Sea level prediction using climatic variables: a comparative study of SVM and hybrid wavelet SVM approaches

Abstract

Climate change is expected to adversely affect the coastal ecosystem in many ways. One of the major consequences of climate change in coastal areas is sea level rise. In order to manage this problem efficiently, it is essential to obtain reasonably accurate estimates of future sea level. This study focuses essentially on the identification of climatic variables influencing sea level and sea level prediction. Correlation analysis and wavelet coherence diagrams were used for identifying the influencing variables, and support vector machine (SVM) and hybrid wavelet support vector machine (WSVM) techniques were used for sea level prediction. Sea surface temperature, sea surface salinity, and mean sea level pressure were observed to be the major local climatic variables influencing sea level. Halosteric effect is found to have a major impact on the sea level. The variables identified were subsequently used as predictors in both SVM and WSVM. WSVM employs discrete wavelet transform to decompose the variables before being input to the SVM model. The performance of both the models was compared using statistical measures such as root mean square error (RMSE), correlation coefficient (r), coefficient of determination (r2), average squared error, Nash–Sutcliffe efficiency, and percentage bias along with graphical indicators such as Taylor diagrams and regression error characteristic curves. Results indicate that the WSVM model predicted sea level with an RMSE of 0.029 m during the training and 0.040 m during the testing phases. The corresponding values for SVM are 0.043 m and 0.069 m, respectively. Also, the other statistical measures and graphical indicators suggest that WSVM technique outperforms the SVM approach in the prediction of sea level.

Introduction

Accumulation of greenhouse gases in the atmosphere is the main cause of global warming and consequent change in the earth’s climate. Climate change is projected to have serious ramifications on the oceans, sea level rise (SLR) being one of the major impacts. The sea level is changing continuously owing to thermal expansion of seawater, changes in salinity, and melting of glaciers and ice sheets. There are many climatic and other variables contributing to the observed changes in sea level. Some of the short-lived anthropogenic greenhouse gases such as methane (CH4), chlorofluorocarbons (CFCs), hydrochlorofluorocarbons (HCFCs), and hydrofluorocarbons (HFCs) contribute to sea level changes which may persist for long time (Zickfeld et al. 2017). The Intergovernmental Panel on Climate Change (IPCC) reported a rise in sea level of about 1.7 mm/year on a global scale over the period 1901–2010 (IPCC 2014). A relatively recent study on sea level rise conducted using altimeter observations reported a rise of about 3 ± 0.4 mm/year after 1993 (Nerem et al. 2018). Sea level rise may bring about coastal flooding, coastal erosion and degeneration of the coastal biome (Chang et al. 2014). Therefore, in the present scenario it is vital to study the changes in sea level and also to obtain a reasonable future estimate of sea level to facilitate implementation of realistic and efficient management strategies in the coastal zone.

Before performing modelling to obtain accurate estimates of sea level, it is essential to identify the climatic variables that influence the sea level. Global and local variations in sea level will be different due to dynamic ocean processes, seabed movements, and redistribution of water mass (IPCC 2014). Dynamical ocean processes include steric effects, thermal expansion, ocean currents, large-scale circulation, melting of glaciers, etc. (Cui et al. 1995). Maddah (2016) developed a linear relationship between sea level and surface temperature. Also, it has been reported that wind forcing and ocean salinity structure play a substantial role in regional steric level changes (Cui et al. 1995; Durack et al. 2014; IPCC 2014). Halosteric contraction caused by salinity enhancement can negate the thermosteric contribution to sea level (Durack et al. 2014; Suzuki and Ishii 2011). Local-scale sea level variations are governed by other climatic variables such as sea surface temperature and sea level pressure (Cui et al. 1995; Heyen et al. 1996). Ishida et al. (2020) developed a model to predict sea level in coastal areas on an hourly time scale using variables such as wind, air temperature, mean sea level pressure, relative positions of sun and moon along with annual global air temperature, which is an indicator of climate change. Karamouz et al. (2013) established a relationship between sea level fluctuations and climatic variables such as sea level pressure and temperature. Naren and Maity (2017) introduced a semi-empirical approach to predict changes in sea level using both atmospheric and oceanic variables.

