A copula approach for sea level anomaly prediction: a case study for the Black Sea

Abstract

Forecasting future sea levels is of great importance in terms of the conservation of coastal areas, monitoring and forecasting coastal ecosystems, and the maintenance and planning of coastal structures. In addition, highly accurate sea level forecasts allow adequate water management policies and coastal infrastructures to be developed. Today, many methods, such as harmonic analyses, artificial neural networks and support vector machines, are used to predict sea level anomalies. In this study, a novel approach based on Copula functions is presented for the prediction of sea level anomalies. The primary purpose of this study is to examine the applicability and capability of Copula-based prediction models in predicting short-term variations in the sea level. The minimum 95% correlations and minimum 22 mm RMSE values in sea level anomaly predictions during the testing period indicate that the Copula approach can be a powerful tool in the prediction of sea level anomalies.

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A copula approach for sea level anomaly
prediction: a case study for the Black Sea
Ahmet Yavuzdoğan & Emine Tanır Kayıkçı
To cite this article: Ahmet Yavuzdoğan & Emine Tanır Kayıkçı (2020): A copula approach
for sea level anomaly prediction: a case study for the Black Sea, Survey Review, DOI:
10.1080/00396265.2020.1816314
To link to this article: https://doi.org/10.1080/00396265.2020.1816314
Published online: 15 Sep 2020.
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A copula approach for sea level anomaly
prediction: a case study for the Black Sea
Ahmet Yavuzdoğan ∗
1
and Emine Tanır Kayıkçı
2
Forecasting future sea levels is of great importance in terms of the conservation of coastal areas,
monitoring and forecasting coastal ecosystems, and the maintenance and planning of coastal
structures. In addition, highly accurate sea level forecasts allow adequate water management
policies and coastal infrastructures to be developed. Today, many methods, such as harmonic
analyses, artificial neural networks and support vector machines, are used to predict sea level
anomalies. In this study, a novel approach based on Copula functions is presented for the
prediction of sea level anomalies. The primary purpose of this study is to examine the
applicability and capability of Copula-based prediction models in predicting short-term
variations in the sea level. The minimum 95% correlations and minimum 22 mm RMSE values in
sea level anomaly predictions during the testing period indicate that the Copula approach can
be a powerful tool in the prediction of sea level anomalies.
Keywords: Sea level anomaly, Time series forecasting, Prediction, Copula, Black Sea, Sea level rise
1. Introduction
Considering that 60% of the world’s population lives in
coastal areas, forecasting future sea levels is of great
importance in terms of the conservation of coastal
areas, monitoring and forecasting coastal ecosystems,
providing estimates of fish harvests, and the maintenance
and planning of coastal structures. In addition, the fore-
casting of sea level changes is crucial for predicting future
climate change and global warming effects. Highly accu-
rate sea level forecasts allow adequate water management
policies and coastal infrastructures to be developed. Fur-
thermore, the sea level is of great importance, as it is used
in geodesy to determine the reference point for vertical
control networks. The accurate determination of the aver-
age sea level provides accurate vertical datum infor-
mation. Therefore, highly accurate sea level prediction is
required.
Instantaneous sea levels are affected temporally and
spatially by some geographical and meteorological vari-
ables, such as temperature, runoff, water salinity, evapor-
ation, precipitation, and the interactions between surface
water and low-lying aquifers (Imani et al. 2014c). More-
over, the effects of these variables are usually non-linear
and may vary from region to region. For this reason, it
is quite difficult to predict the sea level by modelling
each variable that contributes to changes in this level
via linear methods (Talebizadeh and Moridnejad 2011;
Imani et al. 2014c).
Since hydro-meteorological effects are neglected in the
tidal harmonic analysis traditionally used in sea level pre-
dictions, the prediction error between actual measure-
ments and harmonic predictions can reach 30%
(Ghorbani et al. 2010). Depending on the model and
hydro-meteorological conditions, errors can range from
a few centimetres to decimetres (Imani et al. 2018). In
addition, short-term tidal predictions of the sea level
require simultaneous meteorological data that are often
not available (Pashova and Popova 2011; Imani et al.
2018). Traditional methods used in sea level predictions,
such as the Box-Jenkins and autoregressive integrated
moving average (ARIMA) estimates, are often based on
linear assumptions. Yet, contributors to the sea level
form a nonlinear structure. Therefore, a nonlinear sea-
level time series based on these contributors cannot be
represented correctly using linear models. Since sea level
contributors form a non-linear structure that can vary
from region to region, it is very difficult to use them as
input data in sea level models. Instead of using these con-
tributors directly, many models have been developed that
predict future sea levels using past sea level records.
Recently, many machine learning and artificial neural net-
work (ANN)-based methods have been used to predict sea
levels using past records directly (Makarynskyy et al.
2004; Imani et al. 2014c; Zhao et al. 2019). In addition,
many models have been developed to predict short-term
sea level anomalies using hybrid methods (Niedzielski
and Kosek 2009;Fuet al. 2019). Both the ANN and
machine learning-based models and hybrid models have
been used to deal with the complex, nonlinear dependen-
cies affecting sea levels.
Many studies have been conducted in the past related to
sea level forecasting. Irvine and Eberhardt (1992)
1
Department of Geomatics Engineering, Faculty of Engineering and Natural
Sciences, Gümüşhane University, 29100 Gümüşhane, Turkey
2
Department of Geomatics Engineering, Faculty of Engineering, Karadeniz
Technical University, 61000 Trabzon, Turkey
∗Corresponding author, email: yavuzdogan@gumushane.edu.tr
© 2020 Survey Review Ltd
Received 30 April 2020; accepted 25 August 2020
DOI 10.1080/00396265.2020.1816314 Survey Review 2020 1

