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J Geod (2014) 88:975–988 DOI 10.1007/s00190-014-0737-5 ORIGINAL ARTICLE Extracting tidal frequencies using multivariate harmonic analysis of sea level height time series A. R. Amiri-Simkooei · S. Zaminpardaz · M. A. Sharifi Received: 4 December 2013 / Accepted: 28 May 2014 / Published online: 29 June 2014 © Springer-Verlag Berlin Heidelberg 2014 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 observa- tions and extract the main tidal constituents for which effi- cient multivariate methods are to be developed. This contri- bution applies the multivariate least-squares harmonic esti- mation (LS-HE) to the tidal time series of the UK tide- gauge stations. The first 413 harmonics of the tidal con- stituents and their nonlinear components are provided using the multivariate LS-HE. A few observations of the research are highlighted: (1) the multivariate analysis takes informa- tion of multiple time series into account in an optimal least- squares sense, and thus the tidal frequencies have higher A. R. Amiri-Simkooei Section of Geodesy, Department of Surveying Engineering, Faculty of Engineering, University of Isfahan, 81746-73441 Isfahan, Iran A. R. Amiri-Simkooei Chair Acoustics, Faculty of Aerospace Engineering, Delft University of Technology, Kluyverweg 1, 2629 HS Delft, The Netherlands S. Zaminpardaz (B ) · M. A. Sharifi Geodesy Division, Department of Surveying and Geomatics Engineering, Faculty of Engineering, University of Tehran, North-Kargar Ave., Amir-Abad, 1439-5515 Tehran, Iran e-mail: [email protected] detection power compared to the univariate analysis. (2) Dominant tidal frequencies range from the long-term sig- nals to the sixth-diurnal species interval. Higher frequencies have negligible effects. (3) The most important tidal con- stituents (the first 50 frequencies) ordered from their ampli- tudes 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 frequen- cies on five different data sets around the world are com- pared. 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 phenom- enon, 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 frequen- cies. 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 Kudryavt- 123 Downloaded from http://www.elearnica.ir
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  • J Geod (2014) 88:975–988DOI 10.1007/s00190-014-0737-5

    ORIGINAL ARTICLE

    Extracting tidal frequencies using multivariate harmonic analysisof sea level height time series

    A. R. Amiri-Simkooei · S. Zaminpardaz · M. A. Sharifi

    Received: 4 December 2013 / Accepted: 28 May 2014 / Published online: 29 June 2014© Springer-Verlag Berlin Heidelberg 2014

    Abstract This contribution is seen as a first attempt toextract the tidal frequencies using a multivariate spectralanalysis method applied to multiple time series of tide-gaugerecords. The existing methods are either physics-based inwhich the ephemeris of Moon, Sun and other planets areused, or are observation-based in which univariate analysismethods—Fourier and wavelet for instance—are applied totidal observations. The existence of many long tide-gaugerecords around the world allows one to use tidal observa-tions and extract the main tidal constituents for which effi-cient multivariate methods are to be developed. This contri-bution applies the multivariate least-squares harmonic esti-mation (LS-HE) to the tidal time series of the UK tide-gauge stations. The first 413 harmonics of the tidal con-stituents and their nonlinear components are provided usingthe multivariate LS-HE. A few observations of the researchare highlighted: (1) the multivariate analysis takes informa-tion of multiple time series into account in an optimal least-squares sense, and thus the tidal frequencies have higher

    A. R. Amiri-SimkooeiSection of Geodesy, Department of Surveying Engineering,Faculty of Engineering, University of Isfahan, 81746-73441Isfahan, Iran

    A. R. Amiri-SimkooeiChair Acoustics, Faculty of Aerospace Engineering,Delft University of Technology, Kluyverweg 1, 2629 HS Delft,The Netherlands

    S. Zaminpardaz (B) · M. A. SharifiGeodesy Division, Department of Surveying andGeomatics Engineering, Faculty of Engineering,University of Tehran, North-Kargar Ave., Amir-Abad,1439-5515 Tehran, Irane-mail: [email protected]

    detection power compared to the univariate analysis. (2)Dominant tidal frequencies range from the long-term sig-nals to the sixth-diurnal species interval. Higher frequencieshave negligible effects. (3) The most important tidal con-stituents (the first 50 frequencies) ordered from their ampli-tudes range from 212 cm (M2) to 1 cm (OQ2) for the dataset considered. There are signals in this list that are notavailable in the 145 main tidal frequencies of the literature.(4) Tide predictions using different lists of tidal frequen-cies on five different data sets around the world are com-pared. The prediction results using the first significant 50constituents provided promising results on these locations ofthe 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 importantissue for different applications such as safe navigation andhydrographic surveys. Because the tide is a periodic phenom-enon, it can be modeled by a series of periodic functions suchas sinusoidal ones. A reliable tidal analysis and predictionrequires a reliable knowledge on the (main) tidal frequen-cies. Different tidal frequencies have been listed by manyresearchers based on the tidal theory. They usually expand thetide generating potential harmonically using major planets(e.g., Moon and Sun) ephemeris through different methods.We may at least refer to Doodson (1921, 1954), Cartwrightand Tayler (1971), Cartwright and Edden (1973), Büllesfeld(1985), Xi (1987, 1989), Tamura (1987, 1995); Hartmannand Wenzel (1994, 1995), Roosbeek (1996) and Kudryavt-

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  • 976 A. R. Amiri-Simkooei et al.

    sev (2004). These studies are all physics-based becauseno tidal observations were used. The methods assume thatthe tidal frequencies are known but their amplitudes areunknown.

