2. Materials and methods

#### 2.1. Study area

optimal extraction of information from a time series of any length [2]. The continuous wavelet transform (CWT) provides redundant information by mapping a time series into a function of time and frequency. The discrete wavelet transform (DWT) computes the transform for discrete values for time and frequency [3]. Hence, DWT is simple, easy to implement, and has low computational requirements. CWT requires a high computational time; however, it allows a larger freedom in wavelets selection than DWT. Redundant information in pictures from the CWT makes it easier to interpret results from the analysis of dynamic time series data. For an analysis where the main purpose is to reveal patterns or hidden information and data compression is not of concern, then redundancy provided by CWT may be useful [3]. Generally, CWTs are useful for dynamical analyses, while DWTs are better for information compression

This study utilized the CWT rather than DWT. The CWT may use an arbitrary number of daughter wavelets built from mother wavelet to match salinity oscillation periods, as needed for the optimal extraction of information concerning astronomical and meteorological forcing of the salinity intrusion. The daughter wavelets will be complete but not orthogonal. A DWT can be complete and orthogonal, given that it is built from functions with geometrically spaced periods; however, it may not have frequency flexibility necessary for salinity intrusion analysis [2].

Various studies have used wavelet analysis to analyze nonstationary time series water quality, meteorological, and hydrological data. Torrence and Compo provided a practical guide to wavelet analysis and also analyzed time series of the El Niño-Southern Oscillation (ENSO) [4]. Liu et al. ratified the bias problem in the estimation of the wavelet power spectrum and applied wavelet analysis to Niño-3 SST data [5]. Wavelet analysis was also used to study water-quality parameters [6, 7]. Zhang et al. [8], Sovi et al. [9], and Somoza et al. [10] used wavelet analysis to characterize water level variation. A number of studies have analyzed tidal variation and its influence on rivers using wavelet analysis [2, 11–13]. Ideião et al. studied the variability of the total monthly rainfall with the aid of CWT [4, 14]. The cross-wavelet transform (XWT) and wavelet coherence (WTC) techniques were used to analyze geophysical time series, for example, the effects of tidal range and river discharge on the salinity intrusion [15, 16], the impacts of Arctic Oscillation index and ENSO on the Baltic sea ice [17, 18], the relative humidity, and the shortwave radiation dataset [19]. Partial wavelet coherence (PWC) and multiple wavelet coherence (MWC) were used to study the impact of ENSO on the variability of tropical cyclones [20]. Wavelet analysis has also been applied in economics

This study used CWT to analyze the period characteristics of tide level, river discharge, meteorological forcing variables, and seawater flux. It also quantified the relationships between river discharge, tide and meteorological forcing variables, and salinity intrusion, using WTC, XWT, and PWC. Several studies examined the effects of river discharge, tidal range, and meteorological forcing on salinity intrusion [15, 16]. Meteorologically induced sea surface variation (MISSV) and large periodic river discharge are considered to be effective water exchange mechanism between Lakes Nakaumi and Shinji, and the Japan Sea [22]. Though tidal amplitude on the Japan Sea is small, astronomical tides appear to be an effective water exchange mechanism [16]. The study of the influence of external forces on salinity

[2, 3].

216 Wavelet Theory and Its Applications

field [3, 21].

Lakes Shinji and Nakaumi form a coupled brackish lake system in the western part of Japan (Figure 1). Lake Shinji has an average depth of 4.5 m, a surface area of 80 km<sup>2</sup> , and a volume of 0.366 km<sup>3</sup> . Lake Nakaumi has an average depth of 5.4 m, a surface area of 86.2 km<sup>2</sup> , and a volume of 0.47 km<sup>3</sup> . The Ohashi River (7.0 km long) connects the two lakes and the Sakai Channel (7.5 km long) connects Lake Nakaumi to the Japan Sea. The Hii River at the west-end of Lake Shinji supplies the lake system with most of its fresh water. Lake Shinji is a mesohaline lake with an average salinity between 1 and 6 PSU. Lake Nakaumi has a strongly differentiated two-layer system; the salinity of the surface water is 14–20 PSU and that of the bottom layer is 25–30 PSU. Hence, these brackish lakes are stably stratified due to salinity (density) differences, and density gradients have a large impact on water movement in this system [23].

