Investigation of Black Sea Mean Sea Level Variability by Singular Spectrum Analysis

Abstract

The mean sea level has been continuously increasing since the end of the 19th century and will continue to increase in the 21st century. The Intergovernmental Panel on Climate Change (IPCC) states that the sea level will rise by 40-60 cm until 2100. This situation will lead to social and economic problems, especially in coastal areas. For this reason,
studies on sea level determination have great importance in our country. In this paper, we used the singular spectrum analysis (SSA) to investigate mean sea level variability along the coasts of the Black Sea, which is an intercontinental inland sea. This study aimed to determine the trend in sea level change along the coasts of the Black Sea over time. The
mean sea level data from 10 tide gauge stations (Amasra, Batumi, Bourgas, Constantza, Igneada, Poti, Sevastapol, Trabzon II, Tuapse and Varna) are analyzed in this study. The mean sea level data were obtained from the Permanent Service for
Mean Sea Level (PSMSL). SSA was applied to the mean sea level observations at tide gauges stations, and the results were interpreted. According to the analysis results, there are increasing trends at the Batumi, Poti, Tuapse, Constantza, Sevastopol and Varna stations. The obtained trend of Bourgas station is not significant. There is The results of the Amasra, Igneada and Trabzon II tide gauge stations were inadequate in interpreting any change. There were no trends at these stations. Close eigenvalues were computed from the mean sea level at the tide gauge stations. This situation shows that
there is a dominant seasonal component in the time series.

Keywords: Black Sea, Mean Sea Level, Tide Gauge, Singular Spectrum Analysis

Full text

33
International Journal of Engineering and Geosciences (IJEG),
Vol; 5, Issue; 1, pp. 033-041, February, 2020, ISSN 2548-0960, Turkey,
DOI: 10.26833/ijeg.580510
INVESTIGATION OF BLACK SEA MEAN SEA LEVEL VARIABILITY BY SINGULAR
SPECTRUM ANALYSIS
Cansu Beşel1*, Emine Tanır Kayıkçı 1
1Karadeniz Technical University, Engineering Faculty, Department of Geomatic Engineering, Trabzon, Turkey
(cansubesel/etanir@ktu.edu.tr); ORCID 0000-0003-3434-6483, ORCID 0000-0001-8259-5543
*Corresponding Author, Received: 21/06/2019, Accepted: 22/07/2019
ABSTRACT: The mean sea level has been continuously increasing since the end of the 19th century and will continue to
increase in the 21st century. The Intergovernmental Panel on Climate Change (IPCC) states that the sea level will rise by
40-60 cm until 2100. This situation will lead to social and economic problems, especially in coastal areas. For this reason,
studies on sea level determination have great importance in our country. In this paper, we used the singular spectrum
analysis (SSA) to investigate mean sea level variability along the coasts of the Black Sea, which is an intercontinental
inland sea. This study aimed to determine the trend in sea level change along the coasts of the Black Sea over time. The
mean sea level data from 10 tide gauge stations (Amasra, Batumi, Bourgas, Constantza, Igneada, Poti, Sevastapol, Trabzon
II, Tuapse and Varna) are analyzed in this study. The mean sea level data were obtained from the Permanent Service for
Mean Sea Level (PSMSL). SSA was applied to the mean sea level observations at tide gauges stations, and the results were
interpreted. According to the analysis results, there are increasing trends at the Batumi, Poti, Tuapse, Constantza,
Sevastopol and Varna stations. The obtained trend of Bourgas station is not significant. There is The results of the Amasra,
Igneada and Trabzon II tide gauge stations were inadequate in interpreting any change. There were no trends at these
stations. Close eigenvalues were computed from the mean sea level at the tide gauge stations. This situation shows that
there is a dominant seasonal component in the time series.
Keywords: Black Sea, Mean Sea Level, Tide Gauge, Singular Spectrum Analysis