developed autoregressive integrated moving average
(ARIMA) models for Lakes Erie and Ontario that used
average monthly sea level data to predict the lake levels.
Brundrit (1995) used ARIMA models to examine sea
levels at eight tide gauge stations. Vaziri (1997) used the
ARIMA and ANN models to predict the 12-month Cas-
pian Sea level. Meshkani and Meshkani (1997) used
ARIMA models to model the stochastic sea level fluctu-
ations in the Caspian sea. Şen et al. (2000) used simple lin-
ear and periodic nonlinear models to model the
deterministic aspect of lake level records. For the non-
linear modelling of the lake level records, a second-
order Markov model and Fourier series were used.
Makarynskyy et al. (2004) used ANNs to predict one-
month sea level anomalies. Niedzielski and Kosek
(2009) used a polynomial-harmonic model based on the
least-squares (LS) method to forecast the linear trends
in the annual and semi-annual oscillations in the sea
level anomalies. Imani et al. (2013) used the polynomial
interpolation and Holt-Winters exponential smoothing
methods to predict the satellite-derived seasonal Caspian
Sea level anomaly. Imani et al. (2014b) compared ANN
and ARMA methods in terms of predicting sea level
anomalies in the Caspian Sea. Imani et al. (2014a) used
a hybrid model that included principal component analy-
sis and ARIMA to predict sea level anomalies in the Cas-
pian Sea. Imani et al. (2014c) used evolutionary support
vector regressions and gene expression programming to
predict sea level anomalies. Srivastava et al. (2016)inves-
tigated the combined use of quantitative forecasting
methods for rises in sea level using exponential smoothing
state-space models (ESMs) and an ARIMA model. Fu
et al. (2019) presented a hybrid model that combined
empirical mode decomposition, single spectrum analysis,
and least squares for satellite-derived sea level anomaly
prediction. Zhao et al. (2019) used ANNs and
ARMA models to predict sea level anomalies in the Yel-
low Sea.
Today, copulas are used in the modelling of dependency
structures between variables in many fields, including the
biomedical field (Wang and Wells 2000), Hydrology (Bár-
dossy and Li 2008), Climatology (Vogl et al. 2012), Econ-
omics and Finance (Aloui et al. 2013), Hydro-Geodesy
(Modiri et al. 2015), and Geodesy (Modiri et al. 2020;
Yavuzdog
̆an and TanırKayıkçı2020). Since copulas do
not have restrictive assumptions such as the multivariate
normality assumption in linear regression-based methods,
copulas are not affected by the distributions of variables
and errors. The advantages of using copulas in modelling
include: (i) both linear and nonlinear dependencies can be
modelled, (ii) an arbitrary selection of a marginal distri-
bution can be made, and (iii) it is possible to model
extremes (Kumar and Shoukri 2007). Copulas are very
important in measuring dependencies, as they reflect the
joint distributions between variables without having to
consider marginal distributions. Thanks to this feature,
they are very convenient when modelling complex depen-
dencies (Qu and Lu 2019).
Also, variability of sea level from seasonal to interann-
ual time scales is caused by several processes, such as
changes in ocean heat content and circulation, changes
in sea level pressure, and changes in river runoff regimes
(Etcheverry et al. 2015). Therefore, it is difficult to deter-
mine the interannual and interdecadal effects of sea level
in harmonic analysis. Yet since the copulas are the joint
distribution of variables in the range [0, 1] where marginal
distributions are uniform, not all effects need to be mod-
elled separately.
Copulas have been used many times to model complex
dependencies and have achieved significant success in pre-
diction performance (Simard and Rémillard 2015; Arya
and Zhang 2017; Nguyen-Huy et al. 2017; Niemierko
et al. 2019). Hence, this approach may have significant
potential in predicting sea level anomalies. In the present
study, an Archimedean copula approach, which can
model both linear and nonlinear complex dependencies,
was used to predict sea level anomalies from past sea
level records. The primary purpose of this study is to
examine the applicability and capability of copula-based
prediction models in predicting short-term sea level
variations.
In this study, modelling sea level anomaly time series
with complex dependency structures with Archimedean
copulas was investigated. Therefore, the temporal depen-
dencies of the sea level anomaly time series in the Black
Sea are modelled with Archimedean copulas. The sea
level time series were predicted using these models.
Finally, simulated data and observed datawere compared
to assess the performance of the copula-based prediction
model.
2. Methodology
2.1. Dataset and working area
In this study, a Ssalto/DUACS multi-mission altimeter
product was used as the dataset (http://www.marine.
copernicus.eu). DUACS is a processing system that pro-
vides sea level products using satellite altimeters.
DUACS provide along-track (L3) and gridded (L4) sea
level products of two different types: near-real-time and
delayed-time products. In this study, which aims to offer
a novel approach to the prediction of sea level anomalies
in the Black Sea, multi-mission (gridded) satellite altime-
try data were used as distributed by the Copernicus Mar-
ine Environment Monitoring Service (CMEMS). This
multi-mission satellite altimetry data include weekly aver-
age sea level anomalies from 1 January 1993 to 31 Decem-
ber 2016. Delayed-time satellite altimetry data are
gridded (0, 125◦by 0, 125◦), and sea-level anomalies are
calculated according to a twenty-year mean. Sea level
anomaly data are obtained by combining ERS1,
Topex / Poseidon, ERS2, Jason - 1, Envisat, Jason - 2,
Cryosat, Altika, Sentinel - 3A, Jason - 3 altimeter
measurements. Additionally, the quality controls were
established by the data centre, and it made all necessary
atmospheric and geophysical corrections to the sea level
anomaly data. The study area and altimetry grid nodes
are shown in Fig. 1.
2.2. Copula-based prediction (CP) model
The concept of the copula was developed by Sklar (1959).
According to Sklar, in the bivariate case (x,y), a copula
(C)isdefined as a function that connects multivariate dis-
tributions (H(x,y)) with their univariate marginal distri-
butions (Fx(x) and Fy(y)) (Sklar 1959) as follows:
H(x,y)=C(Fx(x), Fy(y)) (1)
The dependency structures can be time-dependent
structures that show the variations in random variables
Yavuzdoğan and Tanir Kayikçi A copula approach for sea level anomaly prediction
2Survey Review 2020

