Chaotic Features of Decomposed Time Series from Tidal River Water Level

Abstract

This study assessed the characteristics of water-level time series of a tidal river by decomposing it into tide, wave, rainfall-runoff, and noise components. Especially, the analysis for chaotic behavior of each component was done by estimating the correlation dimension with phase-space reconstruction of time series and by using a close returns plot (CRP). Among the time series, the tide component showed chaotic characteristics to have a correlation dimension of 1.3. It was found out that the water level has stochastic characteristics showing the increasing trend of the correlation exponent in the embedding dimension. Other components also showed the stochastic characteristics. Then, the CRP was used to examine the characteristics of each component. The tide component showed the chaotic characteristics in its CRP. The CRP of water level showed an aperiodic characteristic which slightly strayed away from its periodicity, and this might be related to the tide component. This study showed that a low water level is mainly affected by a chaotic tide component through entropy information. Even though the water level did not show chaotic characteristics in the correlation dimension, it showed stochastic chaos characteristics in the CRP. Other components showed stochastic characteristics in the CRP. It was confirmed that the water level showed chaotic characteristics when it was not affected by rainfall and stochastic characteristics deviating from the bounded trajectory when water level rises due to rainfall. Therefore, we have shown that the water level related to the chaotic tide component can also have chaotic properties because water level is influenced by chaotic tide and rainfall shock, thus it showed stochastic chaos characteristics.

1. Introduction

Natural behavior of the hydrological system is changing nonlinearly, influenced by climate change, and it does not always show chaotic characteristics. Coastal areas, where more than 50% of the world’s population inhabit, are particularly more vulnerable to storms, typhoons, and floods than inland areas [1]. Tidal river areas where freshwater stream meets the ocean are influenced by the tide that cause higher flood levels and greater damages. From this point of view, predicting tidal river water level is important in flood disaster management. The water level of a tidal river is composed of individual hydrologic components such as tide, wave, and river stage. When it rains, the river stage fluctuates due to the basin runoff discharge, which affects the water level of the tidal river. Therefore, it is necessary to analyze how the time series of each component and rainfall occurrence have affected the water-level time series of the tidal river. To do this, we may need to separate hydrological components which include tidal level, wave height and river stage from water level of a tidal river [2].

Various analysis techniques, which are mostly linear models, have been developed for the predictions of rainfall and runoff which are important for managing water resources and for disaster prevention efficiently. However, a hydrological time series with nonlinear characteristics requires nonlinear models for forecasting and analysis [3,4,5,6,7,8]. In particular, techniques such as the false nearest neighbor algorithm, the Lyapunov exponent method, and the method based on the correlation dimension are used for the chaos analysis to identify the nonlinear dynamics characteristics of hydrological time series [9,10,11,12,13,14].

Sangoyomi et al. (1996) performed chaos analysis using correlation dimension, nearest neighbor dimension, and false neighbor dimension to understand the seasonal variation patterns of the Great Salt Lake (GSL) biweekly volume series [15]. Then they obtained the correlation dimension of 3.4, suggesting there are four factors affecting the GSL volume series. On the other hand, Kim et al. (2001) used a close returns plot (CRP) to analyze the chaotic characteristics of waste water flow in Fort Collins, Colorado, and the GSL volume series in Utah, USA [16]. Sivakumar (2002) assessed the reliability of the correlation dimension of the short-term hydrological time series using phase-space reconstruction and an artificial neural network, arguing that the accuracy of the correlation dimension does not depend on the length of the time series but on its ability to rationally show dynamic changes in the system [17]. Salas et al. (2005) suggested that the increased rainfall duration interferes with the identification of chaos characteristics, finding that the river with the stronger basin storage contribution departs significantly from the behavior of a chaotic system [18]. Because streamflow result from a complex transformation of precipitation that involves accumulating and routing excess rainfall throughout the basin and adding surface and groundwater flows, the result may be that streamflow at the outlet of the basin depart from low dimensional chaotic behavior. This shows that hydrological processes, such as rainfall, storage, and runoff, affect each other and have chaotic characteristics.