Figure 1. Location of Lakes Shinji and Nakaumi, and Nakaura Watergate monitoring station (insert). The red arrow indicates the third eastern-side floodgate, the location of sampling, and measurement equipment. Also shown are meteorological and hydrological monitoring stations (source: [16]).

#### 2.2. Observations

Salinity data used in the study were collected at Nakaura Watergate monitoring station shown in Figure 1. Nakaura Watergate (width 414 m, depth 6.8 m) had five floodgates in the east and in the west (each 32 m long), and three floodgates at the center. On the western pile of the third east-side floodgate (indicated by a red arrow, Figure 1), submerged water pumps were installed for water sampling at 1, 2, 4, and 6 m from the bottom. The water was pumped to acrylic boxes in the floodgate administration building, where water temperature, electrical conductivity, and dissolved oxygen were measured every 30 min using custom-made sensors (Alec Electronics Co., Ltd.). Salinity was calculated from electrical conductivity.

Continuous measurements of salinity over a period of 6 years (February 1998 to March 2004) are available, although there are periods with missing data. Instantaneous seawater flux for the period January 2001 to October 2003 was used in this study. The salinity data were averaged to a 1-h interval to match the intervals of the other data used in the analysis. One-hour interval meteorological data (atmospheric pressure, wind speed, and direction) were collected at Matsue Meteorological Station (available on Japan Meteorological Agency website, http:// www.jma.go.jp/jma/index.html). The wind was treated as a mathematical vector, and the mathematical convention for the direction was used, that is, wind direction was converted from "meteorological direction" to "math direction." The wind vector was resolved into its u (wx) and v (wy) components. Wind from the west was denoted as positive u and from the south a positive v. Tidal data used were recorded at Mihonoseki tide gauge station, and river discharge was recorded in Hii River at Nadabun gauging station (available on Japan's water information system website, http://www1.river.go.jp/).

#### 2.3. Methodology

#### 2.3.1. Salinity transport

Instantaneous advective salt transport (Ms, kg m�1s�1) per unit width of a section, normal to the longitudinal flow of the channel, is given by the following expression [24, 25]:

$$M\_s = \int\_0^h \rho V \mathbf{S} d\mathbf{z} = \overline{\rho V \mathbf{S}} \cdot \mathbf{h} \tag{1}$$

2.3.2. Continuous wavelet transform (CWT)

energy [8, 18].

CWT decomposes a time series into a time-frequency space and determines both the dominant modes of variability and their variation with time [4]. The wavelet is applied as a bandpass filter to the time, stretching it in time by varying its scale(s) and normalizing it to have a unit

A wavelet ψð Þt is a function that oscillates around the t-axis and loses strength as it moves away from the center, behaving like a small wave [4]. Beginning with a mother wavelet ψ, a

where ψð Þt is the mother wavelet, ψτ,sð Þt is the daughter wavelet, t is a nondimensional "time" parameter, s is a scaling or a dilation factor that controls the width of the wavelet, and τ is a

This study used the Morlet wavelet, which consists of a plane wave modulated by a Gaussian or in other words a complex exponential function multiplied by a Gaussian window. Hence, it represents the best compromise between frequency and time localization. A complex wavelet is essential for this study, as it yields a complex transform, with information on both the amplitude and the phase, crucial to study the synchronization of oscillations between different

<sup>ψ</sup>0ðÞ¼ <sup>t</sup> <sup>π</sup>�1=<sup>4</sup> <sup>e</sup>

ð ∞

�∞

functions, that is, the set of functions defined on the real line and satisfying Ð

