Journal of Geodesy

, Volume 88, Issue 10, pp 975–988 | Cite as

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

  • A. R. Amiri-Simkooei
  • S. Zaminpardaz
  • M. A. Sharifi
Original Article

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)

This section briefly reviews an approach—it is the least-squares harmonic estimation (LS-HE)—that extracts the tidal frequencies using tide-gauge records (tidal time series). LS-HE was originally introduced by Amiri-Simkooei (2007) and Amiri-Simkooei et al. (2007) in which its application to GPS position time series was investigated. The goal of LS-HE is to improve an existing linear model of observation equations as
$$\begin{aligned} E(y) = Ax , D(y) = Q_y \end{aligned}$$
(1)
where 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

Harmonic estimation (HE) is used to identify periodic patterns in the functional part of the model. For a given time series, the simplest structure may take into account just a trigonometric term \(y(t) = a_ k \cos \omega _{k} t + b_{k} sin \omega _{k} t\) which is a sinusoidal wave with an initial phase. Therefore, the functional model in Eq. (1) is extended to
$$\begin{aligned} E(y)=Ax + A_k x_{k} , D(y)=Q_{y} \end{aligned}$$
(2)
where the matrix \(A_{k}\) consists of two columns corresponding to the frequency \(\omega _{k}\) of the sinusoidal function. It is of the form
$$\begin{aligned} A_{k} =\left[ {\begin{array}{l@{\quad }l} \cos \omega _{k} t_{1} &{} \sin \omega _{k} t_{1} \\ \cos \omega _{k} t_{2} &{} \sin \omega _{k} t_{2} \\ \vdots &{} \vdots \\ \cos \omega _{k} t_{m} &{} \sin \omega _{k} t_{m} \\ \end{array}} \right] ,\quad x_{k} =\left[ {\begin{array}{l} a_{k} \\ b_{k} \\ \end{array}} \right] \end{aligned}$$
(3)
with \(a_{k} , b_{k} \) and \(\omega _{k}\) being unknown real numbers. The unknown frequency \(\omega _{k}\) in Eq. (3) is identified through the application of LS-HE to the time series. For this purpose, the following null and alternative hypotheses are put forward:
$$\begin{aligned} H_{0}: E(y) = Ax \end{aligned}$$
(4)
versus
$$\begin{aligned} H_{a} : E(y) = Ax + A_{k} x_{k} \end{aligned}$$
(5)
The identification of the frequency \(\omega _{k} \) is completed through the following maximization problem (Amiri-Simkooei et al. 2007):
$$\begin{aligned} \omega _{k} = \mathop {argmax}\limits _{\omega _{j} } P(\omega _{j} ) \end{aligned}$$
(6)
where
$$\begin{aligned} P(\omega _{j}) = {\hat{e}}_{0}^{T} Q_{y}^{-1} A_{j} (A_{j}^{T} Q_{y}^{-1} P_{A}^\bot A_{j} )^{-1} A_{j}^{T} Q_{y}^{-1} {\hat{e}}_{0} \end{aligned}$$
(7)
with \({\hat{e}}_{0} = P_{A}^\bot y\), the least-squares residuals and \(P_{A}^\bot = I-A(A^{{T}} Q_{y}^{-1} A)^{-1} A^{{T}} Q_{y}^{-1} \) an orthogonal projector (Teunissen 2000b); both are given under the null hypothesis. The matrix \(A_{j} \) has the same structure as \(A_{k} \) in Eq. (3); the one that makes \(P(\omega _{j} )\) maximum is set to be \(A_{k} \). Analytical evaluation of this maximization problem is complicated due to the existence of many local maxima. A plot of spectral values \(P(\omega _{j} )\) versus a set of discrete values for \(\omega _{j} \) is used to investigate the contribution of different frequencies in the construction of the original signal. To choose the discrete values of \(\omega _{j} \) for constructing the above-mentioned plot, we refer to Amiri-Simkooei and Tiberius (2007) in which a recursive formula was proposed. The frequency that maximizes \(P(\omega _{j} )\) is considered to be \(\omega _{k} \) from which matrix \(A_{k} \) is constructed using Eq. (3).
The hypothesis \(H_{0}\) should be tested against \(H_{a} \) to see whether or not the spectrum at the detected frequency \(\omega _{k} \) is indeed significant. The test statistic used is:
$$\begin{aligned} T_2 = {\hat{e}}_{0}^{T} Q_{y}^{-1} A_{k} (A_{k}^{T} Q_{y}^{-1} P_{A}^\bot A_{k} )^{-1} A_{k}^{T} Q_{y}^{-1} {\hat{e}}_{0} \end{aligned}$$
(8)
If \(Q_{y} \) is known, the test statistic has a central Chi-square distribution with two degrees of freedom under \(H_{0} \), i.e., \(T_2 \sim {\chi }^{2}(2,0)\) (Teunissen 2000a). On the other hand, if the covariance matrix has the form \(Q_{y} =\sigma ^{2}Q\), with \(\sigma ^{2}\) denoting the unknown variance of the unit weight, the test statistic is of the form (Teunissen et al. 2005)
$$\begin{aligned} T_2 = \frac{{\hat{e}}_{0}^{T} Q^{-1}A_{k} (A_{k}^{T} Q^{-1}P_{ A}^\bot A_{k} )^{-1}A_{k}^{T} Q^{-1} {\hat{e}}_{0} }{2{\hat{{\upsigma }}}_{a}^2 } \end{aligned}$$
(9)
where the estimator for the variance, \({\hat{{\upsigma }}}_{a}^2 \), has to be computed under the alternative hypothesis. Under \(H_{0} \), the test statistic in Eq. (9) has a central Fisher distribution, i.e., \(T_2 \sim F(2,m-n-2k)\). If the null hypothesis is rejected, we can perform the same procedure for finding yet other frequencies.

