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The real world phenomena being observed by sensors are generally non-stationary in nature. The classical linear techniques for analysis and modeling natural time-series observations are inefficient and should be replaced by non-linear techniques of whose theoretical aspects and performances are varied. In this manner adopting the most appropriate technique and strategy is essential in evaluating sensors’ data. In this study, two different time-series analysis approaches, namely least squares spectral analysis (LSSA) and wavelet analysis (continuous wavelet transform, cross wavelet transform and wavelet coherence algorithms as extensions of wavelet analysis), are applied to sea-level observations recorded by tide-gauge sensors, and the advantages and drawbacks of these methods are reviewed. The analyses were carried out using sea-level observations recorded at the Antalya-II and Erdek tide-gauge stations of the Turkish National Sea-Level Monitoring System. In the analyses, the useful information hidden in the noisy signals was detected, and the common features between the two sea-level time series were clarified. The tide-gauge records have data gaps in time because of issues such as instrumental shortcomings and power outages. Concerning the difficulties of the time-frequency analysis of data with voids, the sea-level observations were preprocessed, and the missing parts were predicted using the neural network method prior to the analysis. In conclusion the merits and limitations of the techniques in evaluating non-stationary observations by means of tide-gauge sensors records were documented and an analysis strategy for the sequential sensors observations was presented.

The surface of the sea deforms continuously. Its level, measured relative to an arbitrary datum, is called ‘sea level’ and changes with time and is the most obvious indicator of ocean changes. Changes in sea level are greater in the shallow waters near a coast than in the open sea, and, because a large fraction of the human population resides in coastal areas, variations in sea level have aroused interest for a long time. Knowledge of the near-shore sea-level variations is of great importance for safe navigation, and sea-level observations provide valuable input to ocean science and to geodynamic and geoscience applications [

Time-series analysis is a fundamental issue in evaluating sea-level observations and identifying the tidal components of sea-level changes, as in many other fields of empirical research [

LSSA is a least squares estimation method for computing variance- and power-spectra and suggested by [

Wavelet analysis is another method that can be used to analyze time series that contain non-stationary powers at many different frequencies [

Although CWT is a common tool for analyzing localized intermittent oscillations in time series, it is very often desirable to examine together two time series that are expected to be linked in some way. In particular, it may be useful to examine whether regions in time-frequency space with large common power have a consistent phase relationship and therefore are suggestive of causality between the time series [

The results of this study confirmed the applicability of the employed techniques in analyzing and investigating the sea-level variations recorded by tide-gauge sensors. The LSSA is a very useful technique in spectral analysis for inspecting and clarifying periodic signals hidden in noisy time series with trends. In the prediction of the missing data in sea-level series, the neural-network method worked well, considering the quality measures of the prediction. Because natural series, like sea-level observations, are generally non-stationary, the ability of neural networks to model non-linear processes without any a-priori assumptions about the generating processes provides an advantage in prediction. The significant periodicities revealed by LSSA were confirmed in the results of the wavelet analysis. Furthermore, the correlation between the time series of the two tide gauges was explained using the wavelet tools. Wavelet is a strong method for the time-frequency analysis of non-stationary sequential data and is suggested for investigating sea-level changes.

TUSELS presently consists of a data center in Ankara and a series of operational tide gauges located along the surrounding Mediterranean-, Marmara-, Aegean- and Black-Sea coasts of Turkey (see

Sea-level monitoring studies in Turkey began in 1930s, and the General Command of Mapping (GCM) has the responsibility of establishing and operating the TUSELS tide gauges and distributing their data. The activities of transferring, quality control and analysis of tide gauge-data are carried out at the data center in Ankara. In 1998 and 1999, the tide gauges were modernized and all existing analogous floating type tide-gauge sensors in stilling wells were upgraded to digital and automatic devices by GCM in order to meet the GLOSS (Global Sea-level Observing System) standards [

After modernization the tide-gauge stations, [

In the numerical tests in this investigation, the monthly data from the Antalya-II and Erdek tide gauges, downloaded from [

The autocorrelation functions of sea-level observations recorded at Antalya-II and Erdek (see

In the results of the GCM’s harmonic analysis [

A set of observations or results obtained from a physical process, arranged in a specific manner, is called a data series. If the data series has a chronological ordering, it constitutes a time series [

Spectral analysis techniques permit the identification of periodicities or hystereses in the time-series and their decomposition into periodic signals. In the cases of measurements of small amplitudes and high noise-to-signal ratios, reflecting the superposition of different signals, spectral-analysis techniques provide the best results [