Machine learning algorithms have wide range of applications in the field of climate studies, especially in modelling. Among these methods, artificial neural networks (ANN) have been extensively adopted for prediction and downscaling in the field of water resources and climate change (Dorado et al. 2003; Juan et al. 2017; Yadav and Chandel 2014). Another technique called decision tree, which can be employed for both classification and regression analysis, has also been widely used for prediction (Mangai and Gulyani 2016). Apart from these techniques, there are other competent machine learning approaches; support vector machine (SVM) is one such learning algorithm. Initially, SVM technique was widely employed for classification studies (Elhag et al. 2013; Modaresi and Araghinejad 2014; Zhu and Blumberg 2002). In recent times, SVM has been widely accepted as a promising approach for modelling and prediction applications (Khaledian et al. 2020; Tripathi et al. 2006; Voyant et al. 2017). This technique has also been used to predict water quality (Yunrong and Liangzhong 2009), field hydraulic conductivity (Das et al. 2011), soil moisture, and stream flow (Gill et al. 2006; Lin et al. 2006).

A wide variety of hybrid models are also available that enhances the accuracy of prediction. Fourier transforms and wavelet transforms are the commonly adopted signal denoising techniques. Wavelet transforms have been used in the field of climate change, hydrology as well as for image processing (Koirala and Gentry 2012; Pišoft et al. 2004; Xiao and Zhang 2011). Wavelet neural network (WNN) is the most widely implemented hybrid model, in which wavelet transforms are employed to denoise the input signals before being input to the prediction model. Krishna et al. (2011) and Nury et al. (2017) employed WNN for the prediction of river flow and temperature. These authors reported that WNN model outperformed the standard ANN and other autoregressive (AR) models. Koirala and Gentry (2012) coupled wavelet transform with SWAT to predict future water yield at the outlet of a watershed. Similar to ANN, SVM has also been coupled with wavelet transforms to improve prediction accuracy (Kalteh 2013; Mohammadi et al. 2015). Mohammadi et al. (2015) forecasted horizontal global solar radiation using a hybrid wavelet support vector machine. Kalteh (2013) employed both WNN and WSVM to predict river flow on a monthly time scale; it was observed that WSVM outperformed WNN. Detailed literature review revealed that only few studies have been reported on the application of Artificial Intelligent-based techniques for prediction of sea level. Furthermore, it was found that most of the studies reported considered only thermosteric effect on sea level, which could overestimate predicted sea levels. The objectives of the present study are to identify the local climatic variables that impact sea level the most and to predict sea level using SVM and WSVM. The highlight of this work is that a methodology is proposed for the prediction of sea level using climatic variables employing a hybrid wavelet machine learning approach.

Study location

Data pertaining to climatic variables that were likely to influence sea level were obtained for a location (9.75° N, 76.25° E) near to the Willingdon Island tidal gauge station in Ernakulam, Kerala. Satellite altimeter observations of sea level (ssh), sea surface salinity (ss), and ocean current (oc) were retrieved from Copernicus Marine Environment Monitoring Service (CMEMS) web portal for the period 2000–2014. Similarly, data pertaining to evaporation (e), sea surface temperature (sst), surface pressure (sp), and mean sea level pressure (psl) were retrieved through the web portal of the European Centre for Medium-Range Weather Forecasts (ECMWF) for the period 2000–2014.

Methods

The methodology mainly consists of three distinctive phases. The first phase was to identify the climatic variables (predictors) which impact the sea level using correlation analysis and wavelet coherence diagrams; this was followed by the second phase in which the SVM and WSVM sea level prediction models were developed and computations were performed. The final phase involved comparison of the performance of these two models. In the first phase, correlation analysis was performed between the dependent variable (ssh) and independent climatic variables to identify the predictors of the model. Furthermore, wavelet coherence diagrams were prepared for potential climatic variables based on the results of correlation analysis. Wavelet coherence is an indicator of relationship between variables in a time–frequency frame, with values ranging from 0 (low coherence) to 1 (high coherence). Sea level was modelled with SVM and WSVM techniques using the predictors identified from the correlation analysis and the wavelet coherence diagrams.