Wx;ψð Þ¼ τ;s

where Wx;ψð Þ <sup>τ</sup>;<sup>s</sup> is the CWT of a time series x tð Þ, <sup>L</sup><sup>2</sup>

where ψ0ð Þt is the Morlet wavelet, ω<sup>0</sup> is the fundamental frequency, which gives the number of

x tð Þ <sup>1</sup> ffiffiffiffiffi j j<sup>s</sup> <sup>p</sup> <sup>ψ</sup><sup>∗</sup> <sup>t</sup> � <sup>τ</sup> s

The cross-wavelet transform (XWT), a multiscale signal analytical technique, combines the wavelet transform and cross-spectrum analysis. XWT analyzes multiple time-frequencies of

iω0t e �t

ð Þ ℝ with respect to the wavelet ψ is a function of two variables,

� �; s, <sup>τ</sup> <sup>∈</sup> <sup>ℝ</sup>; s 6¼ <sup>0</sup> (3)

http://dx.doi.org/10.5772/intechopen.75177

219

Use of Wavelet Techniques in the Study of Seawater Flux Dynamics in Coastal Lakes

<sup>2</sup> <sup>=</sup><sup>2</sup> (4)

� �dt (5)

ð Þ ℝ denotes the set of square integrable

∞ �∞

j j x tð Þ <sup>2</sup>

dt < ∞, ψ<sup>∗</sup>

family of "daughter wavelet" ψ τð Þ ;s is computed by scaling and translating ψ [4]:

1 ffiffiffiffiffi j j<sup>s</sup> <sup>p</sup> <sup>ψ</sup> <sup>t</sup> � <sup>τ</sup> s

ψτ,sðÞ¼ t

translation parameter controlling the location of the wavelet.

time series [3, 27]. Morlet wavelet is defined as [4]

oscillations within the wavelet itself.

CWT of a time series x tð Þ<sup>∈</sup> <sup>L</sup><sup>2</sup>

is the complex conjugation of ψ.

2.3.3. Cross-wavelet transform (XWT)

Wx;ψð Þ τ;s [3]:

where r is the density, V is the longitudinal velocity component, and S is the longitudinal salinity. The upper bar denotes averaging over the total depth of the water column, h.

The study lakes are shallow. Pressure variation in shallow lakes is negligible. Therefore, the density of water was calculated using the following approximate density formula neglecting pressure [26]:

$$\rho = 999.83 + 0.808S - 0.0708(1 + 0.068T)T - 0.003(1 - 0.012T)(35 - S)T \tag{2}$$

where T is the temperature in �C and S is the salinity in PSU.

#### 2.3.2. Continuous wavelet transform (CWT)

2.2. Observations

218 Wavelet Theory and Its Applications

2.3. Methodology

pressure [26]:

2.3.1. Salinity transport

Salinity data used in the study were collected at Nakaura Watergate monitoring station shown in Figure 1. Nakaura Watergate (width 414 m, depth 6.8 m) had five floodgates in the east and in the west (each 32 m long), and three floodgates at the center. On the western pile of the third east-side floodgate (indicated by a red arrow, Figure 1), submerged water pumps were installed for water sampling at 1, 2, 4, and 6 m from the bottom. The water was pumped to acrylic boxes in the floodgate administration building, where water temperature, electrical conductivity, and dissolved oxygen were measured every 30 min using custom-made sensors

Continuous measurements of salinity over a period of 6 years (February 1998 to March 2004) are available, although there are periods with missing data. Instantaneous seawater flux for the period January 2001 to October 2003 was used in this study. The salinity data were averaged to a 1-h interval to match the intervals of the other data used in the analysis. One-hour interval meteorological data (atmospheric pressure, wind speed, and direction) were collected at Matsue Meteorological Station (available on Japan Meteorological Agency website, http:// www.jma.go.jp/jma/index.html). The wind was treated as a mathematical vector, and the mathematical convention for the direction was used, that is, wind direction was converted from "meteorological direction" to "math direction." The wind vector was resolved into its u (wx) and v (wy) components. Wind from the west was denoted as positive u and from the south a positive v. Tidal data used were recorded at Mihonoseki tide gauge station, and river discharge was recorded in Hii River at Nadabun gauging station (available on Japan's water