2.2 Multivariate harmonic estimation

If in a linear model, instead of one time series, there exist several (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:
$$\begin{aligned} \begin{aligned}&E(\mathrm{vec}(Y)) = (I_{r} \otimes A) \mathrm{vec}(X) +(I_{r} \otimes A_{k} ) \mathrm{vec}(X_{k} ) ,\\&D(\mathrm{vec}(Y)) = \Sigma \otimes Q \end{aligned} \end{aligned}$$
(10)
where vec and \(\otimes \) are the vec-operator and the Kronecker product, respectively. For more information about the properties of vec-operator and Kronecker product, we refer to Magnus (1988) and Amiri-Simkooei (2007). The \(m\times r\) matrix \(Y=[y_{1} y_{2} \ldots y_{r} ]\)includes observations from the 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.
The power spectrum of the multivariate model has the following form: (Amiri-Simkooei and Asgari 2012)
$$\begin{aligned} P(\omega _{j} ) = \mathrm{tr}({\hat{E}}^{{T}}Q^{-1}A_{j} (A_{j}^{T} Q^{-1}P_{A}^\bot A_{j} )^{-1}A_{j}^{T} Q^{-1}{\hat{E}}\Sigma ^{ -1}) \end{aligned}$$
(11)
with \({\hat{E}} = P_{A}^\bot Y\) the least-squares residuals of the 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:
$$\begin{aligned} T_2 = \mathrm{tr}({\hat{E}}^{{T}}Q^{-1}A_{k} (A_{k}^{T} Q^{-1}P_{A}^\bot A_{k} )^{-1}A_{k}^{T} Q^{-1}{\hat{E}}\Sigma ^{ -1})\nonumber \\ \end{aligned}$$
(12)
which under the null hypothesis has a central Chi-square distribution with \(2r\) degrees of freedom, i.e., Open image in new window provided that both \(\Sigma \) and Q are known and that the original observables are normally distributed.