Support vector machine (SVM)

SVM is a machine learning algorithm, which has been widely accepted as a tool for data mining, especially in modelling. This technique can be employed for performing regression and pattern recognition. The notion behind this algorithm is to separate data points by an optimal hyperplane using kernel trick (Haykin 2009). Even nonlinearly separable cases can be handled with this algorithm. The data points in the input feature space are mapped onto a higher dimensional feature space using a suitable kernel function (Gunn 1998). SVM has the advantage of convex optimization that it does not confront the problem of local minima and also allows a sparse solution to be obtained (Voyant et al. 2017). The SVM learning algorithm can solve problems with small samples; it doesn’t require too many training samples to develop a model (Li and Kong 2014).

The objective function is to minimize

$$J = C\mathop \sum \limits_{n = 1}^{N} E[y(x_{n} ) - y_{dn} ] + \frac{1}{2}\left\| {w^{2} } \right\|,$$
(1)

where C is the inverse weight penalty parameter, E: error function, and \(\frac{1}{2}\left\| {w^{2} } \right\|\): weight penalty term. \(y\left( {x_{n} } \right)\) is the estimator output, \(x_{n}\) is the input data, \(N\) is the total number of training patterns, and \(y_{dn}\) is the target data. In order to preserve the sparseness characteristic of the SVM classifier, E is defined as:

$$E_{\epsilon} \left( z \right) = \left\{ {\begin{array}{*{20}c} {\left| z \right| - \epsilon ,} & {{\text{if}}\;\left| z \right| > \epsilon } \\ 0 & {\text{Otherwise}} \\ \end{array} }. \right.$$
(2)

This is called ϵ-insensitive error function; it disregards errors lesser than ϵ. \(\left| z \right| = \left| {y_{dn} - y\left( {x_{n} } \right)} \right|\). Two slack variables \(\xi_{n}\) and \(\xi^{\prime}_{n}\) are introduced such that \(\xi_{n} \ge 0\) and \(\xi^{\prime}_{n} \ge 0\). \(\xi_{n} > 0\) denotes data points over the ϵ-tube and \(\xi^{\prime}_{n} > 0\) denotes those beneath the ϵ-tube.

$${\text{i}}.{\text{e}}.\;y_{dn} > y\left( {x_{n} } \right) + \epsilon ,{\text{if}}\;\xi_{n} > 0$$
$$y_{dn} < y\left( {x_{n} } \right) + \epsilon ,{\text{if}}\;\xi^{\prime}_{n} > 0.$$

The objective function can be modified as

$${\text{Minimize}}\;J = C\mathop \sum \limits_{n = 1}^{N} (\xi_{n} + \xi^{\prime}_{n} ) + \frac{1}{2}\left\| {w^{2} } \right\|$$
(3)

subject to \(\xi_{n} \ge 0\), \(\xi^{\prime}_{n} \ge 0\), \(y_{dn} \le y\left( {x_{n} } \right) + \epsilon + \xi_{n}\), \(y_{dn} \ge y\left( {x_{n} } \right) - \epsilon - \xi^{\prime}_{n}\).

In order to handle the constraints, Lagrange multipliers are introduced. Support vector regression is performed in a feature space,

$$y\left( x \right) = w^{T} \phi \left( x \right) + w_{0},$$
(4)

where ɸ is the feature map and \(w_{0}\) is the bias.

$$w = \mathop \sum \limits_{n = 1}^{N} (\lambda_{n} - \lambda^{\prime}_{n} )\phi \left( {x_{n} } \right),$$
(5)

where \(\lambda_{n}\) and \(\lambda^{\prime}_{n}\) are the Lagrangian multipliers.

Substituting (5) into (4) gives

$$y\left( x \right) = \mathop \sum \limits_{n = 1}^{N} \left( {\lambda_{n} - \lambda^{\prime}_{n} } \right)K\left( {x,x_{n} } \right) + w_{0}.$$
(6)

\(K\left( {x,x_{n} } \right) = \phi^{T} \left( x \right)\phi \left( {x_{n} } \right)\), which is the kernel function. Support vectors are those data points that contribute to Eq. (6) (Hsieh 2009). First degree polynomial kernel was used for this study. The architecture and flow chart of SVM based model are presented in Figs. 1 and 2, respectively.