International Journal of Engineering and Geosciences (IJEG),
Vol; 5, Issue; 1, pp. 033-041, February, 2020,
34
1. INTRODUCTION
The climate depends on the correlation between the
atmosphere, hydrosphere, cryosphere and geosphere
(Nacef et al.,2016). Sea level rise, which is predicted to
significantly affect coastal areas, is one of the most
important pieces of evidence for climate change.
Identifying and understanding the causes of sea level
change are important in climate change studies at global
and regional scales. Sea-level change occurs at different
rates over a broad time scale depending on the location.
Global mean rate which is estimated satellite altimetry-
based has reported a ± 3 mm/year change over a few
decades (Cazenave et al., 2014). However, this rate varies
across the Earth. For this reason, it is important to predict
sea level changes, determine the areas that will be
affected by these changes and take precautions.
Observations of sea level change are also important for
geodesy. In geodesy, the determination of sea level
change is crucial in terms of identifying the vertical
datum and determining the geoid. In the most applications
require a datum for determining ocean depths (Yilmaz et
al.,2016).
In our country, which is surrounded by the sea on
three sides, sea level monitoring and forecasting studies
are extremely important. Karaca and Ünal (2003) noted
that the sea level increase would cause a loss in our
country’s national income of approximately 10%.
The estimates of sea level change are computed by
satellite altimetry and tide gauge data. Related to sea level,
established sea level monitoring networks and data
centers have been established at global, regional and local
scales. A few of these institutions include the Permanent
Service for Mean Sea Level (PSMSL) and the European
Global Ocean Observing System (EuroGOOS). The
PSMSL serves to provide hundreds of tide gauge data
points at monthly and yearly scales, controlling and
presenting. In our country, tide gauge data are presented
by the Turkish Sea Level Monitoring System (TUDES;
URL 1).
Time series models are widely used in the analysis
and estimation of climate data. Singular spectrum
analysis (SSA) is a powerful technique in time series
analysis. This technique is applied to many problems (see,
Ghil et al.,2002). It is particularly advantageous in
complex seasonal component estimation and the analysis
of non-stationary time series (Hassani et al., 2009). In the
literature, there are many studies on SSA theory and
applications (Golyandina 2001; Hassani et al., 2009;
Golyandina 2010; Golyandina ve Zhigljavsky 2013).
In this study, the mean sea level data from 10 tide
gauge stations (Amasra, Batumi, Bourgas, Constantza,
Igneada, Poti, Sevastapol, Trabzon II, Tuapse and Varna)
were analyzed using SSA. With SSA, the change in sea
level along the coasts of the Black Sea, as well as
harmonic oscillations at these stations and seasonal
effects, will be determined. Concurrently, the sea level
change along the coasts of the Black Sea will be
interpreted through the obtained results.
2. DATA AND METHODS
2.1 Study Area and Tide Gauge Data
In this paper, we used the mean sea level data from
tide gauge records along the Black Sea coast. The Black
Sea is a unique location, as it is the most isolated inland
sea in the Atlantic Ocean system (Goriacikin and Ivanov
2006). There are several studies on sea level change in
this region. In these studies, the sea level of the Black Sea
coast increased rapidly, see Kubryakov and Stanichyni
2013; Vigo et al., 2005; Avşar et al., 2015.
The mean sea level data from 10 tide gauge stations
(Amasra, Batumi, Bourgas, Constantza, Igneada, Poti,
Sevastapol, Trabzon II, Tuapse and Varna) located along
the coasts of the Black Sea are used in this study. The
Amasra, Igneada and Trabzon II tide gauges are in
Turkey; the Batumi and Poti stations are in Georgia; the
Bourgas and Varna stations are in Bulgaria; the
Constantza station is in Romania; and the Sevastopol
station is in the Ukraine (Figure 1).
We used monthly mean sea level time series from the
PSMSL, and the data sets are from the Revised Local
Reference (RLR) in the PSMSL (URL 2). The data spans
for each station are different from each other, and the data
have gaps of 5% and 14% (Table 1).
Figure.1 Tide gauge stations used in this study (https://www.psmsl.org/data/obtaining/map.html)

International Journal of Engineering and Geosciences (IJEG),
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Table 1 Tide gauge stations and geographic coordinates
Station
Country
Latitude
(Degree)
Longitude
(Degree)
Time Span
Amasra
Turkey
41.4333
32.2333
2001 – 2009
Batumi
Georgia
41.6333
41.7000
1882 – 2015
Bourgas
Bulgaria
42.4833
27.4833
1929 – 1996
Constantza
Romania
44.1666
28.6666
1933 – 1997
Igneada
Turkey
41.8833
28.0166
2002 – 2009
Poti
Georgia
42.1666
41.6833
1874 – 2015
Sevastapol
Ukraine
44.6166
33.5333
1910 – 1994
Trabzon II
Turkey
41.0000
39.7333
2002 – 2009
Tuapse
Russia
44.1000
39.0666
1917 – 2017
Varna
Bulgaria
43.1833
27.9166
1929 – 1996
2.2 Singular Spectrum Analysis
Singular spectrum analysis is a powerful filtration
technique. This method has a wide range of applications
in the fields of hydrology, oceanography, medicine,
economy and earth sciences. SSA is a nonparametric
approach that is widely used in time series analysis
(Hassani et al. 2009). This method extracts periodic and
quasi-periodic signals in a time series. A spectrum of
eigenvalues is used to determine these signals.
The SSA technique consists of four steps: embedding,
singular value decomposition, grouping and diagonal
averaging. In the first stage, the original time series is
decomposed, and in the second stage, the original time
series is reconstructed. Embedding and singular value
decomposition belong to the decomposition stage, and the
grouping and diagonal averaging belong to the
reconstruction stage. The steps of this technique are
shown in Figure 2. The main components selected for
reconstruction include information about the trend and
harmonic component. Spectral decomposition and
reconstruction provide a susceptible determination of
trends, seasonal fluctuations, and low frequency
components.
Figure. 2 Singular spectrum analysis steps
The first step in the SSA technique is embedding. In
this step, a one-dimensional series is transferred into a
multidimensional series. Therefore, the Hankel matrix
(trajectory matrix) is taken from the original time series.
Time series , with length N, is
equalized to an L-series vector, as shown in Eq.(1).
 (1)
where L is the window length, or embedding dimension;
; and , where
 (2)
The Hankel matrix of series X,