However, the frequency-domain analysis with traditional spectral techniques assumes that the underlying processes are stationary in time, but many natural signals are non-stationary because of their irregular or time-limited features. In this case, linear analysis approaches, such as Fourier transforms, may not be practical and efficient for analyzing these signals. Therefore, non-linear analysis approaches should be adopted to study non-stationary real-world phenomena. Currently, many advanced analysis techniques, such as wavelet transforms, are widely used to study non-linear behavior of time series [

From an application point of view, unlike the LSSA method, the wavelet transforms accept regularly sampled continuous data as an input for efficient analysis and reliable results. Therefore an unequally sampled time series with data voids requires pre-processing before analysis with wavelet-transform algorithms. In this study, the neural-network method was used to predict the missing values in sea-level signals from the tide-gauge-sensors records (see the missing data in the time-series plots in

In LSSA, an observed time series is considered to be a function of time _{i}_{i}_{i}_{f}

In the least-squares method, the model parameters are determined to minimize the difference between

In the projection theorem,

In spectral analysis, the hidden periodicities, which are expressed in terms of cosine and sine base functions, are inspected. Therefore, if a set of spectral frequencies (_{i}

Let _{1}_{i}_{2}_{i}^{T}_{i}t_{i}t_{i}

_{i}_{i}

It is obvious from

In summary, the observed time series may include trigonometric base functions (see

The sea-level observations were analyzed using LSSA, and the hidden periodicities of the sea-level changes in the investigated span were clarified. The periods, frequencies (cycle/year), amplitudes and phases with their root-mean square-errors and percentage variance levels (%var: a ratio indicating how much of the signal

The neural-network method, based on learning events using available samples

In the heuristic algorithm of this method, the basic element of a neural network is a processing node (

The processing nodes constitute a set of fully interconnected layers, except that there are no interconnections between nodes within the same layer in the standard feed-forward back-propagation algorithm. The structure of a typical MLFB-NN includes three types of layers: input, hidden and output (as seen in

The output of the model (y) with a single hidden and output neuron can be represented by:

A learning algorithm is the most critical part of a neural-network method. Among a number of learning strategies, the feed-forward back-propagation learning algorithm, introduced by [

When training with the LM method, the increment of weights Δ

The performance of the neural-network model is evaluated in terms of the correlation coefficient _{i}_{i}

Prior to the wavelet analysis of sea-level data, the missing data in the time series (see

Wavelet analysis involves a transform from a one-dimensional time series to a diffuse two-dimensional time-frequency image for detecting localized and quasi-periodic fluctuations using the limited time span of the data [^{−2} and ensures that the edge effects are negligible beyond this point [

The CWT of a time series is its convolution with the local basis functions, or wavelets, which can be stretched and translated with flexible resolution in both frequency and time. The CWT of the time series _{X,ψ}^{2}. The complex argument of _{X,ψ}_{0} is the dimensionless frequency and _{0} = 6) (see

The XWT spectrum of two time series (X and Y) with wavelet transforms (W_{X} and W_{Y}) for the analysis of the covariance of two time series is defined by [_{XY}_{XY}_{XY}

The WTC is a measure of the intensity of the covariance of the two series in time-frequency space, unlike the XWT power, which reveals areas with high common power. The WTC of two time series is defined by [^{2}(_{scale}_{time}_{1}_{2}

The time-series data filled by the NN prediction (see in

The XWT of the two time series, Antalya-II and Erdek, is displayed in

Similar to that exploited by the XWT, an alternate way of investigating the phase difference of sea-level variations between the two tide-gauge records was explored through WTC. Regarding applications, whereas the XWT power reveals the areas with high common power of CWTs of two time series, the WTC can show the degree of coherence of the XWT in the time-frequency space. The WTC of the sea-level data sets is shown in

In this study, we applied LSSA and various wavelet-transform techniques, namely CWT, XWT and WTC, to time-frequency analyses of monthly sea-level variations recorded at the Antalya-II (36.8°N, 30.6°E) and Erdek (40.4°N, 27.8°E) tide gauges of TUSELS. The LSSA results clarify the amplitudes, phases, and percentage variance levels of the hidden periodicities. In the LSSA results, the 19-year sea-level observations at Antalya-II reveal significant annual (period of T = 12 month with 8.9 ± 0.4 cm amplitude), semiannual (period of T = 6 month with 2.4 ± 0.4 cm amplitude) and terannual (period of T = 4 month with 1.8 ± 0.4 cm amplitudes) cycles. The spectral analysis of the 10 year-tide gauge records at Erdek shows that the sea-level variations have significant annual (with an amplitude of 5.0 ± 0.5 cm) and semiannual cycles (with an amplitude of 1.9 ± 0.5 cm). The relative mean sea-level changes at Antalya-II and Erdek are found 7.9 ± 1.1 mm/yr and 2.8 ± 0.9 mm/yr, respectively, from the LSSA. Whereas the trend calculated for Antalya-II confirms the harmonic-analysis results of GCM reported by [