Instantaneous advective salt transport (Ms, kg m�1s�1) per unit width of a section, normal to

where r is the density, V is the longitudinal velocity component, and S is the longitudinal

The study lakes are shallow. Pressure variation in shallow lakes is negligible. Therefore, the density of water was calculated using the following approximate density formula neglecting

r ¼ 999:83 þ 0:808S � 0:0708 1ð Þ þ 0:068T T � 0:003 1ð Þ � 0:012T ð Þ 35 � S T (2)

rVSdz ¼ rVS∙h (1)

the longitudinal flow of the channel, is given by the following expression [24, 25]:

0

salinity. The upper bar denotes averaging over the total depth of the water column, h.

Ms ¼ ð h

where T is the temperature in �C and S is the salinity in PSU.

(Alec Electronics Co., Ltd.). Salinity was calculated from electrical conductivity.

information system website, http://www1.river.go.jp/).

CWT decomposes a time series into a time-frequency space and determines both the dominant modes of variability and their variation with time [4]. The wavelet is applied as a bandpass filter to the time, stretching it in time by varying its scale(s) and normalizing it to have a unit energy [8, 18].

A wavelet ψð Þt is a function that oscillates around the t-axis and loses strength as it moves away from the center, behaving like a small wave [4]. Beginning with a mother wavelet ψ, a family of "daughter wavelet" ψ τð Þ ;s is computed by scaling and translating ψ [4]:

$$
\psi\_{\tau,s}(t) = \frac{1}{\sqrt{|s|}} \psi\left(\frac{t-\tau}{s}\right); s, \tau \in \mathbb{R}; s \neq 0 \tag{3}
$$

where ψð Þt is the mother wavelet, ψτ,sð Þt is the daughter wavelet, t is a nondimensional "time" parameter, s is a scaling or a dilation factor that controls the width of the wavelet, and τ is a translation parameter controlling the location of the wavelet.

This study used the Morlet wavelet, which consists of a plane wave modulated by a Gaussian or in other words a complex exponential function multiplied by a Gaussian window. Hence, it represents the best compromise between frequency and time localization. A complex wavelet is essential for this study, as it yields a complex transform, with information on both the amplitude and the phase, crucial to study the synchronization of oscillations between different time series [3, 27]. Morlet wavelet is defined as [4]

$$
\psi\_0(t) = \pi^{-\not\!\!/ } e^{i\omega\_0 t} e^{-r^2/\_2} \tag{4}
$$

where ψ0ð Þt is the Morlet wavelet, ω<sup>0</sup> is the fundamental frequency, which gives the number of oscillations within the wavelet itself.

CWT of a time series x tð Þ<sup>∈</sup> <sup>L</sup><sup>2</sup> ð Þ ℝ with respect to the wavelet ψ is a function of two variables, Wx;ψð Þ τ;s [3]:

$$\mathcal{W}\_{\mathbf{x};\psi}(\tau,\mathbf{s}) = \int\_{-\infty}^{\infty} \mathbf{x}(t) \frac{1}{\sqrt{|\mathbf{s}|}} \psi^\* \left(\frac{t-\tau}{\mathbf{s}}\right) dt \tag{5}$$

where Wx;ψð Þ <sup>τ</sup>;<sup>s</sup> is the CWT of a time series x tð Þ, <sup>L</sup><sup>2</sup> ð Þ ℝ denotes the set of square integrable functions, that is, the set of functions defined on the real line and satisfying Ð ∞ �∞ j j x tð Þ <sup>2</sup> dt < ∞, ψ<sup>∗</sup> is the complex conjugation of ψ.

#### 2.3.3. Cross-wavelet transform (XWT)

The cross-wavelet transform (XWT), a multiscale signal analytical technique, combines the wavelet transform and cross-spectrum analysis. XWT analyzes multiple time-frequencies of two time series from multiple time scale points, thereby exposing regions with a common high power, and further reveals information about the phase relationship in time-frequency space, hence determining correlations [15, 18].

