Feature ArticleMarine Data Analysis Based on Wavelet Transform
By Shengjie Liu
Department of Military Oceanography
Dalian Naval Academy
Survey data offer a practical way to understand and study the ocean, but before analyzing the data, anomaly correction, or signal separation, should be completed. Anomalies in salinity signals can be found and corrected through wavelet coefficients at different scales and through the separation of tidal signals from the waves.
In response to these advantages, the Department of Military Oceanography has researched and tested salinity measurements of anomaly correction and tidal separation based on a wavelet transform. Two measured in-situ examples presented here validate the method the department has developed.
Noise, or anomalies, are avoidable in measured marine data. Most noise can be decreased by traditional ways, such as the smoothing function and Fourier filtering. Traditional methods cannot, however, process anomalies such as data being too positive or too negative because these methods are unable to deal well with instantaneity in the signals.
Because the Fourier transform has only frequency resolution and no time resolution, it cannot be used to analyze signals that contain discontinuities and sharp spikes. To overcome this, using a wavelet transform is becoming more popular for analyzing time series data with localized variations. Both the dominant variability and how it varies in time can be obtained by decomposing a time series into time-frequency space with the wavelet coefficients.
It is clear the wavelet method has advantages over traditional methods in anomaly correction and signal separation for in-situ marine data.
Basic Functions, Coefficients
A wavelet is a function of the Hilbert space L2(R), such that Equation 1 is an orthonormal basis of L2(R).
A signal can be expressed in a perfect way by the wavelet. In practice, the scale function is needed to finish the analysis operation. The wavelet decomposition can be expressed as Equation 2, where the coefficients can be calculated with Equation 3 and Equation 4.
By modifying the coefficients of aj,k and dm,k according to the specific criterion, the signals can be processed through an inverse equation. There are many kinds of orthogonal wavelet base functions, and the Daubechies wavelet series, which was constructed by the famous mathematician Ingrid Daubechies, is very prominent among them. The authors used the second order Daubechies wavelet (db2) for this method.
Anomaly Correction of Salinity Data
The application of a wavelet transform to noise reduction was developed by Stanford University statistician David L. Donoho. By filtering the wavelet coefficients, which are lower than the threshold, denoising can be performed well. Donoho’s method has been applied to many fields as a useful way of signal processing.
For a signal with an anomaly, the wavelet coefficients corresponding to the anomaly are quite big. Based on Donoho’s method, the anomaly can be corrected by dealing with the relative wavelet coefficients.
To test this method, salinity measurements from the North China Sea taken between 1980 to 1999 were used. Due to instrument or manual fault, the data contained a number of anomalies (for example, in the values for January 23, 1990, July 12, 1994, and July 16, 1996). The salinity data with anomalies can be analyzed with the following concept.
By using a db2 wavelet function, the wavelet coefficients of salinity data can be calculated where the wavelet coefficients corresponding to the anomaly are quite positive or negative. It is obvious the anomaly can be corrected by reducing these huge wavelet coefficients.
Suppose that y(ti) and f(ti) are two series of measured and true salinity data in Equation 5: y(ti) = f(ti) + zi, i = 1, 2, ··· , n.
Here, zi is the anomaly at ti, and n is the sampling number. f(ti) can be reconstructed based on the db2 wavelet transform by three steps. First, calculate the wavelet coefficients of db2, aj,k and dm,k, according Equation 3 and Equation 4. The second step is to calculate the variance σm,k directly at each scale because the mean value dm,k of the wavelet coefficients is zero. With the assumption of normal distribution, only 0.3 percent of the wavelet coefficients are beyond threshold value, wj = dm,k 3σm,k, which can be regarded as the anomaly. After setting wj, filter the wavelet coefficients by using the shrinking function shown in Equation 6.
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Gereon Th. Budéus has been involved in high-latitude oceanography and instrument development at Alfred Wegener Institute for Polar and Marine Research since 1992. He received his Ph.D. from the University of Hamburg and leads the Alfred Wegener Institute team developing the Optimare Precision Salinometer, which is partnered with Optimare.