3 Numerical results and discussions

Tide height time series are the data used in the present contribution, which are provided from 45 tide-gauge stations in UK. The UK tide-gauge stations consist of data that span the time domain from 1 January 1993 to 31 March 2011. The stations positions are listed in Appendix A and illustrated in Fig. 1. Two multivariate data sets are made of these time series. We make a comparison between the univariate and multivariate analysis, and extract tidal frequencies from a multivariate time series using the LS-HE method.
Fig. 1

UK tide-gauge stations

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 discrete frequencies at which the power spectrum in Eqs. (7) and (11) are evaluated can be derived using a recursive relation (Amiri-Simkooei and Tiberius 2007):
$$\begin{aligned} T_{{j+1}} =T_{j} (1+\alpha T_{j} /T),\quad \alpha =0.1,\quad j=1,2,\ldots \end{aligned}$$
(13)
with a starting period of \(T_{1} =30\ \mathrm{min}\) (Nyquist period) and T being the total time span (1 year or 18.24 years). Because of using long time series with high sampling rate, i.e., 15 min, the starting period \(T_{1} \) is selected as 2 h. The step size used for periods, \(T_{j} {=2\pi /\omega }_{j} \), is small at high frequencies and gets larger at lower frequencies. The lowest frequency that we checked is \(\omega _{\mathrm{min}} =2\pi /T\), i.e., one cycle over the total time span. It is worthwhile mentioning that for calculating the power spectra, a linear regression model was considered to make the design matrix A under the null hypothesis.
We applied LS-HE to a univariate (one time series) and multivariate (318 time series) 1-year time series to make a comparison between the two analyses. The power spectra for the univariate and multivariate analyses are shown in Fig. 2 [using Eq. (7)] and Fig. 3 [using Eq. (11)], respectively. The spectra are separated into tidal species with the central frequencies of \(n/24\ \mathrm{h}, n=1,2,\ldots ,12\), i.e., there exist many periodic patterns with periods around \(24\ \mathrm{h}/n, n=1,2,\ldots ,12\). Higher harmonics could similarly be seen by choosing \(T_{1} \) = 30 min (Nyquist period) and by increasing the sampling rate. The spectra in Figs. 2 and 3 are in fact very similar. The main difference is indeed in the power of the detected frequencies. In multivariate analysis the common-mode frequencies, contributed to the tide structure, are more obvious and hence they can be easily detected. Fig. 4a, b shows a zoom-in of the ter-diurnal signal of Figs. 2 and 3, respectively. As it can be seen, the multivariate spectrum has higher detection power compared to the univariate spectrum; more frequencies can then be detected.
Fig. 2

Univariate least-squares power spectrum of a (single) 1-year tide height time series with sampling rate of 15 min provided from tide-gauge station at North Shields in UK. The dashed lines indicate period range of diurnal signal and its higher harmonics

Fig. 3

Multivariate least-squares power spectrum of 318 1-year tide height time series with sample rate of 15 min provided from 43 tide-gauge stations in UK. The dashed lines indicate period range of diurnal signal and its higher harmonics

Fig. 4

Zoom-in of ter-diurnal signal in least-squares power spectrum of a a single 1-year time series, b 318 1-year time series

The multivariate LS-HE method is now applied to the eleven 18.24-year time series (second multivariate data set) to obtain the power spectrum of the series and consequently to extract the tidal frequencies. Longer time series (e.g., 18.24 years vs. 1 year) make the possibility to see closer frequencies separated in the spectrum, which may be merged due to the leakage problem when dealing with short time series. Again, the multivariate harmonic estimation (using 11 time series) increases the detection power of the common-mode signals. Figure 5 shows the multivariate least-squares power spectrum of these series, while Fig. 6 shows a zoom-in of the diurnal signals along with their higher harmonics. A wide spectrum of signals ranging from long-term annual signals to the diurnal signals and their higher harmonics can be observed in the spectrum. The spectrum is separated into tidal species with the central frequencies of \(n/24\ \mathrm{h}, n=1,2,\ldots ,12\). There exist series of peaks close to the diurnal signal and its higher harmonics. These are the main tidal constituents.
Fig. 5

Multivariate least-squares power spectrum of eleven 18.24-year tide height time series with sampling rate of 15 min provided from 11 tide-gauge stations in UK

