Global and Planetary Change 100 (2013) 362–370
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Temporal scaling behavior of sea-level change in Hong Kong — Multifractal temporally weighted detrended fluctuation analysis Yuanzhi Zhang a, b,⁎, Erjia Ge c, d a
Yuen Yuen Research Centre for Satellite Remote Sensing, Institute of Space and Earth Information Science, the Chinese University of Hong Kong, Hong Kong Coastal Zone Studies, Shenzhen Research Institute, Yue Xing Road 2, Shenzhen 518057, China Jockey Club School of Public Health and Primary Care, the Chinese University of Hong Kong, Hong Kong d Department of Geography and Resource Management, the Chinese University of Hong Kong, Hong Kong b c
a r t i c l e
i n f o
Article history: Received 6 December 2011 Accepted 27 November 2012 Available online 5 December 2012 Keywords: Sea-level rise Time series Long-range correlation Multi-fractal scaling behavior Hong Kong's coast
a b s t r a c t The rise in global sea levels has been recognized by many scientists as an important global research issue. The process of sea-level change has demonstrated a complex scaling behavior in space and time. Large numbers of tide gauge stations have been built to measure sea-level change in the North Pacific Ocean, Indian Ocean, North Atlantic Ocean, and Antarctic Ocean. Extensive studies have been devoted to exploring sea-level variation in Asia concerning the Bohai Gulf (China), the Yellow Sea (China), the Mekong Delta (Thailand), and Singapore. Hong Kong, however, a mega city with a population of over 7 million situated in the mouth of the Pear River Estuary in the west and the South China Sea in the east, has yet to be studied, particularly in terms of the temporal scaling behavior of sea-level change. This article presents an approach to studying the temporal scaling behavior of sea-level change over multiple time scales by analyzing the time series of sea-level change in Tai Po Kou, Tsim Bei Tsui, and Quarry Bay from the periods of 1964–2010, 1974–2010, and 1986–2010, respectively. The detection of long-range correlation and multi-fractality of sea-level change seeks answers to the following questions: (1) Is the current sea-level rise associated with and responsible for the next rise over time? (2) Does the sea-level rise have specific temporal patterns manifested by multi-scaling behaviors? and (3) Is the sea-level rise is temporally heterogeneous in the different parts of Hong Kong? Multi-fractal temporally weighted de-trended fluctuation analysis (MF-TWDFA), an extension of multi-fractal de-trended fluctuation analysis (MF-DFA), has been applied in this study to identify long-range correlation and multi-scaling behavior of the sea-level rise in Hong Kong. The experimental results show that the sea-level rise is long-range correlated and multi-fractal. The temporal patterns are heterogeneous over space. This finding implies that mechanisms associated with the local ecological environment, hydrodynamic and morphodynamic processes, and human activities might have driven a distinct sea-level rise in Hong Kong. © 2012 Elsevier B.V. All rights reserved.
1. Introduction The global rise of sea level over the past century has been observed and recorded by different measures, including tide-gauge data, satellite radar altimetry, fossil shells, and notches (Church and White, 2006; Smith et al., 2011). Since the 1990s, the sea-level rise, at an average rate of 3.4 mm per year, remains upward (Cabanes et al., 2001; Nerem et al., 2010). Various factors, associated with climate change (IPCC, 2001) and human socio-economic activities (Gornitz et al., 1997), have exacerbated the rising trend of sea level in Southeast Asia in general (Cheng and Qi, 2010) and Hong Kong in particular (Wong ⁎ Corresponding author at: Yuen Yuen Research Centre for Satellite Remote Sensing, Institute of Space and Earth Information Science, the Chinese University of Hong Kong, Hong Kong. E-mail address:
[email protected] (Y. Zhang). 0921-8181/$ – see front matter © 2012 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.gloplacha.2012.11.012
et al., 2003). The Intergovernmental Panel on Climate Change (IPCC), an international organization responsible for the surveillance of climate change, warned that at current trends, the sea surface level will increase by 48 cm by 2100, twice as much as in the preceding 50 years, from 2050 (Solomon et al., 2007). This increasing rate of sea-level rise could lead to serious consequences, including coastal erosion, stormsurge flooding, and loss of human life, especially if the scaling behavior of sea-level change cannot be understood thoroughly and no effective strategies can be provided for the prevention of this kind of calamity. Sea-level change has exhibited a complex scaling behavior with large spatial and temporal variation (Chu et al., 1997; Douglas, 1997; Ding et al., 2001; Douglas et al., 2001; Milne et al., 2009; Morner, 2010). The long-term trend of sea-level change on a global scale has been hypothesized to result from ocean thermal expansion, which is caused when sea water expands due to global warming, thus leading to water exchange between an ocean and other reservoirs, including