Khatibi et al. (2012) used five nonlinear dynamic methods using the runoff time series observed at the Sohutluhan hydrological station in Turkey, and suggested the existence of low-dimensional chaos [11]. Meanwhile, Kim et al. (2015a) analyzed the chaotic behavior of hydro-meteorological processes in the Great Salt Lake and Bear River Basin, USA, by applying wavelet transform and correlation dimension to the precipitation, air temperature, discharge, and storage volume. They found out that the chaotic characteristic of the storage volume which is output is likely a byproduct of the chaotic behavior of the reservoir system itself, rather than that of the input data. In addition, Kim et al. (2015b) analyzed the time series characteristics related to the drought length (DL) using nonlinear dynamic techniques, such as BDS statistic and a close returns test (CRT) [19]. The utility and effectiveness of the methods were first demonstrated on synthetic time series and then tested on real hydrologic time series. They suggested that the daily inflow of the Soyang River in South Korea is different from the DL data in the characteristics. The different characteristics of the original real series and their DLs may be due to the noises from many sources and therefore, noise cancellation may be important for obtaining original property of the time series. On the other hand, Rajagopalan et al. (2019) extracted the dominant quasi-periodic components through the wavelet analysis of the flow rate at the Lees Ferry station in the Colorado River and quantified the trajectory branching in the phase space over time using local Lyapunov exponents [20]. As a result, the flow rate of the Colorado River was revealed to be highly predictable.

Previous studies have mostly used the runoff data of the rivers inland that are influenced by the rainfall-runoff process. However, tidal rivers are influenced by many other factors, such as tidal level, wave height, river water level, and rainfall, which make their properties nonlinear. Tidal rivers are typically affected by backwater, a zone of spatially decelerating flow that is transitional between the stream flow of upstream and the tidal flow of downstream in the river, and it has a complex characteristic. Since water level and runoff of tidal rivers vary with the influence of the tide, the backwater effect during high tides can cause more flooding damages [2]. For example, in 2016, the damages by the typhoon ‘Chaba’ in Ulsan city area of South Korea were aggravated by the tide. During the typhoon ‘Jebi’ in 2018, the maximum tide level has been observed, and the effect of tidal currents did not drain the runoff into the river, causing great damage to the Kansai Airport in Japan [21]. In order to reduce flood damage by tide effect, many studies have been analyzing tides and waves. Siek and Solomatine (2010) predicted storm surges using a nonlinear chaos model and showed that the chaotic characteristics were caused by the tidal level and tidal wave’s complex effects [2]. In addition, Zounemat-Kermani and Kisi (2016) identified the chaotic characteristics of a coastal area’s dynamic system using significant wave height, wave period, and wave direction [6]. They suggested that the wind direction has a low correlation dimension of 4 or less, while the significant wave height and the mean wave period have a high correlation dimension of 5 to 8 in their chaotic characteristics. However, these studies conducted chaos characterization only for storm surges or single hydrological factors, implying a limitation in applying the method to tidal rivers affected by various hydrological components.

As mentioned above, many studies have been analyzed to characterize the nonlinear characteristics of rainfall and runoff. Understanding the dynamics of the rainfall and runoff process is one of the most important problems in hydrology for the management of water resources. However, it is very difficult to understand the dynamics of the water level in tidal river because the tidal river is affected not only by rainfall, but also by ocean components such as tide and wave. None of the studies could provide information considering the possibility of the existence of chaotic behavior by decomposing the components affecting the water level in the tidal river. The purpose of this study is to attempt to address this issue. Therefore, this study separates the hydrological components (tide, wave, rainfall-runoff (river stage by rainfall), and noise levels) using daily water level data of a tidal river observed in Ulsan station. In identifying the relationship between the separated components and the original water-level time series, the study quantifies the information provided by the four components in the water-level time series based on the information entropy theory. In addition, the phase-space reconstruction for the time series of each component was conducted to examine the characteristics based on the attractor. The correlation dimension for the time series of each component is analyzed for chaos characterization and CRP is used to suggest the periodicity as well as its chaotic characteristics.

