariate and multivariate formulation. Also, a comparison is made between the results of this contribution and those obtained using the physics-based methods. It is also worth mentioning that neither of the physics-based nor the observation-based methods can provide the exact (error-free) tide predictions. There is always a difference (prediction error) between the predicted values and the observed tide heights. The meteorological effects are the main factors that limit the precision of tidal prediction. This is in fact the case in most of the areas around the world (see later on the results). From the statistical point of view, it can likely be associated to the unknown colored noise of the instantaneous tide-gauge records, which is the subject for further research.

The objectives of the present contribution may be summarized as follows: (1) we introduce a powerful method to extract tidal frequencies based on the many available tidal observations around the world. The multivariate LS-HE uses common-mode signals to extract such frequencies. LS-HE is neither limited to evenly-spaced data nor to integer frequencies. Further, current spectral analysis methods (e.g., Fourier method) cannot incorporate common signals of multiple series. (2) Using tide-gauge records we present a complete list of tidal frequencies. We then present a list of main tidal constituents (50 frequencies) of which promising results on tide prediction can be obtained in different locations of the world. (3) The effect of different tidal species on tide prediction is then investigated. The only important frequencies belong to the long-term species to the 6th diurnal harmonic. Signals with higher frequencies have negligible effect on tide prediction.

This paper is organized as follows: we review the LS-HE theory and its multivariate formulation in Sect. 2. As an observation-based method, the LS-HE is then applied to the univariate and multivariate tidal time series in Sect. 3. A comparison is made between the univariate and multivariate analysis and the tidal frequencies are extracted. Tide prediction is performed using different lists of tidal constituents in five different locations around the world so as to make several comparisons. Finally, we make some conclusions in Sect. 4.

## 2 Least-squares harmonic estimation (LS-HE)

*E*and

*D*, are the expectation and dispersion operators, respectively,

*A*is the \(m\times n\) design matrix, \(Q_y \) is the \(m\times m\) covariance matrix of the m-vector of observables

*y*, and

*x*is the

*n*-vector of unknown parameters.

The LS-HE method determines an appropriate design matrix A for the functional model through the parameter significance testing (Teunissen 2000a). In fact, this method identifies periodic patterns in terms of harmonic functions in the functional part of the model and thus improves it. To increase the detection power of the tidal frequencies, we use the multivariate formulation of the LS-HE developed by Amiri-Simkooei and Asgari (2012). We aim to detect the common-mode frequencies of multiple tide-height time series taken from various tide-gauge stations in UK. As a generalization of the Fourier spectral analysis, LS-HE is neither limited to evenly-spaced data nor to integer frequencies.

### 2.1 Univariate harmonic estimation

### 2.2 Multivariate harmonic estimation

*r*) time series for which the design matrix A and the covariance matrix \(Q_{ y} \) are the same, then the model is referred to as a multivariate linear model (Amiri-Simkooei 2007, 2009). For a multivariate model, Eq. (2) is generalized to:

*r*series; so do the \(n\times r\) matrices \(X=[x_{1} x_{2} \ldots x_{r} ]\) and \(X_{k} =[x_{{1k}} x_{{2k}} \ldots x_{{rk}} ]\) for the unknown parameters of the

*r*series. The components of the \(r\times r\) matrix \(\Sigma \) and the unknowns in the \(m\times m\) matrix

*Q*can be estimated using a multivariate analysis method (Amiri-Simkooei 2009). The Kronecker structure in Eq. (10), i.e., \(I_{r} \otimes A_{k} \), indicates that there is a common frequency (possibly with different amplitudes and phases) in all of the series which should be detected using the multivariate harmonic estimation.

*r*time series and \(P_{A}^\bot = I-A(A^{{T}}Q^{-1}A)^{-1}A^{{T}}Q^{-1}\) an orthogonal projector of the univariate model. Equation (11) considers all of the time series simultaneously and takes into account the possible cross-correlation through \(\Sigma \) and time correlation through

*Q*in an optimal least-squares sense. The matrix \(\Sigma \) is estimated as \(\hat{\mathop {\Sigma }}=\hat{E}Q^{-1}\hat{\mathop {E}}/(m-n)\) (Teunissen and Amiri-Simkooei 2008, 2009). The test statistic for testing the significance of the detected frequency is:

*Q*are known and that the original observables are normally distributed.

