Aalborg Universitet
Least squares techniques for extracting water level Fluctuations in the Persian Gulf and Oman Sea
Farzaneh, Saeed; Forootan, Ehsan; Parvazi, Kamal
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Earth and Space Physics
DOI (link to publication from Publisher):
10.22059/JESPHYS.2019.269327.1007060
Publication date:
2020
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Publisher's PDF, also known as Version of record Link to publication from Aalborg University
Citation for published version (APA):
Farzaneh, S., Forootan, E., & Parvazi, K. (2020). Least squares techniques for extracting water level Fluctuations in the Persian Gulf and Oman Sea. Earth and Space Physics, 45(4), 99-119.
https://doi.org/10.22059/JESPHYS.2019.269327.1007060
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Journal of the Earth and Space Physics, Vol. 45, No. 4, Winter 2020, P. 99-119 DOI: 10.22059/jesphys.2019.269327.1007060
Least Squares Techniques for Extracting Water Level Fluctuations in the Persian Gulf and Oman Sea
Farzaneh, S.1*, Forootan, E.2 and Parvazi, K.3
1. Assistant Professor, Department of Surveying and Geomatics Engineering, Faculty of Engineering, University of Tehran, Tehran, Iran
2. Lecturer, school of Earth and Ocean Sciences, Cardiff University, Cardiff, United Kingdom 3. Ph.D. Student Department of Surveying and Geomatics Engineering, Faculty of Engineering,
University of Tehran, Tehran, Iran (Received: 28 Nov 2018, Accepted: 1 Oct 2019)
Abstract
Extracting the main cyclic fluctuations from sea level changes of the Persian Gulf and Oman Sea is vital for understanding the behavior of tides and isolating non-tidal impacts such as those related to climate and changes in the ocean-sea circulations. This study compares two spectral analysis methods including: Least Squares Spectral Analysis (LSSA) and Least Squares Harmonic Estimation (LSHE), to analyze satellite altimetry derived sea surface height changes of the Persian Gulf and Oman Sea. SSH data are derived from about 16 years of satellite altimetry observations (1992 to 2008), including the Topex/Poseidon and Jason-1 missions. By analyzing the real data, we extract significant tidal components in the spectrum of LSSA and LS-HE including those with the period of 62.07, 173.3, 58.71, 45.68, 88.86, 364.2 and 117.5 days, which are interpreted as Principal Lunar semi-diurnal, Luni-Solar Diurnal, Principal Solar Semi-diurnal, Principal Lunar Diurnal, GAM2, annual, Solar Diurnal periods are dominant in the level fluctuations. Moreover, some tidal components appear in the spectrum of LSSA and LS-HE, from which the Moon's semi- diurnal component M is dominant. Also, to evaluate the efficiency of these two techniques, we run three experiments in each extracted frequency from LSSA, LS-HE, and astronomical tide tables are separately used to predict the sea level in the Persian Gulf and Oman Sea for three years.
The results of this prediction indicate that RMSE from LSSA, astronomical table, and LS-HE is 0.101 m, 0.093 m, and 0. 086 m, respectively. According to the results LS-HE is found a more efficient technique to analyze cyclic fluctuations from altimetry measurements.
Keywords: Persian Gulf and Oman Sea, Least Square Spectral Analysis (LSSA), Least Square Harmonic Estimation (LS-HE), Satellite Altimetry.
1. Introduction
The description, understanding and quantitative determination of the tides has been an important research topic in geodesy oceanography. Ocean tides, resulting from the gravitational attractions of the moon and the sun, causes more than 80% of the total variability of the sea surface. Tides have strong influence on coastal environment and the protection of its ecosystem, and play a significant role in climate and also needed for the precise treatment of space observations (Fok, 2012).
Advances in satellite radar altimetry technology have enabled a globally sampled record of sea surface and has become an important tool for monitoring global and regional sea surface height (SSH) (Fu and Cazenave, 2001), measuring level fluctuations of inland water bodies (Khaki et
al., 2015), and even soil moisture (Frappart et al., 2015; Papa et al., 2003). Thereby, complementing traditional tide gauge (in situ) measurements, which despite their valuable utility in reflect local sea level fluctuations, they have limitations such as inhomogeneous spatial distribution, inconsistency between reference datum's, and a suboptimal reference to the moving coast (Chelton et al., 2001).
Tide data are frequently used for different applications such as safe navigation and hydrographic surveys. Their identity can be expressed by their frequencies and noise structure. Tidal analysis, which is focused in this paper, tries to condense a long-term record of observations into a brief collection of time-invariant constants. Due to the periodic behaviour of the tide-generating
*Corresponding author: farzaneh@ut.ac.ir
100 Journal of the Earth and Space Physics, Vol. 45, No. 4, Winter 2020
forces (e.g., those resulting from the relative [to Earth] motions of the Moon and Sun), periodicities contained within a tidal record is often extracted in order to describe the tidal displacement at a location as a sum of the associated harmonics (see, e.g., https://tidesandcurrents.noaa.gov/predhist.ht ml). A reliable tidal analysis and prediction requires a reliable knowledge on the (main) tidal frequencies and noise structure. So far, all the proposed methods for determining tidal frequencies have been Theorized based;
in other words, on these methods applied the ephemeris of Moon, Sun and other planets to extract tidal frequencies without any use of tidal observations, as mentioned by Doodson (1954), Tamura (1993), Parvazi et al. (2015), Xi and Hou(1987). These methods assume that the tidal frequencies are known, but their amplitudes are unknown.
To extract the tidal frequencies, many studies have analysed sea level height with different methods such as the Fourier and wavelet.
Historically, Fourier spectral analysis has been used to examine the global energy and frequency distributions of SSH time series (Boashash and Butland, 2003). Its popularity is due to the prowess of the method, as well as its simplicity of application. As a result, the term ’spectrum’ has become almost synonymous with the Fourier transform of time series (Wu et al., 2009). Fourier analysis, however, exhibits some drawbacks in analysing time series, which are unequally sampled or those with data gaps (Rubin, 2002). Filling the gaps with inverted data might be erroneous when large gaps present in the time series, or due to the approximation approach used for interpolation (Papa et al., 2003).