2. Study Area and Methodologies

This study separated a water-level time series of tidal river into tide, wave, rainfall-runoff (or river stage), and noise level components as proposed by Lee et al. (2018) [2]. Additionally, we analyzed using each component time series for searching for chaotic characteristics. The methodologies for a water level decomposition and for the chaos analysis were described in this section. The flowchart and analysis method of this study is shown in Figure 1. First, the water level time series was decomposed into tide, wave, rainfall-runoff (or river stage), and noise components through wavelet analysis, curve fitting, and the filtering technique (see Lee et al., 2018) [2].

Furthermore, the information entropy theory was used to quantify the amount of information provided by four components to the water level time series to determine the relationships between the separated components and the original water level time series. Based on the estimated amount of information, the nonlinear dynamics and chaos analysis were applied to observe the characteristics for each time series of component of the water level time series. The methods we used to analyze the chaotic characteristics of the time series include a metric method called correlation dimension and a topological method called a close returns plot (CRP) [22,23].

2.1. Study Area

The study area is Taehwa river basin in Korea, and the water level time series measured at Ulsan station in the basin is obtained. Taehwa River basin is located in the south-eastern region of the Korean peninsula. The basin area is 646 km² and average slope of the basin is 0.2770. The length of river is 47.54 km and the shape factor is 0.72, which shows a characteristic that peak runoff can be occurred in a short time. The west side of the basin is mountain area, so the river flows from west to east then meets the open sea in the estuary. The estuary is subject both to marine influences such as tides, waves, and influx of saline water and to riverine influences such as flows of freshwater and sediment. There are two Ulsan stations (Figure 2). One is for observing the sea surface level and the other is for observing water level of tidal river. Because the Ulsan sea surface level station is at the downstream end of the station for water level, the study area is deemed suitable for applying and verifying the proposed methodology.

2.2. Study Area Water Level and Sea Surface Level Characteristics

The water level and sea-level time series from Ulsan stations are obtained at a daily scale (data size of 2891) from 1 January 2011 to 31 November 2018. The unit of data is daily data, and daily data have been observed stably without missing data since 2011. The water level time series shows the typical characteristics of a tidal river with high variation in low water level, and is also affected by rainfall (Figure 3a). Meanwhile, the sea level time series have low variation characteristics and are influenced by tides and waves.

2.3. Decomposition and Information Quantification of the Time Series

2.3.1. Decomposition of the Time Series

A tidal river is located in a coastal area and influenced by backflow caused by the mixture of tide and stream. Therefore, the water level and the runoff fluctuate with the tidal impact, and flood damages are aggravated by the backwater effect during high tide. This makes the characteristic of the tidal river water level understand the fluctuation behavior of rainfall, tides, and waves. Consequently, this study decomposed the water level time series as proposed by Lee et al. (2018) [2], where wavelet analysis and curve fitting are used to separate the water level data into the “tide” and “wave” component and the filtering method is used to separate the rest of the components, “rainfall-runoff or river stage” and “noise”.

2.3.2. Information Quantification

Entropy theory is used to quantify the time series effect of the separated components. Shannon and Weaver (1949) defined the limit entropy as Equation (1) [24]:

$$H\left(X\right)=-{\displaystyle \sum }_{n=1}^{N}p\left(x_n\right)logp\left(x_n\right)$$

(1)

where $p\left(x_n\right)$ is the occurrence probability of $x_n$ ($n=1,2,3,\dots ,N$), and $H\left(X\right)$ is the limit entropy indicating the information amount of X. When the variable $y_m\left(m=1,2,3,\dots ,N\right)$ is the information having a relation with the variable $x_n$, the uncertainty of $x_n$ is likely reduced. Based on this theory, the conditional entropy $H\left(X|Y\right)$ can be estimated, as shown in Equation (2), and indicate whether a random variable $y_m$ helps to predict the value of another random variable $y_m$.

$$H\left(X|Y\right)=-{\displaystyle \sum }_{n=1}^N{\displaystyle \sum }_{m=1}^{N}p\left(x_n,y_m\right)lnp\left(x_n|y_m\right)$$

(2)

where $p\left(x_n,y_m\right)$ is a joint probability when $X=\left\{x_n\right\}$ and $Y=\left\{y_m\right\}$. $p\left(x_n|y_m\right)$ is a conditional probability of X at a given Y. Information transferred between X and Y can be explained as Equation (3):

$$T\left(X,Y\right)=H\left(X\right)-H\left(X|Y\right)$$

(3)

The information sent from X to Y is defined as $S\left(X,Y\right)$ in Equation (4):

$$S\left(X,Y\right)=\frac{T\left(X,Y\right)}{H\left(Y\right)}$$

(4)

where $H\left(Y\right)$ is the limit entropy of the single variable Y and $T\left(X,Y\right)$ is the trans-information between X and Y.