Fig. 1
figure1

Architecture of SVM

Fig. 2
figure2

Flow chart of SVM

Wavelet support vector machine (WSVM)

Wavelet transform is an advanced form of Fourier transform. In the case of Fourier transforms, it is not possible to obtain both time and frequency information. But, wavelet transforms can produce both time and frequency information; also, it can handle both stationary and non-stationary data (Kalteh 2013; Santos and da Silva 2014). The wavelet transform is just a waveform that exists over a limited period with a mean value of zero (Misiti et al. 1996). There are mainly two types of wavelet transforms, viz., continuous wavelet transform and discrete wavelet transform (Santos and da Silva 2014). Discrete wavelet transform can be employed as a digital filter bank to get rid of the noise present in a signal (Misiti et al. 1996).

Discrete wavelet transform (DWT)

DWT is a piecewise function; it decomposes the original signal into a number of components. DWT can be obtained by discretizing the continuous wavelet transform (CWT) function. Mathematically, CWT can be expressed as follows (Chan 1996):

$${\text{CWT}}\left( {a,\tau } \right) = \frac{1}{\sqrt a }\int {S\left( t \right)\psi \left( {\frac{t - \tau }{a}} \right){\text{d}}t},$$
(7)

where \(a\) is the scale and \(\tau\) is the shift.

DWT can be derived as given below:

$${\text{DWT}}\left( {m,n} \right) = a_{0}^{{ - \frac{m}{2}}} \int {S\left( t \right)\psi (a_{0}^{ - m} t - n\tau_{0} ){\text{d}}t},$$
(8)

where \(a = a_{0}^{m}\) and \(\tau = n\tau_{0} a_{0}^{m}\), the variables \(m\) and \(n\) are integers.

Mother wavelet

Wavelets are produced from a fundamental wavelet, termed mother wavelet through scaling and translation. The various mother wavelets accessible are Haar, Daubechies, Biorthogonal, Coiflets, Symlets, Morlet, Mexican Hat, and Meyer. Daubechies family wavelets are denoted as dbN, where N indicates the number of vanishing moments (available from db1 to db10); theoretically, it may vary from 1 to infinity. db1 and Haar wavelet are alike. Region of support and number of vanishing moments are the two characteristic properties of the mother wavelet function. Region of support refers to the span of the wavelet and which in succession influences the feature localization abilities of the wavelet. Vanishing moment describes the capability of the wavelet to portray the information or polynomial behaviour in a given signal (Maheswaran and Khosa 2012). For instance, db1 wavelet, with one moment, simply encodes polynomials with one coefficient or constant signal elements. db2 wavelet encodes (2 coefficients) constant and linear signal elements; db3 wavelet encodes constant, linear as well as quadratic signal elements (Maheswaran and Khosa 2012).

In this study, Daubechies wavelets were used, since they are compactly supported orthonormal wavelets with a maximal number of vanishing moments for the specified support; also, it has been employed in many studies (Maheswaran and Khosa 2012; Misiti et al. 1996). db10 is adopted in the present study because of its smoother form, which results in better depiction of time series data (Misiti et al. 1996; Nalley et al. 2012; Santos and da Silva 2014).

Decomposition process

The minimum and maximum number of decomposition levels was calculated using appropriate formulae. According to Nourani et al. (2014), the minimum level of decomposition can be computed using the equation

$$L = \text{int} [\log N_{S} ].$$
(9)

According to Lei et al. (2013), the maximum level of decomposition for a discrete wavelet transform can be computed as

$$L = \text{int} [\log_{2} N_{S} ],$$
(10)

where \(L\) is the decomposition level and \(N_{S}\) is the signal length (number of data points)

Wavelet transform was applied to the original series (time series of input variables), producing two signals, viz., approximations (A) and details (D). Approximations are the high-scale, low-frequency components of the signal. The details are the low-scale, high-frequency components. Approximation is the most important component, which reveals signal identity and detail is the nuance. This decomposition is an iterative process and so the successive approximations are decomposed in turn into approximations and details (Krishna et al. 2011). Each of these decomposed signals was input to the SVM model and the outputs were summed up to get the final predicted signal. The flow chart of WSVM is depicted in Fig. 3.