## 3 Numerical results and discussions

To see the advantage of the multivariate analysis over the univariate analysis, we selected 318 1-year time series among the 45 available series (first multivariate data set). For this purpose, we split long time series of each station into a couple of 1-year time series, making in total 318 1-year series. The time series were evenly spaced, namely, 15 min spaced tide height series, except for having at maximum 6 gaps. In a multivariate linear model, the design matrix A of different time series should be identical. Therefore, the gaps should be located in the same places for all series. In other words, if there are some gaps in one of the series of the multivariate model, one has to omit the data in the location of those gaps in all series. This makes in total 173 common gaps (out of 35,064 samples) within the 318 time series, distributed over 1 year.

Further, to extract the tidal frequencies, a multivariate time series is provided such that its length is suitable to separate close frequencies. The time span of these time series is chosen to be a little more than 18 years. It consists of 11 time series (second multivariate data set) derived from 11 stations, namely stations 6, 8, 10, 13, 16, 17, 29, 31, 43, 44 and 45 with a time interval from 1 January 1993 to 31 March 2011. These time series contain equally spaced data with the sample rate of 15 min, possibly with some gaps. According to the procedure described above, the total number of gaps in the second multivariate data set becomes 190,240, distributed over the 18.24 years.

### 3.1 Harmonic estimation

The tidal frequencies related to 413 important peaks (of Fig. 5), which for long time series in multivariate spectrum are well separated from their neighbors, are listed in Appendix B (Tables 4, 5, 6). Table 4 includes the frequencies that are very close to the 145 main tidal frequencies (available in literature) of which their differences are less than 10\(^{-4}\) cycle/h. Their Darwin’s symbols are also included in the table. Table 5 includes the remaining frequencies. The frequencies listed in these two tables are ordered in an ascending order. The most important tidal constituents (the first 50 constituents), presented in a descending order of their amplitudes, are listed in Table 6. Most of them belong to the main tidal frequencies in Table 4. For the data set considered, the first four important tidal frequencies are M2, S2, N2, and K2 with amplitudes of 212 cm, 74 cm, 41 cm, and 21 cm, respectively. The fiftieth constituent is OQ2 with amplitude of 1 cm. This frequency list is an important list of tidal constituents applicable to many fields of applications such as tide predictions.

Relevant recent studies based on tidal observations do not provide users with a complete list of tidal frequencies covering the whole spectrum of tidal data. Mousavian and Mashhadi-Hossainali (2012) extract at most only 17 tidal constituents through a univariate analysis. Ducarme et al. (2006b), by the use of VAV program, list only 91 tidal constituents ranging from the diurnal to ter-diurnal signal and Ducarme et al. (2006a) list 16 low-tidal frequencies. This simply shows the power of multivariate LS-HE in detecting tidal frequencies, which has led to extracting 413 tidal constituents in the present contribution, ranging from long-term signals to short-term of 1/12 diurnal signal.

To support our last statement two kinds of tide predictions have been performed based on different lists of frequencies on different data sets.

### 3.2 Tide prediction based on multivariate data set

Mean standard deviation of residuals, the ratio of standard deviation/tide height variation of 4-week predicted data compared with observed data, computed for several lists of frequencies (time interval: from 2011/4/1 to 2011/4/28)

Frequency list | Tide-gauge station number | |||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|

Mean residual vector’s standard deviation (cm) | ||||||||||||

The ratio of std/tide height variation | ||||||||||||

6 | 8 | 10 | 13 | 16 | 17 | 29 | 31 | 43 | 44 | 45 | Mean | |

1st list | 14.7 | 27.5 | 13.0 | 23.9 | 12.1 | 21.4 | 12.6 | 9.0 | 9.1 | 7.9 | 17.9 | 15.4 |

0.02 | 0.03 | 0.02 | 0.02 | 0.01 | 0.02 | 0.02 | 0.01 | 0.01 | 0.02 | 0.02 | 0.02 | |

2nd list | 14.3 | 14.0 | 11.8 | 33.3 | 16.3 | 22.1 | 11.7 | 14.2 | 12.2 | 9.8 | 20.3 | 16.4 |