In this paper, we focus on time-invariant base-functions to detect tidal frequencies using tidal observation analysis without predefining these frequencies. For this purpose, we compare the application of the Least Square Spectral Analysis (Vanicek, 1969, 1971) and the Least Squares Harmonic Estimation (LS-HE) developed by Amiri- Simkooei and Asgari (2014), Amiri- Simkooei (2014, 2012), and Amiri-Simkooei (2007). Our motivation to select these techniques is: 1- they are not limited to evenly-spaced data nor to integer frequencies; 2- they allow us to detect
common-modes of signals, in a least squares sense, and thus are very efficient in detecting cyclic patterns; and (3) they can be easily used for univariate and multivariate examples.
Therefore, the mathematical objectives (MOs) of this study include: (MO1) justifying the mathematical (dis-)similarity of these two techniques and their relationship with the commonly used (discrete) Fourier Analysis; (MO2) assessing the accuracy of the extracted frequencies, while evaluating the effect of the noise that contaminate the observations, effect of blunders, impact of missing values in time series; and (MO3) recognizing the accuracy of extraction of nearby frequencies and effect of the data length in extracting the nearby frequencies. A detailed investigation of these objectives has not been provided in previous studies, thus, this assessment complements the literature by evaluating the skill of LSSA and LS-HE techniques for tidal analysis studies.
To assess the ability of LSSA and LS-HE, on a real case study, we apply them at the SSH time series of the Persian Gulf and the Oman Sea derived from the Topex/Poseidon mission (1992-2002), Jason-1 (2003-2008), and Jason-2 (2009- 2014). The Persian Gulf, located in the southwest of the Asian continent is a shallow, semi-enclosed basin in a typical arid zone and is an arm of the Indian Ocean. It is located between the longitude of 48–57° E and the latitude of 24–30° N (Figure 9). This Gulf is connected to the deep Gulf of Oman through the narrow Strait of Hormuz. The Persian Gulf covers an area of approximately 226,000 km2 with a length of 990 km. Its width varies from 56 to 338 km, separating Iran from the Arabian Peninsula with the shortest distance of about 56 km in the Strait of Hormuz. This basin has an average depth of about 35 m, and the deepest water depth is approximately 107 m (Purser and Seibold, 1973).
The remaining part of this study is organized as follows: in Section 2, the datasets of the study are introduced, and the methodology of their analysis is explained in Section 3. The results are reported in Section 4, and finally, the study is summarized and concluded in Section 5.
Least Squares Techniques for Extracting Water Level Fluctuations in the … 101
2. Data
The first studies regarding the use of the satellite for sea level were raised in 1969 in the Williamstown Conference of Solid Earth and Ocean Physics. Four years later in spatial laboratory, the first measures of sea level were performed by altimetry radar S-193 with accuracy of about one meter.
Later, various satellites were used for different space missions (Aviso and Podaac, 2008).
Topex/Poseidon generated SSH measurements with RMSE accuracy better than 5 cm for a single-pass and better than 2 cm at global scale. To eliminate atmospheric effects, this satellite was placed in an orbit with high altitude, 1336 km to perform exact orbit modelling. With the progress of distance with laser, JGM3 (Joint Gravity Model) and non-gravity models reduce the radial error of RMS to about 2 cm. For the first time, seasonal period and other time changes of which ocean were determined as globally with high accuracy.
Jason-1 and Jason-2 were placed in orbit in continuance of the Topex/Poseidon mission and the mission of Topex/Poseidon and Jason-1, respectively. The altimetry measurement accuracy of these two satellites is about 2.5 cm. With the investigation in the data of Topex/Poseidon, Jason-1 and Jason-2, when these satellites were in a similar orbit, had equal coverage and by intermission bias on data of Jason-1, Jason -2 (in data files of these satellites), they were combined with the data of Topex/Poseidon and achieved about 22 years of data on similar passes. The satellite data in this study include Topex/Poseidon satellite data during 1992- 2002, Jason-1 satellite during 2003-2008 and Jason-2 satellite during 2009-2014 in the Persian Gulf and the Gulf of Oman. In fact, this data is data on which the effect of the wet Troposphere error (Δw), dry troposphere error (Δd), ionosphere error (ΔI), polar tide error (ΔpT), inverse pressure effect bias (ΔIB), sea state bias, electromagnetic bias (ΔE) and error of centre of gravity changes of altimeter antenna (Δc); is applied. All of these corrections apply to observations.
These data are available at ftp://avisoftp.cnes.fr/AVISO/pub (Aviso and Podaac, 2008).
3. Methodology
Consider f containing sampled SSH observations, it can be written as a functional model of (Vaníček, 1969, 1971; Amiri- Simkooei, 2007):
= + (1) where f is vector of observations containing both periodic and non-periodic terms;
and refer to non-periodic and periodic part of the model, respectively. In a similar manner, both LSSA and LS-HE methods try to extract the periodical part of the model in Equation (1) by using a series of sinusoidal base-functions (Vanicek, 1969). Both LSSA and LS-HE use a least square approximation to find the amplitudes associated to the base- functions.
In order to create a time-dependent observation vector, we create a vector ( = , … , ) where we have an observation at any time ( = , … , ).
And finally, we come to a time series.
Let be vector space defined by an inner product ( , =< | >=
∑ ∈ ( ) ( )) and as its base- functions, i.e. , i=1,…, m are column- vectors with the same dimension as . The aim of a spectral analysis (LSSA or LS_HE) is to look for unknown coefficients , = 1, … , , which provide the best approximation of in the vector space M:
= ∑ = (2) is a vector containing the coefficients and each base-function is i-th column of the matrix . In a matrix form, the optimum follows the minimization criterion of (Vaníček, 1969, 1971; Amiri-Simkooei, 2007):
∥ − ∥=∥ ̂ ∥↣ (3) Using the least squares adjustment (LSA) with the mean quadratic norm, coefficients are therefore determined as:
= ( ) (4) Considering the periodic pattern of model in
102 Journal of the Earth and Space Physics, Vol. 45, No. 4, Winter 2020
Equation (1), one should a priory select appropriate base-functions to model the sampled observations. As mentioned in Vanicek (1969), LSSA thus uses general trigonometric polynomial of:
( ) = ∑ cos ( − ) (5) where is Frequency, is Amplitude, is Phase component of j.