For two time series xn and yn, their cross-wavelet transform is given by [4, 15]

$$\boldsymbol{W}\_{n}^{\mathcal{X}\boldsymbol{Y}}(\mathbf{s}) = \boldsymbol{W}\_{n}^{\mathcal{X}}(\mathbf{s})\boldsymbol{W}\_{n}^{\mathcal{Y}\*}(\mathbf{s}) \tag{6}$$

"ratio of number of typhoons to number of tropical cyclones" and "large-scale atmospheric

1 23 ð Þ …<sup>p</sup> <sup>¼</sup> <sup>1</sup> � Cd

where C denotes the p � p matrix of all the complex wavelet coherencies ϱij, that is,

<sup>ϱ</sup>1j:qj ¼ � <sup>C</sup><sup>d</sup>

R1j:qj ¼

Cd 11

j1 ffiffiffiffiffiffiffi Cd <sup>11</sup> <sup>q</sup> ffiffiffiffiffi Cd

> Cd j1 � � � � � � ffiffiffiffiffiffiffiffiffiffiffiffi Cd 11Cd

and the squared partial wavelet coherence of x<sup>1</sup> and xj allowing for all the other series is simply

The partial phase delay (phase difference) of x<sup>1</sup> and xj given all the other series is defined as the

the phase, making it suitable to capture oscillatory behavior. Complex partial wavelet coher-

�, and phase angle <sup>ϕ</sup>1j:qj

0 @

A phase difference of zero indicates that the time series moves together at the specified time-

Eð Þ �90; 0 , then xj leads x1. A phase difference of 180 (or �180) indicates an antiphase

¼ Arctan

ence, ϱ1j:qj considered can be separated into its real part, R ϱ1j:qj

� �

ϕ1j:qj

� � �

. A complex wavelet function contains information about both the amplitude and

J ϱ1j:qj � � 1

R ϱ1j:qj � �

Eð Þ 0; 90 , the series moves in a phase and the time series of x<sup>1</sup> leads xj; if

1 23 ð Þ …<sup>p</sup> ) between the series <sup>x</sup><sup>1</sup> and all the other series

http://dx.doi.org/10.5772/intechopen.75177

Use of Wavelet Techniques in the Study of Seawater Flux Dynamics in Coastal Lakes

) of x<sup>1</sup> and xjð Þ 2 ≤ j ≤ p allowing for all the other

) of x<sup>1</sup> and xj allowing for all the other series is defined as

jj <sup>q</sup> (10)

jj <sup>q</sup> (11)

. The phase difference, ϕ1j:qj

� �, and imaginary part,

A (12)

, is given

(9)

221

factors" after removing the effect of El Nino-Southern Oscillation (ENSO) [20].

R2

The squared multiple wavelet coherence (R<sup>2</sup>

, <sup>C</sup><sup>d</sup> <sup>¼</sup> detC.

The partial wavelet coherence (R1j:qj

the absolute value of Eq. (10), that is,

.

� �, or in its amplitude, <sup>ϱ</sup>1j:qj

The complex partial wavelet coherence (ϱ1j:qj

<sup>C</sup> <sup>¼</sup> <sup>ϱ</sup>ij � �<sup>p</sup>

i,j¼1

series is given by [3]

the square of R1j:qj

angle of ϱ1j:qj

as follows [3]:

frequency; if ϕ<sup>x</sup>1xj

J ϱ1j:qj

ϕ<sup>x</sup>1xj

x2,…, xp is given by the following formula [3]:

where W<sup>X</sup> <sup>n</sup> ð Þ<sup>s</sup> and <sup>W</sup><sup>Y</sup> <sup>n</sup> ð Þs are wavelet transforms of xn and yn, respectively, and \* denotes complex conjugation. WXY <sup>n</sup> ð Þ<sup>s</sup> � � � � is the cross-wavelet power. If two time series have background power spectra PX <sup>k</sup> and P<sup>Y</sup> <sup>k</sup> , then their theoretical distribution of the cross-wavelet power is given by [4, 18].

$$D\left(\frac{\left|W\_n^X(s)W\_n^{Y\*}(s)\right|}{\sigma\_X\sigma\_Y} < p\right) = \frac{Z\_v(p)}{v}\sqrt{P\_k^X P\_k^Y} \tag{7}$$

where σ<sup>X</sup> and σ<sup>Y</sup> are the respective standard deviations, Zvð Þp is the confidence level associated with the probability p for a probability distribution function (pdf) defined by the square root of the product of two chi-squared (χ2) distributions.