2.4. Phase Space Reconstruction

2.4.1. Method of Delays

In reality, natural phenomena can be characterized as deterministic, stochastic, or chaotic, and the reconstruction of the phase space and the attractor could be used to identify characteristics of the time series. Packard et al. (1980) and Takens (1981) suggested the method of delays for the phase-space reconstruction of the time series [25,26]. A scalar time series $\left\{x_i\right\},i=1,2,\dots$ is embedded into m-dimensional space by constructing the vectors as Equation (5):

$$\overrightarrow{x_i}=\left(x_i,x_{i+1},\cdots ,x_{i+\left(m-1\right){\tau }_d}\right),\overrightarrow{x_i}\in \mathrm{R}$$

(5)

where ${\tau }_d$ is the delay time and $m$ is the embedding dimension, both of which must be chosen appropriately.

2.4.2. Estimation of Delay Time

There are some methods widely used to determine an appropriate delay time: (1) autocorrelation function (ACF) [27], (2) average mutual information [28], (3) correlation integral [29], and (4) the C-C method [30]. Although ACF is the general method for obtaining the delay time, its high dependence on linearity is not applicable for the nonlinear system. Meanwhile, the average mutual information method can estimate the delay time regardless of linearity or nonlinearity, but has a limitation in that it requires a large amount of data. Ultimately, this study used ACF and average mutual information to estimate the delay time for the phase space reconstruction of each component and then the attractors can be obtained for each time series. Choosing the proper delay time and method of delays can be found in [23,30].

2.5. Correlation Dimension

The correlation dimension is used for the quantitative analysis of the typical geometry and the localized distribution of attractors in a phase space. In obtaining the correlation dimension, Gassberger and Procaccia (1983) proposed a method by estimation of the quantity called correlation integral [31]. This value is the number of phase vector points calculated for the radius, which is located with certain scaling, and the correlation integral [C(r)], which can be obtained as Equation (6):

$$C\left(m,N,r\right)=\frac{2}{M\left(M-1\right)}{\displaystyle \sum }_{1<i<j\le M}\theta \left(r-\Vert \overrightarrow{x_i}-\overrightarrow{x_j}\Vert \right), r>0$$

(6)

where $\theta \left(a\right)=0\mathrm{if}a<0$, $\theta \left(a\right)=1\mathrm{if}a\ge 0$. $\theta$ is the Heaviside step function where zero is applied to real numbers less than zero, and one is applied to real numbers greater than zero. N is for the number of all coordinate points in one-dimensional space, $M=N\left(m-1\right)$ for the number of state vector points in m-dimensional space, and state vector points are expressed by $\overrightarrow{x_i}$ and $\overrightarrow{x_j}$ in the phase space. When the value of the radius r is given, it can be converted to the logarithmic scale to obtain $D_2$, which is called the correlation dimension, as shown in Equation (7):

$$D_2=\underset{r\to 0}{\lim }\frac{log\left[C\left(r\right)\right]}{logr}$$

(7)

The correlation dimension can be defined as the variance of correlation integral from the change in the scale. If the hydrological time series is stochastic, the correlation dimension continues to increase as the embedding dimension m increases. However, if it is chaotic, it is characterized as converging into a certain value, regardless of an increase in the embedding dimension $m$.