Equation (5) consists of two main parts of a signal, the linear trend and the periodic portion. For representing the periodic term.
Therefore, Equation (5) should be defined by considering the minimum quadratic distance from the observed vector , while detecting the major periodic terms with frequencies . Note that the frequencies are assumed to be known beforehand. Therefore, each general trigonometric polynomial expands as below:
( ) = cos ( ) + cos ( ) (6) To achieve Equation (6), two equations of
= + and = 2arctan ( ), are used.
For a specific frequency , matrix with the base-functions , can be written as:
=
( ) ( )
( ) ( )
. .
. .
. .
( ) ( )
(7)
Each spectral value, corresponding to the frequencyj, is defined as the ratio of the length of f projected on the orthogonal base-functions aij and scaled by the length of observations as (Vaníček, 1969, 1971;
Amiri-Simkooei, 2007):
= ( ) (8) Choosing = × ; least square spectrum simplifies as:
= ( ) (9) Equations (8) and (9) are related to the LSSA technique, while Equation (8) is the weighted
case and Equation (9) is a simple equally- weighted case.
The univariate harmonic estimation spectrum is defined as below (Vaníček, 1969, 1971;
Amiri-Simkooei, 2007):
The univariate harmonic estimation power spectrum shown as below:
= ̂ ( ) ̂ (10) where
̂ = (11) Represents least square residuals (Amiri- Simkooei, 2007)
= − ( ) (12) Substituting Equations (11) and (12) into Equation (10), at first, the problem of non- inevitability is solved, and then the final formula is presented as follows:
=
( − ( ) ) (( ( −
( ) ) ) ) ( −
( ) ) (13) Is the orthogonal projector of the univariate model that projects along range space of A, i.e. ( ), onto orthogonal component of range space of A, i.e. ( ). For zero mean stationary random process containing only white noise, linear part of model vanishes ( =0) and the weight matrix of = × , the univariate LS-HE estimated spectrum simplifies as (Amiri-Simkooei and Asgari, 2012; Amiri-Simkooei, 2007):
= ( ) (14) To obtain S(ωj) using several approaches are possible. One of these methods, using analytical relations for the matrix Aj, which is obtained for a maximum value of S(ωj).
However, this method due to the presence of large local amounts is very cumbersome and complicated. Therefore, numerical methods are used to solve the problem. For this purpose, a discrete relationship between ωj and S(ωj) is established and we can obtain the graph of the spectral values of S(ωj) in terms of the discrete values of ωj. In this way, ω which has the maximum value of the spectral value S(ωj) it is chosen as the frequency of interest.
t TjLeast Squares Techniques for Extracting Water Level Fluctuations in the … 103
The difference between the LSSA-derived spectrum Equation (9) and the univariate LS_HE derived spectrum Equation (14) is the scaling (normalization) factor of . In case of existing several time series, the model is referred to as a multivariate linear model which uses all the time series simultaneously with as the cross- correlation matrix of observations.
Multivariate power spectrum obtained as (Vaníček, 1969, 1971; Amiri-Simkooei, 2007):
=
( ( ) ) Σ (15)
where is least squares residual of observation matrix with time series stored in its columns.
= (16) For zero-mean uncorrelated series choosing
Σ = ( , , . . . , ), spectrum
simplifies as (Vaníček, 1969, 1971):
= ∑ ( ) / (17) Spectrums in the Equations (8), (9), (14), (15) and (17) are functions of frequency . These functions may include several local maximums. More significant frequencies will show up with greater spectral values and maximize the spectrum (Amiri-Simkooei and Asgari, 2012; Amiri- Simkooei, 2007; Vanicek, 1969, 1971).
Analytical methods for extracting the local maximums might be complicated. Spectral values, therefore, can be computed for a set of primary frequencies to find the significant ones that appear with maximum spectral values in the spectrums. Equations (8) and (9) can respectively be used to compute spectrums for a weighted and equally- weighted LSSA; Equation (14) corresponds to the univariate LS-HE; and finally Equations (15) and (17) refer to the weighted and equally-weighted multivariate LS-HE approach, respectively.
3-1. Least square harmonic estimation for frequency extraction
Harmonic estimation method is used to introduce periodic patterns in functional model. For a time series, the simplest periodic behavior that can be added to
improve a functional model is to include:
y(t) = a cos ωj jt+ b sin ωj jt (18) This relationship is in fact a sinusoidal wave with a primary phase,aj and bj signal amplitude and ωj frequency. Therefore, the functional model changes as follows:
E(y) = Ax + A xj j (19) After merging Equations 18 and 19, we will have Equation (20):
1 1
2 2
m m
cosω t sinω t cosω t sinω t a
A = , x =
b cosω t sinω t
j j
j j j
j j
j
j j
(20) Since in this model, in addition to aj and bj, the frequency ωjis also unknown; therefore, the least squares harmonic estimation method should be used to solve the problem. For this purpose, the assumptions of zero H and the 0 opposite of Haare defined as follows:
H : E(y) = Ax0
H : E(y) = Ax + A xa j j (21) An important issue here is whether the theory or assumptions used in the model is correct or not. For this purpose, two assumptions are assumed to be zero and the assumption is opposite to. For example, an assumption is that the data are unmistakable. To examine whether these assumptions are valid or not, we oppose the zero hypothesis and consider the opposite hypothesis, so that the variables become clearer for us. In a model that is assumed to have error and error effects absent (H ), 0 H test against 0 Ha informs us that indicates whether additional variable should be added to the calculation or not. The main goal is to solve the problem of finding frequencies that can maximize the following.
In fact, the goal is to find the frequencies that can be obtained for these frequencies by maximizing the amount of power spectrum.
Then, based on this maximum power
104 Journal of the Earth and Space Physics, Vol. 45, No. 4, Winter 2020
spectrum, it is possible to analyze the frequency extraction of important tidal components.