Fig. 6

Zoom-in of diurnal and its higher harmonics in multivariate least-squares power spectrum of eleven 18.24-year time series

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 456). 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

In this section, using different lists of frequencies, tidal heights are predicted and the results are compared with the available data. For tidal prediction, Eq. (2) is used to estimate the coefficients \(a_{k}\) and \(b_{k}\) for a given frequency. These two coefficients can be used to obtain both the amplitude and phase of the sinusoidal function: \(\hbox {amplitude}= \sqrt{(a_{k}^{2} +b_{k}^{2} )}, \hbox {phase}=\hbox {arctan}({b_{k} }/{a_{k} })\). For each frequency list, the coefficients \(a_{k}\)’s and \(b_{k}\)’s are estimated by the least-squares method. The estimated coefficients are then used to predict the tidal heights for time instants outside the time span of the available data. The frequency lists are: CTE, the 1st list; frequencies in Tables 4 and 5, the 2nd list; frequencies in Table 6, the 3rd list; and CTE plus frequencies in Tables 4 and 5 that lie between 4th and 6th harmonics of diurnal signals, the 4th list. Using these frequency lists, tidal heights were separately predicted and then the residual vectors (differences between observations and predicted data) along with their standard deviations in each station were computed. Two kinds of tide predictions were performed in each of the 11 stations, (1) 4 weeks for time interval 2011/4/1 to 2011/4/28, and (2) 8 months for time interval 2011/4/1 to 2011/12/1. The mean standard deviations are listed in Tables 1 and 2, respectively. Based on these values, we did several investigations.
Table 1

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

1st list: CTE, 2nd list: Tables 4 and 5, 3rd list: Table 6, and 4th list: CTE + those in Tables 4 and 5 that lie between 4th and 6th harmonics of diurnal signals

Table 2

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

1st list: CTE, 2nd list: Tables 4 and 5, 3rd list: Table 6, and 4th list: CTE + those in Tables 4 and 5 that lie between 4th and 6th harmonics of diurnal signals

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.

The tide predictions are based on four frequency lists mentioned above. The results are presented in Table 3. A few observations can be highlighted from these results. (1) In all cases, the tide prediction based on the frequencies extracted in this contribution yields better results in comparison with the outcome of the CTE list (first list). (2) The results of the CTE list and those of CTE plus frequencies in Tables 4 and 5 that lie between 4th and 6th harmonics of diurnal signals (4th list) are identical. This is however not the case for the prediction results based on long time series in Tables 1 and 2. This indicates that when using short time series (1-year series) for tide prediction, the higher frequencies than the third harmonic of the diurnal signals have no significant effect on tidal prediction. This is likely due to the presence of colored noise in tidal data. When using long time series (18.24 years) for tidal prediction, because we do not model the colored noise (white noise model is used), the role of higher frequencies become significant in comparison with the low and medium ones. This indicates that when one uses shorter series (1 year) for tidal prediction, the contribution of high frequency signals decreases. (3) The results of the 2nd (with 413 frequencies) and 3rd (with 50 frequencies) lists are nearly identical. This indicates that the 50 tidal constituents of Table 6 are in fact an important list that presents a shorter list of tidal constituents and meets the requirements for many practical applications.
Table 3

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

Table 4

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

   
Table 5

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.

After extracting the frequencies, using different frequency lists, tidal heights were predicted. These frequency lists were: CTE, the 1st list, frequencies in Tables 4 and 5, the 2nd list, frequencies in Table 6, the 3rd list, CTE + frequencies in Tables 4 and 5 that lie between 4th and 6th harmonics of diurnal signals, the 4th list. Using the frequency lists mentioned above, tidal heights were separately predicted for 4 weeks and 8 months for each of the 11 stations for the time intervals 2011/4/1 to 2011/4/28 and 2011/4/1 to 2011/12/1, respectively. Then the prediction error (difference between observed and predicted data) and their mean standard deviations in each station were computed. Based on these values, we could have the following conclusions:
  • 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.

