View of Chaotic Identification of Hourly and Daily Water Level Time Series Data in Different Areas of Elevation at Pahang River

Research Article

Chaotic Identification of Hourly and Daily Water Level Time Series Data in Different

Areas of Elevation at Pahang River

Adib Mashuri1_{, Nur Hamiza Adenan}2*_{, NorSuriya Abd Karim}3

1,2,3_{Department of Mathematics, Faculty of Science and Mathematics, Universiti Pendidikan Sultan Idris,}

Tanjung Malim, Perak, Malaysia hamieza@fsmt.upsi.edu.my2*

Article History: Received: 10 November 2020; Revised: 12 January 2021; Accepted: 27 January 2021; Published online: 05 April 2021

Abstract:This study focused on chaotic analysis of water level data in different elevations located in the highland and

lowland areas. This research was conducted considering the uncertain water level caused by the river flow from highland to lowland areas. The analysis was conducted using the data collected from the four area stations along Pahang River on different time scales which were hourly and daily time series data. The resulted findings were relevant to be used by the local authorities in water resource management in these areas. Two methods were used for the analysis process which included Cao method and phase space plot. Both methods are based on phase space reconstruction that is referring to reconstruction of one dimensional data (water level data) to d-dimensional phase space in order to determine the dynamics of the system. The combination of parameters



and d is required in phase space reconstruction. Results showed that (i) the combination of phase space reconstruction’s parameters gave a higher value of parameters by using hourly time scale compared to daily time scale for different elevation; (ii) different elevation gave impact on the values of phase space reconstructions’ parameters; (iii) chaotic dynamics existed using Cao method and phase space plot for different elevation and time scale. Hence, water level data with different time scale from different elevation in Pahang River can be used in the development of prediction model based on chaos approach.

Keywords: Chaotic analysis, different elevation, different time scale, water level, phase space reconstruction 1. Introduction

Geographical location of Malaysia does not give an impact on natural disaster such as earthquakes, typhoons and volcanoes. However, flood disaster often hit several places in Malaysia (Musa & Shafii, 2012). Flood can usually occurred in upstream areas (Bozorg-Haddad, Hamedi, Fallah-Mehdipour, & Loáiciga, 2018) and downstream area (Barman & Choudhury, 2015). In addition, different elevations areas also contributes to flood (Kourgialas & Karatzas, 2011). In this research, different elevations are referred to highland and lowland areas along the river that contribute to flood disaster in located at Pahang River. The analysis was done in different elevation areas since the drainage system in highland area gives pressure to the river flow in lowland area which can cause destruction due to uncertain water level; hence, it may lead to flood phenomenon (Marcinkowski & Grygoruk, 2017; Tare et al., 2017). This uncertain water level may greatly influenced by rain that made a pattern for its intensity (Anika & Kato, 2019). Hence, in this study, the comparative chaotic dynamics identification for water level time series data at different elevations involving highland and lowland was conducted in order to develop prediction model based on chaos approach.

Chaotic dynamics identification using time series was first discovered by (Lorenz, 1963). Since 1963, the study on this approach has been widely investigated in various field such as in atmospheric (Ruslan & Hamid, 2019) and hydrology (Mashuri, Adenan, & Hamid, 2019). As the world is changing into a modern world, vast knowledge and applications on chaos approach have been developed including the implementation of chaos in science and engineering as well as in hydrology area. Through literature studies, the chaotic analysis on water level systems is widely applied in many countries such as in China (Huang, Huang, Jiang, & Zhou, 2017), Singapore (Wang & Babovic, 2014) and Australia (Tongal & Berndtsson, 2014). Literally, different water areas have different geographical characteristics which may contribute to the accuracy of the prediction using chaotic approach. Therefore, it is a crucial need to conduct chaotic identification on water level time series data in Malaysia in order to examine the chaotic dynamics existence and perform prediction by developing the prediction model.