  
 
  
  (3)
The second step in SSA is the singular value
decomposition (SVD). Then, the computed eigenvalues
and eigenvectors of the matrix  and SVD are
applied to the Hankel matrix X.

 (4)
where , is the eigenvalue of the S matrix 
. According to SSA theory, close eigenvalues
indicate the existence of seasonal components in the time
series (Khelifa vd.,2016). In Eq. 4, the ( )
components are referred to as the eigentriple of the X
matrix, where  is the normalised eigenvector
corresponding to the eigenvalues.
When the eigenvalues are found, the decomposition
stage is completed and proceeds to the reconstruction
stage. The grouping step is first applied in this stage. The
X matrix is split into several groups, and the matrices in
each group are summed. Let,
 (5)
 (6)
Here,  represents eigentriple grouping.
Then, each matrix   is transformed into a
new series with length N, where d is the rank of a matrix.
This step is referred to as diagonal averaging.
Data
Decomposition Embedding
Singular Value
Decomposition
Reconstruction Grouping
Diagonal
Average

International Journal of Engineering and Geosciences (IJEG),
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Let the Y matrix be computed. Y is the lengt
matrix, and  is an element of . As a
result, Y is converted to series , and the
reconstruction elements are written as:









 








  (7)
where  ,
 and . If , 
 and

; otherwise, the choice  gives ,
and for ,  (Moreno and
Coelho 2018; Osmanzade 2017; Hassani and Thomakos
2010; Golyandina et al., 2001).
3. APPLICATION
In this paper, we aim to determine sea level change
along the coasts of the Black Sea using SSA. With the
SSA, the trend and seasonal effect on the mean sea level
can be determined. The mean sea level time series
recorded at the tide gauge stations used in this study are
presented in Figure 3. At the 10 tide gauge locations along
the coasts of the Black Sea, mean sea level data in the
period 1930-2017 showed similar changes.
Figure. 3 Tide gauge station time series graphics
The window length (L) is determined for the
decomposition stage of the SSA. Determining the
window or embedding dimension (L) is one of the most
critical steps of this method. The main components
decompose better when the L value is large (Hassani et
al., 2009). However, there is no strict rule for determining
window length. Hassani et al. (2009) suggested that L is
selected as the median of the 1,2,…,N values. Golyandina
(2010) proposed that L≤N / 2 be selected. In this study,
the window length L for each station was selected as N /
2. Thus, we computed the L×L Hankel matrix and L
eigentriples.
According to the eigenvalues, we observed that there
are close eigenvalue groups. The close eigenvalues of the
stations are identified as harmonic components. At the
same time, the eigenvalues are close to each other,
showing the presence of a seasonal component in the time
series. There are close eigenvalues at the Batumi,
Constantza, Poti, Sevastapol, Trabzon II, Tuapse and
Varna stations, as presented in Figure 4. The first
harmonic eigenvectors are shown in Table 2.
Table 2 Harmonic eigenvectors
Station Name
Harmonic eigenvectors
Batumi
6-7
Constantza
5-6, 15-16,17-18,21-22
Poti
3-4
Sevastapol
1-2, 7-8,9-10
Trabzon II
7-8, 10-11
Tuapse
5-6
Varna
21-22, 24-25, 28-29
Figure. 4 Eigenvalues of Batumi, Constantza, Poti,
Sevastapol, Trabzon II, Tuapse and Varna stations.
Then, the reconstruction stage is achieved using Eq.
(5), Eq. (6) and Eq. (7). The difference trend solutions are
computed as a result of different combinations of
eigentriples. For this, the eigentriples must be selected, as
they are the best representation of the data. In this study,
the first five reconstruction components (RCs) were
selected for each station. Because, according to the RCs
graphics, the first five RCs contain practically all trend
and seasonal components of time series. RCs are shown
in the time series graphics (Figure 5). The RCs that best
represented the trend were found in the time series for
each station.