over time or structures that indicate the variations in two
different variables relative to one another. Additionally,
one of the important features of copulas is that they reflect
both linear and nonlinear relationships between variables.
Time series prediction is a technique that is used for
forecasting events across time. The technique is used
across many fields of study, from Geology to Behavioural
Studies to Economics (https://whatis.techtarget.com/
definition/time-series-forecasting, 13 January 2020).
One of the important features of copulas is that they
can generate sample data with the same dependency
structure as that of the modelled dependencies.
Hence, the time-dependent dependency structures can
be modelled by using the Copula approach, and time
series predictions can then be made through these
models. The following steps are used to generate
copula-based simulation data for time series predic-
tions in the bivariate case (Laux et al. 2011;Modiri
et al. 2018; Vogl et al. 2012):
(1) Transformation of the time series into independent,
identical distributions (iid variables)
(2) Calculation of the marginal distributions of the
dependent (y) and independent (x) variables
(Fx(x), Fy(y))
(3) Converting time series input data into rank space
using the best-fitting marginal distribution (ui,vi)
(4) Calculate the copula parameters for all candidate
copulas
(5) Select the best-fitting copula via the goodness-of-fit
test
(6) Generate sample random data from the conditional
copula cumulative distribution function (CDF) in rank
space
(7) Generate copula-based prediction values via inverse
marginal distributions.
2.2.1. Fitting the marginal distribution
The first step in CP is to model the distributions of each of
the dependent and independent variables with theoretical
distribution functions. In this study, the Generalized
Extreme Value distribution, Normal (Gaussian)
distribution, and Logistic distribution, as listed in Table
1, were considered for the marginal distributions of the
variables.
The theoretical marginal distribution that best fits the
variables is selected by using the Akaike and Bayesian
Information Criteria (AIC and BIC) as follows (Akaike
1974;Witet al. 2012):
AIC =2k−ln (B) (2)
BIC =k(ln(n)−2ln(B)) (3)
where nis the number of data points, kindicates the num-
ber of parameters in the theoretical model, and Bindi-
cates the maximum value of the likelihood function for
the model. The best-fitting theoretical marginal distri-
bution is the one with the lowest AIC and BIC values.
2.2.2. Archimedean copula
The Archimedean copula approach was the first proposed
by Kimberling (1974). Archimedean copulas can be
derived easily using a generator function. Therefore,
they are commonly used in dependence modelling appli-
cations. In the bivariate case, an Archimedean copula
(C(u,v)) can be formed by using the following equation:
C(u,v)=
w
−1
w
(u)+
w
(v),
u