2. Data

Pahang River is the longest river in Peninsular Malaysia. It crosses several towns in Pahang State such as Jerantut, Temerloh, Maran, Bera, and Pekan which then flows to the South China Sea. This river also crosses some tourist areas in Pahang such as Endau Rompin National Park, Cameron Highland and Genting Highland (Khairul et al., 2015). Many areas in Pahang are gazetted as agricultural areas and have become one of the

economy activities in this state. Along Pahang River, about 745,000 hectares of land are cultivated by many agricultural activities of many types of crops and plants such as paddy, rubber and oil palm. As flood phenomenon may occur and can severely affect these economy activities areas, therefore it is important to predict the water level at these areas in order to help the water resource management so that the government revenue sources can be (Weng & Mokhtar, 2010).

Historically, flood disaster recorded in 1926 was considered as the worst flood phenomenon in Malaysia that affected most areas in Peninsular Malaysia (Ghani, Chang, Leow, & Zakaria, 2012). Then, another flood happened in 1971 which affected Pahang severely and caused damaged for about 300km2_{. In 2007, the flood}

again occurred in Pahang River with estimated loss of 86 million dollars by the Department of Irrigation and Drainage Malaysia (Ghani et al., 2012).

Different elevations may give impact on speed, water level and flow (Chapman, 1996). Therefore, Pahang River is chosen as the location of this study since the characteristics of the catchment area of the river that consists of lowland and highland areas along the river fit the aim of this research that is to determine the chaotic dynamics of water level in different elevation. Hence, this study focuses on its tributaries that flow to Pahang River catchment from highland to lowland area. As the third largest state in Malaysia, Pahang has many economy activities as well as that lies along Pahang River. The highland area of the Pahang River acts as a water catchment and serves as a major hydropower source for the lowland. It also provides water resources for agriculture, industrial and domestic use downstream (Razali, Syed Ismail, Awang, Mangala Praveena, & Zainal Abidin, 2018).

Four different tributaries were selected for the analysis where two rivers were located at highland areas and another two rivers were located on lowland areas. Data for this study were collected at four different stations located at these tributaries. The details of the water level time series data are shown in Table 1. Meanwhile, Figure 1 and Figure 2 show the trends of data that were analysed according to different time scale. Referring to both figures, time series data in this analysis showed fluctuation and uncertainties. Therefore, the analysis need to be conducted based on chaos approach in order to give information about the tributaries and the development of prediction model based on chaos approach.

Table 1.Data for the chaotic analysis existence at different elevation areas in Pahang River with different time scales

Elevation Station Hourly Daily

Station Code Total data (hours) Station Code Total data (days) Highland Areas Jelai River at JeramBugor SPT01h 3529 SPT01d 4458

Jelai River at Kuala Medang SPT02h 3529 SPT02d 4458 Lowland Areas Pahang River at Temerloh SPR01h 3529 SPR01d 4458 Pahang River at LubokPaku SPR01h 3529 SPR01d 4458

The selection of time scale is important for the suitability of the study as to provide useful input for the local authority for early warning disaster preparation and water resource management. Furthermore, the need for preliminary information for early warning disaster is the main reason of time scale selection (Pandey & Srinivas, 2015). As proven, chaotic behaviour analysis using hourly time series data is more effective for prediction in flood areas (Sivakumar & Wallender, 2005). Meanwhile, daily time series data is suitable for water resource management for analysis and prediction (Adenan & Noorani, 2016). Hence, the selection of time scale is very important according to the need of the certain areas.

a) _b)

c) d)

Figure 1. Hourly water level time series data in Pahang River for station (a) SPT01h, (b) SPT02h, (c) SPR01h and (d) SPR02h

a) b)

c) d)

Figure 2. Daily water level time series data in Pahang River for station (a) SPT01d, (b) SPT02d, (c) SPR01d dan (d) SPR02d

3. Methodology

3.1. Chaotic Behaviour Analysis

There are various methods used in determining chaotic dynamics for hydrological time series data such as Cao method (Mashuri et al., 2019), phase space plot (Yildirim, Hacinliyan, Akkaya, & Ikiel, 2016), correlation dimension (Albostan & Önöz, 2015) and Lyapunov exponent (Mihailović et al., 2019). Each method gives different result in analysing the existence of chaotic dynamics (Khatibi et al., 2012) In this study, Cao method and phase space plots were used to identify the chaotic dynamics. The phase space plot is widely used in hydrological water level studies. Meanwhile, Cao method can calculate embedding dimension parameters (d) as well as identifying the presence of chaotic behaviour. Cao (1997) stated that this method shows advantages