0.02 | 0.02 | 0.02 | 0.02 | 0.02 | 0.02 | 0.02 | 0.02 | 0.02 | 0.02 | 0.02 | 0.02 | |

3rd list | 14.1 | 14.8 | 11.8 | 33.7 | 16.5 | 22.4 | 11.7 | 14.1 | 12.0 | 9.7 | 20.2 | 16.4 |

0.02 | 0.02 | 0.02 | 0.02 | 0.02 | 0.02 | 0.02 | 0.02 | 0.02 | 0.02 | 0.02 | 0.02 | |

4th list | 11.2 | 10.8 | 8.4 | 20.9 | 11.2 | 10.5 | 6.9 | 8.5 | 8.4 | 7.0 | 12.5 | 10.6 |

0.02 | 0.01 | 0.01 | 0.02 | 0.01 | 0.01 | 0.01 | 0.01 | 0.01 | 0.01 | 0.01 | 0.01 |

Mean standard deviation of residuals, the ratio of standard deviation/tide height variation of 8-month predicted data compared with observed data, computed for several lists of frequencies (time interval: from 2011/4/1 to 2011/12/1)

Frequency list | Tide-gauge station number | |||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|

Mean residual vector’s standard deviation (cm) | ||||||||||||

The ratio of std/tide height variation | ||||||||||||

6 | 8 | 10 | 13 | 16 | 17 | 29 | 31 | 43 | 44 | 45 | mean | |

1st list | 24.8 | 30.3 | 15.0 | 33.0 | 21.2 | 26.4 | 16.6 | 16.9 | 18.2 | 16.5 | 30.2 | 22.7 |

0.04 | 0.04 | 0.03 | 0.02 | 0.03 | 0.02 | 0.02 | 0.03 | 0.03 | 0.04 | 0.03 | 0.03 | |

2nd list | 23.9 | 21.0 | 15.5 | 43.9 | 23.8 | 29.7 | 15.8 | 20.3 | 19.9 | 17.2 | 31.7 | 23.9 |

0.03 | 0.03 | 0.03 | 0.03 | 0.03 | 0.02 | 0.02 | 0.03 | 0.03 | 0.04 | 0.03 | 0.03 | |

3rd list | 24.0 | 21.6 | 15.5 | 44.0 | 24.0 | 29.8 | 15.9 | 20.3 | 19.8 | 17.2 | 31.7 | 24.0 |

0.03 | 0.03 | 0.03 | 0.03 | 0.03 | 0.02 | 0.02 | 0.03 | 0.03 | 0.04 | 0.03 | 0.03 | |

4th list | 23.0 | 17.8 | 12.7 | 31.4 | 20.9 | 18.8 | 13.0 | 16.7 | 17.8 | 16.2 | 27.5 | 19.6 |

0.03 | 0.02 | 0.02 | 0.02 | 0.02 | 0.02 | 0.02 | 0.03 | 0.03 | 0.03 | 0.03 | 0.02 |

We first investigate the importance of different tidal species in tidal observation structure. In most recent studies, the most contributing tidal frequencies are considered to be those frequencies that lie between the long-term signals and at most the 6th diurnal species. The results provided in Table 1 confirm this issue. When we compare the mean residuals standard deviation of the 1st list and 4th list, we observe that by adding the frequencies between the 4th diurnal species and 6th diurnal species to the CTE list, the tide prediction gets better about 30 %. The CTE list, proposed by the studies of Cartwright and Tayler (1971) and Cartwright and Edden (1973), considers the tidal frequencies up to and including the 3rd harmonics. This improvement confirms the significant contribution of the frequencies at the 4th, 5th and 6th diurnal species. We note, however, that the contribution of the higher frequencies than the 6th diurnal species is negligible for the data set considered in this study.

We listed 413 tidal frequencies using the observed sea level height, which are presented in Tables 4 and 5 in Appendix B (2nd list); the most important tidal constituents (the first 50 constituents) are presented in Table 6 (3nd list). The results in Tables 1 and 2 show that the standard deviation of the residuals are approximately the same for these two lists, i.e. 16.4 cm (Table 1) and \(\sim \)24.0 cm (Table 2).