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glaciers, ice caps, and ice sheets (Church, 2001; Davis et al., 2005; IPCC, 2007). On the other hand, tidal activities, atmospheric pressure variation, sea surface wind, storm surge, and coastal land reclamation have led to regional short-term fluctuations of sea-level change (Yim, 1993a). It has been suggested that the process of sea-level change, in terms of time, is self-similar and seasonal, with random variation (IPCC, 2007). However, there are very few studies on the formal analysis of its temporal scaling behavior. To shed some more light on the behavior of sea-level change, its temporal process needs to be unraveled. In particular, one needs to study the complex dynamics of sea-level change with reference to long-range correlation and self-similarity over time. That is, the scaling behavior of sea-level change should be understood. Accurate measurement of this scaling behavior provides an approach to modeling temporal processes and to formulating physical mechanisms of complex dynamics (Feder, 1988), such as sea-level rise. Studying the temporal scaling behavior of sea-level change over multiple time scales can enable the detection of the long-range correlation and underlying multi-fractality that characterize this intricate process. A large number of tide gauge stations have been set up to measure sea-level change across the world, with many studies in Asia concerning the Bohai Gulf (China), the Yellow Sea (China), the Mekong Delta (Thailand), and Singapore (Smith et al., 2011). Studies on the multiscale characteristics, for example, of the upper sea layer temperature in the West Philippines (Chu and Hsieh, 2007), the stream flow and flood/drought of the Yangtze River, China (Zhang et al., 2008; Zhang et al., 2010a, 2010b), and the temporal variation of Arctic sea ice (Agarwal et al., 2012), have made progress towards understanding the sea-level change in different places in the world. However, Hong Kong, which is a specific area situated in the mouth of the Pear River Estuary on the north coast of the South China Sea, has yet to be rigorously studied at both global and local scales. Hong Kong has become the second largest shipping port in the world next to Singapore due to its container traffic. With a population of over 7 million, it is about 1100 km 2 in area, of which approximately 10% is coastal reclamation (Peart and Yim, 1992). In Hong Kong, over 60% of the population is concentrated in low-lying land that was created in the past 100 years through coastal land reclamation, since higher altitude areas are difficult to develop and are more prone to landslides (Yim, 1993a). The rise of sea level in this densely populated and low-lying metropolis has posed a serious risk to human lives and to the world economy. Moreover, during 1954 and 2003, the rate of sea-level rise in the South China Sea was generally higher than the global mean of 1.0 to 2.0 mm per year (IPCC, 2001). Hong Kong, in comparison to the whole of the South China Sea, has had a higher average sea-level rise, recorded as an annual rate of 7.6± 3.8 mm in the Quarry Bay tide gauge station and 3.4 ± 3.1 mm in the Tai Po Kou tide gauge station from 1993 to 2003 (Wong et al., 2003). However, this rising trend, similar to the South China Sea, was not maintained and fell sharply in 2001 and 2003 (Wong et al., 2003; IPCC, 2007). Understanding the temporal scaling behavior of sea-level changes provides a direct approach to explore and formulate the long-range correlation and multi-fractality of sea-level fluctuation in Hong Kong. In addition, the temporal patterns of sea-level change can be compared to show whether the rising behavior is spatially heterogeneous, such as at Tai Po Kou, Tsim Bei Tsui, and Quarry Bay in eastern, western, and southern Hong Kong, respectively. The purpose of this study is to determine the temporal scaling behavior of sea-level change in Hong Kong. Specifically, the study seeks to answer the following questions in a quantitative manner: (1) Is the rise of sea level self-similar and long-range correlated over time? (2) Does the sea-level change have any specific periodicity, such as daily, seasonal, or annual patterns? (3) Does the sea-level change exhibit multi-fractal behavior? (4) Does the temporal behavior of sealevel changes vary across the east, west, and southeast parts of Hong Kong? To facilitate the discussion, we first give a description of
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the tide gauge data in Section 2. Relevant methods related to time series analysis, including fractal analysis, rescale range analysis, multi-fractal de-trended fluctuation analysis, and multi-fractal temporally weighted de-trended fluctuation analysis are then briefly introduced in Section 3. In Section 4, analysis results and interpretation are examined and provided. Finally, we conclude the paper with a summary and outlook for future research. 