0 500 1000 1500 2000 2500 3000 3500 55 56 57 58 59 Hour W at er L ev el ( m ) 0 500 1000 1500 2000 2500 3000 3500 80 81 82 83 84 Hour W a te r L e v e l (m ) 0 500 1000 1500 2000 2500 3000 3500 23.5 24 24.5 25 25.5 Hour W at er L ev el ( m ) 0 500 1000 1500 2000 2500 3000 3500 12 12.5 13 13.5 14 14.5 15 Hour W a te r L e v e l (m ) 0 500 1000 1500 2000 2500 3000 3500 4000 54 56 58 60 Day W at er L ev el ( m ) 0 500 1000 1500 2000 2500 3000 3500 4000 79 80 81 82 83 84 85 86 Day W at er L ev el ( m ) 0 500 1000 1500 2000 2500 3000 3500 4000 24 26 28 30 32 34 Day W at er L ev el ( m ) 0 500 1000 1500 2000 2500 3000 3500 4000 12 14 16 18 20 22 Day Wate r Le vel (m)

compared to other methods since Cao method does not contain any parameters except time delay,



and it does not depend on the number of data.

3.2. Phase Space Plot

Phase space reconstruction involves a collection of time series data that is observed in one dimensional time series where the time series will be reconstructed to d-dimension phase space. The Y time series is recorded as follow:

𝑌 = {𝑦1, 𝑦2, 𝑦3, … , 𝑦𝑁}

1 2 1,

{ ,

,...,

N N

}

Y



y y

y



y

(1)

Y is referred toas time series and it will be used to determine the presence of the chaotic dynamics as well as to calculate the required parameter. According to Takens (1981), phase space delay time  , embedding dimension d and phase space d-dimension is in the form of d

i Z as in the following: 2 ( 1) ( i, i , i ,..., ) d i d Z  y y y y   (2)

with i = 1, 2,…,

N

 

(

d

1)



whereby 𝑍𝑖∈ 𝑅𝑑.In order to obtain the value of



, the average mutual

information (AMI) is used in calculation to determine the chaotic behaviour using phase space plot and Cao method only. The AMI method can be calculated by using the following equation:





2

_{ }



_



_

1

1

,

log

N a a T a a T a a a T

p u

u

AMI

p u u

N

p u

p u

   























(5)

where p u

 

_a and p u



_{a T}



is the probabilityof finding the value of ua and ua T given time series Y,

respectively, and p u



_au_{a T}



is the probability of p u

 

a andp u



a T



.

Cao method gives a practical way in determining the value of d where this value gives optimum value (d_opt) for scalar time series (Cao, 1997). The value of d_opt by using Cao Method is calculated by

 



_{ }

1



1

E d

E d

E d





, and (6)

 

1 1 1 1 N d d d n jj d d n _{n jj} Z Z E d N d Z Z         



 (7)

with referring to the Euclidean distance and d jj

Z referring to the neighbouring value for d n

Z . If E1

 

d does not change when the value d is larger than d0, hence d01 is the optimum dimension value (Cao, 1997).

Besides identifying the embedding dimension value (d), this method could help to determine chaotic dynamics of a system. If the value E1(d) increases continuously with d, hence the time series is random. If the chaotic dynamic exists, graph of E2(d) versus d will show at leastE2( )d 1. The calculation of E2(d) is given as follow:

 



_{ }

1



2 E d E d E d     , (9)

 

1

1

N d NN i d i d i

E

d

x

x

N

d

  



    









. (10)

4. Results and Discussion

In this section, the results are discussed in three parts. The first part presents the results on the chaotic analysis of hourly time scale at different elevations while the second part is on the chaotic analysis of daily time scale at different elevations. After that, all are discussed critically in the latter part.

Part 1: Hourly Time Scale

The combination of value (



, d) for the base of the analysis using phase space reconstruction is as shown in Table 5.