(4)
where
w
denotes the generator function of the Archime-
dean copula, and θis the copula parameter.
Table 1 Generalized extreme value, normal, and logistic
distributions
Parameters Formula
Location μ, Scale σ,
Shape ξf(x;
m
,
s
,
j
)=1+
j
x−
m
s

−1/
j
if
j
=0
e−(x−
m
)/
s
if
j
=0

Location μ, Scale σf(x;
m
,
s
)=1
s

2
p
√e−1
2
x−
m
s
()
2
Location μ, Scale sf(x;
m
,s)=e−(x−
m
)/s
s(1+e−(x−
m
)/s)2
1 Working area and satellite altimetry grid points
Yavuzdoğan and Tanir Kayikçi A copula approach for sea level anomaly prediction
Survey Review 2020 3

In this study, three Archimedean copulas (Clayton,
Frank, Gumbel) are used since they have wide copula par-
ameter spaces and can reflect both positive and negative
dependencies. Furthermore, these Archimedean copulas
can reflect upper- and lower-tail dependencies. Tail depen-
dency measures the correlation amount of low or high
extremes in a data set. One of the most important features
of copulas is that it reflects tail dependencies. As time-
dependent correlations are examined within the scope of
this study, even though tail dependency is not used, it is
important for other dependence modelling studies.
.Clayton copula
The Clayton copula, which is an asymmetric Archime-
dean copula, can reflect positive dependencies and
local dependencies in the lower tail (Fig. 2). It is
appropriate to model dependencies with Clayton
copulas when the variables have high correlations
with small values. The generator function of the Clay-
ton copula is:
w
Cl (x)=1
u
(t−
u
−1) (5)
The CDF of the Clayton copula is written as:
C
u
(u,v)=max [(u−
u
+v−
u
−1), 0]−1
u
(6)
where θdenotes the Copula parameter, which is in the
interval [ −1, 1)(Joe1997). When θ= 0, the variables
are independent, and when θconverges to infinity,
there is a positive dependency and the lower-tail
dependency increases.
.Frank copula
The Frank copula has a symmetrical structure but does
not reflect tail dependencies. The generator function of
Frank copula is:
w
Fr(t)=−ln e−
u
t−1
e−
u
−1
 (7)
where θdenotes the copula parameter of the Frank
copula. It has a range of ( −1,1), where −1indi-
cates a complete negative dependency, ∞indicates a
complete positive dependency, and 1 indicates inde-
pendent variables (Joe 1997). The CDF of the Frank
copula is:
C
u
(u,v)=1
u
ln 1 +(e−
u
u−1)(e−
u
v)
e−
u
−1