The neural-network method was used to preprocess the sea-level data sets, and the missing parts in the time series were predicted with a feed-forward back-propagation algorithm. In the end, the quality of the prediction, as evaluated

The time series preprocessed with the neural network were analyzed with wavelet transforms to observe the localized intermittent periodicities as high-power regions in the spectra with CWT by expanding the time series into time-frequency space and to inspect the common power and relative phase of the two time series in time-frequency space using XWT. We also used WTC between two CWTs to find significant coherence in the parts having low common power between the time series. The CWTs of the sea-level data sets reveal annual, semiannual and terannual periodic cycles for Antalya-II and Erdek. In the CWT images, the large-scale periodicities (annual cycles) are recognized as the full data span, whereas the smaller-scale oscillations (semiannual and terannual cycles) are partly along the spectra. The results from the CWTs of the sea-level variations confirm the LSSA findings.

The XWT of the two CWTs shows that the Antalya-II and Erdek time series has a high common spectral power at the annual-cycle periodic belt in full span and partly at the semiannual cycle. Considering the relative phase relationships derived from the XWT, the sea-level changes recorded at the Antalya-II tide gauge lead the sea-level changes recorded at the Erdek tide gauge by 20° pointing straight-up arrow (nearly in-phase). These results on the coherence of the Antalya-II and Erdek sea-level variations were confirmed and strengthened by the WTC results.

In the results of this study, we see that the LSSA has strong features in the frequency-domain analysis of the time series, especially in evaluating unequally spaced data with gaps, spikes, datum shifts and trends, such as sea-level observations. However, when series preprocessing is required for analysis in other methods (such as the wavelet-transform methods here) the neural-network method works well for predictions. As a principle advantage of the neural-network method that it is capable of approximating any continuous function, so adopting a hypothesis about the underlying structure is not required [

The author thanks S. Pagiatakis for his valuable time and information on the spectral-analysis method, A. Grinsted for the wavelet-tools Matlab codes used in this research, and the anonymous reviewers and the editor for the detailed comments, which helped improving the quality of the manuscript.

TUSELS tide-gauge stations in Turkey [

The structure of a digital TUSELS tide-gauge station [

The sea-level data of Antalya-II and Erdek tide gauges for the considered time span:

Plots of the autocorrelation functions of the sea-level observations for the

Original sea-level data (

Example of neural network applications to time series predictions (e.g., using a (4-4-1)-multi-layer with four input neurons for observations

Time series of Antalya-II and Erdek tide gauges by fill by MLFB-NN method.

Scatter plots of target and output data: the correlations between the observations and NN outputs for the

Morlet wavelet function, depending on the changes in translation (

CWT power spectra of the monthly sea-level observations at the Antalya-II and Erdek tide gauges. The thick black contours indicate the 95% confidence level, and the region below the thin solid line indicates the cone of influence (COI), beyond which edge effects may distort the picture.

Specifications of Antalya-II and Erdek tide gauges [

| ||
---|---|---|

Station name | ||

| ||

Location (latitude, longitude) | 36°50′N, 30°37′E | 40°23′N, 27°51′E |

PSMLS country/station code | 310/052 | 310/038 |

Spanning of the used data | 1986–2005 | 1995–2005 |

Acoustic gauge sensor | Aquatrak 4100 | Aquatrak 4100 |

New acoustic systems installation year | 1998 | 1999 |

The LSSA results of the sea-level data of Antalya-II and Erdek tide gauges.

Periodic constituent | ANNUAL | 1.000 | 0.089 | 0.004 | 95.853 | 0.255 | YES |

Periodic constituent | SEMI-ANNUAL | 0.500 | 0.024 | 0.004 | 326.282 | 0.249 | YES |

Periodic constituent | TER-ANNUAL | 0.333 | 0.018 | 0.004 | 358.778 | 0.250 | YES |

Periodic constituent | ANNUAL | 1.000 | 0.050 | 0.005 | 113.433 | 0.286 | YES |

Periodic constituent | SEMI-ANNUAL | 0.500 | 0.019 | 0.005 | 245.112 | 0.284 | YES |

Periodic constituent | TER-ANNUAL | - | - | - | - | - | - |

Summary of the adopted parameters in the NN prediction of the sea-level time series.

Feed-forward backpropagation network | ||

Each layer only receives inputs from previous layers | ||

Changes in a network’s weights and biases are due to the intervention of Levenberg-Marquardt algorithm | ||

weights and biases are adjusted by error-derivative vectors backpropagated through the network | ||

Function that maps a neuron’s (or layer’s) net output | ||

Mean Square Error (MSE=E^{T}E/N, RMSE=sqrt(MSE)) |