2.6. Close Returns Plot

A close returns plot (CRP) is used for analyzing the chaotic characteristics of a time series in this study. The chaos system is characterized by complex and strange attractors on a phase space with the repeated occurrence of a periodic orbit. The analysis is based on the rule that, after a random time, the chaos system passes through the neighborhood point again [16,32]. For this, $\mathsf{\delta }$, indicated by the difference of the points in the time series, is used as in Equation (8):

$$\mathsf{\delta }=\left|x_i-x_j\right|$$

(8)

If the system is close to the state t = j at t = 1 in the phase space, then the necessary condition is $\mathsf{\delta }=\left|x_i-x_j\right|\to 0$, and $\mathsf{\delta }$ could be drawn diagonally. In addition, the same diagonal line is drawn repeatedly at i = j. The figure is drawn as in Equation (9):

$$\mathsf{\delta }=\left|x_i-x_j\right|,\{\begin{array}{c}BlackPoint,if\delta <r\\ WhitePoint,otherwise\end{array}$$

(9)

where $r=\left(0.01~0.1\right)\times \left|x_{max}-x_{min}\right|$ is the radius at an arbitrary point, and $x_{max}$ and $x_{min}$ are the maximum and minimum values of the time series, respectively. The range of $r$ was suggested in [16].

3. Characteristics of Water Level and Decomposed Time Series

3.1. Decomposition of Water Level Time Series

The water level time series from Ulsan gauging station was decomposed into four components of tide, wave, noise, and rainfall-runoff using the wavelet analysis and filtering technique proposed by [2]. The decomposition results are shown in Figure 4 and

Table 1. Basic statistics for the decomposed time series of water level from Ulsan station.

Component	Water Level	Tide	Wave	Noise	Rainfall-Runoff
Number of data (daily)	2891 (1 January 2011~31 November 2018)
Mean (m)	1.28	1.17	0.12	0.02	0.01
Standard deviation (m)	0.18	0.10	0.12	0.05	0.12
Coefficient of skewness (m)	8.61	0.14	0.22	2.00	23.47
Coefficient of variation	0.15	0.08	1.01	−1.87	15.08

. Tide and wave components were separated through a sea surface level time series, and rainfall-runoff and noise components are defined as components excluding tide and wave components from the water level time series. A high pass filter is applied to separate these two components. In other words, the rainfall-runoff component and noise component were separated based on a certain threshold value, which reflects the characteristic that the water level rises rapidly when rainfall occurs. Therefore, rainfall-runoff shows a value of 0 in the no rainfall period, and has a certain value when the water level rises rapidly in the rainfall period.

The water level time series has similar statistical characteristics with those of the tide component, but has a higher coefficient of skewness and a coefficient of variation due to the rainfall (Table 1). For the tide, it was identified that the mid-period component was well extracted in approximately 15 days for the spring and neap tide. In addition, compared to the other components, the tide shows the narrower ranges for standard deviation and coefficient of variation. On the other hand, the remaining component after the tide component was extracted from the sea-level time series is considered as the wave component, which shows a high coefficient of variation of 1.01 indicating a wide range (Table 1). The rainfall-runoff component occurs by rainfall and so it has zero values if there is no rainfall. Additionally, it corresponds to the sudden rises in a water level time series and has very high variability and skewness as shown in Table 1. The noise component is the last component.

3.2. Water Level and Sea Surface Level Characteristics

The information entropy theory was applied to identify the relationship among the separated four components as shown in Equations (1)–(4). In addition, the amount of information provided to the water level time series was quantified, and the results are summarized in the following

Table 2. Rate of influence for each component.

Component	Entropy Information	Influence Rate (%)
Tide	2.22	48
Wave	1.14	24
Rainfall-runoff	0.80	17
Noise	0.51	11
${\displaystyle \sum }$	4.67	100

.

${\displaystyle \sum }$ The sum of the four information entropies obtained using Equations (1)–(4) is 4.67. The value of the tide component is 2.22, indicating that it has the most information on the original water level data. This was followed by the wave component with 1.14, the rainfall-runoff component with 0.80, and the noise component having the least information with 0.51.

4. Chaotic Features Analysis of Decomposed Time Series

4.1. Estimation of Delay Time and Attractor

4.1.1. Delay Times for the Decomposed Time Series

To identify the chaotic characteristics of each component, phase space for each time series is reconstructed. For the reconstruction of the phase space, the delay time for each time series is calculated using the autocorrelation function (ACF) and average mutual information. The ACF for each component is shown in Figure 5.

