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Accelerometers, which can be installed inside a floating platform on the sea, are among the most commonly used sensors for operational ocean wave measurements. To examine the non-stationary features of ocean waves, this study was conducted to derive a wavelet spectrum of ocean waves and to synthesize sea surface elevations from vertical acceleration signals of a wave buoy through the continuous wavelet transform theory. The short-time wave features can be revealed by simultaneously examining the wavelet spectrum and the synthetic sea surface elevations. The

Wind-generated gravity waves are among the most significant phenomena on the ocean. However, the mechanics and features of these types of waves are highly complex and random because of the combined influences of meteorological, hydrological, oceanographic and topographical factors. Studies of wind-generated gravity waves have continued ever since their contributions to water wave mechanics were recognised more than a century ago. To increase our practical knowledge of wind-generated gravity waves field measurements must be performed, but most measurement sensors are only suitable for use nearshore or in shallow water areas. Apart from remote sensing devices, moored buoys and vessels are the only platforms suitable for wave measurement in deep water areas [

A wave buoy floats on the sea surface and moves up and down with the waves. Different sensors, such as GPS and accelerometers, have been developed and installed on buoys for measuring the waves. Although some studies have proved the practicability of GPS sensors for measuring waves [

To obtain a spectrum from acceleration records, a Fourier transform of the spectral transform has often been used in the past. The theory of the Fourier transform assumes the signal is stationary. As a result, the wave spectrum estimated by Fourier transform provides enough information to describe the sea-surface elevations as a stationary and Gaussian process. However, in Nature, most real signals are non-stationary, as are wave signals. Liu [

The application of the continuous wavelet transform has become increasingly common since its inception in the early 1980s. Compared to the Fourier transform which is based on the concept of frequency, the continuous wavelet transform is based on the concept of time-frequency localization. Wavelet transforms are capable of obtaining orthonormal basis expansions of signals using time-frequency atoms that enable us to localize the signals in time and frequency domains. Due to the feature of time-frequency localization, the wavelet theory has been applied successfully to solve various geophysical problems [

Up to now, most of the related studies on wavelet spectrum have been applied to sea elevation signals. However, most of the

Based on the continuous wavelet transform (CWT) theory, the acceleration signal can be broken into various wavelets that are scaled and shifted versions of a pre-chosen mother wavelet function. The acceleration signal _{c}_{b}_{,}_{a}_{AC}^{2}, derived from _{b}_{,}_{a}_{b}_{,}_{a}

In most cases, this condition may be reduced to the (only slightly weaker) requirement that

The relationship between the wavelet function and the mother wavelet function in the Fourier (spectral) space can be expressed as follows:
_{AC}

To implement

In these equations, _{0} is a constant that forces the admissibility condition, as shown in _{a}_{0} is also the peak frequency of the mother Morlet function in the frequency domain. After transformation, the new peak frequency of the Morlet wavelet function becomes _{0} and

_{AC}^{2} can be expressed as |_{AC}^{2}, which represents the spectral information at different positions (times) _{AC}^{2}, the spectral information of the sea surface elevations |_{η}^{2} can be obtained using the equations discussed above if the acceleration signal _{c}

To estimate the time series of the sea surface elevations _{η}^{2} is the wavelet scalogram of the sea surface elevations, which is related to |_{AC}^{2}. Based on the theory discussed above, the wavelet scalogram of the acceleration signal |_{AC}^{2} can be estimated. To implement _{η}^{2}. Tucker [_{AC}^{2} and the spectrum of the sea surface elevations |_{η}^{2}:

Because the Morlet wavelets are Gaussian-modulated sine/cosine functions,

As discussed above, we can transform the scalogram of acceleration signal |_{AC}^{2} into the wavelet spectrum |_{AC}^{2} based on _{η}^{2} which represents the frequency spectra at different time

Then we can transform |_{η}^{2} into a scalogram of sea surface elevations |_{η}^{2}. Finally we can synthesize the sea surface elevations _{η}^{2} through

To verify the numerical accuracy of computing

The estimated wavelet spectrum |_{η}^{2}, calculated by the ^{−4} on |_{AC}^{2}. The very-low-frequency bins mentioned in this paper are the frequency bins that are lower than the common frequency bins of wind-generated waves. Many studies considers this energy density in the very-low-frequency band as noise [

To verify the accuracy of estimated wavelet spectra of sea surface elevations, which are derived from

In _{c}^{2} are wavelet spectra of 1,500 time-series records of observational sea surface elevations, and |^{2} are estimated wavelet spectra of sea surface elevations derived from 1,500 sets of synthetic acceleration data. Note that the synthetic wave acceleration data were derived from the observational sea surface elevations records by double forward difference. To convert the wavelet spectra of synthetic accelerations into wavelet spectra of sea surface elevations |^{2}, the transfer function in _{x}_{y}^{2} and |^{2}, respectively; _{x}_{y}^{2} and |^{2}, respectively.