2. Time series of sea-level change in Hong Kong Hong Kong is an international city, situated close to the mouth of the Pearl River estuary on the northern coast of the South China Sea (see Fig. 1). Its western side is open to the Pearl River estuary and influenced by fresh water discharge and sedimentation distribution, with an average rate of 50 or 60 m/year of development (Yim, 1993b). On the other hand, the eastern side suffers from the long-term influence of coastal cold currents from the East China Sea along the Taiwan Strait to the south China coast in the winter, and warm currents from the Pacific across Luzon Strait to the South China Sea in the spring (Wong et al., 2003). The southern part of Hong Kong is predominantly oceanic. In addition, Victoria Harbor, the very narrow strait between Kowloon and Hong Kong Island, has decreased dramatically in size over the past 150 years, due to intensive reclamations. Fig. 1 shows the locations of Hong Kong and tide gauge stations responsible for recording sea-level change data. Given such a complex geographical configuration, this study employed three daily tidal series from the tide gauge stations of Tai Po Kou, Tsim Bei Tsui, and Quarry Bay Island, respectively, to study the temporal scaling behavior of sea-level rise in eastern, western, and southern Hong Kong (see Fig. 1). The Hong Kong Observatory runs eight tide gauge stations, which are Chep Lap Kok, Kwai Chung, Quarry Bay, Shek Pik, Tai Miu Wan, Tai Po Kau, Tsim Bei Tsui, and Waglan Island, covering all areas of the territory. The three stations, Tai Po Kau, Tsim Bei Tsui, and Quarry Bay, were selected for the present study due to their specific geographical locations and data quality. Over 80% of the hourly reading records of the three stations can be obtained from the Hong Kong Observatory (Yim, 1991; Wong et al., 2003). To facilitate this study, small gaps in the time series were filled with a linear interpolation (Wong et al., 2003). The three series sampled at hourly tide gauge readings from the 1960s comprise of greater than 400,000 data points in each series. Such a long time series leads to a heavy workload of data analysis and makes the study impossible due to the limitations of computational power. Moreover, this study is mainly concerned with the temporal scaling behavior of sea-level change on daily, weekly, seasonal, annual, and decade intervals. Therefore, hourly tide gauge data were grouped into 24 h to create three time series of daily average sea-levels for Tai Po Kau, Tsim Bei Tsui, and Quarry Bay. Fig. 2 shows the daily tide gauge readings of the three stations: (a) Tai Po Kau (1964–2010); (b) Tsim Bei Tsui (1974–2010); and (c) Quarry Bay (1986–2010) for the periods listed in the parentheses. In this study, tide height is in centimeters (cm) above the Chart Datum used by the Hong Kong Observatory. The Chart Datum is 0.146 meters (m) below the Principal Datum about 0.88 m below the Yellow Sea Datum. 3. Method Sea-level change appears to be a complex dynamic that varies intermittently with interwoven periods where the sea level rises and falls sharply. This intricate process may be long-range correlated and multi-fractal. To explore the temporal scaling behavior of sea-level rise, multi-fractal temporally weighted de-trended fluctuation analysis is used to study its dynamics, particularly its long-range correlation and self-similarity. To facilitate our presentation, we first give a brief description of fractal and rescaled range analyses (R/S analysis). Then we will explain why multi-fractal de-trended fluctuation analysis
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Fig. 1. Location of Tide Gauge Stations in Hong Kong.
(MF-DFA) in general and multi-fractal temporally weighted de-trended fluctuation analysis in particular are needed to study the complex dynamic of sea-level rise over time. 3.1. Fractal and R/S analyses Fractal analysis provides a mathematical formalism describing extremely complex and irregular objects whose dimension is fractional (Mandelbrot, 1983; Feder, 1988). Self-similarity is an underlying concept of fractals, characterizing objects that repeat themselves at finer and finer scales/stages ad infinitum. The process of sea-level fluctuations appears to possess self-similarity when viewed on different scales. The degree of self-similarity is used to measure the fractal dimension, a non-integer value beyond conventional Euclidean dimensions. The time series of sea-level rise, for example, can be recursively subdivided into self-similar components. The size of a fractal object can be measured by the fractal dimension: −s
Mδ→0 ðF Þ∝cδ
trends in the dataset. The rescaled range analysis (R/S analysis) was proposed by Hurst (1950). This method has been used extensively to study the scaling behaviors of complex processes in general and the temporal dependency of time series in particular. The Hurst exponent, H, indicates the degree of correlation, estimated by measuring the data series over multiple time scales. Experimentally, when the Hurst exponent H is larger than 0.5, the time series is long-range correlated. On the other hand, when H is less than 0.5, it means that the time series is long-range anti-correlated. If H is equal to 0.5, then it indicates that no correlations exist and the time series is considered as random. The Hurst exponent H is closely associated with the general scaling exponent, h(q), used to identify long-range correlation and multi-fractality of time series in multi-fractal de-trended fluctuation analysis and its modified version, multi-fractal temporally weighted de-trended fluctuation analysis, as will be discussed in the Subsections 3.2 and 3.3.