(8)
.Gumbel copula
The Gumbel copula is suitable for modelling variables
when there is a high correlation structure between
large values and its asymmetric structure (Fig. 2). In
addition, it can reflect only positive dependencies
and upper-tail dependencies.
w
Gu(t)=(−ln (t))
u
(9)
where θindicates the copula parameter of the Gumbel
copula, which has the range [1, 1) (Nelsen 2007).
Here, 1 indicates independence, and ∞indicates a
complete positive dependence between variables. The
CDF of the Gumbel copula is:
C
u
(u,v)=e−((−ln (u)
u
)(−ln (v)
u
))
1
u
(10)
Figure 2 illustrates the densities of the Archimedean
copulas. A copula density exhibits its dependency
structure. For example, Fig. 2ashows the asymmetric
structure of the Clayton copula and its ability to cap-
ture dependencies in the lower tail. Similarly, Fig. 2c
shows the asymmetric dependency structure that cap-
tures the dependencies in the upper tail of the Gumbel
copula.
2.2.3. Copula parameter estimation
In the bivariate case, the copula parameter is a coefficient
with different ranges for each copula that captures the
dependency between the variables. This coefficient is
important in terms of showing the direction and amount
of dependence. Parametric and non-parametric esti-
mation methods are available for the estimation of the
copula parameter. The Inference Functions for Margins
Estimation method was developed by Joe and Xu in
1996 to estimate the copula parameter using the Maxi-
mum Likelihood (ML) method (Joe and Xu 1996). In
this study, Inference Functions for Margins Estimation
was used to estimate the copula parameters. In Maximum
Likelihood Estimation, the parameters of the marginal
distributions and the copula parameters are estimated
simultaneously. The density function of the copula is writ-
ten as:
c(u,v)=∂C(u,v)
∂u∂v(11)
where c(u,v) indicates the density function of the copula.
2 The densities of the Clayton, Frank, and Gumbel Archimedean copulas (Yavuzdoğan and Tanır Kayıkçı2020). aClayton
copula
u
=5. bFrank copula
u
=10 and cGumbel copula
u
=5
Yavuzdoğan and Tanir Kayikçi A copula approach for sea level anomaly prediction
4Survey Review 2020

Let (X1,X2,X3,...,Xn) be a sample dataset. The log-
likelihood function is written as:
l(
u
)=
T
i=1
log c(F1(x1), F2(x2), ...,Fn(xn);
u
)

(12)
where T,F, and θindicates amount of data, the marginal
distribution function, and the Copula parameter, respect-
ively. The copula parameter can be estimated by maximis-
ing the log-likelihood equation:
ˆ
u
MLE =argmax(l(
u
)) (13)
2.2.4. The goodness-of-fit test
One of the most important problems in dependency
modelling studies is the selection of the best copula
for modelling the dependence between the variables.
In this study, AIC and BIC selection criteria were
used to select the best copula. For the bivariate case,
the AIC and BIC are:
AIC =−2
n
i=1
ln [c(ui,1,ui,2 );
u
]+2k(14)
BIC =−2
n
i=1
ln [c(ui,1,ui,2 );
u
]+ln (nk)(15)
where c(ui,1 ,ui,2 ) denotes the density function of a can-
didate copula, nrepresents the sample size, and kis the
number of estimated model parameters. AIC and BIC
are based on a likelihood principle, and the preferred
copula has the lowest AIC and BIC values (Gülöksüz
2016).
2.2.5. The conditional copula functions
Conditional copula functions for the Clayton, Frank, and
Gumbel copulas are given below (Joe 1997; Modiri et al.
2018).
CClayton
V=v(u,v)=u−
u
−1−1+u−
u
+v−
u

−1
u
−1
()(16)
CFrank
V=v(u,v)=e−u
u
−1+e−v
u

−1+e−
u

1+−1+e−u
u

−1+e−v
u

−1+e−
u

(17)
CGumbel
V=v(u,v)=−ln u
u
−1

ln v
u
−ln v
u
()
1
u
−1
()

ue −ln u
u
+ln v
u
()
1
u

(18)
Once the conditional Copula function has been identified
in a modelling problem, the sample data are converted to
the data space via the inverses of their marginal
distributions.
2.3. Model performance criteria
The root mean square error (RMSE), mean average error
(MAE), and the correlation coefficient (r) are used to
assess the performance of bivariate models. These
performance criteria are calculated as:
RMSE =
n
i=1yi−yp
i