Fig. 3
figure3

Flow chart of WSVM

Model evaluation

The performance of both SVM and WSVM models was evaluated using statistical measures such as root mean squared error (RMSE), correlation coefficient (r), coefficient of determination (r2), average squared error (ASE), Nash–Sutcliffe efficiency (NSE), and percentage bias (PBIAS) (Elshorbagy et al. 2000; Koirala and Gentry 2012; Moriasi et al. 2007; Tao et al. 2018). In addition to the above measures, graphical indicators such as Taylor diagrams and regression error characteristic (REC) curves were constructed to visually aid in identification of the outperforming model. Formulas used for the computation of the statistical measures are as follows:

$${\text{RMSE}} = \sqrt {\frac{1}{N}\mathop \sum \limits_{i = 1}^{N} \left( {Y_{i}^{O} - Y_{i}^{P} } \right)^{2} }$$
(11)
$$r^{2} = \left[ {\frac{{\mathop \sum \nolimits_{i = 1}^{N} \left( {Y_{i}^{O} - \overline{{Y^{O} }} } \right)\left( {Y_{i}^{P} - \overline{{Y^{P} }} } \right)}}{{\sqrt {\mathop \sum \nolimits_{i = 1}^{N} \left( {Y_{i}^{O} - \overline{{Y^{O} }} } \right)^{2} } \sqrt {\mathop \sum \nolimits_{i = 1}^{N} \left( {Y_{i}^{P} - \overline{{Y^{P} }} } \right)^{2} } }}} \right]^{2}$$
(12)
$${\text{ASE}} = \frac{1}{N}\mathop \sum \limits_{i = 1}^{N} \left( {Y_{i}^{O} - Y_{i}^{P} } \right)^{2}$$
(13)
$${\text{NSE}} = 1 - \left[ {\frac{{\mathop \sum \nolimits_{i = 1}^{N} \left( {Y_{i}^{O} - Y_{i}^{P} } \right)^{2} }}{{\mathop \sum \nolimits_{i = 1}^{N} \left( {Y_{i}^{O} - \overline{{Y^{O} }} } \right)^{2} }}} \right]$$
(14)
$${\text{PBIAS}} = \left[ {\frac{{\mathop \sum \nolimits_{i = 1}^{N} \left( {Y_{i}^{O} - Y_{i}^{P} } \right) \times 100}}{{\mathop \sum \nolimits_{i = 1}^{N} Y_{i}^{O} }}} \right],$$
(15)

where \(Y_{i}^{O}\) and \(Y_{i}^{P}\) are the observed and predicted ith value of sea level, \(\overline{{Y^{O} }}\) and \(\overline{{Y^{P} }}\) are the mean of observed and predicted values of sea level, and \(N\) is the total number of data points.

Results and discussion

Identification of climatic variables

Sea level is influenced by many local climatic variables; the identification of the most influencing variables (predictors) will enable more accurate predictions of sea level. The variables chosen for the correlation analysis to identify the predictors were sea level (ssh), sea surface salinity (ss), ocean current (oc), evaporation (e), sea surface temperature (sst), surface pressure (sp), and mean sea level pressure (psl). Correlation analysis was performed between the sea level (predictand) and the other independent variables. The results of the analysis are presented in Table 1. From Table 1, it can be observed that variables such as mean sea level pressure, ocean current, surface pressure, sea surface salinity, and sea surface temperature possess very good correlation with sea level. Since sea surface salinity has a good correlation with sea level, it is clear that halosteric effect is significant. Furthermore, surface pressure holds strong relation with mean sea level pressure; besides that, ocean current exhibits good relation with mean sea level pressure, surface pressure, and sea surface temperature. These inter-correlated climatic variables such as ocean current and surface pressure were removed from the predictors list. Thus, climatic variables such as sea surface temperature, sea surface salinity, and mean sea level pressure were chosen as the predictors in this study and hence it can be said that both thermosteric and halosteric effects are accounted for in the prediction of sea level. Wavelet coherence diagrams constructed between the predictand and predictors are presented in Figs. 4, 5, and 6. The Wavelet coherence diagram depicts the correlation between two variables in a time–frequency frame. In all three diagrams (Figs. 4, 5, and 6), the thick black lines bounding the red coloured regions reflect 95% confidence level and the thin solid line represents the cone of influence. The faded region beyond it should be carefully treated due to potential edge effects. The colour scale ranges from blue (zero correlation) to red (perfect correlation). From Figs. 4, 5, and 6, it is evident that the red region is more predominant, indicating that there is a strong correlation between the sea level and the climatic variables considered. In addition, the prominent orientation of the arrows in the first two figures (Figs. 4 and 5) is to the right, indicating a positive correlation. Mean sea level pressure and sea surface temperature are therefore positively correlated with the sea level. In Fig. 6, the orientation of the arrows is to the left, indicating a negative correlation between sea level and sea surface salinity. It is also clear from the wavelet coherence diagram (Fig. 6) that sea level is significantly influenced by the halosteric effect, as red coloured regions are more in the diagram. It is, therefore, evident from these figures that there exists strong correlation between the predictand and predictors.