3.2. Multi-fractal de-trended fluctuation analysis ;
s∈R
ð1Þ
where F is a set, Mδ(F) is the measurement of F with dimension s, δ is the scale, and c is the s-dimensional measure of F. It can be interpreted as the number of covers, for instance, fixed-size boxes or open balls, required to cover the set F. In short, the required number of covers scales with the dimension. For a time series, the properties of a fractal structure can be examined by the correlation function (Kantelhardt et al., 2001). However, Chen et al. (2002) found that direct calculation of correlation is inappropriate when noise is superimposed on time series and underlying
The correlation of time series is often masked by trends, mostly caused by external influences, for example seasonal impacts on sealevel rise (Kantha and Clayson, 2000). Effectively removing these trends is thus crucial to identifying the actual correlation of a time series. Multi-fractal de-trended fluctuation analysis (MF-DFA) proposed by Kantelhardt et al. (2002) provides a method to characterize longrange correlations and multi-fractal properties of non-stationary time series. This method has been widely used in different disciplinary studies, such as earthquake prediction (Telesca et al., 2005; Telesca and Lapenna, 2006), hydrology (Kantelhardt et al., 2003), sunspot
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In the second step, we divided the profile Y (i) into Ns ≡ int(N/s) non-overlapping segments of equal length s. Since the length N of the series is often not a multiple of the time scales s, there may remain a short part at the end of the profile. In order not to ignore this part of the series, the same procedure is repeated starting from the opposite end. Thereby, 2Ns segments are obtained altogether. The third step calculates the local trends for each of the 2Ns segments by a least square fit of the series. Then the variance is obtained as: 2
F ðs; vÞ ¼
s 1X 2 fY ½ðv−1Þs þ i−yv ðiÞg ; s i¼1
ð3Þ
for each segment v, v = 1, … , Ns and 2
F ðs; vÞ ¼
s 1X 2 fY ½N−ðv−Ns Þs þ i−yv ðiÞg ; s i¼1
ð4Þ
for v = Ns + 1, … , 2Ns. Here, yv(i) is the fitting polynomial which is regarded as the local trend in segment v. The MF-DFA can be reckoned as MF-DFA m depending on the order m, for example, linear, quadratic, cubic, or higher order. There are two main reasons for using linear trends in the study of sea-level rise. First, in standard MF-DFA, linear trends are eliminated to explore the essence of the scaling behavior of a time series. Based on this, the general Hurst exponent h(q) of MF-DFA and the Hurst exponent of rescaled range analysis (Hurst, 1950) were mathematically associated to detect long-range correlation of a time series. Second, with little understanding of a time series, a linear trend is typically a default for de-trending. Therefore, for simplicity, only linear trends are considered in this paper. In the fourth step, the qth-order fluctuation function is defined as follows: (
2N i 1 Xs h 2 q=2 F q ðsÞ≡ F ðs; vÞ 2N s v¼1
)1=q :
ð5Þ
Here q is the mathematical moment which takes any value except zero. The Fq(s) is required to satisfy s ≥ m + 2 as defined in Eq. (4). Finally, the multiple scaling exponents h(q) in hðqÞ
F q ðsÞ∝s
Fig. 2. Daily sea-level change series of Tai Po Kou, Tsim Bei Tsui, and Quarry Bay from 1964, 1974, and 1986, respectively, to 2010 in Hong Kong.
activities (Movahed et al., 2006), and avian-influenza epidemic (Leung et al., 2011). The generalized MF-DFA procedure is mainly composed of five steps. Suppose that {xk} is a series of length N with compact support. In the first step, we determine the profile as: i X ½xk −hxi; i ¼ 1; 2; …; N: Y ðiÞ≡
ð2Þ
k¼1
The mean 〈x〉 can be regarded as a global trend of the time series {xk}. The subtraction of the mean 〈x〉 is not compulsory, since it would be eliminated by the de-trending procedure in the third step.
ð6Þ
can be obtained from analyzing the fractal plots of each moment of q. Here, h(q) is the general Hurst exponent which can be used to characterize the scaling behavior of time series. For stationary time series, H(2) is equal to the Hurst exponent H discussed above. For non-stationary time series, on the other hand, H =h(2) − 1. Grounded on the relationship between H and h(2) (Kantelhardt et al., 2002; Movahed et al., 2006), time series can be identified as long-range correlated (if H >0.5), uncorrelated (if H = 0.5), or long-range anticorrelated (if 0 b H b 0.5). The Hurst exponent H is thus a key index for long-range correlation of time series. In addition, for monofractal time series, h(q) is independent of q. This is because the scaling behaviors of the variances F2(s, v) are identical for all segments v, and the averaging procedure in Eq. (5) gives the same scaling behaviors for all values of q. In contrast, this is not the case for multi-fractal time series. In addition, for positive values of q, the segments v with large variance F2(s, v) (i.e., large deviation from the linear fit) dominate the average Fq(s). In this case, h(q) describes the scaling behavior of the segments having large fluctuations. On the other hand, negative values of q make the small variance F 2(s, v) dominate the average Fq(s). So h(q) here depicts the scaling behavior of the segments with small fluctuations.