The goal is to find the frequency ωj by solving the following minimization problem:
j ωj
ωj= argmax P(ω) (22)
= || [ ] || = arg min || ̂ ||
= || ̅ || . ̅ =
̅ = ̅ ( ̅ ̅ ) ̅
= argmax ̂ ( ̅ ) ̂
= argmax P(ω )
̂ = ̅
T -1 T -1 -1 T -1
j ˆ0 y j j y A j j yˆ0
P(ω ) = e Q A (A Q P A ) A Q e (23)
0 A
ˆe P y
(24) In the above relationship, ̅ = [ … ] . || . || (.) (.) and ̂
the remainder of the least squares is subject to the opposite.Aj is obtained using Equation (20). The first two columns of this matrix include a linear behavior of time series and the following columns for finding periodic behavior in time series is used. ˆe0 The least squares residue vector and PA is the orthogonal projector. These parameters are obtained under the assumptionH . 0 In order to find the matrix Aj in (23), for different ωj bands, different matrices are tested to find the maximum value of P(ωj) The matrixes Aj have the same matrix structure as in (20). The matrix Aj, the maximum value ofP(ωj) , obtained by it, and is chosen as the matrix Aj.
To obtain P(ωj) using several approaches are possible. One of these methods, using analytical relations for the matrix Aj, which is obtained for a maximum value of P(ωj).
However, this method due to the presence of large local amounts is very cumbersome and complicated. Therefore, numerical methods are used to solve the problem. For this
purpose, a discrete relationship between ωj and P(ωj) is established, and we can obtain the graph of the spectral values of P(ωj) in terms of the discrete values of ωj. In this way, ω which has the maximum value of the spectral value P(ωj) is chosen as the frequency of interest.
The choice of different ωj is done by Equation (25):
j+1 j j
T =T (1+αT /T), α=0.1, j=1,2,...
j j
ω = 2π/T (25) where is the Nyquist period and T is the total length of the time series. The value of in Equation (25) is equivalent to 1 hour.
Considering the value of 0.1 for alpha is because, first, this coefficient must be chosen so that the algorithm can extract all the different frequencies. Second, this factor should be such that the volume of calculations does not last long. Therefore, for the above two goals, the appropriate value for this coefficient is 0.1.
After finding ωj and also the maximum value of P(ωj), it is necessary to test the selected frequencies with statistical assumptions. The statistics used for this test are as follows:
T -1 T -1 -1 T -1
2 ˆ0 y k k y A k k yˆ0
T = e Q A (A Q P A ) A Q e (26) The statistics under the assumption , has a Chi Squared distribution function with two degrees of freedom.
4. Simulation
The JASON-1 and missions are T/P’s follow- on that overfly almost the same reference T/P ground tracks from 2002 to 2008 (continued on the Topex satellite mission), respectively (Picot et al., 2003). The two satellites fly 254 ground tracks (passes) in 9.9156-day repeat cycle to measure topography changes of the sea surface (measure the height of the sea surface). The time series of observations are constructed according to satellite measurements when crossing a point (Sharifi et al., 2013). In fact, there are missing values, high amplitude noise, outliers, and blunders in the SSH time series caused by altimeter failures due to, e.g. a
T 1 -1 T 1
PAI-A(A Q A) A Q
Least Squares Techniques for Extracting Water Level Fluctuations in the … 105
raining weather, ice and land coverage etc.
The mentioned conditions are considered within the simulation stage, when LSSA and LS-HE are applied to analyse the SSH time series.
In order to simulate the time series, we used a periodic signal with a constant trend which was simulated using Equation (27):
( ) =
+ + ∑ [ ( ) + ( )]
(27) where SSH(t) is the water altitude at time moment of t, is the mean time series (water level), j=1,2,3,…,k of the components number, K is determined based on the number of applied components. and finally, is the rate of change.
Here, we assumed that the simulated time series contain a superposition of seven different sinusoidal signals with the frequencies of 62.07-1; 173.3-1; 58.71-1; 45.68-1; 88.86-1; 364.2 -1 and 117.5-1 cycle/day with different amplitudes and phases. Sampling rate equals to that of TOPEX/POSIDON (T/P) and Jason-1 missions, i.e. 9.9156 days. The simulation was done by MATLAB software using Equation (27). Considering the seven main tidal components that play the most role in producing a periodic signal. White noise is also used to consider and add noise to the time series.
4-1. Effect of noise
Similar to the most of spectral analysis methods, LSSA and LS-HE are also sensitive to noise. Therefore, when the time series are noisy, some undesirable effects would appear (consider the effect of noise on the desired time series), for instance, the peak behavior of the estimated spectrum.
To investigate a vast possibility of noisy behavior, temporally random noise with a standard normal distribution were added to the simulated time series with 16- year length. The simulated noise amplitudes differ from 1 to 30 mm. The results show that
∆ ratio, in which ∆ is the difference of the estimated frequency and its actual simulated value and is the simulated frequencies, corresponding to the annual frequency became more than 5 times greater when the noise amplitude increased from 1 to 30 mm. This is also combined with decreasing the amplitude of the significant peaks in the spectrum. For instance, the effect of noise on the accuracy of the extracted annual cycle and its spectrum is illustrated in Figure (1). Figure (1-a) indicates that when the amplitude of the noise is greater than 14 mm, a bias of 9 hours appears in the estimation of the annual cycle.
Figure (1-b) shows that the power of the simulated significant frequency, decreased when the noise presented. For those simulations containing noise amplitude greater than 15 mm, the power of the annual frequency reduced to 0.3, which might even fail a statistical significant test. The graph in Figure (1-b) indicates that the relationship between the noise amplitude and the estimated power of the peak is not linear.
Note that the significance of the peaks, estimated by LSSA and LS-HE, can be tested statistically (Sharifi et al., 2013; Farzaneh and Parvazi, 2018). When a high magnitude noise is present in the time series, subsequently, the probability of the rejection of statistical test increases. Discussing a reliable testing method or a proper significant threshold is behind the scope of the current study.