The box-whisker plot in ^{−4} in ^{−4} at very-low-frequency bins are more evident than electronic noise when the wavelet spectrum is estimated by

Since the estimated wavelet spectra of sea surface elevation records have be derived, the synthesized data sets of sea surface elevations can be also imitated through the inverse CWT algorithm of the estimated scalogram of the sea surface elevation record. It is necessary to measure the accuracies of synthetic sea surface elevation records. To obtain the synthetic sea surface elevations, _{η}_{d}_{s}

The obvious normalised differences shown at very-low-frequency bins indicate that a high-pass filter is necessary to eliminate the noises caused by the effect of transfer function ^{−4} when we apply _{e}

The results of this analysis indicated that to obtain an accurate estimate of the sea surface elevations from the wavelet spectrum, the cut-off frequency should not be less than 16/512 Hz. Finally 22/512 Hz was chosen as the cut-off frequency for our following treatments because its corresponding root-mean-square-error of the 75th percentile of data set is lowermost in

To evaluate the width of the marginal area, we need to consider the windows of the wavelet functions that indicate the effective zone of the wavelet functions. Outside this window, the amplitude of the wavelet function is weak enough to be neglected. In this analysis, we define a window that stops at the positions given by −_{t}_{0} and the standard deviation (second moment) _{t}

The |^{2} of the Morlet mother wavelet can be simplified based on the Pythagorean identity:

^{2} is an even function. In other words, the _{0} value of the Morlet mother wavelet is equal to 0 and the _{0} value of the Morlet wavelet function only depends on the translation parameter

^{3/2}. Combining

In addition to the wavelet spectrum, the influence of signal leakage on synthesizing the sea surface elevations should be considered, as well. The width of the marginal area is also related to the wave frequency. According to wave theory, the sea surface elevations of irregular waves can be constructed by adding a large number of sinusoidal waves (component waves) with different amplitudes, frequencies, and phases [_{n}_{sj}

The maximum wave period of our wave datasets is approximately 10 s. The window width of wavelet functions (3_{t}

After confirming the feasibility of applying the aforementioned wavelet-based algorithm to observational wave records of a pile station,

One record of observational accelerations was chose to derive the wavelet spectrum of sea surface elevations (as shown in

The energy distribution of _{i}_{p}_{p}

To detect wave nonlinearity at the spectral peak of frequency 2_{p}_{p}_{p}_{p}_{p}_{r}_{p}_{r}_{p}

To avoid the overlap of energy density between _{p}_{p}_{r}_{p}_{p}_{p}

A time-averaged wavelet spectrum, estimated by integrating the time domain of wavelet spectrum in _{p}_{p}_{p}

After finding the nonlinear features that existed in a wavelet spectrum like the one shown in _{p}_{p}_{p}_{f}_{2} (as shown in _{p}_{1} (as shown in _{p}

In theory, the two components from _{p}_{p}_{a}_{n}_{n}

After reviewing 8,000 wavelet spectra derived from _{f}_{a}_{a}_{a}_{f}_{a}_{a}_{f}

Accelerometers, which are commonly used in the airline industry, are also among the most useful sensors for ocean wave measurements. So far, the accelerometer-equipped buoy is one of the most popular tools to obtain wave information. The FFT-based algorithm is undoubtedly a suitable method to derive the wave spectrum and also sea surface elevations from buoy acceleration signals, however, the non-stationary or nonlinear phenomenon of wave energy may be concealed in the wave spectrum derived by Fourier transform. Hence a wavelet-based algorithm was developed to examine the local characteristics of wave nonlinearity in both the time and frequency domains. Although a large number of studies have applied the wavelet-based algorithm to examine the

To accurately synthesize the sea surface elevations from the wavelet spectrum, it is necessary to reduce the low-frequency noises caused by the transfer function which transforms the wavelet spectrum of accelerations to a wavelet spectrum of sea surface elevations. A high-pass filter with a rectangular window was introduced to diminish the evident noises at very low frequencies. The ideal cut-off frequency of high-pass filter was then examined and determined for our cases. We also verified an ideal margin width of synthetic wave signals because the effect of spectral leakage always causes inaccurate estimations at the beginning and the end of time.