    To extract the tidal frequencies, many studies have ana-lyzed sea level height with different methods such as theFourier and wavelet. Flinchem and Jay (2000) and Jay andKukulka (2003) considered tide times series to be non-stationary and introduced the continuous wavelet transform(CWT) method, a complementary to harmonic analysis andFourier methods, to extract tidal information. Ducarme et al.(2006a) used a method based on the maximum likelihood—itis the Akaike Information Criterion (AIC) method (Sakamotoet al. 1986)—to find non-tidal components in tidal residuesobtained from reduction of all estimated tides throughthe program VAV (Venedikov et al. 2001, 2003, 2005).Pytharouli and Stiros (2012) applied spectral analysis tothe time series of the astronomical tide (smoothed tide timeseries) based on the NormPeriod code (Pytharouli and Stiros2008). Capuano et al. (2011) adopted independent compo-nent analysis (ICA) (Hyvarinen et al. 2001) to obtain nonlin-ear independent tidal constituents.

    As an observation-based method, this study is also basedon tidal observations of which the frequencies, the ampli-tudes, and the phases are assumed unknown to be estimated.We aim to estimate the tidal frequencies based on a math-ematical and statistical approach, namely the least-squaresharmonic estimation (LS-HE) developed by Amiri-Simkooeiet al. (2007) and Amiri-Simkooei (2007). LS-HE has alreadybeen used by Mousavian and Mashhadi-Hossainali (2012)to extract the tidal frequencies in which at most only 17tidal constituents were extracted using a univariate analy-sis. We now apply the multivariate formulation of LS-HEdeveloped by Amiri-Simkooei and Asgari (2012) to multipletime series. After successful applications of LS-HE and itsmultivariate formulation to many GNSS data series such asAmiri-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 andMashhadi-Hossainali (2013) in which different periodic pat-terns were identified in the GNSS series, we now consideranother application of LS-HE in the geodetic community,namely, tidal time series analysis. The time series employedin this contribution concern the tide-gauge records (heights)obtained from the UK tide-gauge stations sampled at the rateof 15 min.

    Apart from the above-mentioned methods such as theFourier method, wavelet method and VAV—they are all uni-variate analysis methods—this contribution presents the mul-tivariate LS-HE method. An important advantage of thismethod over the observation-based and/or the physics-basedmethods is that it enables one to detect the main shallowwater

    tidal constituents (non-linear tides) using common-mode sig-nals in multiple series. In this study, we managed to find aconsiderable number of major (Appendix B, Table 4) andminor (Appendix B, Table 5) shallow water components, andinvestigated their effects on tidal predictions.

    The available tide-gauge data sets over the last centuryaround the world make valuable data that need to be prop-erly processed in the presence of modern computing tech-niques. In addition, proper analysis methods are to be devel-oped for estimating the tidal frequencies and amplitudes inan appropriate manner. As an efficient method, this contribu-tion uses the LS-HE method in its univariate and multivariateformulation. Also, a comparison is made between the resultsof this contribution and those obtained using the physics-based methods. It is also worth mentioning that neither of thephysics-based nor the observation-based methods can pro-vide the exact (error-free) tide predictions. There is alwaysa difference (prediction error) between the predicted valuesand the observed tide heights. The meteorological effects arethe 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 ofthe instantaneous tide-gauge records, which is the subject forfurther research.

    The objectives of the present contribution may be sum-marized as follows: (1) we introduce a powerful method toextract tidal frequencies based on the many available tidalobservations around the world. The multivariate LS-HE usescommon-mode signals to extract such frequencies. LS-HEis neither limited to evenly-spaced data nor to integer fre-quencies. Further, current spectral analysis methods (e.g.,Fourier method) cannot incorporate common signals of mul-tiple series. (2) Using tide-gauge records we present a com-plete list of tidal frequencies. We then present a list of maintidal constituents (50 frequencies) of which promising resultson tide prediction can be obtained in different locations ofthe world. (3) The effect of different tidal species on tide pre-diction is then investigated. The only important frequenciesbelong to the long-term species to the 6th diurnal harmonic.Signals with higher frequencies have negligible effect on tideprediction.

    This paper is organized as follows: we review the LS-HE theory and its multivariate formulation in Sect. 2. Asan observation-based method, the LS-HE is then applied tothe univariate and multivariate tidal time series in Sect. 3. Acomparison is made between the univariate and multivariateanalysis and the tidal frequencies are extracted. Tide predic-tion is performed using different lists of tidal constituentsin five different locations around the world so as to makeseveral comparisons. Finally, we make some conclusions inSect. 4.