References

  1. Amiri-Simkooei AR (2007) Least-squares variance component estimation: theory and GPS applications. Ph.D. thesis, Mathematical Geodesy and Positioning, Faculty of Aerospace Engineering, Delft University of Technology, DelftGoogle Scholar
  2. Amiri-Simkooei AR, Tiberius CCJM, Teunissen PJG (2007) Assessment of noise in GPS coordinate time series: methodology and results. J Geophys Res 112:B07413. doi:10.1029/2006JB004913 Google Scholar
  3. Amiri-Simkooei AR, Tiberius CCJM (2007) Assessing receiver noise using GPS short baseline time series. GPS Solut 11(1):21–35CrossRefGoogle Scholar
  4. Amiri-Simkooei AR (2009) Noise in multivariate GPS position time series. J Geod Berlin 83:175–187CrossRefGoogle Scholar
  5. Amiri-Simkooei AR, Asgari J (2012) Harmonic analysis of total electron contents time series: methodology and results. GPS Solut 16(1):77–88CrossRefGoogle Scholar
  6. Büllesfeld FJ (1985) Ein Beitrag zur harmonischenDarstellung des gezeitenerzeugenden Potentials. Reihe C, Heft 314, Deutsche GeodätischeKommission, MünchenGoogle Scholar
  7. Capuano P, De Lauro E, De Martino S, Falanga M (2011) Waterlevel oscillations in the Adriatic Sea as coherent selfoscillations inferred by independent component analysis. Progr Oceanogr 91:447460CrossRefGoogle Scholar
  8. Cartwright DE, Edden AC (1973) Corrected tables of tidal harmonics. Geophys J R Astron Soc 33:253–264CrossRefGoogle Scholar
  9. Cartwright DE, Tayler RJ (1971) New computation of the tide generating potential. Geophys J R AstronSoc 23:45–74CrossRefGoogle Scholar
  10. Doodson AT (1921) The harmonic development of the tide generating potential. Proc R Soc Lond A 100:305–329CrossRefGoogle Scholar
  11. Doodson AT (1954) Re-print of above, with minor corrections, same title. Znt Hydrog Rev 31:11–35Google Scholar
  12. Ducarme B, Venedilkov AP, de Mesquita AR, Costa DS, Blitzkow D, Vieira R, Freitas SRC (2006a) New analysis of a 50 years tide guage record at Cananéia (SP-Brazil) with the VAV tidal analysis program. Dynamic Planet, Cairns, Australia, 22–26 August, 2005. Springer, IAG Symposia 130:453–460Google Scholar
  13. Ducarme B, Venedikov AP, Arnoso J, Vieira R (2006b) Analysis and prediction of ocean tides by the computer program VAV. In: Proceedings of 15th international symposium on earth tides, Journal of Geodynamics 41:119–127Google Scholar
  14. Flinchem EP, Jay DA (2000) An introduction to wavelet transform tidal analysis methods. Estuar Coast Shelf Sci 51:177200CrossRefGoogle Scholar
  15. Hartmann T, Wenzel HG (1994) The harmonic development of the earth tide generating potential due to the direct effect of the planets. J Geophys Res Lett 21:1991–1993CrossRefGoogle Scholar
  16. Hartmann T, Wenzel HG (1995) The HW95 tidal potential catalogue. J Geophys Res Lett 22:3553–3556CrossRefGoogle Scholar
  17. Hyvarinen A, Karhunen J, Oja E (2001) Independent component analysis. Wiley, New YorkCrossRefGoogle Scholar
  18. Jay DA, Kukulka J (2003) Revising the paradigm of tidal analysis the uses of nonstationary. Ocean Dyn 53:110125CrossRefGoogle Scholar
  19. Kudryavtsev SM (2004) Improved harmonic development of the earth tide-generating potential. J Geod 77:829–838CrossRefGoogle Scholar
  20. Magnus JR (1988) Linear structures. Oxford University Press, London School of Economics and Political Science, Charles Griffin & Company LTD, LondonGoogle Scholar
  21. Mousavian R, Mashhadi-Hossainali M (2012) Detection of main tidal frequencies using least squares harmonic estimation method. J Geod Sci 2(3):224–233Google Scholar