Because Table 6 consists of tidal frequencies that their amplitudes are more than 1 cm, it indicates that including tidal constituents that have amplitudes less than 1 cm do not significantly improve tide prediction. This seems to be an important conclusion as it presents a shorter list of tidal constituents that meet the requirements for many practical applications. The difference between the 4-week (Table 1) and 8-month (Table 2) predictions is because the prediction is a kind of extrapolation. As expected the precision of the results decreases for long term prediction. For example, the seasonal variations can be one of the influencing factors.

Based on the results presented in Tables 1 and 2, one can conclude that tide prediction error can never become smaller than a certain value. This conclusion is valid for all frequency lists used. This is mainly due to the uncertainties in the instantaneous water level in general and errors in the measured tide-gauge records in particular. This indicates that error sources in the measured water levels are subject to colored noise that needs to be investigated in future research.

### 3.3 Tide prediction based on other tide-gauge data sets

To see the performance of the detected frequencies at work in a more global scale, five other data sets are used. They include: (1) UK tide-gauge stations of which the data at those stations were not used when extracting the tidal frequencies, (2) a few tide-gauge stations in Persian Gulf, (3) a few stations in Indian Ocean, (4) a few stations in Pacific Ocean, and (5) two stations in Canada. For the UK stations, 103 1-year tide series from different stations except those listed in Tables 1 and 2 (station numbers 6, 8, 10, 13, 16, 17, 29, 31, 43, 44 and 45) with sampling rate of 15 min were used. In Persian Gulf, we considered eight 1-year time series in Bandar Abbas, Kangan, and Chabahar stations. The data in these stations are hourly. The data in Indian Ocean were chosen from Davis, Esperance, and Masirah stations. These data were sampled hourly from which we selected 19 1-year tide series. Regarding Pacific Ocean, we used the data from Cook Islands, Marshall Islands, Fiji, Kiribati, Nauru and FSM. Eleven 1-year hourly time series in these stations were employed. In Canada, we used four 1-year time series recorded in the stations, Queen Charlotte and Vancouver, with sampling rate of 1 h.

Mean standard deviation of residuals, the ratio of standard deviation/tide height variation of 1-month predicted data compared with observed data, computed for several lists of frequencies; UK tide-gauge stations (second column), Persian Gulf stations (third column), Indian Ocean stations (fourth column), Pacific Ocean stations (fifth column), Canada Stations (sixth column)

Frequency list | Tide-gauge station | ||||
---|---|---|---|---|---|

Mean residual vector’s standard deviation (cm) | |||||

The ratio of std/tide height variation | |||||

UK tide-gauge stations | Persian Gulf tide-gauge stations | Indian Ocean tide-gauge stations | Pacific Ocean tide-gauge stations | Canada tide-gauge stations | |

1st list | 31.9 | 21.8 | 12.9 | 10.9 | 31.5 |

0.04 | 0.06 | 0.06 | 0.05 | 0.05 | |

2nd list | 15.7 | 13.6 | 9.6 | 4.8 | 17.9 |

0.02 | 0.04 | 0.05 | 0.02 | 0.03 | |

3rd list | 15.1 | 13.5 | 9.3 | 4.7 | 17.8 |

0.02 | 0.04 | 0.05 | 0.02 | 0.03 | |

4th list | 31.9 | 21.8 | 12.9 | 10.9 | 31.5 |

0.04 | 0.06 | 0.06 | 0.05 | 0.05 |

Main tidal frequencies detected by least-squares power spectrum of 18.24-year multivariate time series (11 series used)

No. | Frequency (cycle/h) | Darwin’s symbol | No. | Frequency (cycle/h) | Darwin’s symbol |
---|---|---|---|---|---|