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  • Extracting tidal frequencies using multivariate harmonic analysis 977

    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 tidalfrequencies using tide-gauge records (tidal time series). LS-HE was originally introduced by Amiri-Simkooei (2007) andAmiri-Simkooei et al. (2007) in which its application to GPSposition time series was investigated. The goal of LS-HE isto improve an existing linear model of observation equationsas

    E(y) = Ax, D(y) = Qy (1)

    where E and D, are the expectation and dispersion operators,respectively, A is the m × n design matrix, Qy is the m × mcovariance matrix of the m-vector of observables y, and x isthe n-vector of unknown parameters.

    The LS-HE method determines an appropriate designmatrix A for the functional model through the parametersignificance testing (Teunissen 2000a). In fact, this methodidentifies periodic patterns in terms of harmonic functions inthe functional part of the model and thus improves it. Toincrease the detection power of the tidal frequencies, weuse the multivariate formulation of the LS-HE developedby Amiri-Simkooei and Asgari (2012). We aim to detectthe common-mode frequencies of multiple tide-height timeseries taken from various tide-gauge stations in UK. As a gen-eralization of the Fourier spectral analysis, LS-HE is neitherlimited to evenly-spaced data nor to integer frequencies.

    2.1 Univariate harmonic estimation

    Harmonic estimation (HE) is used to identify periodic pat-terns in the functional part of the model. For a given timeseries, the simplest structure may take into account just atrigonometric term y(t) = ak cos ωk t + bksinωk t which isa sinusoidal wave with an initial phase. Therefore, the func-tional model in Eq. (1) is extended to

    E(y) = Ax + Ak xk, D(y) = Qy (2)

    where the matrix Ak consists of two columns correspondingto the frequency ωk of the sinusoidal function. It is of theform

    Ak =

    ⎡⎢⎢⎢⎣

    cos ωk t1 sin ωk t1cos ωk t2 sin ωk t2...

    ...

    cos ωk tm sin ωk tm

    ⎤⎥⎥⎥⎦ , xk =

    [akbk

    ](3)

    with ak, bk and ωk being unknown real numbers. Theunknown frequency ωk in Eq. (3) is identified through the

    application of LS-HE to the time series. For this purpose, thefollowing null and alternative hypotheses are put forward:

    H0 : E(y) = Ax (4)

    versus

    Ha : E(y) = Ax + Ak xk (5)

    The identification of the frequency ωk is completed throughthe following maximization problem (Amiri-Simkooei et al.2007):

    ωk = argmaxω j

    P(ω j ) (6)

    where

    P(ω j ) = êT0 Q−1y A j (ATj Q−1y P⊥A A j )−1 ATj Q−1y ê0 (7)

    with ê0 = P⊥A y, the least-squares residuals and P⊥A =I − A(AT Q−1y A)−1 AT Q−1y an orthogonal projector (Teu-nissen 2000b); both are given under the null hypothesis. Thematrix A j has the same structure as Ak in Eq. (3); the one thatmakes P(ω j ) maximum is set to be Ak . Analytical evaluationof this maximization problem is complicated due to the exis-tence of many local maxima. A plot of spectral values P(ω j )versus a set of discrete values for ω j is used to investigatethe contribution of different frequencies in the constructionof the original signal. To choose the discrete values of ω j forconstructing the above-mentioned plot, we refer to Amiri-Simkooei and Tiberius (2007) in which a recursive formulawas proposed. The frequency that maximizes P(ω j ) is con-sidered to be ωk from which matrix Ak is constructed usingEq. (3).

    The hypothesis H0 should be tested against Ha to seewhether or not the spectrum at the detected frequency ωk isindeed significant. The test statistic used is:

    T2 = êT0 Q−1y Ak(ATk Q−1y P⊥A Ak)−1 ATk Q−1y ê0 (8)

    If Qy is known, the test statistic has a central Chi-squaredistribution with two degrees of freedom under H0, i.e.,T2 ∼ χ2(2, 0) (Teunissen 2000a). On the other hand, if thecovariance matrix has the form Qy = σ 2 Q, with σ 2 denot-ing the unknown variance of the unit weight, the test statisticis of the form (Teunissen et al. 2005)

    T2 = êT0 Q

    −1 Ak(ATk Q−1 P⊥A Ak)−1 ATk Q−1ê02σ̂2a

    (9)

    where the estimator for the variance, σ̂2a , has to be computedunder the alternative hypothesis. Under H0, the test statistic in

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  • 978 A. R. Amiri-Simkooei et al.

    Eq. (9) has a central Fisher distribution, i.e., T2 ∼ F(2, m −n − 2k). If the null hypothesis is rejected, we can performthe same procedure for finding yet other frequencies.

    2.2 Multivariate harmonic estimation

    If in a linear model, instead of one time series, there existseveral (r) time series for which the design matrix A andthe covariance matrix Qy are the same, then the model isreferred to as a multivariate linear model (Amiri-Simkooei2007, 2009). For a multivariate model, Eq. (2) is generalizedto:

    E(vec(Y )) = (Ir ⊗ A)vec(X) + (Ir ⊗ Ak)vec(Xk),D(vec(Y )) = � ⊗ Q (10)

    where vec and ⊗ are the vec-operator and the Kroneckerproduct, respectively. For more information about the prop-erties of vec-operator and Kronecker product, we refer toMagnus (1988) and Amiri-Simkooei (2007). The m × rmatrix Y = [y1 y2 . . . yr ]includes observations from the rseries; so do the n × r matrices X = [x1x2 . . . xr ] andXk = [x1k x2k . . . xrk] for the unknown parameters of ther series. The components of the r × r matrix � and theunknowns in the m × m matrix Q can be estimated usinga multivariate analysis method (Amiri-Simkooei 2009). TheKronecker structure in Eq. (10), i.e., Ir ⊗ Ak , indicates thatthere is a common frequency (possibly with different ampli-tudes and phases) in all of the series which should be detectedusing the multivariate harmonic estimation.