Table 5.The resulted values of



and d using hourly time series data

River Area Highland Lowland

( , d) SPT01h SPT02h SPR01h SPR02h (19,6) (8,6) (23,5) (32,5) a) b) c) d)

Figure 3. Chaotic analysis by using Cao method for hourly water level time series data at different elevations in Pahang River: (a) SPT01h, (b) SPT02h, (c) SPR01h and (d) SPR02h

a) b)

c)

d)

Figure 4. Chaotic analysis by using phase space plot for hourly water level time series data at different elevations in Pahang River: (a) SPT01h, (b) SPT02h, (c) SPR01h and (d) SPR02h



1 2 3 4 5 6 7 8 9 10 1 Embedding Dimension (d) E 2 (d ) 1 2 3 4 5 6 7 8 9 10 1 Embedding Dimension (d) E 2 (d )

Minimum embedding dimension using Cao's method

1 2 3 4 5 6 7 8 9 10 1 Embedding Dimension (d) E 2 (d ) 1 2 3 4 5 6 7 8 9 10 1 Embedding Dimension (d) E 2 (d ) 55 55.5 56 56.5 57 57.5 58 58.5 59 59.5 60 55 55.5 56 56.5 57 57.5 58 58.5 59 59.5 60 x(t) x (t + 1 9 ) 79.5 80 80.5 81 81.5 82 82.5 83 83.5 84 79.5 80 80.5 81 81.5 82 82.5 83 83.5 84 x(t) x (t + 8 ) 23.5 24 24.5 25 25.5 23.5 24 24.5 25 25.5 x(t) x (t + 2 3 ) 12 12.5 13 13.5 14 14.5 15 15.5 12 12.5 13 13.5 14 14.5 15 15.5 x(t) x( t+ 32 )

Part 2: Daily Time Scale

The combination of value (



, d) for the base of the analysis using phase space reconstruction is shown in Table 6.

Table 6.The resulted values of



and d using hourly time series data

River Area Highland Lowland

(



, d) SPT01d SPT02d SPR01d SPR02d

(11,6) (14,6) (8,5) (9,5)

a) b)

c) d)

Figure 5. Chaotic analysis by using Cao method for daily water level time series data at different elevations in Pahang River: (a) SPT01d, (b) SPT02d, (c) SPR01d and (d) SPR02d

a) b)

c) d)

Figure 6. Chaotic analysis by using phase space plot for daily water level time series data at different elevations in Pahang River: (a) SPT01d, (b) SPT02d, (c) SPR01d and (d) SPR02d

1 2 3 4 5 6 7 8 9 10 1 Embedding Dimension (d) E 2 (d ) 1 2 3 4 5 6 7 8 9 10 1 Embedding Dimension (d) E 2 (d ) 1 2 3 4 5 6 7 8 9 10 1 Embedding Dimension (d) E 2 (d ) 1 2 3 4 5 6 7 8 9 10 1 Embedding Dimension (d) E 2 (d ) 52 53 54 55 56 57 58 59 60 61 62 52 53 54 55 56 57 58 59 60 61 62 x(t) x (t + 1 1 ) 78 79 80 81 82 83 84 85 86 87 78 79 80 81 82 83 84 85 86 87 x(t) x (t + 1 4 ) 23 24 25 26 27 28 29 30 31 32 33 34 23 24 25 26 27 28 29 30 31 32 33 34 x(t) x (t + 8 ) 12 14 16 18 20 22 24 12 14 16 18 20 22 24 x(t) x (t + 9 )

Part 3: Discussion

Different land elevation may give different values of parameter (Aeschbacher et al., 2009). Therefore, the values of parameters for the two elevations under study which are lowland and highland areas are being compared. Table 5 shows the result of time delay,



that was calculated by using AMI method for each station located at highland and lowland areas using hourly time series data. The values of



at highland areas were 19 and 8 at station SPT01h and SPT02h, respectively, while at lowland area, the values of



obtained were 23 and 32 at SPR01h and SPR02h, respectively. The result on values showed that



was higher at lowland area compared to highland areas. This is because the water level at highland areas gives pressure to the river flow at lowland areas. Thus, the water level at these areas may be uncertained and unpredictable.