After confirming the feasibility of the wavelet-based algorithm, we applied the algorithm to synthesizing wavelet spectra of sea surface elevations from the observational acceleration signals of a wave buoy. The wavelet spectra showed that individual nonlinear wave features are instantaneous or short-time events. Most of these short-time nonlinear wave events occur for only several seconds. This suggests that the energies at different harmonic frequency bins might have been averaged in the Fourier-type wave spectrum. It also explains why the energy at the harmonic frequency is not noticeable in the wave spectrum, although it is apparent in the wavelet spectrum. In addition to the nonlinear features that are extracted from the wavelet spectrum, the wave profiles of individual nonlinear wave events were also examined in this study. By analysing the short-time wave profiles of sea surface elevation records, which were derived from acceleration signals of a wave buoy by the wavelet-based algorithm, we confirm the local wave vertical asymmetry is related to short-time wave nonlinearity. Hence, the wavelet spectrum and sea surface elevations are both significant to explore the short-time wave nonlinearity.

In previous studies on analysing nonlinear wave data, most observational data are sea surface elevation records and were measured in shallow waters because nonlinear wave phenomena are common and easily observed there. However, nonlinearity can also occur in waters of deep or intermediate depth, where

This work was supported by the National Science Council (NSC 98-2923-I-006-001-MY4 and NSC 102-2221-E-006-078-MY2) in Taiwan. The

The authors declare no conflict of interest.

Scale parameter

Translation parameter

_{c}

Acceleration signal

_{n}

Amplitude of nonlinear wave profile

_{ψ}

Admissibility constant

_{c}

Correlation coefficient of the wavelet spectra

_{1}

Energy density at major peak frequency of wave spectrum

_{2}

Energy density at 2_{p}

_{n}

Total energy of signal

_{i}

Scalar frequency of the

_{p}

Major peak Frequency of wave spectrum

Gravitational acceleration

_{n}

Wave height of nonlinear wave profile

_{s}

Significant wave height synthesized from observational wave records by the zero-up-crossing method

_{sj}

The significant wave height of wave case

Number of total samples number of total samples for each wave record

Total number of wave cases

_{d}

Normalised differences between

_{e}

Root mean square error

_{n}

Non-dimensional root mean square error

_{f}

Ratio of the energy density at 2_{p}_{p}

_{0}

Centre of the wavelet function

_{a}

Local wave vertical asymmetry

_{AC}

^{2}

Scalogram of acceleration signa

_{AC}

^{2}

Wavelet spectrum of acceleration signal

_{η}

^{2}

Scalogram of sea surface elevations

_{η}

^{2}

Wavelet spectrum of sea surface elevations

^{2}

Wavelet spectra of observational sea surface elevation data

^{2}

Estimated Wavelet spectra of sea surface elevation data, which are derived from synthetic acceleration data

_{b,a}

Transformed wavelet function

_{b,a}

The complex conjugate of wavelet function

Mother wavelet function

Fourier space of

_{0}

A constant that forces the admissibility condition

Angular frequency

_{x}

Mean values of |^{2}

_{y}

Mean values of |^{2}

_{t}

Standard deviation of wavelet function

_{x}

Standard deviations of |^{2}

_{y}

Standard deviations of |^{2}

Synthetic sea surface elevations

Observational sea surface elevations

Sampling interval of the signals

The location and surrounding bathymetry of the Qigu pile station. The numbers in Figure 1b stand for water depths. The pile station is located approximately 3 km from the western coast of Taiwan, where the water depth is 15 m.

The normalised differences between two wavelet spectra of sea surface elevations. One was directly calculated from an observational sea surface elevation record, and the other wavelet spectrum was estimated from the synthetic acceleration data which was derived from the same observational record of sea surface elevations by means of double forward difference. To estimate the wave spectrum of sea surface elevations from the synthetic acceleration data, the transfer function in

A box-whisker plot of correlation coefficients for paired wavelet spectra of 1,500 sea surface elevation records. One set of wavelet spectra was directly calculated from observational sea surface elevation records, and the other set was estimated from the synthetic acceleration data which were derived from observational records of sea surface elevations by means of double forward difference. The top and bottom of each box are the 25th and 75th percentiles of the samples, respectively. The line in the middle of each box is the sample median. The upper and lower whiskers present the highest and lowest results from the samples.

The normalised differences between the synthetic sea surface elevations and the observational sea surface elevations.

Simulation errors of sea surface elevations for different cut-off frequencies. The box-whisker plot shows that the estimated results are more accurate if the cut-off frequency is larger than 16/512 Hz.

The non-dimensional root-mean-square errors between observational sea surface elevations and synthetic sea surface elevations from 1,500 wave cases. The errors are obvious and unstable at initial and final 70 s durations of a wave record.

The location and surrounding bathymetry of buoy station. The numbers inside the figure stand for water depths. The water depth of buoy location is 30 m.

(

An instantaneous spectrum extracted from the wavelet spectrum of

Probabilities of sustained time durations of the spectral peak at 2_{p}

A comparison between a wave spectrum derived by Fourier transform and a time-averaged wavelet spectrum derived by wavelet transform.

(_{f}_{a}_{a}_{f}_{f}_{p}_{p}_{a}