    The power spectrum of the multivariate model has thefollowing form: (Amiri-Simkooei and Asgari 2012)

    P(ω j ) = tr(ÊT Q−1 A j (ATj Q−1 P⊥A A j )−1 ATj Q−1 Ê�−1)(11)

    with Ê = P⊥A Y the least-squares residuals of the r timeseries and P⊥A = I − A(AT Q−1 A)−1 AT Q−1 an orthogonalprojector of the univariate model. Equation (11) considersall of the time series simultaneously and takes into accountthe possible cross-correlation through � and time correlationthrough Q in an optimal least-squares sense. The matrix � isestimated as �̂ = Ê Q−1 Ê/(m − n) (Teunissen and Amiri-Simkooei 2008; Amiri-Simkooei 2009). The test statistic fortesting the significance of the detected frequency is:

    T2 = tr(ÊT Q−1 Ak(ATk Q−1 P⊥A Ak)−1 ATk Q−1 Ê�−1)(12)

    which under the null hypothesis has a central Chi-squaredistribution with 2r degrees of freedom, i.e., T2 ∼ χ2(2r, 0)provided that both � and Q are known and that the originalobservables are normally distributed.

    3 Numerical results and discussions

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

    To see the advantage of the multivariate analysis over theunivariate analysis, we selected 318 1-year time series amongthe 45 available series (first multivariate data set). For thispurpose, we split long time series of each station into a coupleof 1-year time series, making in total 318 1-year series. Thetime series were evenly spaced, namely, 15 min spaced tideheight series, except for having at maximum 6 gaps. In amultivariate linear model, the design matrix A of differenttime series should be identical. Therefore, the gaps shouldbe located in the same places for all series. In other words,if there are some gaps in one of the series of the multivariatemodel, one has to omit the data in the location of those gapsin all series. This makes in total 173 common gaps (out of35,064 samples) within the 318 time series, distributed over1 year.

    Further, to extract the tidal frequencies, a multivariate timeseries is provided such that its length is suitable to separateclose frequencies. The time span of these time series is cho-sen to be a little more than 18 years. It consists of 11 timeseries (second multivariate data set) derived from 11 stations,namely stations 6, 8, 10, 13, 16, 17, 29, 31, 43, 44 and 45 with

    Fig. 1 UK tide-gauge stations

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  • Extracting tidal frequencies using multivariate harmonic analysis 979

    a time interval from 1 January 1993 to 31 March 2011. Thesetime series contain equally spaced data with the sample rateof 15 min, possibly with some gaps. According to the proce-dure described above, the total number of gaps in the secondmultivariate data set becomes 190,240, distributed over the18.24 years.

    3.1 Harmonic estimation

    The discrete frequencies at which the power spectrum inEqs. (7) and (11) are evaluated can be derived using a recur-sive relation (Amiri-Simkooei and Tiberius 2007):

    Tj+1 = Tj (1 + αTj/T ), α = 0.1, j = 1, 2, . . . (13)

    with a starting period of T1 = 30 min (Nyquist period) and Tbeing the total time span (1 year or 18.24 years). Because ofusing long time series with high sampling rate, i.e., 15 min,the starting period T1 is selected as 2 h. The step size usedfor periods, Tj= 2π/ω j , is small at high frequencies andgets larger at lower frequencies. The lowest frequency thatwe checked is ωmin = 2π/T , i.e., one cycle over the totaltime span. It is worthwhile mentioning that for calculatingthe power spectra, a linear regression model was consideredto make the design matrix A under the null hypothesis.

    We applied LS-HE to a univariate (one time series) andmultivariate (318 time series) 1-year time series to make acomparison between the two analyses. The power spectrafor the univariate and multivariate analyses are shown inFig. 2 [using Eq. (7)] and Fig. 3 [using Eq. (11)], respec-tively. The spectra are separated into tidal species with thecentral frequencies of n/24 h, n = 1, 2, . . . , 12, i.e., thereexist many periodic patterns with periods around 24 h/n, n =1, 2, . . . , 12. Higher harmonics could similarly be seen bychoosing T1 = 30 min (Nyquist period) and by increasing the

    Fig. 2 Univariate least-squares power spectrum of a (single) 1-yeartide height time series with sampling rate of 15 min provided from tide-gauge station at North Shields in UK. The dashed lines indicate periodrange of diurnal signal and its higher harmonics

    Fig. 3 Multivariate least-squares power spectrum of 318 1-year tideheight time series with sample rate of 15 min provided from 43 tide-gauge stations in UK. The dashed lines indicate period range of diurnalsignal and its higher harmonics

    sampling rate. The spectra in Figs. 2 and 3 are in fact verysimilar. The main difference is indeed in the power of thedetected frequencies. In multivariate analysis the common-mode frequencies, contributed to the tide structure, are moreobvious and hence they can be easily detected. Fig. 4a, bshows a zoom-in of the ter-diurnal signal of Figs. 2 and 3,respectively. As it can be seen, the multivariate spectrum hashigher detection power compared to the univariate spectrum;more frequencies can then be detected.