The values of



calculated by using AMI method using daily time series data at four different stations in lowland and highland areas are shown in Table 6. The results on the values of



for station SPT01d, SPT02d, SPR01d and SPR02d were 11, 14, 8 and 9, respectively. Comparing the values obtained, data in highland areas gave a higher value of



compared to lowland areas with a slight difference.

The value of E1(d) was calculated to determine dimension parameter, d by using the corresponding value of



. The resulted values of d by using hourly time series data in each station are shown in Table 5 with the values of d were 6 and 5 in highland and lowland areas, respectively. The values of d by using daily time series data are presented in Table 6 with the values of d were 6 and 5 in highland and lowland areas, respectively. The value of d obtained showed that the data on highland areas gave a higher value of d than in the lowland area.

Cao method can also determine the system dynamics of time series data either random or deterministic through the determination of parameter E2(d). For both time scale analysis, the presence of chaotic dynamics in the time series of Pahang River for different elevation can be determined through Figure 3 and Figure 5 where there is at least one value of E2(d) ≠ 1. Both figures showed that each of the data involved in this analysis gave thevalues of E(d) asnot equal to 1.Therefore, hourly and daily water level time series data for each elevationare chaotic dynamics by using Cao method. In fact, chaotic analysis using hourly time series data is more effective in the prediction of certain emergency cases scenario such as floods (Sivakumar&Wallender, 2005).

Phase space plot for hourly time scale is shown in Figure 4. The dynamics of the data that started from one original point through a trajectory and moved in one space can be seen. There are some isolated points away from space. Noise disturbances may arise from other impurities that are also scattered together when activity records the water level time series data. However, most of the points are in the trajectory of the space and there exists a region of attraction in the phases of space which suggests chaotic dynamics has been determined (Sivakumar, 2002). Phase space plot shows that the trajectory can be seen more saturated and accumulated in Figures 4(a) and Figure 4(b) which involves highland area compared to Figure 4(c) and Figure 4(d) which involves time series data from lowland areas.

Figure 6 shows phase space plot for corresponding embedding dimension phase (d) using the value of



obtained for daily time series data from different elevation. There exists an attraction region in the center of the phase space plot for all stations involved. The phase space plot shows there exist isolated point but there are still in a region of trajectory that is in the middle of the phase space. With the existence of the attraction region, the chaotic dynamics are present in the daily time series data in Pahang River (Sivakumar, 2002).

5. Conclusion

Analysis on the existence of chaotic dynamical system on water level time series data at different elevation using hourly and daily data found that the combinations of phase space reconstruction’s parameters give higher values by using hourly time scale compared to daily time scale Besides, this study also found that different elevation gives impact on the values of phase space reconstructions’ parameters where the data on the highland areas gives higher values than the lowland areas. Therefore, in consequence, chaotic dynamics exist at different elevations in Pahang River on different time series data using Cao method and phase space plot. Hence, water level data from different elevation and time scale in Pahang River can be used in the development of prediction model based on chaos approach.

The time series data involved in this study are hourly and daily time series data. The analysis on the phase space reconstruction’s parameters shows that it gives higher values by using hourly time scale compared to daily time scale for different elevation. The difference can be seen in the determination of the value of



as well as the

value of d in different elevation area. In addition, the difference in elevation affects the parameter's determination due to geographical differences.

Significant differences can be seen in the chaotic analysis of the time series data. Even though the time scale is different, however, they are collected on the same river but in different elevation. Data observation on hourly time series data influences the accuracy of parameters value,



and d rather than daily time series data. This is due to the detailed observation in the hourly time series data since the data are collected for every hour.

Therefore, this study provides important information on the difference in determining the parameter and chaotic analysis of water level at highland and lowland areas where it gives different results based on elevation differences. Therefore, it is important to choose appropriate time scale in order to give better water level prediction.

6. Acknowledgement

The researchers would like to extend their gratitude to the Malaysian Department of Irrigation and Drainage for their cooperation and information on the time series data needed in conducting this study that are located at four stations in Pahang River. Besides, the researchers would like to extend their acknowledgement to the Ministry of Education (MOE) Malaysia (FRGS/1/2018/STG06/UPSI/02/3) for providing financial aid in this study.

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