As shown in Figure 5a, the ACF of the water level time series is decreased exponentially until 200 days since the beginning. This exponential decrease indicates the presence of chaotic characteristics in the water level time series [33]. In addition, the small variability in the water level time series can be explained by the autocorrelation coefficients of the tide and wave components, which shows periodicity under tidal influence. For the noise and rainfall-runoff components, the delay time can be chosen from the first minimum and zero crossing points.

On the other hand, Figure 6 shows the variability of average mutual information. Similar to the results of the ACF, the water level time series, tide, and wave components have a pattern showing regular fluctuation which shows the periodicities, and the noise and rainfall-runoff components show a little fluctuation around zero value. From the ACF and average mutual information, the delay time can be estimated (i) at the point where the value of each component becomes 0, (ii) at the first minimum point that the ACF has before passing 0, or (iii) at the point corresponding to 1/e, if the graph shows an exponential decrease. Considering the type of each ACF, the delay times are estimated as shown in

Table 3. Estimated delay times by the autocorrelation function(ACF)and average mutual information.

Component	Delay Time (Day)
	ACF	Average Mutual Information
Water level	9 (first minimum)	9 (first minimum)
Tide	7 (zero crossing)	6 (first minimum)
Wave	8 (zero crossing)	7 (first minimum)
Noise	6 (first minimum)	4 (first minimum)
Rainfall-runoff	5 (zero crossing)	9 (first minimum)

.

As mentioned, the results of the two methods are almost identical for the original water level time series, the tide component, and the wave component with periodicity. However, for the rainfall-runoff component, the results were different from each method. Such difference was caused by the ACF, which does not reflect the nonlinear characteristics and the average mutual information requiring a large amount of data [34].

4.1.2. Attractors for the Decomposed Time Series

The estimated delay times are used to reconstruct the attractors in the two-dimensional phase space, as shown in Figure 7. At the low water level with no rainfall, the attractor is located within the spatial boundary. However, when the rainfall occurs, the increasing water level leads the trajectory out of the periodic boundary, causing a spark (Figure 7a). On the other hand, the tide component slightly deviates from the periodicity as it moves with its own boundary (Figure 7b). In fact, since tides fluctuate with a cycle of 24 h and 25 min, drawing an attractor using daily data has a limitation in that it partially deviates from the circular orbit. Meanwhile, the wave component also shows periodicity, but it deviates from the trajectory more than the tidal component. Noise component shows a more sensitive response to rainfall compared to the tide component (Figure 7d). Because the rainfall-runoff component only increases with rainfall (see Figure 4d), it is not possible to draw an attractor with a certain boundary.

4.2. Estimation of Correlation Dimension

Correlation dimensions were estimated to identify the hydrological factors that affect the water level data of the water level station. The correlation estimation requires calculation of the slope of the local area and the delay time. If the correlation dimension and embedding dimension diverges in the graph, the data can be considered as stochastic. On the other hand, when it converges to a value, the data can be considered as chaotic. Figure 8 shows the correlation integral described by the relation between log C(r) and log r for each data when the embedding dimension (m) changes from 2 to 25.

Casaleggio et al. (1995) divided the graphs of logC(r) and log r into three regions (noise, scaling, and saturation regions), and suggested that the correlation dimension can be estimated using the slope of the scaling region that has the linearity [35]. In this study, we estimated the relationship between the correlation dimension and the embedding dimension values for each dimension in the scaling region of Figure 8, where the accurate estimation of the correlation dimension can be made, and the results are shown in Figure 9. Especially, the correlation integral of rainfall-runoff component shows different characteristics from others. As mentioned above, the rainfall-runoff component usually shows a value of 0 and has a certain value when the water level rises rapidly. Therefore, the number of time series existing in the circle is not large, even though the dimension and the radius of circle are increased (see Figure 8e).

The result shows that the tide component has chaotic characteristics, given the convergence to a certain value of 1.30. However, the water level data and wave, noise, and rainfall-runoff components have stochastic characteristics as it showed a diverging pattern. Di et al. (2018) confirmed that correlation dimension of rainfall data at a specific rainfall station is diverging even though the dimension increases [36]. Through the above study, we can estimate that the water level time series shows stochastic characteristics because the correlation dimension diverged without converging to a specific value. As for the tide component, the correlation dimension of 1.30 indicates the two factors having influence on the component. They are likely to have the periodic characteristic caused by the moon that orbits around the Earth in approximately 30 days and by the Earth’s changing position as it orbits around the sun in approximately one year. Tide component shows fluctuations in the short-cycle (orbit of the earth), the mid-cycle (gravity of Earth and moon), and the long-cycle (gravity of earth and sun). However, since this study used daily data, only the effects on the mid-cycle and long-cycle components could be considered. In this case, the tide component has aperiodic and nonlinear characteristics as they are gradually deviating from the periodicity, rather than having an accurate period.