    The multivariate LS-HE method is now applied to theeleven 18.24-year time series (second multivariate data set)to obtain the power spectrum of the series and consequentlyto extract the tidal frequencies. Longer time series (e.g.,18.24 years vs. 1 year) make the possibility to see closer fre-quencies separated in the spectrum, which may be mergeddue to the leakage problem when dealing with short timeseries. Again, the multivariate harmonic estimation (using 11time series) increases the detection power of the common-mode signals. Figure 5 shows the multivariate least-squarespower spectrum of these series, while Fig. 6 shows a zoom-in of the diurnal signals along with their higher harmon-ics. A wide spectrum of signals ranging from long-termannual signals to the diurnal signals and their higher har-monics can be observed in the spectrum. The spectrum isseparated into tidal species with the central frequencies ofn/24 h, n = 1, 2, . . . , 12. There exist series of peaks closeto the diurnal signal and its higher harmonics. These are themain tidal constituents.

    The tidal frequencies related to 413 important peaks (ofFig. 5), which for long time series in multivariate spectrumare well separated from their neighbors, are listed in Appen-dix B (Tables 4, 5, 6). Table 4 includes the frequencies that arevery close to the 145 main tidal frequencies (available in lit-erature) of which their differences are less than 10−4 cycle/h.

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  • 980 A. R. Amiri-Simkooei et al.

    Fig. 4 Zoom-in of ter-diurnal signal in least-squares power spectrumof a a single 1-year time series, b 318 1-year time series

    Fig. 5 Multivariate least-squares power spectrum of eleven 18.24-yeartide height time series with sampling rate of 15 min provided from 11tide-gauge stations in UK

    Their Darwin’s symbols are also included in the table. Table 5includes the remaining frequencies. The frequencies listed inthese two tables are ordered in an ascending order. The mostimportant tidal constituents (the first 50 constituents), pre-sented in a descending order of their amplitudes, are listed in

    Fig. 6 Zoom-in of diurnal and its higher harmonics in multivariateleast-squares power spectrum of eleven 18.24-year time series

    Table 6. Most of them belong to the main tidal frequencies inTable 4. For the data set considered, the first four importanttidal frequencies are M2, S2, N2, and K2 with amplitudes of212 cm, 74 cm, 41 cm, and 21 cm, respectively. The fiftiethconstituent is OQ2 with amplitude of 1 cm. This frequencylist is an important list of tidal constituents applicable to manyfields of applications such as tide predictions.

    Relevant recent studies based on tidal observations donot provide users with a complete list of tidal frequen-cies covering the whole spectrum of tidal data. Mousavianand Mashhadi-Hossainali (2012) extract at most only 17tidal constituents through a univariate analysis. Ducarmeet al. (2006b), by the use of VAV program, list only 91tidal constituents ranging from the diurnal to ter-diurnal sig-nal and Ducarme et al. (2006a) list 16 low-tidal frequen-cies. This simply shows the power of multivariate LS-HEin detecting tidal frequencies, which has led to extracting413 tidal constituents in the present contribution, rangingfrom long-term signals to short-term of 1/12 diurnal sig-nal.

    To support our last statement two kinds of tide predictionshave been performed based on different lists of frequencieson different data sets.

    3.2 Tide prediction based on multivariate data set

    In this section, using different lists of frequencies, tidalheights are predicted and the results are compared withthe available data. For tidal prediction, Eq. (2) is used toestimate the coefficients ak and bk for a given frequency.These two coefficients can be used to obtain both the ampli-tude and phase of the sinusoidal function: amplitude =√

    (a2k + b2k ), phase = arctan(bk/ak). For each frequencylist, the coefficients ak’s and bk’s are estimated by the least-squares method. The estimated coefficients are then used to

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  • Extracting tidal frequencies using multivariate harmonic analysis 981

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

    Frequency list Tide-gauge station numberMean 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 diurnalsignals

    Table 2 Mean standard deviation of residuals, the ratio of standard deviation/tide height variation of 8-month predicted data compared withobserved data, computed for several lists of frequencies (time interval: from 2011/4/1 to 2011/12/1)

    Frequency list Tide-gauge station numberMean 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 diurnalsignals

    predict the tidal heights for time instants outside the time spanof the available data. The frequency lists are: CTE, the 1stlist; frequencies in Tables 4 and 5, the 2nd list; frequenciesin Table 6, the 3rd list; and CTE plus frequencies in Tables 4and 5 that lie between 4th and 6th harmonics of diurnal sig-nals, the 4th list. Using these frequency lists, tidal heightswere separately predicted and then the residual vectors (dif-ferences between observations and predicted data) along withtheir standard deviations in each station were computed. Twokinds of tide predictions were performed in each of the 11stations, (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. Themean standard deviations are listed in Tables 1 and 2, respec-tively. Based on these values, we did several investigations.