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3.3. Multi-fractal temporally weighted de-trended fluctuation analysis The conventional MF-DFA, which ignores the association of point locations in a time series, may not reflect temporally (spatially) non-stationary phenomena in real life, such as the sea-level rise series in this study. This implies that the method might fail to capture local effects because of proximity in time (space). Moreover, the removal of local trends by discontinuous polynomial fitting in MF-DFA can lead to oscillations in the fluctuation function, particularly in both the beginning and the ending of a local window of a time series (see Eqs. (3) and (4) (Alvarez-Ramirez et al., 2005). In addition, this uniformity might be able to cause significant errors in crossover points that separate distinct regimes having different fractal scaling behaviors. To avoid such problems, this study applies multi-fractal temporally weighted de-trended fluctuation analysis (MF-TWDFA) (Zhou and Leung, 2010) to the time series of sea-level rise in Hong Kong. This method allows for local relationships of points in time series in the de-trending procedure of MF-DFA. Similar to the conventional MF-DFA, the MF-TWDFA consists of five steps. The first and last two steps are in line with those of the MF-DFA. Two procedures, including a sliding window and geographically weighted regression, are introduced in the third step discussed above. First, to capture local effects, a sliding window procedure is applied to move a fixed-size window along the series for de-trending. Specifically, yv(i) Eqs. (3) and (4) can be determined using time points close to point i. Y(i) is estimated by a fitting polynomial for each local window. Then Y(i) and the corresponding Y^ ðiÞ can be obtained using the moving window, defined as {j : |i − j| ≤ s}. Replacing yv(i) by Y^ ðiÞ in the MF-DFA Eqs. (3) and (4), the de-trending procedure can be redefined as follows:
2
F ðs; vÞ ¼
s n o2 1X Y ½ðv−1Þs þ i−Y^ ððv−1Þs þ iÞ ; s i¼1
ð7Þ
for each segment v, v = 1, … , Ns and
2
F ðs; vÞ ¼
s n o2 1X Y ½N−ðv−Ns Þs þ i−Y^ ðN−ðv−Ns Þs þ iÞ ; s i¼1
ð8Þ
for v = Ns + 1, … , 2Ns, where Y^ ðiÞ is the value of the fitting polynomial of the moving window at point i. Second, a varying-parameter regression model (Leung and Mei, 2003), such as a geographically-weighted regression (GWR) that allocates different weights to points based on distance between two events in space, can also be employed to capture local effects due to proximity in time. For a time series, a fitting polynomial, within a moving window, can be expressed as follows: Y^ ðiÞ ¼ β0 ðiÞ þ β1 ðiÞi þ ε;
ð9Þ
and β = (β0(i),β1(i))T can be calculated by −1
βðiÞ ¼ ðT ′W ðiÞT Þ
T ′W ðiÞY;
where T is a 2 × N matrix: 0
1 B1 T¼B @ ⋮ 1
1 1 2 C C; ⋮ A N
ð10Þ
and 0
wi1 B 0 B W ¼@ ⋮ 0
0 wi2 ⋮ 0
⋯ ⋯ ⋱ ⋯
1 0 0 C C : ⋮ A wiN NN
For GWR, the type of weighted function in comparison with the width of the weighted function often does not have significant impact on the regression model. The weighted function wij can be defined as follows: 8 2 2 > < 1− i−j ; s wij ¼ > : 0;
if ji−jj ≤ s; otherwise:
In this study, s is a temporal scale and |i − j| is the distance between time data points i and j in time series of sea-level changes in Hong Kong. 4. Analytical results and interpretation The MF-TWDFA analysis result indicates that sea-level changes in Hong Kong have different long-range correlation and multi-fractality across various temporal scales. Long-range correlation indicates a tendency of sea-level rise (or fall) persistent over multiple scales of time. That is, a sea-level rise (or fall) at this instant has influence and is likely to lead to another rise (or decrease) in a subsequent moment. The different multi-fractalities of Tai Po Kou, Tsim Bei Tsui, and Quarry Bay in Hong Kong indicate that the dynamics of sea-level changes are self-similar at all moments q, but spatially heterogeneous. These multi-fractalities are probably caused by the long-range correlation of sea-level changes. 4.1. Long-range correlation Fig. 3 shows the MF-TWDFA results of the sea-level change of (a) Tai Po Kou, (b) Tsim Bei Tsui, and (c) Quarry Bay, Hong Kong. It suggests different long-range correlations and various multi-scaling behaviors of the three areas because the values of scaling exponents h(2) and crossover points detected are different. This variation in scaling behavior and spatial heterogeneity might be due to either distinct ecological environments or human socio-economic behavior. Tai Po Kou exhibits long-range correlation and multi-scaling behavior of different time scales between 3 and 234 days, 367 and 1066 days, and 1067 and 8955 days (see Fig. 3(a)). That is, when the sea-level rise occurs at this present moment, another rise can be expected to follow at the next moment in time. As shown in Fig. 1, Tai Po Kou lies in Tolo Harbor and opens to the South China Sea (SCS). Due to its specific geographical location, the long-range correlation identified can reflect the self-similarity of the sea-level rise of the SCS. For small scales from 3 to 234 days, the sea-level rise is strongly long-range correlated, as the value of h(2) is up to 0.99. Such long-range correlation in the South China Sea implies an approximate semi-annual pattern of sea-level rise, associated with the seasonal cycle (Ho et al., 2000; Park and Oh, 2007; Zhuang et al., 2010) in