#### 2.3.4. Wavelet coherence (WTC)

Wavelet coherence (WTC) between two CWTs can find significant coherence even though the common power is low and show how confidence levels against red noise backgrounds are calculated. This can be thought of as a local correlation between two time series in the timefrequency space. It finds locally phase-locked behavior. The significance level of the WTC is determined using Monte Carlo methods [15]

$$R\_n^2(s) = \frac{\left| \mathbf{S} \left( \mathbf{s}^{-1} \boldsymbol{W}\_n^{\mathrm{XY}}(\mathbf{s}) \right) \right|^2}{\mathbf{S} \left( \mathbf{s}^{-1} \left| \boldsymbol{W}\_n^{\mathrm{X}}(\mathbf{s}) \right|^2 \right) \cdot \mathbf{S} \left( \mathbf{s}^{-1} \left| \boldsymbol{W}\_n^{\mathrm{Y}}(\mathbf{s}) \right|^2 \right)} \tag{8}$$

where S is a smoothing operator.

#### 2.3.5. Partial wavelet coherence

CWT is increasingly being used in the analysis of marine sciences time series data. However, most of the CWT analyses have been limited to univariate and bivariate analyses, that is, the wavelet power spectrum, the wavelet coherency, and the wavelet phase difference [3]. Wavelet analysis tools have already been extended to allow for multivariate analyses [3, 20]. PWC and PPD are the examples of recent wavelet analysis techniques. The PWC technique is similar to partial correlation and it identifies the resulting wavelet coherence between two time series after eliminating the influence of their common dependence [20]. The applicability of PWC to geophysics was demonstrated during the study of the "stand-alone" relationship between the "ratio of number of typhoons to number of tropical cyclones" and "large-scale atmospheric factors" after removing the effect of El Nino-Southern Oscillation (ENSO) [20].

two time series from multiple time scale points, thereby exposing regions with a common high power, and further reveals information about the phase relationship in time-frequency space,

<sup>n</sup> ð Þ<sup>s</sup> <sup>W</sup><sup>Y</sup><sup>∗</sup>

<sup>n</sup> ð Þs are wavelet transforms of xn and yn, respectively, and \* denotes

<sup>k</sup> , then their theoretical distribution of the cross-wavelet power is given

<sup>¼</sup> Zvð Þ<sup>p</sup> v

� is the cross-wavelet power. If two time series have background

q

ffiffiffiffiffiffiffiffiffiffiffi PX <sup>k</sup> PY k

<sup>n</sup> ð Þs (6)

<sup>2</sup> � � (8)

(7)

For two time series xn and yn, their cross-wavelet transform is given by [4, 15]

<sup>n</sup> ð Þ¼ <sup>s</sup> <sup>W</sup><sup>X</sup>

�

< p

where σ<sup>X</sup> and σ<sup>Y</sup> are the respective standard deviations, Zvð Þp is the confidence level associated with the probability p for a probability distribution function (pdf) defined by the square

Wavelet coherence (WTC) between two CWTs can find significant coherence even though the common power is low and show how confidence levels against red noise backgrounds are calculated. This can be thought of as a local correlation between two time series in the timefrequency space. It finds locally phase-locked behavior. The significance level of the WTC is