1 | 0.0001150 | SA | 14 | 0.0432930 | J1 |

2 | 0.0001511 | SSA | 15 | 0.0443753 | 2PO1 |

3 | 0.0012733 | MSM | 16 | 0.0446036 | SO1 |

4 | 0.0014283 | MM | 17 | 0.0448311 | OO1 |

5 | 0.0028216 | MSF | 18 | 0.0733555 | ST36 |

6 | 0.0030503 | MF | 19 | 0.0746397 | 2NS2 |

7 | 0.0359108 | SIG1 | 20 | 0.0748673 | ST37 |

8 | 0.0372188 | Q1 | 21 | 0.0759496 | OQ2 |

9 | 0.0387306 | O1 | 22 | 0.0761772 | EPS2 |

10 | 0.0402562 | NO1 | 23 | 0.0763798 | ST2 |

11 | 0.0416693 | S1 | 24 | 0.0774621 | O2 |

12 | 0.0415524 | P1 | 25 | 0.0774871 | 2N2 |

13 | 0.0417806 | K1 | 26 | 0.0776897 | MU2 |

27 | 0.0789996 | N2 | 45 | 0.1192424 | MO3 |

28 | 0.0792009 | NU2 | 46 | 0.1207668 | M3 |

29 | 0.0802832 | OP2 | 47 | 0.1207811 | NK3 |

30 | 0.0803976 | H1 | 48 | 0.1220641 | SO3 |

31 | 0.0805114 | M2 | 49 | 0.1222924 | MK3 |

32 | 0.0806252 | H2 | 50 | 0.1248859 | SP3 |

33 | 0.0807396 | MKS2 | 51 | 0.1251141 | SK3 |

34 | 0.0818213 | LDA2 | 52 | 0.1566886 | ST8 |

35 | 0.0820239 | L2 | 53 | 0.1579985 | N4 |

36 | 0.0831056 | 2SK2 | 54 | 0.1582011 | 3MS4 |

37 | 0.0832193 | T2 | 55 | 0.1595109 | MN4 |

38 | 0.0833331 | S2 | 56 | 0.1597129 | ST9 |

39 | 0.0835607 | K2 | 57 | 0.1607946 | ST40 |

40 | 0.0848456 | MSN2 | 58 | 0.1610228 | M4 |

41 | 0.0861555 | 2SM2 | 59 | 0.1612516 | ST10 |

42 | 0.0863831 | SKM2 | 60 | 0.1623327 | SN4 |

43 | 0.0876674 | 2SN2 | 61 | 0.1625353 | KN4 |

44 | 0.1177299 | NO3 | 62 | 0.1638446 | MS4 |

63 | 0.1640721 | MK4 | 93 | 0.2817893 | M7 |

64 | 0.1653570 | SL4 | 94 | 0.2830873 | ST16 |

65 | 0.1666663 | S4 | 95 | 0.2833149 | 3MK7 |

66 | 0.1668951 | SK4 | 96 | 0.2861366 | ST17 |

67 | 0.1982414 | MNO5 | 97 | 0.3190211 | ST18 |

68 | 0.1997533 | 2MO5 | 98 | 0.3205336 | 3MN8 |

69 | 0.1999821 | 3MP5 | 99 | 0.3207361 | ST19 |

70 | 0.2012914 | MNK5 | 100 | 0.3220454 | M8 |

71 | 0.2025757 | 2MP5 | 101 | 0.3233553 | ST20 |

72 | 0.2028033 | 2MK5 | 102 | 0.3235829 | ST21 |

73 | 0.2056256 | MSK5 | 103 | 0.3248678 | 3MS8 |

74 | 0.2058532 | 3KM5 | 104 | 0.3250954 | 3MK8 |

75 | 0.2084468 | 2SK5 | 105 | 0.3263803 | ST22 |

76 | 0.2372001 | ST11 | 106 | 0.3276895 | ST23 |

77 | 0.2385094 | 2NM6 | 107 | 0.3279178 | ST24 |

78 | 0.2387120 | ST12 | 108 | 0.3608028 | ST25 |

79 | 0.2400219 | 2MN6 | 109 | 0.3623141 | ST26 |

80 | 0.2402245 | ST13 | 110 | 0.3638259 | 4MK9 |

81 | 0.2413062 | ST41 | 111 | 0.3666483 | ST27 |

82 | 0.2415344 | M6 | 112 | 0.4010452 | ST28 |

83 | 0.2428443 | MSN6 | 113 | 0.4025571 | M10 |

84 | 0.2430719 | MKN6 | 114 | 0.4038670 | ST29 |

85 | 0.2441279 | ST42 | 115 | 0.4053789 | ST30 |

86 | 0.2443561 | 2MS6 | 116 | 0.4069182 | ST31 |

87 | 0.2445837 | 2MK6 | 117 | 0.4082012 | ST32 |

88 | 0.2458686 | NSK6 | 118 | 0.4471594 | ST33 |

89 | 0.2471779 | 2SM6 | 119 | 0.4830688 | M12 |

90 | 0.2474061 | MSK6 | 120 | 0.4858899 | ST34 |

91 | 0.2787530 | ST14 | 121 | 0.4874281 | ST35 |

92 | 0.2802912 | ST15 |

Total tidal frequencies, except for main ones, detected by least-squares power spectrum of 18.24-year multivariate time series (11 series used)