Table 1 Results of correlation analysis
Fig. 4
figure4

Wavelet coherence between sea level and mean sea level pressure

Fig. 5
figure5

Wavelet coherence between sea level and sea surface temperature

Fig. 6
figure6

Wavelet coherence between sea level and sea surface salinity

Sea level prediction

In this phase of the study, a relationship was established between the predictand (sea level) and the predictors (sea surface temperature, sea surface salinity, and mean sea level pressure) using the SVM and WSVM models. Predictors such as sea surface temperature, sea surface salinity, and mean sea level pressure were the input variables for the formulation of models and sea level was the output or target variable. The entire data set was divided into two sets, such as training data set and testing data set. About 70% of the data (January 2000–June 2010) were employed for training and the remaining 30% data were used for testing.

SVM model

In the course of training, the SVM model parameters such as \(C\) and \(\epsilon\) were optimized through trial and error, and the values obtained are 10 and 0.001, respectively. The trained model was then tested by predicting the values of sea level using the remaining 30% data and the results were compared with the observations of a satellite altimeter. A comparison of sea level predicted by the SVM model with satellite altimetry observations, during both training and testing phases, is presented (Fig. 7). It is clear from the figure that there is a reasonably good match between the predicted sea level and the altimetry observations during both training and testing phases. Furthermore, most of the peaks match closely. It is hence evident that the SVM based model performs reasonably well in predicting sea level.

Fig. 7
figure7

Comparison of altimeter and SVM predicted sea level

WSVM model

In this model, the dependent variable sea level and all the independent variables such as sea surface temperature, sea surface salinity and mean sea level pressure were decomposed using DWT. The mother wavelet adopted was db10. The minimum and maximum levels of decomposition were calculated using Eqs. (9) and (10), respectively, and these were 2 and 7, respectively. Decomposition of a signal at a level greater than the optimum may disturb the characteristic properties of the data set (Pramanik et al. 2011). Therefore, the uppermost level of decomposition was avoided and a decomposition level of 6 was selected. The decomposition process yields six details (D1, D2, D3, D4, D5, D6) and one approximation (A6). The original signal can be represented as \(S = D1 + D2 + D3 + D4 + D5 + D6 + A6\). The details and approximations obtained for each climatic variable are depicted in Figs. 8 and 9.

Fig. 8
figure8

DWT decomposed components of a sea level b mean sea level pressure

Fig. 9
figure9

DWT decomposed components of a sea surface temperature b sea surface salinity

Each of these decomposed components was segregated and given as input to the SVM model. A total of seven output series were obtained from the above seven segregated components. The final output (sea level) was obtained by adding all the above outputs. The results obtained during training and testing are depicted in Fig. 10. It is evident from the figure that there is a good match between the observed and WSVM predicted sea levels during, both training and testing periods.

Fig. 10
figure10

Comparison of altimeter (observed) and WSVM predicted sea level

Comparison between SVM and WSVM models

In order to identify the best sea level prediction model, the statistical parameters of both SVM and WSVM models were compared (Table 2). The model performance evaluation criteria, based on various statistical parameters as specified in Moriasi et al. (2007), are presented in Table 3.