    We first investigate the importance of different tidalspecies in tidal observation structure. In most recent stud-

    ies, the most contributing tidal frequencies are consideredto be those frequencies that lie between the long-term sig-nals and at most the 6th diurnal species. The results pro-vided in Table 1 confirm this issue. When we compare themean residuals standard deviation of the 1st list and 4th list,we observe that by adding the frequencies between the 4thdiurnal species and 6th diurnal species to the CTE list, thetide prediction gets better about 30 %. The CTE list, pro-posed by the studies of Cartwright and Tayler (1971) andCartwright and Edden (1973), considers the tidal frequen-cies up to and including the 3rd harmonics. This improve-ment confirms the significant contribution of the frequenciesat the 4th, 5th and 6th diurnal species. We note, however,that the contribution of the higher frequencies than the 6thdiurnal species is negligible for the data set considered in thisstudy.

    123

  • 982 A. R. Amiri-Simkooei et al.

    Table 3 Mean standard deviation of residuals, the ratio of standarddeviation/tide height variation of 1-month predicted data compared withobserved data, computed for several lists of frequencies; UK tide-gauge

    stations (second column), Persian Gulf stations (third column), IndianOcean stations (fourth column), Pacific Ocean stations (fifth column),Canada Stations (sixth column)

    Frequency list Tide-gauge stationMean residual vector’s standard deviation (cm)The ratio of std/tide height variation

    UK tide-gaugestations

    Persian Gulftide-gauge stations

    Indian Oceantide-gauge stations

    Pacific Oceantide-gauge stations

    Canada tide-gaugestations

    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

    We listed 413 tidal frequencies using the observed sealevel height, which are presented in Tables 4 and 5 in Appen-dix B (2nd list); the most important tidal constituents (thefirst 50 constituents) are presented in Table 6 (3nd list). Theresults in Tables 1 and 2 show that the standard deviation ofthe residuals are approximately the same for these two lists,i.e. 16.4 cm (Table 1) and ∼24.0 cm (Table 2).

    Because Table 6 consists of tidal frequencies that theiramplitudes are more than 1 cm, it indicates that includingtidal constituents that have amplitudes less than 1 cm donot significantly improve tide prediction. This seems to bean important conclusion as it presents a shorter list of tidalconstituents that meet the requirements for many practicalapplications. The difference between the 4-week (Table 1)and 8-month (Table 2) predictions is because the predic-tion is a kind of extrapolation. As expected the precisionof the results decreases for long term prediction. For exam-ple, the seasonal variations can be one of the influencingfactors.

    Based on the results presented in Tables 1 and 2, onecan conclude that tide prediction error can never becomesmaller than a certain value. This conclusion is valid for allfrequency lists used. This is mainly due to the uncertain-ties in the instantaneous water level in general and errorsin the measured tide-gauge records in particular. This indi-cates that error sources in the measured water levels are sub-ject to colored noise that needs to be investigated in futureresearch.

    3.3 Tide prediction based on other tide-gauge data sets

    To see the performance of the detected frequencies at workin a more global scale, five other data sets are used. They

    include: (1) UK tide-gauge stations of which the data at thosestations were not used when extracting the tidal frequencies,(2) a few tide-gauge stations in Persian Gulf, (3) a few stationsin Indian Ocean, (4) a few stations in Pacific Ocean, and (5)two stations in Canada. For the UK stations, 103 1-year tideseries from different stations except those listed in Tables 1and 2 (station numbers 6, 8, 10, 13, 16, 17, 29, 31, 43, 44 and45) with sampling rate of 15 min were used. In Persian Gulf,we considered eight 1-year time series in Bandar Abbas, Kan-gan, and Chabahar stations. The data in these stations arehourly. The data in Indian Ocean were chosen from Davis,Esperance, and Masirah stations. These data were sampledhourly from which we selected 19 1-year tide series. Regard-ing 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. InCanada, we used four 1-year time series recorded in the sta-tions, Queen Charlotte and Vancouver, with sampling rateof 1 h.

    The tide predictions are based on four frequency lists men-tioned above. The results are presented in Table 3. A fewobservations can be highlighted from these results. (1) In allcases, the tide prediction based on the frequencies extractedin this contribution yields better results in comparison withthe outcome of the CTE list (first list). (2) The results of theCTE list and those of CTE plus frequencies in Tables 4 and 5that lie between 4th and 6th harmonics of diurnal signals (4thlist) are identical. This is however not the case for the predic-tion results based on long time series in Tables 1 and 2. Thisindicates that when using short time series (1-year series)for tide prediction, the higher frequencies than the third har-monic of the diurnal signals have no significant effect on tidalprediction. This is likely due to the presence of colored noise

    123

  • Extracting tidal frequencies using multivariate harmonic analysis 983

    in tidal data. When using long time series (18.24 years) fortidal prediction, because we do not model the colored noise(white noise model is used), the role of higher frequenciesbecome significant in comparison with the low and mediumones. This indicates that when one uses shorter series (1 year)for tidal prediction, the contribution of high frequency sig-nals decreases. (3) The results of the 2nd (with 413 frequen-cies) and 3rd (with 50 frequencies) lists are nearly identical.This indicates that the 50 tidal constituents of Table 6 arein fact an important list that presents a shorter list of tidalconstituents and meets the requirements for many practicalapplications.