4.3. Close Returns Plot for the Decomposed Time Series

The attractor and correlation dimension show the nonlinearity of the water level time series and the three components. In addition, a close returns plot was used to determine whether the nonlinear time series has chaotic or stochastic behavior. Regular horizontal line segments in CRP indicates that the data has the characteristics of a periodic function. On the other hand, horizontal line segments are drawn at a certain period with irregularity to indicate chaotic characteristics, and random dot segments indicate stochastic characteristics (Figure 10). Figure 10a shows the pattern of CRP for the periodic model with the repeated appearance of regular line. Figure 10b shows the CRP of the Hénon map indicating a typical example of chaotic characteristics which have irregular horizontal line segments. Meanwhile, Figure 10c shows a time series with stochastic characteristics. In the Hénon map, the chaotic characteristics are seen in the irregular horizontal lines in the uniformed pattern. However, it is found for the stochastic time series that the random dots appear without a regular trajectory, forming a random distribution.

For the water level time series, tide, wave, and rainfall-runoff components, the CRPs were drawn, as shown in Figure 11. Figure 11a shows the CRP using the time series of Figure 12, and the enlarged part in Figure 11a is the square area of Figure 12. Here, the x-axis means the i^th time series, and the y-axis means the distance (T) from the i-th time series. For the period when there was no rainfall, the Ulsan water level data showed the periodic characteristics, with the irregular horizontal segments appearing similar to those of the tide component. However, when rainfall occurred, the data had stochastic characteristics (top right of Figure 11a). As can be seen from time series, the water level periodically fluctuated and maintained the trajectory when the rainfall did not occur. However, they show a diagonal shape deviating from the orbit when the water level rises due to the rainfall. As for the tide component, there was aperiodic recurrence of irregular horizontal line segments, indicating chaotic characteristics (Figure 11b). For wave height, noise, and rainfall-runoff components, there is no horizontal line and it has random points, which means stochastic characteristics (Figure 11c–e).

5. Summary and Discussions

This study reconstructed the phase space, analyzed the correlation dimension, and a CRP to identify the characteristics of the water level time series of the tidal river. Because the water level time series of a tidal river shows complex hydrological characteristics affected by tides, waves, and other factors, this study decomposed the water level time series into four components to examine each of their hydrological characteristics. In quantifying the effects of the decomposed components on the water level time series, entropy information theory was used. As a result, the tide component was identified to have the most influence on the original Ulsan water level time series by 48%, followed by the wave component at 24%, the rainfall-runoff component at 17%, and the noise component at 11%. This indicates the possibility of the tide component with a periodic fluctuation, but not exact periodic, as the major factor of water level change that leads to chaotic characteristics.

Furthermore, the delay time was calculated using the ACF and average mutual information to examine characteristics of each component. For data such as the tide component and the original time series of the water level, the same delay time was calculated. However, the data with random characteristics, such as the rainfall-runoff component, had different delay times due to the difference in the linearity and nonlinearity. The calculated delay times were used to draw the attractors on two-dimensional phase space. As a result, the water level time series, tide component, and wave component had the trajectory within a certain boundary. However, the noise component presented a random trajectory.

In determining whether each component has stochastic or chaotic characteristics, the correlation dimension was estimated. With the correlation dimension of 1.30, the tide component was influenced by two variables (mid-period and long period). As shown in Figure 9, only the tide component had a chaotic characteristic as it converged into a certain value, and the other components showed stochastic characteristics as they diverged. These results are significant in determining whether a linear model or a nonlinear model should be used to predict tidal river water levels with complex hydrological characteristics.