  22. Mousavian R, Mashhadi-Hossainali M (2013) Geodetic constraints on the period of episodic tremors and slips using least squares harmonic estimation method with application to cascadia subduction zone. Acta Geophys. doi:10.2478/s11600-013-0173-6
  23. Pytharouli S, Stiros S (2008) Spectral analysis of unevenly spaced or discontinuous data using the ‘normperiod’ code’. Comput Struct 86:190–196Google Scholar
  24. Pytharouli S, Stiros S (2012) Analysis of short and discontinuous tidal data: a case study from the Aegean Sea. Surv Rev. doi:10.1179/1752270611Y.0000000035
  25. Roosbeek F (1996) RATGP95: a harmonic development of the tide-generating potential using an analytical method. Geophys J Int 126:197–204Google Scholar
  26. Sakamoto Y, Ishiguro M, Kitagawa G (1986) Akaike information criterion statistics. D. Reidel Publishing Company, Tokyo 290 ppGoogle Scholar
  27. Sharifi MA, Sam Khaniani A (2013) Least-squares harmonic estimation of the tropopause parameters using GPS radio occultation measurements. Meteorol Atmos Phys 120(1–2):73–82CrossRefGoogle Scholar
  28. Sharifi MA, Safari A, Masoumi S (2012) Harmonic analysis of the ionospheric electron densities retrieved from FORMOSAT-3/COSMIC radio occultation measurements. Adv Space Res 49(10):1520–1528 Google Scholar
  29. Tamura Y (1987) A harmonic development of the tide-generating potential. Bull Inf Mar Terrest 99:6813–6855Google Scholar
  30. Tamura Y (1995) Additional terms to the tidal harmonic tables. In: Hsu HT (ed) Proceedings of 12th international aymposium earth tides. Science Press, Beijing, pp 345–350Google Scholar
  31. Teunissen PJG (2000a) Testing theory: an introduction. Series on Mathematical Geodesy and Positioning. Delft University Press. http://www.vssd.nl
  32. Teunissen PJG (2000b) Adjustment theory: an introduction. Series on Mathematical Geodesy and Positioning, Delft University Press. http://www.vssd.nl
  33. Teunissen PJG, Amiri-Simkooei AR (2008) Least-squares variance component estimation. J Geod 82(2):65–82. doi:10.1007/s00190-007-0157-x
  34. Teunissen PJG, Simons DG, Tiberius CCJM (2005) Probability and observation theory, Faculty of Aerospace Engineering, Delft University, Delft University of Technology (lecture notes AE2-E01)Google Scholar
  35. Venedikov AP, Arnoso J, Vieira R (2001) Program VAV/2000 for tidal analysis of unevenly spaced data with irregular drift and colored noise. J Geodetic Soc Jpn 47(1):281–286Google Scholar
  36. Venedikov AP, Arnoso J, Vieira R (2003) VAV: a program for tidal data processing. Comput Geosci 29:487–502CrossRefGoogle Scholar
  37. Venedikov AP, Arnoso J, Vieira R (2005) New version of the program VAV for tidal data processing. Comput Geosci 31:667–669CrossRefGoogle Scholar
  38. Xi QW (1987) A new complete development of the tide-generating potential for the epoch J2000.0. Bull Inf Mar Terrest 99:6766–6812Google Scholar
  39. Xi QW (1989) The precision of the development of the tidal generating potential and some explanatory notes. Bull Inf Mar Terrest 105:7396–7404Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2014

Authors and Affiliations

  • A. R. Amiri-Simkooei
    • 1
    • 2
  • S. Zaminpardaz
    • 3
  • M. A. Sharifi
    • 3
  1. 1.Section of Geodesy, Department of Surveying Engineering, Faculty of EngineeringUniversity of IsfahanIsfahanIran
  2. 2.Chair Acoustics, Faculty of Aerospace EngineeringDelft University of TechnologyDelftThe Netherlands
  3. 3.Geodesy Division, Department of Surveying and Geomatics Engineering, Faculty of EngineeringUniversity of TehranTehranIran

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