Table 2 Performance statistics
Table 3 Performance evaluation criteria

During the training period, the SVM model yields \(r^{2} > 0.5\) (acceptable), \(0.75 < {\text{NSE}} \le 1\) (very good) as well as \({\text{PBIAS}} < \pm 10\) (very good), whereas during the testing period \({\text{NSE}} > 0.5\) (satisfactory) and \(\pm 10 \le {\text{PBIAS}} < \pm 15\) (good). Therefore, overall, the model performance can be rated as good. Similarly, for the WSVM model, during both training and testing periods, \(r^{2} > 0.5\), \(0.75 < {\text{NSE}} \le 1\) as well as \({\text{PBIAS}} < \pm 10.\) Hence, the overall performance of this model can be rated as very good. In addition to this, RMSE and ASE values are relatively small for the WSVM model compared to the SVM model. Both the models are acceptable; however, based on the performance statistics, WSVM model outperforms the SVM model.

Taylor diagrams for both training and testing data sets were constructed (Figs. 11 and 12) by plotting the results of the SVM and WSVM models against the reference (observed data). Each model in the Taylor diagram was represented with three major statistical parameters, viz., correlation coefficient, standard deviation, and centred RMS difference. It is evident from the diagram that the WSVM model performs better than the SVM model as the point representing the WSVM model is close to the reference.

Fig. 11
figure11

Taylor diagram for the training data set

Fig. 12
figure12

Taylor diagram for the testing data set

In addition, REC curves for the SVM and WSVM models during training and testing period were plotted (Fig. 13). The REC curve determines the cumulative distribution function of the error; the area above the curve is a metric associated with the error of the prediction model. It is apparent from the figure that the WSVM model outperforms the SVM model as the area over the curve is small for this model.

Fig. 13
figure13

REC curves

Conclusions

Climatic variables have a significant influence on the sea level. Regional sea level can be better analysed by identifying the local climatic variables which drive sea level fluctuations. From the correlation analysis and wavelet coherence diagrams, it was observed that sea surface temperature, sea surface salinity, and mean sea level pressure are the major driving forces that influence sea level. Also, it was observed that mean sea level pressure and sea surface temperature are positively correlated with sea level, whereas sea surface salinity is negatively correlated with sea level.

In this study, sea level was predicted with both the SVM and WSVM models. Satellite altimetry observations of sea level and data on climatic variables such as mean sea level pressure, sea surface temperature, and sea surface salinity, retrieved from the CMEMS and ECMWF web portals, were used for training and testing of the models. The SVM based model was developed using the variables in its original form, whereas the WSVM based model was developed using decomposed elements of the original variables. The performance of both the models was compared using statistical parameters. Furthermore, graphical indicators such as Taylor diagrams and REC curves were used for visual identification of the best performing model. It was found that the WSVM based model was superior to the SVM model in terms of its predictions.

One of the important findings from the study is that the impact of halosteric effects on sea level is very significant and it has to be taken into account in sea level prediction. In addition, machine learning techniques like SVM and WSVM can be effectively used to predict sea level. The methods suggested in this research can be used to obtain reasonably accurate estimates of sea level. The findings of this research would enable scientists to predict sea level based on climatic variables and to evaluate the impact of climate change on rising sea levels. Scientifically sound and accurate information pertaining to the sea level is very much needed for developing appropriate coastal management schemes under changing climate. Comparison of this approach with other machine learning techniques can be set as the future scope of this research.

Availability of data and material

All data can be accessed from European Centre for Medium-Range Weather Forecasts (ECMWF) and Copernicus Marine Environment Monitoring Service (CMEMS) web portals.

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Sithara, S., Pramada, S.K. & Thampi, S.G. Sea level prediction using climatic variables: a comparative study of SVM and hybrid wavelet SVM approaches. Acta Geophys. 68, 1779–1790 (2020). https://doi.org/10.1007/s11600-020-00484-3

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Keywords

  • Climate change
  • Modelling
  • Sea level
  • SVM
  • WSVM