    4 Summary and conclusions

    The importance of the research carried out relies on thetwo goals we followed in the present contribution. First, weextracted the tidal frequencies using the univariate and mul-tivariate harmonic analysis applied to the tidal time series.Second, the extracted frequencies were used to predict thetidal heights. We applied LS-HE to a single 1-year timeseries and a multivariate time series consisting of 318 1-year series. The data for these series were obtained from the43 tide-gauge stations in UK. The spectra of univariate andmultivariate series show that multivariate analysis is advanta-geous over the univariate analysis as it increases the detectionpower of the tidal frequencies. We then adopted the multi-variate LS-HE method to a multivariate time series includ-ing eleven 18.24-year data series of the tide-gauge stationsin UK and extracted 413 tidal frequencies by examinationof their power spectrum. These frequencies included long-term frequencies and the 1st to 12th harmonics of the diurnalsignals.

    After extracting the frequencies, using different frequencylists, tidal heights were predicted. These frequency listswere: 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 and6th harmonics of diurnal signals, the 4th list. Using thefrequency lists mentioned above, tidal heights were sepa-rately predicted for 4 weeks and 8 months for each of the11 stations for the time intervals 2011/4/1 to 2011/4/28and 2011/4/1 to 2011/12/1, respectively. Then the predic-

    tion error (difference between observed and predicted data)and their mean standard deviations in each station were com-puted. Based on these values, we could have the followingconclusions:

    • We investigated the importance of different tidal speciesin tidal observation structure. The only important frequen-cies belong to the long-term species to the 6th diurnal har-monic. The role of the frequencies between 7th diurnalspecies and the 12th diurnal species is negligible.

    • We also found that the tide prediction using the 413extracted frequencies could nearly provide identical resultswith those provided based on the most important con-stituents (50 frequencies). This conclusion was also ver-ified when different tidal time series were used over fivedifferent areas on the world.

    • When comparing the 4-week prediction with 8-month pre-diction, it follows that short-term prediction could providesuperior results than long-term prediction. This is what wewould expect as the precision of predicted results get worsewhen the time span increases.

    • The tidal constituents were made using the data set inUK. We note that if we use different data sets aroundthe world, the power of the detected frequencies willchange and some of the detected signals cannot bedetected and new ones will appear. A more reliablelist of detected frequencies will likely benefit from thetide-gauge data all around the world in which use ismade of multiple time series distributed uniformly. How-ever, although the amplitudes of signals are expected tochange in a different area, the main contributing fre-quencies will likely be unchanged and only the orderof their magnitudes will change. This can be expectedbecause promising results were obtained over five differ-ent locations and using the list of 50 tidal frequencies.Further, we note that site effects such as small basinsand river estuaries can also affect tide predictions. Thiswas however not the subject of discussion in the presentcontribution.

    Acknowledgments We would like to acknowledge the BritishOceanographic Data Center (BODC) for its free tide data we used in thispaper. Useful comments of the editor-in-chief and anonymous reviewersare kindly acknowledged.

    123

  • 984 A. R. Amiri-Simkooei et al.

    Appendix A: UK tide-gauge stations’ namesand coordinates

    No. St. name Latitude (◦) Longitude (◦) No. St. name Latitude (◦) Longitude (◦)

    1 Aberdeen 57.1440 −2.0803 24 Milford Haven 51.7064 −5.05142 Avonmouth 51.5108 −2.7151 25 Millport 55.7496 −4.90583 Bangor 54.6648 −5.6695 26 Moray firth 57.5992 −4.00024 Barmouth 52.7193 −4.0450 27 Mumbles 51.5703 −3.97495 Bournemouth 50.7143 −1.8749 28 Newhaven 50.7818 0.05706 Cromer 52.9343 1.3016 29 Newlyn 50.1030 −5.54287 Devonport 50.3684 −4.1853 30 Newport 51.5500 −2.98748 Dover 51.1144 1.3225 31 North Shields 55.0074 −1.43989 Felixstowe 51.9577 1.3466 32 Port Ellen 55.6276: −6.189910 Fishguard 52.0137 −4.9832 33 Port Erin 54.0852 −4.768111 Harwich 51.9480 1.2921 34 Port Patric 54.8426 − 5.120012 Heysham 54.0318 −2.9203 35 Portrush 55.2068 −6.656813 Hinkley point 51.2153 −3.1345 36 Portsmooth 50.8026 −1.111814 Holyhead 53.3139 −4.6206 37 St Mary 49.9185 −6.316515 Ilfracombe 51.2109 −4.1111 38 Sheerness 51.4456 0.743416 Immingham 53.6310 −0.1868 39 Stornoway 58.2070 −6.388917 St. Helier 49.1833 −2.1167 40 Tobermory 56.6229 −6.064018 Kinlochbervie 58.4567 −5.0504 41 Ullapool 57.8953 −5.158119 Leith 55.9898 −3.1817 42 Waymouth 50.6085 −2.447920 Lerwick 60.154 −1.1403 43 Whitby 54.4894 −0.619921 Liverpool 53.4497 −3.0181 44 Wick 58.4410 −3.086522 Llandudno 53.3317 −3.8252 45 Workington 54.6508 −3.567823 Lowestoft 52.4820 1.7516

    Appendix B: different lists of tidal frequencies

    See Appendix Tables 4, 5 and 6.