general and steric components, including the changes of water temperature and salinity (Chambers, 2006; Fang et al., 2006; Ishii et al., 2006; Cheng and Qi, 2010) in particular. Unlike the extratropical Pacific Ocean, the SCS has been mainly affected by the seasonal cycle of the East Asian monsoon in winter and summer (Liu et al., 2000; Zhuang et al., 2010). Specifically, the winter monsoon during spring leads to a strong northeasterly flow, covering the whole of the SCS. In contrast, the summer monsoon is first established in the southern and central SCS, extending across the entire sea, and then replaced by the northeasterly monsoon in fall (Liu et al., 2000; Park and Oh, 2007). The two seasonal wind stresses derive different surface dynamics causing ocean circulation (Liu et al., 2000; Qu, 2000; Park and
Y. Zhang, E. Ge / Global and Planetary Change 100 (2013) 362–370
Fig. 3. MF-TWDFA results of (a) Tai Po Kou, (b) Tsim Bei Tsui, and (c) Quarry Bay in Hong Kong.
Oh, 2007). This can affect steric sea-level change and can contribute to the sea-level change of the SCS (Chu et al., 1997; Cheng and Qi, 2010; Zhuang et al., 2010). In comparison, the long-range correlation of the temporal scales between 367 and 1066 days becomes weaker and the value of h(2) is 0.63. The crossover time scale around 1066 days (2.9 years) may reflect a distinct dynamic mechanism affecting the sea-level rise under the EI Niño-Southern Oscillation (ENSO). The ENSO is a climatic feature which recurs every two to five years. It often has a negative association with the sea-level variation of the SCS (Fang et al., 2006; Chang et al., 2008; Han and Huang, 2009). This study shows that the ENSO might have been a factor that weakened
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the long-range correlation of sea-level rise. For larger scales from 1067 to 8955 days, this rise retains a strong long-range correlation, with an h(2) =0.92, and persists for more than 24.6 years, longer than the 18.6-year tidal periodicity reported by Ding et al. (2001) from the combined time series data of North Point and Quarry Bay. These detected long-range correlations and multiple crossover time scales, essentially, provide a quantitative description for the multiscale behavior of the SCS and the dynamic mechanisms that have underpinned the sea-level rise of Tai Po Kou, Hong Kong, in particular. Fig. 3(b) shows the MF-TWDFA result of the sea-level change in Tsim Bei Tsui. It shows a complex temporal scaling behavior with the different long-range (anti-)correlations and multiple crossover time scales. These crossover time scales identified at around 26, 186, 400, and 7107 days suggest monthly, semi-annual, annual, and 19.5-year tidal patterns, respectively. Tsim Bei Tsui is on the western side of Hong Kong and open to the Pearl River estuary. The sea water of Tsim Bei Tsui, unlike Tai Po Kou and Quarry Bay, is thus constantly influenced by fresh water discharge and sedimentation distribution (Zhang et al., 2010a, 2010b). The crossover time scales at around 186 and 400 days interpret semi-annual and annual patterns due to the summer and winter monsoons sweeping the entire SCS. The sea level of Tsim Bei Tsui, unlike Tai Po Kou, exhibits long-range anti-correlation, with the general scaling exponent h(2) beyond 1. The (anti-)interaction of sea and river flood, along with a semi-annual cycle, was reported to severely affect the sea surface close to Hong Kong (Huang et al., 2004). Consistent with the observations of Li and Qiao (1982), this long-range anti-correlation essentially formulates the multi-scale behavior of sea-level change in Tsim Bei Tsui under both the SCS oceanic process and the inter-tidal activity of the Pearl River estuary. On the other hand, the long-range correlation of sea-level changes persist at large scales from 400 days and upward, but this is not as strong as that of the small scales between 3 and 186 days. This result implies that hydrodynamic and morphogenetic processes, in association with intensive socio-economic human activities, have affected the Pearl River estuary (Zhang et al., 2010a, 2010b) and therefore the sea-level rise of Tsim Bei Tsui in recent decades. In addition, the 19.5-year time scale appears to suggest a rising tidal cycle as a result of oceanic and hydrodynamic processes and human effects from, for example, land reclamation and construction of engineering infrastructures. Due to the influence of multiple complex processes, it has been reckoned that the long-range correlation of temporal series at the small scale of 3 and 26 days is difficult to define. In-depth analysis on the impact of the Pearl River estuary on sea-level changes in the west of Hong Kong is required to develop further understanding and an explanation of this phenomenon. Unlike Tai Po Kou and Tsui Bei Tsui, Quarry Bay is situated in Victoria Harbor, a very narrow strait between Kowloon and Hong Kong Island. The Quarry Bay has sea water mixed from the SCS in the east and freshwater from the Pearl River estuary in the west. Our analysis result indicates that the sea-level change of Quarry Bay is long-range correlated across multiple temporal scales (see Fig. 3(c)). The sea-level change demonstrates strong long-range correlations at two time regions between 3 and 21 days and 404 and 5710 days, respectively. The general Hurst exponents, h(2), of the two regions are 0.74 