2
n
(19)
MAE =1
n
n
i=1
yp
i−yi
(20)
r=n
i=1yi−
y

yp
i−
yp


n
i=1yi−
y

2n
i=1yp
i−
yp

2
(21)
where yiand yp
iare the observed and predicted
mean sea level anomalies in the ith, week, respect-
ively; nis the amount of weekly sea level anomaly
data; and 
yand 
ypare the mean value of the
observed weekly sea level anomaly data and the
mean value of the predicted weekly sea level
anomaly data, respectively.
3. Results and discussion
In this study, a copula-based time series prediction
method is proposed as a new method for modelling
and forecasting sea level anomalies. The main task is
to assess the potential of the CP model in this endea-
vour. The flowchart of the CP model algorithm is
given in Fig. 3.
3.1. Pre-processing
Thanks to satellite radar altimeters, sea level data can
be obtained from all over the globe. However, it is dif-
ficult to determine the sea level in coastal areas due to
the distortion of wave forms and some auxiliary infor-
mation used in some data corrections and errors (Vig-
nudelli et al. 2019). Radar altimetry wave forms can be
distorted by any inhomogeneities in the properties of
the observed surface (Gómez-Enri et al. 2010). When
making range calculations, serious errors may occur
as a result of ignoring these effects. It is well known
that standard altimetry errors can increase in coastal
zones for two main reasons: (1) the direct effect of
the land on the measurement system; and (2) the
reduction in scale, both temporal and spatial, of both
oceanographic and atmospheric processes in the coastal
zone (Vignudelli et al. 2019). For these reasons, sea
level predictions in coastal areas may not be reliable.
Therefore, the altimetry grid nodes in the coastal
areas (in Fig. 1 red dots) were removed from the
dataset.
The Ssalto/DUACS multi-mission altimeter product
coverage of the Black Sea provides data for 3132 grid
nodes at 0.125-degree intervals. Over a time period of
23 years, each grid node records 1272 sea level anomaly
records in a weekly time series. In this study, each time
series was divided into two parts: 75% training data
(1993–2010 - 936 weeks) and 25% testing data (2011–
2016 - 312 weeks). The training data was used to develop
the CP model, and the testing data was used to investigate
the accuracy of the model. In Fig. 4, blue and red time
series show the training and testing data, respectively.
Yavuzdoğan and Tanir Kayikçi A copula approach for sea level anomaly prediction
Survey Review 2020 5

3.2. Developing the CP model
A time series is required to develop a CP model. A Mean
Train Time Series (MTTS) was obtained by averaging the
time series across all grid nodes in the training period and
used to develop the CP model. In the averaging process,
the SLA time series obtained from each grid node are
summed with each other and divided by the total number
of grids. In Fig. 5, yellow and red lines show the training
time series and Mean Train Time Series (MTTS),
respectively.
3.2.1. Fitting marginal distributions
The first step in developing the CP model is to fit the mar-
ginal distribution of the MTTS to theoretical distribution
functions. In this study, the marginal distribution of the
MTTS was fitted to the Generalized Extreme Value,
Logistic, and Normal (Gaussian) distribution functions.
The best-fitting theoretical distribution functions was
selected via the AIC and BIC performance criteria. The
best-fitting theoretical distribution function was then
used to develop the CP model. The results of the model
4 Training and testing time series
3 CP model algorithm
Yavuzdoğan and Tanir Kayikçi A copula approach for sea level anomaly prediction
6Survey Review 2020

performance tests for the marginal distributions are given
in Tables 2 and 3((∗) denotes the best-fitting theoretical
distribution).
The Generalized Extreme Value and Logistic distri-
butions are chosen as the most appropriate theoretical dis-
tributions for the Time and Sea Level Anomaly (SLA)
variables since they have the lowest AIC and BIC values.
3.2.2. Estimating the copula parameters and goodness of
fit test
The next step in developing the CP model is to estimate
copula parameters. The copula parameters for the Clay-
ton, Frank, and Gumbel copulas were calculated via the
Inference Functions for Margins Estimation method.
The best-fitting copula function to dependent structure
of MTTS was selected using the goodness-of-fit test.
The copula parameters, results of the goodness-of-fit
test, and the best-fitting copula are given in Table 4.(
∗)
denotes the best-fitting copula function.
The Clayton copula performs better than the Frank
and Gumbel Copulas in terms of representing the depen-
dence structure of the MTTS according to the goodness-
of-fit test. Therefore, the CP model was developed using
Generalized Extreme and Logistic distribution functions
as well as the Clayton copula (
u
=0.1583).
Table 2 Model performance test of the marginal distribution
for time
Distribution name AIC BIC
Generalized extreme value∗1309.60 1310.12
Logistic 1314.88 1315.56
Normal 1322.49 1323.18
Table 3 Model performance test of the marginal distribution
for SLA
Distribution name AIC BIC
Generalized extreme value −2165.91 −2151.38
Logistic∗−2165.97 −2156.28
Normal −2143.43 −2133.75
5 Training time series and mean train time series
Table 4 Copula parameters and the goodness-of-fit test
Parameter AIC BIC RMSE
Clayton∗0.1583 −4655.73 −4650.89 2.54
Frank 1.725 −4417.91 −4413.07 2.88
Gumbel 1.1943 −4480.04 −4475.19 2.79
6 Observed and predicted MTTS a, QQ-plot of prediction b
Yavuzdoğan and Tanir Kayikçi A copula approach for sea level anomaly prediction
Survey Review 2020 7