    Table 4 Main tidal frequencies detected by least-squares power spec-trum of 18.24-year multivariate time series (11 series used)

    No. Frequency(cycle/h)

    Darwin’ssymbol

    No. Frequency(cycle/h)

    Darwin’ssymbol

    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

    Table 4 continued

    No. Frequency(cycle/h)

    Darwin’ssymbol

    No. Frequency(cycle/h)

    Darwin’ssymbol

    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

    123

  • Extracting tidal frequencies using multivariate harmonic analysis 985

    Table 4 continued

    No. Frequency(cycle/h)

    Darwin’ssymbol

    No. Frequency(cycle/h)

    Darwin’ssymbol

    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 seriesused)

    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

    Table 5 continued

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

    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

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  • 986 A. R. Amiri-Simkooei et al.

    Table 5 continued

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

    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

    Table 5 continued

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

    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

    123

  • Extracting tidal frequencies using multivariate harmonic analysis 987

    Table 6 Tidal frequencies (most important constituents) detected by least-squares power spectrum of 18.24-year multivariate time series (11 seriesused) that have approximately the same effect as tide prediction using the entire 413 frequencies in Tables 4 and 5

    No. Frequency(cycle/h)

    Meanamplitude(m)

    Darwin’ssymbol

    No. Frequency(cycle/h)

    Meanamplitude (m)

    Darwin’ssymbol

    1 0.0805114 2.122 ± 2e−004 M2 26 0.1207668 0.018 ± 2e−004 M32 0.0833331 0.735 ± 2e−004 S2 27 0.0802832 0.017 ± 2e−004 OP23 0.0789996 0.405 ± 2e−004 N2 28 0.0012733 0.017 ± 5e−004 MSM4 0.0835607 0.207 ± 2e−004 K2 29 0.0000186 0.017 ± 3e−004 ____5 0.0387306 0.108 ± 3e−004 O1 30 0.0863831 0.016 ± 2e−004 SKM26 0.0417806 0.103 ± 2e−004 K1 31 0.0001511 0.016 ± 4e−004 SSA7 0.1610228 0.100 ± 2e−004 M4 32 0.0000400 0.016 ± 4e−004 ____8 0.0820239 0.100 ± 2e−004 L2 33 0.0748673 0.016 ± 2e−004 ST379 0.0776897 0.087 ± 3e−004 MU2 34 0.0807396 0.015 ± 3e−004 MKS210 0.0792009 0.083 ± 2e−004 NU2 35 0.0028216 0.014 ± 2e−004 MSF11 0.0001150 0.071 ± 5e−004 SA 36 0.0806252 0.013 ± 2e−004 H212 0.1638446 0.064 ± 2e−004 MS4 37 0.1222924 0.013 ± 1e−004 MK313 0.0818213 0.045 ± 2e−004 LDA2 38 0.1582011 0.013 ± 2e−004 3MS414 0.1595109 0.039 ± 2e−004 MN4 39 0.0030503 0.012 ± 4e−004 MF15 0.0372188 0.035 ± 3e−004 Q1 40 0.1625353 0.011 ± 1e−004 KN416 0.0415524 0.034 ± 2e−004 P1 41 0.1192424 0.011 ± 2e−004 MO317 0.0861555 0.033 ± 2e−004 2SM2 42 0.0763798 0.010 ± 3e−004 ST218 0.0848456 0.025 ± 2e−004 MSN2 43 0.0000889 0.010 ± 4e−004 ____19 0.0774621 0.024 ± 2e−004 O2 44 0.0000480 0.010 ± 2e−004 ____20 0.0014283 0.022 ± 5e−004 MM 45 0.1666663 0.010 ± 2e−004 S421 0.0761772 0.021 ± 3e−004 EPS2 46 0.1623327 0.010 ± 2e−004 SN422 0.2443561 0.021 ± 1e−004 2MS6 47 0.2400219 0.010 ± 1e−004 2MN623 0.0803976 0.021 ± 1e−004 H1 48 0.1597129 0.010 ± 1e−004 ST924 0.2415344 0.018 ± 1e−004 M6 49 0.0402562 0.009 ± 2e−004 NO125 0.1640721 0.018 ± 2e−004 MK4 50 0.0759496 0.009 ± 2e−004 OQ2

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    Extracting tidal frequencies using multivariate harmonic analysis of sea level height time seriesAbstract 1 Introduction2 Least-squares harmonic estimation (LS-HE)2.1 Univariate harmonic estimation2.2 Multivariate harmonic estimation

    3 Numerical results and discussions3.1 Harmonic estimation3.2 Tide prediction based on multivariate data set3.3 Tide prediction based on other tide-gauge data sets

    4 Summary and conclusionsAcknowledgmentsAppendix A: UK tide-gauge stations' names and coordinatesAppendix B: different lists of tidal frequenciesReferences


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