and 0.77, respectively. This result suggests that the tendency of sea-level rise remains and repeats over different scales from monthly to a 15.6-year period. On the other hand, the sea-level change exhibits a long-range anticorrelation between 21 and 181 days, and the value of h(2) is 1.17, which suggests that sea-level rise at this moment is likely to be followed by sea-level fall at the next moment. This long-range anti-correlation may be associated with the seasonal cycle of the East Asian monsoon (in winter and summer) from the SCS (Liu et al., 2000; Zhuang et al., 2010). As shown in Fig. 3(c), three cross-over time scales are identified at 181, 404, and 5710 days, suggesting semi-annual, annual, and 15.6-year patterns of sea-level change, respectively, in Victoria Harbor. This semi-annual and annual periodicity depicts the complex dynamic of the sea-level changes, which is affected by the bi-seasonal monsoon
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of the SCS and simultaneously by a mixture of oceanic and hydrodynamic processes from the west part of Hong Kong. In addition, coastal and land reclamation influenced by long-term ground settlement has caused a local relative sea-level rise in Quarry Bay, Victoria Harbor (Yim, 1993a, 1993b, 1995; Ding et al., 2001). With over 130,000 people per km2, Victoria Harbor is the most densely populated part in Hong Kong (Yim, 1995). Intensive development, such as the construction of Hong Kong International Airport in the 1990s, has led to the disappearance of natural coastlines in the Harbor (Yim, 1995). Regarding both oceanic and human effects, the analysis suggests that the rise of the sea level is a complex phenomenon with different long-range correlations and multi-scale temporal behavior. To summarize, the study has identified different long-range correlations of the sea-level changes across the east, west, and middle parts of Hong Kong, namely Tai Po Kou, Tsim Bei Tsui, and Quarry Bay, respectively. The analysis indicates that the mechanisms driving this sealevel rise are spatially heterogeneous under oceanic, hydrodynamic, and morphodynamic processes due to the different ecological environments of the three locations. These regions need to devise and take measures appropriate to their own situations in regards to the risk of sea-level rise. The crossover time scales detected in this study indicate that the mechanism of sea-level rise varies at the different time scales, implying semi-annual, annual, and greater than 15-year periodicities of the rise. This finding is also instrumental in the design of strategies for the prevention of such risks with respect to different time scales. 4.2. Multi-fractality In this study, we have identified the multi-fractality of the time series of sea-level rise in Tsim Bei Tsui and Quarry Bay, and the monofractality in Tai Po Kou. This property indicates that the multi-fractal behavior of sea-level change can extend over different mathematical moments q. This means that the sea-level rise exhibits diverse selfsimilarities at different moments. The property of a data set is often statistically evaluated in terms of mean, variance, and even higher moments. Similarly, the property of the self-similarity of a time series can be measured by the distinct moment q. The multi-fractal property of time series can be mathematically described by the generalized Hurst exponent, h(q), the mass function,τ(q), or the generalized fractal dimension, D(q). The mass function,τ(q) is derived from the multifractal analysis (Grassberger and Procaccia, 1983) to characterize the spatial heterogeneity of the theoretical and experimental fractal patterns in general. The generalized fractal dimension, D(q), which has a relationship with τ(q) (Anh et al., 2000; Yu et al., 2009; Leung, 2010), is also used here to examine the multi-fractal behavior of sea-level rise in Hong Kong. Fig. 4(a), (b), and (c) shows the dependence of h(q) on q, and the weaker, nonlinear relationship betweenτ(q) and D(q). The generalized Hurst exponent, h(q), is the dimension of the measure, Fq(s), in Eq. (6). The h(q) has been employed to characterize the scaling behavior and multi-fractality of time series. If the time series is multi-fractal, h(q) will vary with the change of q, depicting fractal properties from different moments q. If the time series is mono-fractal, the value of h(q) will not change with different moments, characterizing a single form of self-similarity over time. Alternatively, the multi-fractal property can be reflected by τ(q) and D(q). The mass function, τ(q), discussed by Halsey et al. (1986), especially the value of τ(q) at q = 0, 1, and 2, is applied to describe the degree of multi-fractality over multiple moments q. When the measure is multi-fractal, the mass function, τ(q), must be a nonlinear function of q. Similarly, the generalized fractal dimension, D(q), having a relationship withτ(q) defined in Grassberger and Procaccia (1983), shows another indication of multi-fractality in terms of different moments. D(q), for instance, can be interpreted as a fractal dimension in which q is 1. When the values of q are 2 and 3, D(q) represents the information dimension of entropy and the correlation dimension of a time series,
Fig. 4. The multifractal property of the sea level change in Tai Po Kou, Tsim Bei Tsui, and Quarry Bay of Hong Kong: (a) h(q), (b)τ(q), and (c) D(q).