> S s�<sup>1</sup>WXY <sup>n</sup> ð Þ<sup>s</sup> � � � � �

CWT is increasingly being used in the analysis of marine sciences time series data. However, most of the CWT analyses have been limited to univariate and bivariate analyses, that is, the wavelet power spectrum, the wavelet coherency, and the wavelet phase difference [3]. Wavelet analysis tools have already been extended to allow for multivariate analyses [3, 20]. PWC and PPD are the examples of recent wavelet analysis techniques. The PWC technique is similar to partial correlation and it identifies the resulting wavelet coherence between two time series after eliminating the influence of their common dependence [20]. The applicability of PWC to geophysics was demonstrated during the study of the "stand-alone" relationship between the

S s�<sup>1</sup> W<sup>X</sup> <sup>n</sup> ð Þ<sup>s</sup> � � � �

<sup>2</sup> � �

� 2

∙S s�<sup>1</sup> W<sup>Y</sup> <sup>n</sup> ð Þ<sup>s</sup> � � � �

WXY

σXσ<sup>Y</sup>

!

hence determining correlations [15, 18].

<sup>k</sup> and P<sup>Y</sup>

<sup>n</sup> ð Þ<sup>s</sup> � � �

D

root of the product of two chi-squared (χ2) distributions.

R2 <sup>n</sup>ð Þ¼ s

W<sup>X</sup> <sup>n</sup> ð Þ<sup>s</sup> <sup>W</sup><sup>Y</sup><sup>∗</sup> <sup>n</sup> ð Þ<sup>s</sup> � � �

<sup>n</sup> ð Þ<sup>s</sup> and <sup>W</sup><sup>Y</sup>

complex conjugation. WXY

220 Wavelet Theory and Its Applications

2.3.4. Wavelet coherence (WTC)

where S is a smoothing operator.

2.3.5. Partial wavelet coherence

determined using Monte Carlo methods [15]

where W<sup>X</sup>

by [4, 18].

power spectra PX

The squared multiple wavelet coherence (R<sup>2</sup> 1 23 ð Þ …<sup>p</sup> ) between the series <sup>x</sup><sup>1</sup> and all the other series x2,…, xp is given by the following formula [3]:

$$R\_{1(23\dots p)}^2 = 1 - \frac{\mathbf{C}^d}{\mathbf{C}\_{11}^d} \tag{9}$$

where C denotes the p � p matrix of all the complex wavelet coherencies ϱij, that is, <sup>C</sup> <sup>¼</sup> <sup>ϱ</sup>ij � �<sup>p</sup> i,j¼1 , <sup>C</sup><sup>d</sup> <sup>¼</sup> detC.

The complex partial wavelet coherence (ϱ1j:qj ) of x<sup>1</sup> and xjð Þ 2 ≤ j ≤ p allowing for all the other series is given by [3]

$$\mathcal{Q}\_{1\not{q}\_{\not{p}}} = -\frac{\mathbf{C}\_{\not{p}1}^{d}}{\sqrt{\mathbf{C}\_{11}^{d}}\sqrt{\mathbf{C}\_{\not{p}}^{d}}}\tag{10}$$

The partial wavelet coherence (R1j:qj ) of x<sup>1</sup> and xj allowing for all the other series is defined as the absolute value of Eq. (10), that is,

$$\mathcal{R}\_{1j.q\_j} = \frac{\left| \mathbf{C}\_{j1}^d \right|}{\sqrt{\mathbf{C}\_{11}^d \mathbf{C}\_{jj}^d}} \tag{11}$$

and the squared partial wavelet coherence of x<sup>1</sup> and xj allowing for all the other series is simply the square of R1j:qj .

The partial phase delay (phase difference) of x<sup>1</sup> and xj given all the other series is defined as the angle of ϱ1j:qj . A complex wavelet function contains information about both the amplitude and the phase, making it suitable to capture oscillatory behavior. Complex partial wavelet coherence, ϱ1j:qj considered can be separated into its real part, R ϱ1j:qj � �, and imaginary part, J ϱ1j:qj � �, or in its amplitude, <sup>ϱ</sup>1j:qj � � � � � �, and phase angle <sup>ϕ</sup>1j:qj . The phase difference, ϕ1j:qj , is given as follows [3]:

$$\phi\_{1j,q\_j} = \arctan\left(\frac{\mathfrak{J}\left(\varrho\_{1j,q\_j}\right)}{\mathfrak{R}\left(\varrho\_{1j,q\_j}\right)}\right) \tag{12}$$