106
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series becom e series diffe an 50% o positions of d randomly. T
are sampled 16 years, of observatio
series with mis that contain 52
Earth and Spac
ntaining blunde adding blunder me series value ated in spectrum
advantage o e series with
analyzed. To ed time serie
number o ssing value me sparse. In
rs to one tha of possible
the missing Therefore, in with the rate there should ons. For ou
ssing values. G 29, 471, 353, 17
ce Physics, Vol
ers; the red-line rs. There are 6 es. b) Power of m. All these fre
f h o s f s n at e g n e d ur
inv tim 89 20%
sam tim Com spa 88.
low sam wer acc rati
Graphs that are s 77 and 89 sampl
l. 45, No. 4, Wi
e represents the 62 values repl f spectrum of th equencies are es
estigations, w me series cont
SSH values
%, 40%, 7 mples, respe me series are s
mputing the arse time seri
910-1; and 11 wer for the t mples. The O re, however curacy in all io of frequen
shown from top les, respectively
inter 2020
(b)
e simulation wit aced in primar he simulated tim
stimated with a
we assumed taining 529, s, which cor 70%, and 8 ectively. The shown in Fig spectrum u es, those mo 17.520-1) bec time series O , K , M a r, extracted cases. Figur ncies in each
p to the bottom y.
)
thout blunders ry time series me series in Fig an accuracy bett
unequally s 471, 353, 17 rrespond to 85% of m e mentioned gure 4.
using LSSA derate (365.2 came conside with 177 an and S freque d with suff re 5 shows t
case.
of the figure re and the
that is gure (3-
ter than
spaced 77 and 10%, missing d five
to the 259-1; erably nd 89 encies ficient the ∆
epresent
Figure
Figure observ 3), d) series
e 5. An overvie Figure 6. T time series with consid
e 6. a) Power s vations (corresp Power spectru with 89 observ
Least Squa
ew of the LSSA The cases 1, 2, that are not sp derable number
(a)
(c)
spectrum of a t ponding to case m of a time se ations (correspo
ares Technique
A results when i 3, 4 and 5 corr parse (case 1, 2 r of missing valu
)
time series with e 2), c) Power s
eries with 177 onding to case
es for Extractin
t is applied to f respond to 10%
2, and 3), the de ues (case 4 and
(e)
h 529 observati pectrum of a ti observations (c 5).
ng Water Leve
five simulated t
%, 20%, 40%, 7 esired frequenc d 5,) the estimat
ions (correspon me series with corresponding t
el Fluctuations
time series with 70%, and 85%
cies were estim ted frequencies
(b)
(d)
nding to case 1 353 observatio to case 4), e) P
s in the …
h missing values missing values mated accurately were less accur
), b) time serie ons (correspond Power spectrum
109
s, shown in s. For those y. For those
rate.
es with 471 ding to case m of a time
110 Journal of the Earth and Space Physics, Vol. 45, No. 4, Winter 2020
The corresponding power spectrums of all the cases in Figure 4 are displayed in Figure (6-a) to (6-e). In case 1, 2 and 3 spectrums, one can easily find six dominant frequencies, and therefore, the missing values do not cause a considerable impact on the estimation. In cases 4 and 5, only four dominant frequencies were detectable.
Figure 6 shows an overview of the estimated spectrums, corresponding to the time series of Figure 4. In each graph, the x-axis refers to the period of change in (days), and the y- axis refers to the spectral value.
Next simulation shows the effect of gap in the time series. We assumed five years of gap (i.e. 25% of the data missed) in the time series of case 1 to 5, shown in Figure 4.
The ratio of frequencies is recomputed for the new time series with a prolonged gap.
The results are summarized in Figure 7.
Comparing Figure 7 with Figure 5, it can be seen that a single gap, which is 5 years in our case, makes slight changes in the accuracy of the simulation in case 1, 2, and 3. In case 4 and case 5, moderated length changes are estimated less accurate in comparison with the results in Figure (5). The estimated spectrum of the time series with gaps were similar to those of Figure 6, therefore, they are not shown here. When the missing values occur with a periodic occurrence, the results of spectral analysis might completely change.
To cover this aspect, we simulated a time series that exhibits 70 days of a year without any observation. Therefore, the mentioned gaps were considered for the time series of Figure 4. The results of spectral analysis indicated that the long periods can be extracted almost with the same accuracy of Figure 6. In the case 5 (with 85% missing values), however, the results of the LSSA became significantly biased. Moreover, there were several artificial frequencies emerging in the spectrum with high spectral values, making detection of the real periodic
component more difficult. For brevity, the results are not shown here.
4-4. Detection of nearby frequencies
Detection of the nearby frequencies is important for analyzing SSH time series since there are other different physical movements of water level at sea, which might be represented by close spectral structure, e.g. the annual tidal frequency and annual fluctuations caused by river flows. Our simulation results show that detecting the nearby frequencies depends strongly on both the frequency of interest as well as the length of time series. For instance, with a 16-year SSH time series, the two low frequency fluctuations of 0.00273786 and 0.00281690 cycle/day were not separable, however, one can separate them when the length of observation was 30 years. Detected numerical values of the frequencies were 0.002717 and 0.002838, respectively. For the nearby semi-annual and monthly frequencies, the situation was different. Using time series with the length of 16 years, the nearby semi-annual frequencies of 182.6-1 and 188-1 cycle/day, as well as those of monthly "32-1 and 31.82-1" cycle/day were successfully separated in the power spectrum. Estimated frequencies corresponding to the nearby semi-annual frequencies were 181-1 and 188.5-1 cycle/day, and corresponding to the nearby monthly frequencies, they were "32-1 and 31.81-1
"cycle/day.
5. Implementation of LSSA and LS-HE on the observed SSH time series over the Persian Gulf and Oman Sea
5-1. Building the SSH Time series
Considering the temporal coverage of the two satellite altimetry missions Topex/poseidon and Jason-1, SSH time series with the length of ~16 years (1992-2008) can be built. From 254 satellite passes, twelve ground tracks over the Persian Gulf and Oman Sea were selected in this study (see Figure 9).