Last, this study analyzed the chaotic characteristics of each component using CRP to understand the recurring trajectory in a graph. When regular horizontal line segments repeatedly appear in the CRP, it can be assumed as the periodic data that the corresponding trajectory is continuously visited. On the other hand, the irregular horizontal line segments indicate chaotic characteristics, while random distribution without lines implies stochastic characteristics. As a result, the original water level time series showed aperiodic characteristics caused by tidal effects during no rainfall (Figure 11a). However, when rainfall occurs, it had oblique line segments indicating stochastic characteristics. Therefore, this time series might be mixed with chaotic and stochastic characteristics, which can be called stochastic-chaos. For the tide component, irregular horizontal line segments appear similar to those of the aperiodic function and so it indicated chaotic characteristics by the uncertainty of natural phenomena. For the other components, the segments are oblique lines, indicating their stochastic characteristics.

Table 4. Summary for characteristics by component.

Component	Attractor	Correlation Dimension	Close Returns Plot (CRP)	Characteristics
Water level	Aperiodic	Stochastic	Stochastic chaos	Aperiodic-stochastic chaos
Tide	Aperiodic	Chaotic	Chaotic	Aperiodic-chaotic
Wave	Aperiodic	Stochastic	Stochastic	Aperiodic-stochastic
Rainfall-runoff	Aperiodic	Stochastic	Stochastic	Aperiodic-stochastic
Noise	Aperiodic	Stochastic	Stochastic	Aperiodic-stochastic

shows the characteristics of each of the components. The water level time series were influenced by various hydrological factors and showed stochastic characteristics in the correlation dimension, but the characteristics of aperiodic-stochastic chaos in CRP. Salarieh et al. (2009) have suggested that the behaviors related to stochastic chaos characteristics can be identified using the white noise in the deterministic chaos system [37,38,39].

To confirm this behavior, the time series of Gaussian noise showing stochastic characteristics and the Hénon map showing chaotic characteristics were generated and combined as shown in Figure 13. Then, the correlation dimension is estimated, and CRP is drawn for investigating chaotic characteristics of the combined time series. As we can see in Figure 13, the time series shows stochastic property in the correlation dimension but stochastic chaos characteristics in CRP. Figure 13c is showing the stochastic phase on the left side of the yellow line, but the chaotic phase on the right side of the line. Therefore, we can confirm the stochastic chaos characteristics for the combined time series, and the water level time series also shows the same stochastic chaos characteristics. In other words, the water level time series showed chaotic characteristics when there was no rainfall but presented stochastic characteristics when rainfall occurred (Figure 11a). As for the tide component, which affects the water level time series by 48 h, it shows aperiodic chaotic characteristics, while the other components were characterized as aperiodic stochastic data.

6. Conclusions

This study conducted chaos characterization analysis using the tidal river water level time series of the Ulsan station in Taehwa river basin, Korea. For the analysis, the data from January 2011 to 31 October 2018 were collected as shown in Figure 3. We considered the effects of increasing water level on tidal rivers which were caused by various hydrological factors such as tides, waves, and rainfall. We identified the components that influence tidal river water-levels and then analyzed the chaotic characteristics for each of them, namely (1) the tide component, (2) the wave component, (3) the rainfall-runoff component, and (4) the noise component. As a result of the water level decomposition, tide component was found to have the most influence on the water level time series. In addition, chaos analysis was performed for each component, which revealed that the tide component has chaotic characteristics while the other components have stochastic characteristics. Additionally, water level time series showed stochastic chaos characteristics in its CRP, and we showed that the combined time series of Gaussian and the Hénon map series also has stochastic chaos properties. This means that the chaotic tide component and stochastic rainfall shock are combined and make the stochastic chaos time series.

Although the results from this study could not allow one to conclude with a definitive answer regarding the existence of chaotic dynamics of the rainfall-runoff process, the study indicates that such an existence should not be excluded. It is necessary, therefore, to continue the investigation to confirm the existence of a chaotic component in the rainfall-runoff process. One notable point is that the stochastic chaos behavior discovered through the combined time series using Gaussian and the Hénon map is similar to that of the water level time series, which suggests a new approach to define the nonlinear behavior. This finding may be helpful for nonlinear modeling for predicting the tidal river water level time series, which can be separated as tide and rainfall components.