International Journal of Engineering and Geosciences (IJEG),
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Figure. 5 Reconstruction components of the tide gauges
The slowly varying eigenvalues present the trend
(Hassani, 2007).
It is realized that the sum of the reconstructed
components present initial time series. The sum of the
reconstruction components that best represent the trend
for each station is shown on the time series graphics
(Figure 6, Figure 7 and Figure 9).
At the Batumi, Poti and Tuapse stations, most data
records have increasing trends in the mean sea level. The
trends show that the mean sea level at Batumi station
increased from -0.11 mm to 0.85 mm during the period of
1882-2015, that at the Poti station increased from -1.62
mm to 1.82 mm during the period of 1874-2015, and that
at the Tuapse station increased from -1.09 mm to 1.39 mm
during the period of 1917-2017. The results are presented
in Figure 6.

International Journal of Engineering and Geosciences (IJEG),
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Figure. 6 Batumi, Poti and Tuapse station time series and reconstruction components
The mean sea level at the Bourgas station has no
significant change from -0.13 mm to -0.21 mm during
the period of 1929-1996. That at the Constantza station
increased from -0.06 mm to 0.14 mm during the period
of 1933-1997, and that at the Sevastopol station
increased from 0.002 mm to 0.17 mm during the period
of 1910-1994. The mean sea level at the Varna station
changed from -1.10 mm to -0.16 mm during the period
of 1929-1996. The results are presented in Figure 7. The
increasing trend stations are shown in Figure 8.
Figure. 7 Bourgas, Constantza, Sevastopol and Varna station time series and reconstruction components

International Journal of Engineering and Geosciences (IJEG),
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39
Figure. 8 Increasing trend stations in the Black Sea coasts ( denotes increasing trends)
The Amasra, Igneada and Trabzon II tide gauge stations
have inadequate data records. These stations have short
time series. Thus, the mean sea level change could not be
clearly determined. The mean sea level at the Amasra
station changed from 0.44 mm to 0.02 mm during the
period of 2001-2009, that at the Igneada station changed
from 0.11 mm to 0.54 mm during the period of 2002-
2009, and that at the Trabzon-II station changed from -
0.10 mm to 0.37 mm during the period of 2002-2009
(Figure 9).
Figure. 9 Amasra, Igneada and Trabzon II station time series and reconstruction components

International Journal of Engineering and Geosciences (IJEG),
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40
4. CONCLUSION
In this study, we used the SSA method to detect mean
sea level change at 10 tide gauge stations along the coasts
of the Black Sea. In addition, we detected the seasonal
influence in the time series. To this end, we discuss the
results of the analysis.
In the analysis results, there are increasing trends at
the Batumi (Georgia), Poti (Georgia), Tuapse (Russia),
Constantza (Romania), Sevastapol (Ukraine) and Varna
(Bulgaria) stations.
The Amasra, Igneada and Trabzon II tide gauge
stations in Turkey were inadequate in interpreting this
change. The analysis results are shown in Table 3.
Table 3 Trends at the tide gauge stations
The results showed that for the Batumi (Georgia)
station, the station values of the trend in the 133-year
period are from -0.11 mm to 0.85 mm, and for the Poti
(Georgia) station, the station values of the trend in the
141-year period are from -1.62 mm to 1.82 mm. For the
Bourgas (Bulgaria) and Varna (Bulgaria) stations, the
trend values for the same period are from -0.13 mm to -
0.21 and from -1.10 mm to -0.26 mm, respectively.
If the eigenvalues are close to each other, this
situation showing the presence of a seasonal component
in the time series. According to the eigenvalue results, the
Batumi, Constantza, Sevastopol, Poti, Tuapse, Trabzon II
and Varna stations have close eigenvalues. This situation
indicates the presence of a seasonal component in the
time series of related stations. No seasonal components
found at the other stations (Amasra, Igneada and
Bourgas).
ACKNOWLEDGEMENTS
The authors are grateful to Permanent Service for Mean
Sea Level (PSMSL) for providing the mean sea level data.
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Station
Trend
(mm)
Mean
Mean Square
Error
(mm)
Data Record
Span
(year)
Batumi
[-0.11, 0.85]
0.26
0.50
133
Constantza
[-0.06, 0.14]
-0.08
0.30
64
Sevastopol
[0.02, 0.17]
0.02
0.31
64
Varna
[-1.10, -0.26]
-0.14
0.28
67
Poti
[-1.62, 1.82]
0.005
0.93
141
Tuapse
[-1.09, 1.39]
0.004
0.62
100
Igneada
[0.11, 0.54]
-0.12
0.60
7
Amasra
[0.44, 0.02]
-0.02
0.35
8
Trabzon II
[-0.10, 0.37]
-0.04
0.51
7
Bourgas
[-0.13, -0.21]
0.002
0.37
67

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