Figure
Figure
SSH of (Sh Jason differ them ideal conta 1992 outlie time imple samp over t 5-2. S From series paper gener altim For e
e 7. an overview The time se of 5 years.
e 8. Topex/pos SSH meas
time series a harifi et al., 2 n-1 observa
rent mission as a simple SSH time ain more than
to 2008. H ers, blunders
series w ementation, ple points of
the Persian G Spectral An m the availa s are selecte r (More th rated using metry in the P
example, fou
Least Squa
w of the LSSA eries of case 1 t
eidon and Jaso surements are b
are compute 2013), with ations. The
s was remov shift functio series, in o n 576 observ However, du s and missing were deriv 626-time s f more than Gulf and Om
alysis based able time se ed to be pr han 600-tim observation Persian Gulf ur time series
ares Technique
A results when i to case 5 are sim
n-1ground trac uilt.
d similar to also conside bias betw ved by adjus on. Therefore our case, sho vations cove ue to the no
g values, sho ved. In series with
530 were b man Sea.
d on LSSA eries, four t
resented in me series w
ns of satel f and Oman S
s are selecte
es for Extractin
it is applied to f milar to those o
ks over the Per
that ring ween sting e, an ould ring oise, orter this the built
time this were lites Sea.
ed at
ng Water Leve
five simulated of Figure 7; how
rsian Gulf and
several poin Oman Sea an article). The series are sh in Table 1.
time series positions of focus on frequencies Persian Gul motivated b attempts th estimate tid observations Sea. The re frequencies Table 2 sum the main ce from astro 1993). The in Table 2
el Fluctuations
time series with wever, they als
Oman Sea. 626
nts in the P nd are presen e positions hown in Figu Figure 11 ordered w Table 1. In
the LS and their lf and Oma by the fact hat have b dal amplitu s over the Pe esults corres are reported mmarizes the
elestial tidal onomical s
aliased valu 2 are comp
s in the …
h missing value o contain a pro
6 time series w
Persian Gulf nted in the te of the selec ure 9 and sum 1 shows the with respect
this section, SSA-extracte amplitude f an Sea. Thi
that there a been undert udes from ersian Gulf an ponding to
in Tables 2 t theoretical v
frequencies studies (Ca es of the fre puted follow
111
es and gap.
olonged gap
with reliable
f and the ext of the cted time mmarized
selected t to the , we only d tidal from the s aim is are fewer taken to
satellite nd Oman the tidal to 4.
values of s derived artwright, equencies wing the
112
empirica while c 9.915625 and thei and Om series of 3 and frequenc time seri tidal co annual, with M
Figure 10
Jo
al formula i considering
5 days. The ir amplitude man Sea, co
f Figure 9, a 4, reports cies corresp ies of Figure omponents, m
semiannua , S (solar
Figure 9
0. Time series o
ournal of the E
in (Sharifi e the sampli e main tidal s over the P omputed fro are summariz the amplitu ponding to
e 11. Among mean ampli al solar t
and lunar
9. Locations of t
of the four selec
Earth and Spac
et al., 2013 ing rate o l frequencie Persian Gul om the time zed in Table ude of tida the selected g 22 detected itude of the tides, along semi-diurna
the four selecte
cted sample poi
ce Physics, Vol
) f s f e s al d d e g al
com com sho freq Sea
Tab
Stati Lat Lon
d sample time s
ints. Position of
l. 45, No. 4, Wi
mponents) a mponent) we ould be con
quencies of a.
ble 1. Positions 9, subject ion. no Poin titude 29.8 ngitude 49.6
series, selected
f the time series
inter 2020
and k (lu ere greater th
nsidered as the Persian
of four selected ted to spectral a nt-1 Point-2 808 25.502 651 53.203
for spectral ana
s are according
uni-solar d an 1 cm, thu the main n Gulf and O
d time series in analysis.
Point-3 24.515 58.416
alysis.
ly shown in Fig
diurnal us they tidal Oman
n Figure
Point-4 26.645 56.580
gure 8.
Least Squares Techniques for Extracting Water Level Fluctuations in the … 113
Table 2. Main tidal components and their aliased frequencies. The first column indicates the sign of tidal frequencies, the third column lists the real value of the tidal frequencies derived from astronomical studies (c.f. [5]), the fourth column shows the corresponding real period, fifth column lists the computed frequencies with respect to 9.915625 days sampling rate, and the sixth column lists their corresponding aliased periods.
Sign Description Frequency
(cycle/day) Period (day) Aliased Frequency (cycle/day)
Aliased Period (day) SA Annual Solar Tide 0.0027378 365.2599774 0.0027378 365.2599774 SSA Semi-annual Solar Tide 0.0054758 182.6211838 0.0054758 182.6211838
MSM Solar monthly 0.0314347 31.8119340 0.0314347 31.8119340 MM Lunar Monthly 0.0362916 27.5545492 0.0362916 27.5545492 MSF Luni-solar Synodic
Fortnightly 0.0677264 14.7652927 0.0331245 30.1891006 MF Lunar Fortnightly 0.0732022 13.6607901 0.0276487 36.1680374 Q1 Larger Lunar Elliptic Diurnal 0.8932441 1.1195148 0.0144143 69.3755724
O1 Principal Lunar Diurnal 0.9295357 1.0758059 0.0218773 45.7094201 P1 Principal Sonar Diurnal 0.9972621 1.0027454 0.0112472 88.9109452 S1 Solar Diurnal 1.0000001 0.9999999 0.0085092 117.5202863 K1 Luni-Solar Diurnal 1.0027379 0.9972696 0.0057714 173.2683888 O2 Side, half-Daily Sun 1.8590714 0.5379030 0.0437547 22.8546975 N2 Larger Lunar Elliptic Semi-
diurnal 1.8959820 0.5274312 0.0201857 49.5400487 M2 Principal Lunar semi-diurnal 1.9322736 0.5175250 0.0161060 62.0888194
S2 Principal Solar Semi-diurnal 2.0000000 0.5000000 0.0170186 58.7592316
K2 Luni-Solar Semi-diurnal 2.0054758 0.4986348 0.0115428 86.6341944 M3 Lunar ter-diurnal 2.8984104 0.3450167 0.0262665 38.0712527 M4 Shallow water over tides of
principal lunar 3.8645472 0.2587625 0.0322119 31.0444328 S4 Shallow water over tides of
principal solar 4.0000000 0.2500000 0.0340372 29.3796365 M6 Shallow water over tides of
principal lunar 5.7968208 0.1725084 0.0483179 20.6962834 S6 Shallow water over tides of
principal solar 6.0000000 0.1666667 0.0497951 20.0822785 M8 Shallow water eighth diurnal 7.7290945 0.1293813 0.0364271 27.4520769
Table 3. Extracted tidal frequencies from sample points of Figure 10. Frequencies are reported in (cycle/day).