No. | Frequency (cycle/h) | No. | Frequency (cycle/h) |
---|---|---|---|

1 | 0.0000186 | 11 | 0.0236874 |

2 | 0.0000400 | 12 | 0.0241075 |

3 | 0.0000480 | 13 | 0.0255531 |

4 | 0.0000814 | 14 | 0.0315666 |

5 | 0.0000889 | 15 | 0.0372094 |

6 | 0.0186536 | 16 | 0.0372276 |

7 | 0.0196252 | 17 | 0.0387394 |

8 | 0.0211314 | 18 | 0.0390626 |

9 | 0.0214003 | 19 | 0.0415436 |

10 | 0.0224419 | 20 | 0.0735580 |

21 | 0.0761509 | 69 | 0.1595022 |

22 | 0.0761859 | 70 | 0.1595197 |

23 | 0.0776809 | 71 | 0.1610140 |

24 | 0.0776984 | 72 | 0.1610316 |

25 | 0.0779179 | 73 | 0.1638358 |

26 | 0.0820151 | 74 | 0.1638539 |

27 | 0.0820326 | 75 | 0.1651545 (2SNM4) |

28 | 0.0866663 | 76 | 0.1654427 |

29 | 0.0889773 (3S2M2) | 77 | 0.1954197 |

30 | 0.0892055 (2SK2M2) | 78 | 0.1967289 |

31 | 0.1134188 | 79 | 0.1967383 |

32 | 0.1162181 | 80 | 0.1969315 |

33 | 0.1164194 | 81 | 0.1982139 |

34 | 0.1166495 | 82 | 0.1982327 |

35 | 0.1177212 | 83 | 0.1982502 |

36 | 0.1177387 | 84 | 0.1984434 |

37 | 0.1179325 | 85 | 0.1995251 |

38 | 0.1179469 | 86 | 0.1997627 |

39 | 0.1179569 | 87 | 0.1997808 |

40 | 0.1181595 | 88 | 0.2010638 (NSO5) |

41 | 0.1192336 | 89 | 0.2012664 |

42 | 0.1192512 | 90 | 0.2012764 |

43 | 0.1194706 (2MP3) | 91 | 0.2014940 |

44 | 0.1207524 | 92 | 0.2025669 |

45 | 0.1220554 | 93 | 0.2027945 |

46 | 0.1222830 | 94 | 0.2030321 |

47 | 0.1225200 | 95 | 0.2040881 (NSK5) |

48 | 0.1235754 | 96 | 0.2043157 (3MQ5) |

49 | 0.1235891 | 97 | 0.2053974 (MSP5) |

50 | 0.1238042 (2MQ3) | 98 | 0.2071375 |

51 | 0.1238173 | 99 | 0.2083336 |

52 | 0.1250003 (S3) | 100 | 0.2330679 |

53 | 0.1251054 | 101 | 0.2341489 |

54 | 0.1251416 | 102 | 0.2343784 |

55 | 0.1253411 | 103 | 0.2345810 |

56 | 0.1491218 | 104 | 0.2356627 (5MKS6) |

57 | 0.1525569 | 105 | 0.2356883 (2(MN)S6) |

58 | 0.1538668 | 106 | 0.2358896 (5M2S6) |

59 | 0.1540694 | 107 | 0.2374021 (3MnuS6) |

60 | 0.1550736 | 108 | 0.2397937 (2MSNK6) |

61 | 0.1551498 | 109 | 0.2417632 (3MKS6) |

62 | 0.1553787 (4MS4) | 110 | 0.2430469 (4MN6) |

63 | 0.1566798 | 111 | 0.2456660 (2SN6) |

64 | 0.1568912 (2MnuS4) | 112 | 0.2486904 (2(MS)N6) |

65 | 0.1579747 (3MK4) | 113 | 0.2502285 |

66 | 0.1580078 | 114 | 0.2759307 |

67 | 0.1580279 | 115 | 0.2772406 |

68 | 0.1581923 | 116 | 0.2774425 |

117 | 0.2789562 | 161 | 0.3200571 |

118 | 0.2789813 | 162 | 0.3202022 |

119 | 0.2802649 (4MK7) | 163 | 0.3203047 (3MSNK8) |

120 | 0.2802774 | 164 | 0.3205248 |

121 | 0.2804925 | 165 | 0.3205423 |

122 | 0.2815754 | 166 | 0.3218172 (4MSK8) |

123 | 0.2818024 | 167 | 0.3222742 (4MKS8) |

124 | 0.2830273 | 168 | 0.3235585 |

125 | 0.2830785 | 169 | 0.3246389 (3M2SK8) |

126 | 0.2831992 | 170 | 0.3292020 (2SML8) |

127 | 0.2833055 | 171 | 0.3294302 (MSKL8) |

128 | 0.2846004 | 172 | 0.3564411 |

129 | 0.2846116 | 173 | 0.3566706 |

130 | 0.2846248 | 174 | 0.3577516 |

131 | 0.2848274 | 175 | 0.3579792 |

132 | 0.2848386 | 176 | 0.3592641 (3MNO9) |

133 | 0.2859090 | 177 | 0.3607678 |

134 | 0.2860222 | 178 | 0.3607766 |