respectively. However, it is difficult to find appropriate physical meanings of D(q) as the value of q goes beyond 3. Mathematically, multifractality can be completely determined by the entire moments of τ(q) and D(q). The experimental results indicate distinct multi- and monofractalities of sea-level change in Hong Kong, which appears to suggest that the spatial heterogeneity of the variation is probably due to the diversity of ecological environments and complexity of social behaviors.
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Unlike Tsim Bei Tsui and Quarry Bay, Tai Po Kou, for example, is predominantly oceanic and exhibits mono-fractality in the sea-level rise. This finding implies that different hydrological, morphological, and oceanic processes can trigger distinct multi- and mono-fractality of the sea-level change over space. The self-similarity and multifractality of sea-level change appears to be first discovered by this study. 5. Conclusion In this study, we have investigated the multi-scale behavior of sealevel change in the east, west, and south parts of Hong Kong over time. Through the application of MF-TWDFA analysis to the time series of sea-level change in Hong Kong, we have detected the long-range correlation and multi-fractality in Tsim Bei Tsui and Quarry Bay, and the mono-fractality in Tai Po Kou. This tendency for sea levels to rise can persist and extend across multiple time scales of days, months, years, and decades. Furthermore, multiple crossover time scales separating distinct sea-level change regimes have been unraveled. In particular, semi-annual, annual, and 15.6-year patterns have been identified. Comparison of the scaling behavior of the sea-level change in the three locations, however, shows that the mechanism of sea-level change in Hong Kong is spatially heterogeneous. This is probably because of the distinct local ecological environments, hydrological dynamics, and intensive human behaviors. Our study supports the existing evidence of the temporal and spatial heterogeneity of rising sea levels. This variation exists not only across large regions, but also across small areas like Hong Kong. The result suggests that an effective strategy allowing for global oceanic and local morphodynamic processes is necessary for the prevention of sea-level rise globally and in Hong Kong in particular. However, this study is also limited due to the availability of more time-series data. With more datasets available in the near future, we can achieve further understanding of the underlying mechanisms driving multi-scale temporal behavior in sea-level change globally and locally. To obtain a complete picture, the next step is to extend the study to the scaling behavior of sea-level change in a wider space covering Asia, Europe, Africa, and North America. In addition, human activities can be incorporated to estimate and explore their effect on spatio-temporal processes of sea-level change. Such studies will advance regional analyses of sea-level rise and provide practical guidelines for policy making. Acknowledgements Local tidal gauge data from Hong Kong Observatory (HKO) are highly appreciated. This research is jointly supported by an Open Fund at the State Key Laboratory of Urban and Regional Ecology (SKLURE2010-2-3), the CUHK Direct Grants (2021081), the National Science Foundation of China (41271434), GRF (CUHK 459210 and 457212), ITF (GH/002/11GD), and the National Key Technologies R&D Program in the 12th-Five-Year Plan of China (Applied Remote Sensing Monitoring System for Water Quality and Quantity in Guangdong, Hong Kong and Macau, 2012BAH32B03). References Agarwal, S., Moon, W., Wettlaufer, J.S., 2012. Trends, noise, and re-entrant long-term persistence in Arctic sea ice. Proceedings of the Royal Society A. http://dx.doi.org/ 10.1098/rspa.2011.0728. Alvarez-Ramirez, J., Rodriguez, E., Echeverria, J.C., 2005. Detrending fluctuation analysis based on moving average filtering. Physica A: Statistical Mechanics and its Applications 354, 199–219. Anh, V., Lam, K., Leung, Y., Tieng, Q., 2000. Multifractal analysis of Hong Kong air quality data. Environmetrics 11, 139–149. Cabanes, C., Cazenave, A., Provost, C.L., 2001. Sea level rise during past 40 years determined from satellite and in situ observations. Science 294, 840–842. Chambers, D.P., 2006. Observing seasonal steric sea level variation with grace and satellite altimetry. Journal of Geophysical Research 111, 1–13. Chang, C.W.J., Hsu, H.H., Wu, C.R., 2008. Interannual model of sea level in the South China Sea and the roles of EI Niño and EI Niño Modoki. Geophysical Research Letters 35.
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