Sign Point-1 Point-2 Point-3 Point-4 Sign Point-1 Point-2 Point-3 Point-4 SA 0.0027459 0.0027289 0.0027968 0.0027798 O2 0.0436363 0.0436363 0.0436533 0.0436363 SSA 0.0055495 0.0055325 0.0055325 0.0055155 N2 0.0201347 0.0201518 0.0201688 0.0201859 MSM 0.0312703 0.0313385 0.0314067 0.0314920 M2 0.0161113 0.0160943 0.0161113 0.0161113 MM 0.0363017 0.0360971 0.0363590 0.0363700 S2 0.0170318 0.0169636 0.0170489 0.0170318 MSF 0.0332487 0.0332487 0.0330099 0.0330781 K2 0.0115948 0.0114756 0.0115437 0.0115608 MF 0.0277059 0.0275012 0.0275865 0.0276377 M3 0.0261540 0.0258641 0.0261199 0.0263245 Q1 0.0144068 0.0144238 0.0143727 0.0143897 M4 0.0324130 0.0320719 0.0322083 0.0322083 O1 0.0218909 0.0218909 0.0218909 0.0218909 S4 0.0340503 0.0340674 0.0340674 0.0340844 P1 0.0112541 0.0112370 0.0112541 0.0112370 M6 0.0482761 0.0483443 0.0482931 0.0483443 S1 0.0085115 0.0085625 0.0086136 0.0085966 S6 0.0497261 0.0497772 0.0498796 0.0497431 K1 0.0057707 0.0057707 0.0057707 0.0057707 M8 0.0359342 0.0364723 0.0363529 0.0362368
114
5-3. Spe Before d should univariat investiga section series of This mea pseudo-t observat observat low fre leakage The ma negligibl annual, s (the effe annual a Howeve effect impleme time ser temporal series. T scaled b series to these c spectrum which univariat the mult HE is di For a b signals, daily, an Figure 1
Figure 11
Jo
ectral Analy discussing th
note that, te LS-HE ation of mu
considers a f SSH (Mean
ans that the A tide. With t tions simila tions in offs equency var
to the LS-H agnitude of le for tidal semi-annual ect of leaka and semi-ann r, this pheno on sma enting the m
ries were te l mean value The centered by the stan
be standard circumstance m would sim
is a linea te LS-HE s tivariate LS isplayed in F better compa
a magnific nnual and an
2.
1. a) Spectrum Spectrum of
ournal of the E
sis based on he results of since the
are equiv ultivariate LS all the 626 ning Pseudo- Altimeter tim the differenc ar to the hore.). Exist riations mig HE extracted
the leakage frequencies and seasona age phenom nual period i omenon has ll periods multivariate
emporally c e was reduce d time serie ndard deviat dized and uni s, multivar mplify as E ar combinat spectrums. T -HE and un Figures (11-a arison betwe cation signa nnual-half is
(a)
of frequencies f frequencies of
Earth and Spac
n LS-HE f LS-HE, we
LSSA and valent. Ou S-HE in thi virtual time -Tide Gauge me series is a
ce, we have Tide-Gauge ting of those ght cause a frequencies is howeve s as well a al frequencie enon on the is negligible
a significan s). Before
LS-HE, the centered, i.e
ed from time es were then tion of time it-less. Unde riate LS-HE Equation 16 tion of the The result o
nivariate LS a) and (11-b) een the two al daily-half
displayed in
s of multivariat univariate LS-H
ce Physics, Vol
e d ur s e e:
a e e e a s.
r s s e e.
nt e e e.
e n e r E 6, e f - ).
o f, n
Com uni use How freq freq seri com cor epo freq pur extr ana In resu In cha com assu met sam amp can and test imp uni Am Sim (20 The Tim Sea ind are
te LS-HE appl HE applied on
l. 45, No. 4, Wi
mparing fre variate and m e one time se
wever, som quencies m quencies may ies of times mbining se rresponding f ochs, one c quency dete rpose of t
raction usin alysis.
the LSSA ults refer to fact, all of anges, whic mmon sou umption in thod is that me frequenc plitudes and n be generali d Oman Sea ted in th plementing
variate and miri-Simkooe mkooei (200 017).
e main tidal me Series in a are prese dicated 60 si summarized
lied on SSH tim SSH time serie
inter 2020
quency extr multivariate eries to extra metimes th may be to
y be small e s can be de everal ser frequency oc can increas ection. Ther this researc g univariate and the u the single o time series ch have urces. Th n the mul t in several cies presen phases. The zed to the w . Results are he multivar
the statistic d multivaria ei and Asga
07), Amiri- frequencies, n the Persian ented the d ignificant fr d in Table 5.