135 | 0.2863636 | 179 | 0.3607866 |

136 | 0.2876491 | 180 | 0.3610042 |

137 | 0.2889590 | 181 | 0.3620865 |

138 | 0.2891866 | 182 | 0.3622009 |

139 | 0.2913243 | 183 | 0.3622991 |

140 | 0.2920096 | 184 | 0.3625166 |

141 | 0.31314500 | 185 | 0.3635896 |

142 | 0.31337693 | 186 | 0.3635983 |

143 | 0.3146618 | 187 | 0.3637128 |

144 | 0.3148894 | 188 | 0.3649082 |

145 | 0.3150939 | 189 | 0.3651108 |

146 | 0.3161987 (3M2NS8) | 190 | 0.3651239 |

147 | 0.3164000 | 191 | 0.3651358 |

148 | 0.3175092 | 192 | 0.3653390 |

149 | 0.3177024 | 193 | 0.3664201 |

150 | 0.3177112 (4MNS8) | 194 | 0.3665364 |

151 | 0.3179144 | 195 | 0.3668759 |

152 | 0.3189986 | 196 | 0.3679319 |

153 | 0.3190123 | 197 | 0.3681602 |

154 | 0.3190298 | 198 | 0.3694701 |

155 | 0.3190417 | 199 | 0.3696970 |

156 | 0.3190498 | 200 | 0.3749998 |

157 | 0.3191330 | 201 | 0.3752274 |

158 | 0.3192024 | 202 | 0.3938886 |

159 | 0.3192237 (5MS8) | 203 | 0.3951729 |

160 | 0.3192493(2(MN)KS8) | 204 | 0.3954011 |

205 | 0.3967110 | 249 | 0.4443364 |

206 | 0.3969130 | 250 | 0.4454193 |

207 | 0.3969392 | 251 | 0.4456225 |

208 | 0.3980203 | 252 | 0.4456463 |

209 | 0.3981953 | 253 | 0.4469324 |

210 | 0.3982229 (5MNS10) | 254 | 0.4472269 |

211 | 0.3984254 | 255 | 0.4484443 |

212 | 0.3995328 (3M2N10) | 256 | 0.4486719 |

213 | 0.3997347 | 257 | 0.4486981 |

214 | 0.4008170 (4MSNK10) | 258 | 0.4499818 |

215 | 0.4010365 | 259 | 0.4502087 |

216 | 0.4010540 | 260 | 0.4514943 |

217 | 0.4012478 (4Mnu10) | 261 | 0.4556691 |

218 | 0.4012728 | 262 | 0.4744016 |

219 | 0.4023295 (5MSK10) | 263 | 0.4759128 |

220 | 0.4023545 | 264 | 0.4787346 (6MNS12) |

221 | 0.4027584 | 265 | 0.4789372 |

222 | 0.4027859 | 266 | 0.4800451 |

223 | 0.4036388 | 267 | 0.4802471 (7MS12) |

224 | 0.4040952 (3MNK10) | 268 | 0.4813287 (5MSNK12) |

225 | 0.4051506 | 269 | 0.4815563 (5MN12) |

226 | 0.4056071 (4MK10) | 270 | 0.4828656 (3M2SN12) |

227 | 0.4066888 (2(MS)N10) | 271 | 0.4841511 |

228 | 0.4068913 | 272 | 0.4843781 (4MSN12) |

229 | 0.4084295 (3MSK10) | 273 | 0.4843875 |

230 | 0.4095111 | 274 | 0.4846063 (4MNK12) |

231 | 0.4097131 | 275 | 0.4856617 |

232 | 0.4097406 | 276 | 0.4858818 |

233 | 0.4099419 | 277 | 0.4858987 |

234 | 0.4110230 (3S2M10) | 278 | 0.4859168 |

235 | 0.4112512 (2(MS)K10) | 279 | 0.4861182 (5MK12) |

236 | 0.4125349 | 280 | 0.4872005 (3M2SN12) |

237 | 0.4127637 | 281 | 0.4874031 |

238 | 0.4371829 | 282 | 0.4887036 |

239 | 0.4376612 | 283 | 0.4887123 (4M2S12) |

240 | 0.4382633 | 284 | 0.4887211 |

241 | 0.4397758 | 285 | 0.4889406 (4MSK12) |

242 | 0.4400021 | 286 | 0.4900229 |

243 | 0.4410857 | 287 | 0.4902248 |

244 | 0.4412883 | 288 | 0.4904530 |

245 | 0.4412989 | 289 | 0.4915341 (3(MS)12) |

246 | 0.4425976 | 290 | 0.4917623 (3M2SK12) |

247 | 0.4428264 | 291 | 0.4930472 |

248 | 0.4441088 | 292 | 0.4932754 |