A phase difference of zero indicates that the time series moves together at the specified timefrequency; if ϕ<sup>x</sup>1xj Eð Þ 0; 90 , the series moves in a phase and the time series of x<sup>1</sup> leads xj; if ϕ<sup>x</sup>1xj Eð Þ �90; 0 , then xj leads x1. A phase difference of 180 (or �180) indicates an antiphase relationship; if ϕ<sup>x</sup>1xj Eð Þ 90; 180 , then xj leads x1; if ϕ<sup>x</sup>1xj Eð Þ �180; �90 , then x<sup>1</sup> leads xj (see Figure 2). Phase difference can be converted into instantaneous time lag between two time series by dividing the phase difference, ϕ1j:qj by the angular frequency corresponding to the scale s, ωð Þs .

3. Results and discussion

3.1.1. Variability of seawater flux

3.1.2. Variability of tide level

3.1.3. Variability of sea level atmospheric pressure

3.1. Analysis of period characteristics

Figure 3 shows the seawater flux per unit width at Nakaura Watergate and its CWT coefficient chart. Positive values of the time series indicate seawater flux toward the Japan Sea and negative toward Lake Nakaumi. The CWT coefficient chart for seawater flux has stable period characteristics, with high power oscillations in the 12-h and 1-day period band throughout the analysis period. Both the red color and the black contour indicate that cycles are strong and

Use of Wavelet Techniques in the Study of Seawater Flux Dynamics in Coastal Lakes

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Figure 4 shows time series plot and CWT coefficient chart for the tide level. The high-power tide level oscillations have statistically significant periods of 12 h and 1 day. This implies considerable power spreads throughout the semi-diurnal and diurnal bands throughout the analysis period. The oscillations indicate spring-neap tidal variations since they appear twice a month. Also observed is a relatively strong statistically significant, though not regular, 2–6-day period cycle that occurs mainly in winter (December to March). Tide level and atmospheric pressure are negatively correlated. The time series of both shows that as the atmospheric

The CWT coefficient chart for the atmospheric pressure (Figure 5) shows continuous statistically significant high power 64-day period cycles from April 2002. There are also 128-day to 1-year period cycles throughout the analysis period. There are some discontinuous and irreg-

Figure 3. (a) The time series of salinity flux (salt flux (kg/m/s)) and (b) its wavelet power spectrum for the Jan 2001 to Oct 2003 period. Period is in days. The red color designates high power oscillations whilst blue is low power oscillations. The black contour designates 95% confidence level, using red noise as the background spectrum. White regions on either end

statistically significant at 95% confidence level (hereinafter statistically significant).

pressure increases, the tide level decreases and vice versa (Figures 4 and 5).

ular high power oscillations in the 1-day and 2–32-day period cycles.

indicate the "cone of influence" where edge effects become important.

#### 2.3.6. Significance tests

It is important to assess the statistical significance of the wavelet, cross-wavelet power and the wavelet coherence. The assessment of the statistical significance levels and confidence intervals against red noise backgrounds was done using direct Monte Carlo simulations.

#### 2.3.7. Wavelet packages and parameters used

Ng and Kwok provided the software for CWT, WTC, and XWT, which is available at http:// www.cityu.edu.hk/gcacic/wavelet. It is used in collaboration with the software provided by Grinsted, which is available at http://www.glaciology.net/wavelet-coherence. The software for PWC was provided by Aguiar-Conraria and Soares and is available at http://sites.google.com/ site/aguiarconraria/joanasoares-wavelets. Some of the parameters used in the analysis are as follows: the mother wavelet function—Morlet; the sampling time (dt)—1 h; the spacing between discrete scales (dj)—0.5 h; the level of significance—5%; and the number of Monte Carlo simulations used to assess statistical significance—1000. Default values were taken for other parameters.

Figure 2. Phase difference circle.