3.2.3. SLA prediction from the CP model
The first step in prediction is to generate marginal CDF
data for both the dependent and independent variables
by using the best-fitting theoretical distributions. In this
study, fitted marginal CDF data are calculated using the
Logistic and Generalized Extreme Value distribution
functions for the dependent and independent variables
of the targeted time series. The second step is to compute
the conditional copula function for the Clayton copula
(equation (16)). The conditional copula function is then
used to generate random sample data from the copula.
The final stage of prediction involves the transformation
of the random sample data into the data space from the
copula space via the inverses of the selected theoretical
distribution functions. In this study, all time series for
each grid during both the training and testing periods
and MTTS were predicted with the CP model. The pre-
diction of the MTTS and statistical information about
the prediction of the MTTS is given in Fig. 6 and
Table 5, respectively.
In the prediction of the MTTS, although the observed
and predicted MTTS values are similar, there are notable
differences between the observed values and the predic-
tions in the early stages of the time series.
The predictions of the training and testing time series
for grid node 2000 are given in Fig. 7. Statistical infor-
mation about the prediction of the time series conducted
using the best-fitting CP model during the training and
testing periods for all grid points are given in Table 6.
As shown in Table 6, both training and testing time series
were predicted with high precision (with at least a 95%
correlation).
Strong correlations between the predicted and observed
SLA values within the testing data demonstrate the capa-
bilities of the CP model in the prediction of the average
SLA. Since the predicted SLA data are highly correlated
with the observations, the mean SLA data are also
obtained with a high spatial correlation. The average
observed and predicted SLA’s for the Black Sea in the
training and testing periods are given in Fig. 8. The
residuals between the average observed and predicted
SLA’s are given in Fig. 9.
Table 5 CP model performance in the prediction of the MTTS
Correlation
(R)
MAE
(m)
RMSE
(m)
Statistics for the CP
model
0.96 0.0136 0.0261
7 Predictions for grid node 2000. aTraining time series for grid node 2000 and btesting time series for grid node 2000
Table 6 The prediction performance of the CP model for SLA
time series for all grids
Correlation MAE (m) RMSE (m)
Min Max Min Max Min Max
Training 0.956 0.972 0.012 0.015 0.014 0.025
Testing 0.966 0.988 0.018 0.027 0.022 0.041
Yavuzdoğan and Tanir Kayikçi A copula approach for sea level anomaly prediction
8Survey Review 2020

9 Residuals between the mean observed and predicted SLA’s. aResiduals for training and bresiduals for testing
8 Mean sea level anomalies and their predictions for the Black Sea. aMean SLA (training period), bmean SLA (testing period),
cprediction of mean SLA (training period) and dprediction of mean SLA (testing period)
Yavuzdoğan and Tanir Kayikçi A copula approach for sea level anomaly prediction
Survey Review 2020 9