## 4 Summary and conclusions

The importance of the research carried out relies on the two goals we followed in the present contribution. First, we extracted the tidal frequencies using the univariate and multivariate harmonic analysis applied to the tidal time series. Second, the extracted frequencies were used to predict the tidal heights. We applied LS-HE to a single 1-year time series and a multivariate time series consisting of 318 1-year series. The data for these series were obtained from the 43 tide-gauge stations in UK. The spectra of univariate and multivariate series show that multivariate analysis is advantageous over the univariate analysis as it increases the detection power of the tidal frequencies. We then adopted the multivariate LS-HE method to a multivariate time series including eleven 18.24-year data series of the tide-gauge stations in UK and extracted 413 tidal frequencies by examination of their power spectrum. These frequencies included long-term frequencies and the 1st to 12th harmonics of the diurnal signals.

We investigated the importance of different tidal species in tidal observation structure. The only important frequencies belong to the long-term species to the 6th diurnal harmonic. The role of the frequencies between 7th diurnal species and the 12th diurnal species is negligible.

We also found that the tide prediction using the 413 extracted frequencies could nearly provide identical results with those provided based on the most important constituents (50 frequencies). This conclusion was also verified when different tidal time series were used over five different areas on the world.

When comparing the 4-week prediction with 8-month prediction, it follows that short-term prediction could provide superior results than long-term prediction. This is what we would expect as the precision of predicted results get worse when the time span increases.

The tidal constituents were made using the data set in UK. We note that if we use different data sets around the world, the power of the detected frequencies will change and some of the detected signals cannot be detected and new ones will appear. A more reliable list of detected frequencies will likely benefit from the tide-gauge data all around the world in which use is made of multiple time series distributed uniformly. However, although the amplitudes of signals are expected to change in a different area, the main contributing frequencies will likely be unchanged and only the order of their magnitudes will change. This can be expected because promising results were obtained over five different locations and using the list of 50 tidal frequencies. Further, we note that site effects such as small basins and river estuaries can also affect tide predictions. This was however not the subject of discussion in the present contribution.

## Notes

### Acknowledgments

We would like to acknowledge the British Oceanographic Data Center (BODC) for its free tide data we used in this paper. Useful comments of the editor-in-chief and anonymous reviewers are kindly acknowledged.

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