(b)
me series and s and tested sta
raction using methods can act the frequ he number o high or enough that o
etected. Thu ries when
ccurs in the e the pow refore, the ch is freq e and multiv univariate LS
observation demonstrate originated he fundam
ltivariate L l time serie nt with dif erefore, the r whole Persian e also statist riate case.
al tests for ate LS-HE ari (2012), A
-Simkooei e , based on th n Gulf and O derived spe requencies, w
tested statistica atistically.
g two n only uency.
r of r the only a us, by
the same wer of
main quency variate S-HE, point.
e SSH from mental LS-HE s, the fferent results n Gulf tically For r both
, see Amiri-
et al.
he 626 Oman ectrum which
ally. b)
Figur
Table
Sig SA SSA MSM
MM MS
MF Table period of ch [day]
1750.8065 1269.6996 1047.1476 677.2367 601.5809 541.0684 491.5759 446.9415 409.7146 364.1820 262.0998 239.6834 192.5925 186.4834 180.7495 173.2891 166.4194 117.4887 112.7509 96.92740
re 12. magnifica 4. Extracted Frequencie
n Alia
Frequ (cycle A 0.002 A 0.005 M 0.031 M 0.036 F 0.033 F 0.027 5. 60 significan hanges Fr
[c
5641 0
6009 0
6005 0
7535 0
9809 0
4803 0
9325 0
718 0
6016 0
0676 0
441 0
4004 0
023 0
4825 0
450 0
677 0
4269 0
7903 0
9923 0
052 0
Least Squa
ation signal of s tidal frequenc es are reported ased uency e/day) S
27459 O
55325 N
13555 M
62506
32316 K
74160 M
nt frequencies e requency of
changes cycle/day]
0.0005712 0.0007876 0.0009550 0.0014766 0.0016623 0.0018482 0.0020343 0.0022374 0.0024407 0.0027459 0.0038153 0.0041722 0.0051923 0.0053624 0.0055325 0.0057707 0.0060089 0.0085114 0.0088691 0.0103170
ares Technique
spectrum of fre cies of multiva in (cycle/day).
Sign Alia Frequ (cycle O2 0.044 N2 0.020 M2 0.016 S2 0.017 K2 0.011 M3 0.026 extracted of mu period of c [day 88.8568 86.4994 77.1718 74.2425 69.4940 65.1710 64.3842 63.6853 63.0014 62.0682 61.1622 60.5311 58.7135 51.9991 50.1753 49.5815 48.9606 46.5882 45.6810 45.1882
es for Extractin
equencies of LS ariate LS-HE
ased uency e/day) Sig 43698 Q 01688 O 61113 P 70318 S 15608 K 64268 ultivariate LS-H changes y]
F
8160 4894 8132 5116 0375 0508 2958 3601 4284 2245 2492 1415 5803 1749 3457 5178 6698 2494 0504 2794
ng Water Leve
S-HE applied on applied on SS
gn Alia
Frequ (cycle 1 0.014 1 0.021 1 0.011 1 0.008 K1 0.005
HE for 626 time Frequency of
changes [cycle/day]
0.0112540 0.0115608 0.0129581 0.0134693 0.0143897 0.0153442 0.0155318 0.0157023 0.0158728 0.0161114 0.0163500 0.0165205 0.0170317 0.0192311 0.0199302 0.0201686 0.0204244 0.0214648 0.0218909 0.0221298
el Fluctuations
n SSH time seri SH time series
ased uency e/day) Si
43897 M
18909 S
12541 M
85115 S
57707 M
series in Persia period of
[da 39.101 36.497 35.374 32.728 31.015 29.353 27.585 26.240 25.846 23.800 23.448 22.537 21.615 21.441 21.262 20.988 20.809 20.655 20.333 20.110
s in the …
ies and tested st s and tested s
ign Alia Frequ (cycle M4 0.032 S4 0.034 M6 0.048 S6 0.049 M8 0.036
an Gulf and Om f changes ay]
19403 78205 48777 89862 50451 36008 57826 00202 66994 05614 83384 78643 54456 15156 26469 85916 97787 57978 33910 01790
115
tatistically.
statistically.
ased uency e/day) 22424 40844 84296 97261 63032
man Sea.
Frequency of changes [cycle/day]
0.0255741 0.0273988 0.0282686 0.0305539 0.0322425 0.0340669 0.0362503 0.0381098 0.0386892 0.0420150 0.0426476 0.0443695 0.0462642 0.0466374 0.0470301 0.0476449 0.0480538 0.0484121 0.0491811 0.0497265 f
116
5-4. Pre extracte In the fo three (Astrono the tide this rese LSHE frequenc altimetry frequenc a list of based on principal through based on above-m frequenc are three The tota surface i
Figure 13
Jo
edictions ba ed frequency ollowing, to c
samples omical-LSSA
is predicted earch, two have been cies in the y observatio cies through f important f n each meth l frequenci
astronomic n the output mentioned
cies of the as e frequency l al time interv
in this resear
3. a) A compari frequency of samples of ex
ournal of the E
ased on thr y
check the eff extracted A-Multivaria d for a period
methods of used to time series ons. After ex
the above t frequencies i hod. On the ies are als cal methods
t frequencie methods tronomical m lists available val of observ rch is from 19
ison between th LSSA. b) A co xtracted frequen
Earth and Spac
ree samples fectiveness o
frequency te LSHE) d of time. In f LSSA and extract the s of satellite xtracting the wo methods is considered e other hand
so provided s. Therefore
s of the two and the method, there
e.
vation of sea 992 to 2008.
he data known omparison betw ncy of astronom
ce Physics, Vol
s f y ), n d e e e s, d d, d e, o e e a
In per surf usin was the wit 201 The sam in F wer mu Lis take firs met astr m.
freq acc
(a)
(b) and predicted ween the data k mical, based on
l. 45, No. 4, Wi
order to pre riod 2006 to
face observa ng three fre s conducted sea surface th actual d 17).
e results of p mples of extr
Figure 13. T re extracted
ltivariate LS t is also us en for these st method ha
thod has ronomical fr
the results quency of mu curacy and re
data for the po known and pred
samples of extr
inter 2020
edict the se 2008, the p ations in the equency list d. According e is calculat
ata (Amiri- predicted tid racted freque Two frequen from both m SHE. Astron sed. Based o
three freque as RMSE 0.1
RMSE 0.0 requencies h
showed th ultivariate L eliability are
oint-1, based on dicted data for racted frequenc
ea surface fo prediction fo e two-year p
mentioned gly, the heig ted and com -Simkooei e des based on ency are pres ncy samples methods LSSA
omical Freq on the pred ency sample 101 m, the s 86 m and has RMSE hat to obtai SHE method higher.
n samples of ex the point-1, ba y of LSHE.
or the or sea period,
above ght of mpared
et al., n three
sented listed A and quency diction es, the econd d use
0.093 in the ds, the
xtracted ased on