As can be seen in Fig. 9, the training residuals range
between about −2to+2 mm, and the test residuals
range from about −15 to about −10 mm.
In addition, comparing the results of the CP model
with other prediction methods is crucial for understand-
ing the CP model’s sea level prediction capability. There-
fore, the performances of sea level prediction methods
and the CP model performance are compared in Table 7.
4. Conclusion
In the present study, a novel method based on the copula
approach was presented to predict sea level anomalies for
the Black Sea. Satellite-based weekly average sea level
altimetry data containing 24 years (1993–2016) of
anomalies were used for prediction. During the training
period, 18 years of sea level anomaly data (1993–2010)
was used for developing the copula-based prediction
(CP) model. Six years of sea level anomaly data (2011–
2016) was used to assess the performance of the CP
model during the testing period. As a result of predictions
made via the CP method in the testing data set, it was
found that the correlation between the observed and pre-
dicted SLA was at least 96% and that the RMSE values
were between 22 and 41 mm. In particular, the high corre-
lation between the observed and predicted SLA data
shows that the CP model reflects the spatial correlation
of the SLA’s with high accuracy. When the residuals for
the testing dataset were examined, it was found that the
residuals ranged between about −16 and −10 mm. The
study results show that the capability of the copula func-
tions in predicting sea levels is satisfactory. Compared to
previous studies, the sea level anomaly prediction per-
formance of the CP model was better than those of
many ANN methods. In future studies, a hybrid predic-
tion model can be created for sea level prediction by com-
bining the CP model with ANN methods. Furthermore,
using the CP model, the SLA data can be predicted
directly without the need for any meteorological or topo-
graphic information.
Notes on contributors
Research Assistant Yavuzdog
̆an has studied in Geomatics
Engineering Department at Karadeniz Technical Univer-
sity since 2017. He was awarded master degree in Geo-
matics Engineering department at Karadeniz Technical
University (Trabzon/Turkey) under Associte Professor
Emine TanırKayıkçıin 2019. Since 2019, he has been
continuing his doctoral studies at Karadeniz Technical
University. Since 2017, he has been working as a research
assistant in the Department of Geomatics Engineering at
Gümüşhane University (Gümüşhane, Turkey). His
research priorities are in analysis of sea level change,
time series analysis of geodetic data/products, copula
based dependence modelling, machine learning and arti-
ficial neural networks.
Associate Professor Doctor Emine TanırKayıkçıhas
studied in Geomatics Engineering Department at Kara-
deniz Technical University since 1998. She was awarded
master degree in Geomatics Engineering department at
Karadeniz Technical University (Trabzon/Turkey) under
Professor Mualla Yalçınkaya in 2001 and doctorate
degree in Advanced Geodesy division of the I
̇nstitute of
Geodesy and Geophysics at Technical University (Aus-
tria) of Vienna under Professor Harald Schuh in 2008.
In 2009, she took up a full-time position as assistant pro-
fessor in Geomatics Engineering department at Karade-
niz Technical University (Trabzon/Turkey). In 2017, she
was appointed associate professor of Geodesy. Since
2017, she was a vice-head of Geomatics Engineering
department of Karadeniz Technical University. She has
been a head of Geodesy and Navigation Commission of
Chamber of Survey and Cadastre Engineers (CSCE)
under the Union of Chambers of Turkish Engineers and
Architects (UCTEA) since 2018. She has been an associ-
ate member of International VLBI Service for Geodesy
and Astrometry (IVS) since 2009 and responsible person
of IVS Analysis Center KTU-GEOD since 2009. She is
member of working group/management committe of cur-
rently running COST Action projects (CA18235, CA
17109, CA17134) and was a working group member of
past COST Action ES1206 from 2014 to 2017). Her
research priorities are in analysis of geodetic VLBI obser-
vations, GNSS meteorology/tomography, sea level moni-
toring, time series analysis of geodetic data/products. At
present, Emine TanırKayıkçısupervises six doctor theses
(topics: GNSS meteorology, GNSS tomography, GNSS
Reflectometry, GNSS Radio Oscultation, Sea Level
Monitoring, Time Series Analysis, Machine Learning
and Copula Correlation). She supervised five Master of
Science theses since 2013 (topics: goodness of fit tests
for VLBI products, time series analysis of coordinate
time series at IGS stations, interpolation of meteorologi-
cal parameters, copula based correlation of sea level and
meteorological parameters).
ORCID
Ahmet Yavuzdog
̆an http://orcid.org/0000-0002-9898-4946
Emine Tanır Kayıkçıhttp://orcid.org/0000-0001-8259-5543
Table 7 Comparison of the SLA prediction methods
Study Method
RMSE
(mm)
Prediction
length
Makarynskyy et al.
(2004)
ANN 151 1 Month
Niedzielski and
Kosek (2009)
LS 35 1 Month
LS+AR 26
Imani et al. (2013) HWES 70 3 Years
ANN 60
Imani et al. (2014b) MLP 54 3 Years
RBF 42
GRNN 59
ARMA 119
Imani et al. (2014a) PCA
+ARIMA
51 3 Years
Imani et al. (2014c) SVR 35 6 Years
Imani et al. (2018) ELM 35 4 Years
RBF 36
RVM 34
SVM 35
Fu et al. (2019) EMD+SSA
+LS
13 3 Years
Zhao et al. (2019) MLFF 18 5 Years
RBF 16
GRNN 16
ARMA 28
This study CP model 22 6 Years
Yavuzdoğan and Tanir Kayikçi A copula approach for sea level anomaly prediction
10 Survey Review 2020

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